Process Dynamics, Modeling, and Control

process dynamics, mo de lin g, an d control TOPICS IN CHEMICAL ENGINEERING A Series of Textbooks and Monograph s SERI...

Author: Babatunde A. Ogunnaike | W. Harmon Ray

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process dynamics, mo de lin g, an d control

TOPICS IN CHEMICAL ENGINEERING A Series of Textbooks and Monograph s SERIES EDITOR

ASSOCIATE EDITORS

KEITH E. GUBBINS Cornell University

MARK A. BARTEAU University of Delaware

KLAVS F. JENSEN Massachusetts Institute of Technology DOUGLAS A. LAUFFENBURGER University of lllinois MANFRED MORARI California Institute of Technology

W. HARMON RAY University of Wisconsin WILUAM B. RUSSEL Princeton University SERIES TITLES Receptors: Models for Binding, Trafficking, and Signalling D. Lauffenburger and f. Lindennan Process Dynamics, Modeling, and Control

B. Ogunnaike and W. H. Ray

process dynamics, modeling, and control BABATUNDE A. OGUNNAIKE E. I. DuPont de Nemours, Experimental Station, and Adjunct Professor, Department of Chemical Engineering University of Delaware

W. HARMON RAY Department of Chemical Engineering University of Wisconsin

New York Oxford OXFORD UNIVERSITY PRESS

1994

Oxford University Press Oxford New York Athens Auckla nd Bangkok Bombay Calcutt a Cape Town Dares Salaam Delhi Florence Hong Kong Istanbu l Karachi Kuala Lumpu r Madraa Madrid Melbourne Mexloo City Nairob i Paris Singap ore Taipei Tokyo Toront o and aasoda ted compan ies in Berlin Ibadan

Copy right@ 1994 by Oxford University Press, Inc. Publish ed by Oxford Univer sity Press, Inc., 200 Madiso n Avenue , New York, New York 10016 Oxford is a registe red tradem ark of Oxford Unlvem ty Press

All righta reserve d. No part of this publica tion may be reprodu ced, stored In a retrieva layatem , or transm itted, In any form or by any means, electronic, mec:hanlcal, photoco pying, reoordlng, or othenv lae, withou t the prior permis sion of Oxford Univer sity Press. Ubrary of Congress Cataloglng-ln-Publlcalion Data Ogunn alke, Babatu nde A. (Babatu nde Ayodeji) Proceas dynami cs, modeli ng, and contro l/ Babatu nde A Ogunnllike, W. Harmo n Ray. p. em. -(Topi cs in chemical enginee ring) Includes Indexes. 1SBN 0-19-509119-1 1. Chemic al process control. I. Ray, W. Harmo n (Wiilis Harmo n), 1940U. Title. m. Series: Topics In chemical enginee ring (Oxford Univer sity Press) TP155.75.036 1994 660'.2815---dc20 94-28307

1 3 5 7 9 8 6 4 2 Printed in the United States of America on add-fre e paper

To Anna and "the boys" (Damini and Deji), ... Agbaja owo ni n'gberu d'ori; and To decades of superb graduate student teachers, ... the heart of process control at Wisconsin.

CCONTJENTS

part I INTRODUCTION

Chapter 1. Introductory Concepts of Process Control 1.1 1.2 1.3 1.4 1.5 1.6

The Chemical Process An Industrial Perspective of a Typical Process Control Problem Variables of a Process The Concept of a Process Control System Overview of Control System Design Summary

Chapter 2. Introduction to Control System Implementation

2.1 Introduction 2.2 Historical Overview 2.3 Basic Digital Computer Architecture 2.4 Data Acquisition and Control 2.5 Some Examples 2.6 Summary

vii

5 5 8 13 15 19 30

35 35 36 38 46 56 61

viii

CONTEN TS

par tll PROCE SS DYNAMICS

Chapter 3. Basic Elements of Dynamic Analysis 3.1 3.2 3.3 3.4 3.5

Introduction Tools of Dynamic Analysis The Laplace Transform Characteristics of Ideal Forcing Functions Summary

Chapter 4. The Prcxess Model 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

The Mathematical Description of Chemical Processes Formulating Process Models State-Space Models Transform-Domain Models Frequency-Response Models Impulse-Response Models Interrelationships between Process Model Forms The Concept of a Transfer Function Summary

Chapter 5. Dynamic Behavior of Linear Low Order-Systems 5.1 5.2 5.3 5.4 5.5 5.6

First Order Systems Response of First-Order Systems to Various Inputs Pure Gain Systems Pure Capacity Systems The Lead/La g System Summar y

Chapter 6. Dynamic Behavior of Linear Higher Order Systems 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Two First-Order Systems in Series Second-Order Systems Response of Second-Order Systems to Various Inputs N First-Order Systems in Series The General Nth-Order System Higher Order Systems with Zeros Summary

67 67 68 71 80 86 89 89 92 104 108 110 111 115 128 134 139 139 142 153 157 161 166 175 175 183 188 200 204 205 211

ix

CONTENTS

Chapter 7. Inverse-Response Systems

225

225 7.1 Introduction 229 7.2 Inverse Response in Physical Processes 7.3 Dynamic Behavior of Systems with Single, Right-Half Plane 231 Zeros 7.4 Dynamic Behavior of Systems with Multiple, Right-Half 237 Plane Zeros 240 7.5 Summary Chapter 8. Time-Delay Systems 8.1 8.2 8.3 8.4 8.5 8.6 8.7

An Introductory Example The Pure Time-Delay Process Dynamic Behavior of Systems with Time Delays The Steam-Heated Heat Exchanger Rational Transfer Function Approximations Model Equations for Systems Containing Time Delays Summary

Chapter 9. Frequency-Response Analysis 9.1 9.2 9.3 9.4 9.5 9.6

Introduction A General Treatment Low-Order Systems Higher Order Systems Frequency Response of Feedback Controllers Summary

VChapter 10. 10.1 10.2 10.3 10.4 10.5 ,' Chapter 11. 11.1 11.2 11.3 11.4 11.5 11.6

245 245 249 252 257 261 265 267 275 275 277 288 295 305 306

Nonlinear Systems

311

Introduction: Linear and Nonlinear Behavior in Process Dynamics Some Nonlinear Models Methods of Dynamic Analysis of Nonlinear Systems Linearization Summary

312 313 314 320 326

Stability

333

Introductory Concepts of Stability Stability of Linear Systems Stability of Nonlinear Systems Dynamic Behavior of Open-Loop Unstable Systems . Stability of Dynamic Systems under Feedback Control Summary

333 337 343 349 353 355

X

CONTENTS

partlll PROCESS MODELING AND IDENTIFICATION

Chapter 12. 12.1 12.2 12.3 12.4 12.5 12.6 Chapter 13. 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Theoretical Process Modeling

363

Introduction Development of Theoretical Process Models Examples cf Theoretical Model Formulation Parameter Estimation in Theoretical Models Validation of Theoretical Models Summary

363 366 368 380 399 401

Process Identification: Empirical Process Modeling

409

Introduction and Motivation Principles of Empirical Modeling Step-Response Identification Impulse-Response Identification Frequency-Response Identification Issues for Multivariable Systems Summary

409 412 417 422 436 447 450

xi

CONTENTS

part IV PROCESS CONTROL

PART IVA: Chapter 14. 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Chapter 15. 15.1 15.2 15.3 15.4 15.5 15.6 15.7 Chapter 16. 16.1 16.2 16.3 16.4 16.5 Chapter 17. 17.1 17.2 17.3 17.4 17.5

SINGLE-WOP CONTROL Feedback Control Systems

461

The Concept of Feedback Control Block Diagram Development Classical Feedback Controllers Closed-Loop Transfer Functions Closed-Loop Transient Response Closed-Loop Stability The Root Locus Diagram Summary

461 464 467 468 472 486 496 504

Conventional Feedback Controller Design

513

Preliminary Considerations Controller Design Principles Controller Tuning with Fundamental Process Models Controller Tuning using Approximate Process Models Controller Tuning using Frequency-Response Models Controller Tuning without a Model Summary

513 519 525 532 541 555 557

Design of More Complex Control Structures

565

Processes with Significant Disturbances Processes with Multiple Outputs Controlled by a Single Input Processes with a Single Output Controlled by Multiple Inputs Antireset Windup Summary

565 581

(~8~) 585 586

Controller Design for Processes with Difficult Dynamics

599

Difficult Process Dynamics Time-Delay Systems Inverse-Response Systems Open-Loop Unstable Systems Summary

599 603 608 617 619

xii

CONTE NTS

Chapt er 18. 18.1 18.2 18.3 18.4 18.5 Chapt er 19. 19.1 19.2

..J ~ 19.4 19.5 19.6

Contro ller Design for Nonlin ear Systems

625

Nonlin ear Contro ller Design Philos ophies Linear ization and the Classical Appro ach Adapt ive Contro l Princi ples Variab le Transf ormati ons Summ ary

625 626 628 632 638

Model-Based Contro l

645

Introduction Contro ller Design by Direct Synthesis Intern al Model Contro l Generic Model Contro l Optim ization Appro aches Summ ary

645 648 665 671 674 675

PAR T IVB: MUL TIVA BIAB LE PRO CESS CON TROL Chapt er 20. 20.1 20.2 20.3 20.4 20.5 20.6 Chapt er 21. 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10

Introd uction to Multiv ariable System s The Nature of Multiv ariable System s Multiv ariable Process Model s Open- Loop Dynam ic Analy sis in State Space Multivariable Transf er Functions and Open- Loop Dynam ic Analy sis Closed-Loop Dynam ic Analy sis SUIIIII\aJ.'}' Interac tion Analy sis and Multip le Single Loop Design s Introduction Prelim inary Consid eration s of Interac tion Analy sis and Loop Pairing The Relative Gain Array (RGA) Loop Pairing Using the RGA Loop Pairing for Nonlin ear Systems Loop Pairing for Systems with Pure Integr ator Modes Loop Pairing for Nonsq uare Systems Final Comm ents on Loop Pairing and the RGA Controller Design Proced ure Summ ary

683 683 688 694 702 712 716 723

723 724 728 735 742 748 750 754 759 764

CONTENTS

Chapter 22. 22.1 22.2 22.3 22.4 22.5 22.6

xiii

Design of Multivariable Controllers

773

Introduction Decoupling Feasibility of Steady-State Decoupler Design Steady-State Decoupling by Singular Value Decomposition Other Model-Based Controllers for Multivariable Processes Summary

773 776 792 805 806 807

PART IVC: COMPUTER PROCESS CONTROL Chapter 23. 23.1 23.2 23.3 23.4 23.5 23.6 23.7

J

Chapter 24. 24.1 24.2 24.3 24.4 24.5 Chapter 25. 25.1 25.2 25.3 25.4 25.5 Chapter 26. 26.1 26.2 26.3 26.4 26.5 26.6 26.7

Introduction to Sampled-Data Systems

821

Introduction and Motivation Sampling and Conditioning of Continuous Signals Continuous Signal Reconstruction Mathematical Description of Discrete-Time Systems Theoretical Modeling of Discrete-Time Systems Empirical Modeling of Discrete-Time Systems Summary

821 825 830 832 835 838 843

Tools of Discrete-Time Systems Analysis

847

Introduction The Basic Concepts of z-Transforms Inverting z-Transforms Pulse Transfer Functions Summary

847 848 860 867 877

Dynamic Analysis of Discrete-Time Systems

883

Open Loop Responses Characteristics of Open-Loop Pulse Transfer Functions Block Diagram Analysis for Sampled-Data Systems Stability Summary

883 893 917 933 943

Design of Digital Controllers

951

Preliminary Considerations The Digital Controller and Its Design Discrete PID Controllers from the Continuous Domain Other Digital Controllers Based on Continuous Domain Strategies Digital Controllers Based on Discrete Domain Strategies Digital Multivariable Controllers Summary

951 953 960 964 970 979 979

xiv

CONTENTS

partV SPECIAL CONTROL TOPICS

Chapter 27. 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9

Chapter 28. 28.1 28.2 28.3 28.4 28.5 28.6 28.7

Chapter 29. 29.1 29.2 29.3 29.4 29.5

29.6 29.7

Model Predictive Control

991

Introduction General Principles of Model Predictive Control Dynamic Matrix Control Model Algorithmic Control (MAC) Commercial Model Predictive Control Schemes: A Qualitative Review Academic and Other Contributions Nonlinear Model Predictive Control Closing Remarks Summary

1017 1020 1023 1027

Statistical Process Control

1033

Introduction Traditional Quality Control Methods Serial Correlation Effects and Standard Process Control When is Traditional SPC Appropriate? Stochastic Process Control More Advanced Multivariate Techniques Summary

1034 1036 1046 1048 1050 1058 1059

Selected Topics in Advanced Process Control

1063

Control in the Absence of Good On-line Measuremen tsState Estimation Control in the Face of Process Variability and Plant/Model Mismatch- Robust Controller Design Control of Spatial Profiles - Distributed Parameter Controllers Control in a Changing Economic Environmen t- On-line Optimization Control in the Face of Component Failure Abnormality Detection, Alarm Interpretation , and Crisis Management Some Emerging Technologies for Advanced Process Control Concluding Remarks

992 996 1000 1008

1011

1063 1069 1083 1086

1088 1089 1094

XV

CONTENTS

Chapter 30.

Process Control System Synthesis- Some Case Studies

1097

30.1 30.2 30.3 30.4 30.5 30.6

Control of Distillation Columns Control of Catalytic Packed Bed Reactors Control of a Solution Polymerization Process Control of an Industrial Terpolymerization Reactor Guidelines for Characterizing Process Control Problems Summary

1097 1105 1113 1122 1133

1145

part VI APPENDICES

Control System Symbols used in Process and Instrumentation Diagrams

1149

Complex Variables, Differential Equations, and Difference Equations

1155

Appendix C.

Laplace and Z..Transforms

1177

Appendix D.

Review of Matrix Algebra

1209

Appendix E.

Computer-Aided Control System Design

1249

Appendix A.

Appendix B.

Author Index

1251

Subject Index

1255

preface Over the last two decades there has been a dramatic change in the chemical process industries. Industrial processes are now highly integrated with respect to energy and material flows, constrained ever more tightly by high quality product specifications, and subject to increasingly strict safety and environmental emission regulations. These more stringent operating conditions often place new constraints on the operating flexibility of the process. All of these factors produce large economic incentives for reliable, high performance control systems in modem industrial plants. Fortunately, these more challenging process control problems arise just at the time when inexpensive real time digital computers are available for implementing more sophisticated control strategies. Most new plants in the chemical, petroleum, paper, steel, and related industries are designed and built with a network of mini- and microcomputers in place to carry out data acquisition and process control. These usually take the form of commercially available distributed control systems. Thus digital computer data acquisition, process monitoring, and process control ·are the rule in process control practice in industry today. Because of these significant changes in the nature of process control technology, the undergraduate chemical engineer requires an up-to-date textbook which provides a modem view of process control engineering in the context of this current technology. This book is directed toward this need and is designed to be used in the first undergraduate courses in process dynamics and control. Although the most important material can be covered in one semester, the scope of material is appropriate for a two-semester course sequence as welL Most of the examples are taken from the chemical process industry; however, the text would also be suitable for such courses taught in mechanical, nuclear, industrial, and metallurgical engineering departments. Bearing in mind the limited mathematical background of many undergraduate engineers, all of the necessary mathematical tools are reviewed in the text itself. Furthermore, the material is organized so that modem concepts are presented to the student but the details of the most advanced material are left to later chapters. In this

xviii

PREFACE

way, those prefering a lighter treatment of the subject may easily select coherent, self-consistent material, while those wishing to present a deeper, more comprehensive coverage, may go further into each topic. By providing this structure, we hope to provide a text which is easy to use by the occasional teacher of process control courses as well as a book which is considered respectable by the professor whose research specialty is process control. The text material has been developed, refined, and classroom tested by the authors over many years at the University of Wisconsin and more recently at the University of Delaware. As part of the course at Wisconsin, a laboratory has been developed to allow the students hands-on experience with measurement instruments, real-time computers, and experimental process dynamics and control problems. The text is designed to provide the theoretical background for courses having such a laboratory. Most of the experiments in the Wisconsin laboratory appear as examples somewhere in the book. Review questions and extensive problems (drawn from many areas of application) are provided throughout the book so that students may test their comprehension of the material. The book is organized into six parts. In Part I (Chapters 1-2), introductory material giving perspective and motivation is provided. It begins with a discussion of the importance of process control in the process industries, with simple examples to illustrate the basic concepts. The principal elements of a modem process control scheme are discussed and illustrated with practical process examples. Next, a rudimentary description of control system hardware is provided so that the reader -can visualize how control schemes are implemented. This begins with a discussion of basic measurement and computer data acquisition methodology. Then the fundamentals of digital computers and interfacing technology are presented in order to introduce the basic concepts to the reader. Finally, control actuators such as pumps, valves, heaters, etc. are discussed. The purpose of the chapter is to provide some practical perspective, before beginning the more theoretical material which follows. Part II (Chapters 3-11) analyzes and characterizes the various types of dynamic behavior expected from a process and begins by providing an introduction to the basic mathematical and analysis tools necessary for the engineering material to be studied. This is followed by a discussion of various representations and approaches in the formulation of dynamic models. The emphasis is on learning how to select the model formulation most appropriate for the problem at hand. The essential features of state-space, transformdomain, frequency-response, and impulse-response models are presented and compared. Then comes a discussion of the fundamental dynamic response of various model types. Processes with time delays, inverse response, and nonlinearities are among the classes considered in some detail. The fundamentals of process stability analysis are then introduced and applied to the models under discussion. Methods for constructing process models and determining parameters for the model from experimental data are discussed in Part Ill (Chapters 12-13). Both theoretical and empirical models are discussed and contrasted. Complementing the usual material on step, pulse, or frequency response identification methods, is a treatment of parameter estimation for models represented by difference and differential equations. Sufficient examples are provided to allow the student to see how each method works in practice.

PREFACE

xix

In Part IV we begin the treatment of control system design. Part IVA (Chapters 14-19) deals with single loop control systems and introduces the basic principles of controller structure (e.g. feedback, feedforward, cascade, ratio, etc.) and controller tuning methodology. The choice of controller type is discussed for processes having the various types of process dynamics described in Part II. Physical examples are used to illustrate the control system design in practical engineering terms. Control system design for multivariable processes having interactions is introduced in Part IVB (Chapters 20-22). Methods of characterizing loop interactions, choosing loop pairing, and designing various types of multivariable controllers are presented and illustrated through physical process examples. While not bringing the reader to the frontiers of research, this section of the book acquaints one with the most important issues in multivariable control and provides approaches to control system design which will work adequately for the overwhelming majority of practical multivariable control problems encountered in practice. Part IVC (Chapters 23-26) introduces the principles of sampled-data process control. This begins with the modeling and analysis of discrete-time systems, develops stability analysis tools, and finally provides control system design methods for these dynamic systems. In Part V (Chapters 27-30), we provide the reader an overview of important special topics which are too advanced to be covered in great depth in this introductory book Among the subjects included are model predictive control, statistical process control, state estimation, robust control system design, control of spatial profiles (distributed parameter systems), on-line intelligence, and computer-aided-design of control systems. The practicing control engineer will find these approaches already in place in some industrial control rooms and thus needs to be aware of the basic concepts and jargon provided here. The last chapter in Part V consists of a series of case studies where the reader is led through the steps in control system synthesis for some representative chemical processes and then shown the performance of the process after employing the controller. Through these more involved example applications, which draw upon a variety of material from earlier chapters, the reader will have a glimpse of how modem process control is carried out by the practicing engineer. An important aspect of the book is the substantial material provided as Appendices in Part VI. Appendix A is devoted to a summary of modem instrumentation capabilities and P & I diagram notation. Appendix B provides a basic review of complex variables and solution methods for ordinary differential and difference equations. Appendix C provides a summary of important relations and transform tables for Laplace transforms and z-transforms. Matrix methods are reviewed in Appendix D. Finally in Appendix E, existing computer packages for computer-aided control system design are surveyed. There are many who contributed their efforts to this book. Undergraduate students at Wisconsin, Delaware, Colorado, and I.I.T. Kanpur provided extensive feedback on the material and helped find errors in the manuscript. Many graduate students, (especially the teaching assistants) tested the homework problems, caught many of the manuscript errors and contributed in other ways to the project. Special thanks go to Jon Debling, Mike Kaspar, Nolan Read, and Raymond Isaac for their help. We are indebted to Derin

XX

PREFACE

Adebeknn, Doug Cameron, Yuris Fuentes, Mike Graham, Santosh Gupta, and Jon Olson who taught from the manuscrip t and provided many helpful suggestions. Also we are grateful to Dave Smith who read the manuscri pt and provided detailed comment s, to Rafi Sela who provided help with examples , and to l-Lung Chien who contrihnh• cl tn {h,.rt."'" ?7 W~ a~ indc"btcd tc Jcc Mill.-::r, Jill• Trainham, and Dave Smith of Dupont Central Science and Engineering for their support of this project. Our thanks to Sally Ross and others at the U.W. Center for Mathema tical Sciences for their hospitalit y during several years of the writing. The preparati on of the camera-re ady copy for such a large book required an enormous amount of work. We are grateful to Andrea Baske, Heather Flemming, Jerry Holbus, Judy Lewison, Bill Paplham, Stephanie Schneider, Jane Smith, and Aimee Vandehey for their contributions along the way. Special thanks to Hana Holbus, who gave her artistic talents to the figures and provided the diligence and skill required to put the manuscri pt in final form. Thanks also to David Anderson who did the copy editing. The book could not have been completed without the patience, support, and forbearance of our wives, Anna and Nell; to them we promise more time. Finally, the authors credit the atmosphere created by their colleagues at the University of Wisconsin and at the Dupont Company for making this book possible. May 15,1994

Babatunde A. Ogunnaike W. Harmon Ray

process dynamics, modeling, and control

part I KN1fJ RODU C1fliO N

I ~

CHAPTER l.

Introductory Concepts of Process Control

CHAPTER2.

Introduction to Control System Implement ation

Kif we couUfirst k_now wfiere we ar, antl wliitfier we are tencfing, we couU 6etter jucfge wfiat to d arui fww to tfo it. Abraham Lincoln (1809-1865)

part I KNTRODUCTKON In embarking upon a study of any subject for the first time, the newcomer is quite likely to fmd the unique language, idioms, and peculiar "tools of the trade" associated with the subject matter to be very much like the terrain of an unfamiliar territory. This is certainly true of Process Dynamics, Modeling, and Control - perhaps more so than of any other subject matter within the broader discipline of Chemical Engineering. It is therefore frequently advantageous to begin a systematic study of such a subject with a panoramic survey and a general introduction. The panoramic survey provides perspective, indicating broadly the scope, extent, and constituent elements of the terrain; an initial introduction to these constituent elements in turn provides motivation for the subsequent more detailed study. Part I, consisting of Chapters 1 and 2, provides just such an orientation tour of the Process Dynamics, Modeling, and Control terrain before the detailed exploration begins in the remaining parts of the book.

CHAP1rJEJR

1 INTRODUCTORY CONCEPTS OF PROCESS CONTROL A formal introduction to the role of process control in the chemical process industry is important for providing motivation and laying the foundation for the more det~ed study of Process Dynamics, Modeling, and Control c~ntained in the iiJkbmin~apters. Thus this chapter is an introductory oveAAew of process control and how it is practiced in the chemical process industry.

1.1

THE CHEMICAL PROCESS cc••luo~J'ZII0..£...:41Vf

In the chemical process industry, the primary objective is to combine chemical processing units, such as chemical reactors, distillation columns{; extractors, rl'e> IW.AJf•'~~ evaporators, heat exchangers, etc., integrated in a rational fas ion into a chemical process in order to transform J;awf'ma~~I:ials and input"eh"ergy into finished products. This concept is illustrated in Figure 1.1 and leads to the definition: 4'$.$~!r!.f..!EE...E~o~sing

unit, _or combinations of processi~g ~nits, used for the converstofiroj raw matenals (through any combznahon of chemical, physical, mechanical, or thermal changes) into finished products, is a chemical process. m-."tD.t:r~,Q·KR~u q~~t.t~...

,~;..:·::;~-:.;,·.;:~:;;·, , .

A concrete example of these somewhat abstract ideas is the crude fractionation section of a typical oil refinery illustrated in Figure 1.2. Here, the raw material (in this case,cf'Jde"':,iif"is numped from the "tank farms," ~•l¥_rAUo~{q.,.~

1.

.. It ~s dafi[~f!~ (o operate !he processing units safely. ThiS ~eanCJnat no Uhlt should be operated at, or near, cond1tions rc;;rrsid'e'rtcl _t~"'be potentially dangerous either to the he~!_t~,£!..!P;t; -'llu<w,· ··' operators ()t>to34t!;t!t~.. of.,!l!e~guipment. The safe~}?!Jhe .~~~iat{ as well as the remote, e!)-YJ!.~i:Ullent also comes intofconsr8era~ere. Process operating conditions th.(lJ,J!\ay lead to the ~lahon of ... Ql w ::f~ environmenta l regulations must be .!!-Voided. Q

'c;;Aitoltt()H,Hw'i

2.

Specified production rates must be maintained. The amount of product output required of a plant at any point in time is usually dictated by market requirements . Thus, production rate specifications must be met and maintained, as much as possible.

3. Product quality specifications must be maintained. tions must either be specif~"c Protiucts not meeting the required qualitv !f>(a.fow.Mii :A ~f~.J:~~ o'& • 0 ~-'6 ~u.&q.!t> 1 ~.-i~()~-'1. ~J$1Uv1 I · mscarged as waste, ~r~. ~~~£.~ poss1ble, !.~.If[~__!L§e at extra cost. The ut!1ization of resources therefore provides the need for economic Crqr.-'ji,CJJ J-':f~~uJ:.~I f.n motivation for striving to satisfy product quality specifications . ;:,.t-..-a'"'

For the process shown in Figure 1.2, some operating constraints mandated by safety would be that the furnace tubes should not~~~ their metallurgical temperature limit and the fractionation unit should not exceed its pressure rating.

CHAP 1 INTRODUCTORY CONCEPTS OF PROCESS CONTROL

7

AirFuelGas

CRUDE OIL FEED STORAGE TANKS

ne"'!• 1 TOn Y-O

FURNACE

PIPESTILL FRACTIONATOR

Figure 1.2. The upstream end of an oil refinery. ~t-•V' point upw-!t9s. for a product such as naptha or gas oil, one could produce more of it, but it would have a lower quality (i.e., more high-boiling materials). Now, chemical processes are, by nature, dynamic, by which we mean that their variables are always changing with time. It is clear, therefore, that to achieve the above noted objectives, there is the need to monitor, and be able to ~"'f:U~t"~nge in, those key process variables that are related to safety, prifduction rates, and product quality. nr11.~oo or44'jo.

This dual task of:

1. 2.

Monitoring certain process condiJ!on i[ldicator variables, and, Inducing changes in the app~(r~ process variables in order to improve process conditions a~r~"l'~'"'»t

is the job of the control system. To achieve good designs for these control ~~ 'or..tfMT:'systems one must embarl< on llie study of a new field, defined as follows: '"' b·· k

Process Dynamics and Control is that aspect of chemical engineering coJCetfzt'tt'1.vith the analysis, design, and implementation of control systems that fa'?i'fi~~6i~T the uchievement of specified objectives of process safety, production rates, and product quality.

INTRODUCTION

8

Residue

Ilea\"y Gas Oil Boiling Point

Light Gas Oil

Naptha

0

Figure 1.3.

1.2

Cumulative Percent Boiled Off

Crude oil boiling point curve illustrating the product distribution of a light crude oil and a heavy crude oil.

AN INDUSTRIAL PERSPECTIVE OF A TYPICAL PROCESS CONTROL PROBLEM

The next phase of our presentation of introductory concepts involves the definition of certain terms that are routinely used in connection with various components of a chemical process, and an introduction to the concept of a process control system. This will be done in Sections 1.3 and 1.4. To motivate the industrial discussion, however, let us first examine, in this section, a typical"01"\61tlrktt(*Tifl~) to well he '! typical attempt P rocess control problem, and present what may .!j,!L&'o,o~lHMq between discussion solve such problems, by following a simulated, but phiusible, r 1>'l;obf ~- ,, .,,..o;o>8~ a plant engineer and a control engineer. to svstems go, this particular example is dplihrahi!l}rrhosen, As industrial :J.'"'Yl~' w(>o~e--u JIJ.t'e""'fatf• o11.tt)::lf[;nr,. be simple, yet po~ing enough important problematic features to cap!uri on how high the furnace tube temperatures can get So, of your two process outputs, Fo- and T, the former is set externally by the fractionator, while the latter is the one you are concerned about controlling? Yes. Your control objective is therefore to regulate the process output T as well as deal with the servo problem of set-point changes every 2-3 days? Yes. Of your input variables which ones do you really pave control over? Only the air flowrate, and the fuel flowrate; and even then, we usually preset the air flowrate and change only the fuel flowrate when necessary. Our main control variable is the air-to-fuel ratio. The other input variables, the crude oil feed rate F, and inlet temperature T;, are therefore disturbances? Yes. Any other process variables of importance that I should know of? Yes, the fuel supply pressure Pp and the fuel's heat content A.F; they vary significantly, and we don't have any control over these variations. They are also

disturbances. CE:

What sort of instrumentation do you have for data acquisition and control action implementation?

PE:

We have thermocouples for measuring the temperatures T and T;; flow meters for measuring F, QF; and a control valve on the fuel line. We have an optical pyrometer installed for monitoring the furnace tube temperature. An alarm is tripped if the temperature gets within a few degrees of the upper limit constraint.

Phase2 CE: PE:

CE: PE:

Do you have a process model available for this furnace? No; but there's an operator who understands the process behavior quite well. We have tried running the process on manual (control) using this operator, but the results weren't acceptable. The record shown below, taken off the outlet temperature strip-chart recorder, is fairly representative. This is the response to a step increase in the inlet feedrate F. (See Figure 1.5). Do you have an idea of what might be responsible? Yes. We think it has to do with basic human limitations; his anticipation of the effect of the feed disturbance is ingenious, but imperfect, and he just couldn't react

T*

Temp.

·. No Control

Time

Figure 1.5. System performance under manual control


CE:

11

fast enough, or accurately enough, to the influence of the additional disturbanc e effects of variations in fuel supply pressure and heat content. Let's start with a simple feedback system then. Let's install a temperature controller that uses measureme nts of the furnace outlet temperatur e T to adjust the fuel flowrate QF accordingly [Figure 1.6(a)}. We will use a PID controller with these controller parameter values to start with (proportional band = 70%, reset rate = 2 repeats/mi n, derivative time = 0). Feel free to retune the controller if necessary. Let's discuss the results as soon as you are ready.

Phase 3 PE:

CE:

The performance of the feedback system [see Figure 1.6(b)], even though better than with manual control, is still not acceptable; too much low-tempe rature feed is sent to the fractionator during the first few hours following each throughput increase. (After a little thought) What is needed is a means by which we can change fuel flow the instant we detect a change in the feed flowrate. Try this feedforwarrl control strategy by itself first (Figure 1.7); augment this with feedback only if you find it necessary (Figure 1.8(a)).

Phase4 PE:

PE:

CE:

With the feedforward strategy by itself, there was the definite advantage of quickly compensating for the effect of the disturbance, at least initially. The main problem was the nonavailability of the furnace outlet temperature measurement to the controller, with the result that we had offsets. Since we can't afford the persistent offset, we had to activate the feedback system. As expected the addition of feedback rectified this problem (Figure 1.8(b)). We have one major problem left the furnace outlet temperatur e still fluctuates, sometimes rather unacceptab ly, whenever we observe variations in the fuel delivery pressure. In addition, we are pretty sure that the variations in the fuel's heat content contributes to these fluctuations, but we have no easy way of quantitatively monitoring these heat content variations. At this point, however, they don't seem to be as significant as supply pressure variations. Let's focus on the problem caused by the variations in fuel supply pressure. It is easy to see why this should be a problem. The controller can only adjust the valve on the fuel line; and even though we expect that specific valve positions should correspond to specific fuel flowrates, this will be so only if the delivery pressure is constant. Any fluctuations in delivery pressure means that the controller will not get the fuel flowrate it asks for. We must install an additional loop to ensure that the temperatur e controller gets the actual flowrate change it demands; a mere change in valve position will not ensure this. We will install a flow controller in between the temperatur e controller and the control valve on the fuel line. The task of this inner loop controller will be to ensure that the fuel flowrate demanded by the temperatur e controller is actually delivered to the furnace regardless of supply pressure variations. The addition of this cascade control system should work well. (See Figure 1.9 for the final control system and its performance.)

Having overheard the successfu l design and installatio n of a control system, let us now continue with our introducti on to the basic concepts and terminology of process control.

INTRODUCTION

12 T

No Control

Time (b)

(a)

Figure 1.6. The feedback control system. T Flow Measurement

Feedforward Controller

No Control

Time (b)

(a)

Figure 1.7. The feedforward control system.

T T* ••••• •••

·...... ... No Control

Time (b)

(a)

Figure 1.8. The feedforward/ feedback control system.

T

T" •••••

··..... ········.... No Control Time

(a)

(b)

Figure 1.9. The final control system (feedforward/feedback-plus-cascade).


1.3

13

VARIABLES OF A PROCESS

The state of affairs within, or in the immediate environment of, a typical processing unit is usually indicated by such quantities as temperature, flow,rates in and out of containing vessels, pressure, composition, etc. Th~s~_are_feg;edJQ, as the variables of the process, or process variables. ' J.f&p.l ~. It is clislQma y to classify these variables according to wlj~~J;er they simply provide information about process conditions, or whether they are rn~c~vfl.!!Je influencing process conditions. On the first level, therefore, there ar~~b categories of process variables: input and output variables.

ft

Input variables are tlwse thnt independently stimulate the system and can thereby induce change in the internal conditions of the process. Output variables are those by which one obtains information about the internal state of the process . ....oifD3..q~tW ,CDoT,c•lcr~t:tsr<J

It is !1J?£WPriate, at this point, to introduce what is called a state variable and di~fm~1sh it from an output variable. State variables are generally WlJ 01f1S.t.4

recogruzea s:

That mzmmum set of variables essential for completely describing the internal state (or condition) of a process. The state variables are therefore the true indicators of the internal state of the process system. The actual manifestation of these internal states by measurement is what yields an output. Thus the output variable is, in actual fact, some measurement either of a single state variable or a combination of state variables. On a second level, it is possible to further classify input variables as follows: 1.

Those input variables that are at our disposal to manipulate freely as we choose are called manipulated (or control) variables.

2.

Those over which we have no control (i.e., those whose values we are in no position to decide at will) are called disturbance variables..

Finally, we must note that some process variables (output as well as input variables) are directly available for measurement while some are not. Those process variables whose values are made available by direct on-line measurement are classified as measured variables; the others are called unmeasured variables (see Figure 1.10.) Although output variables are defined as measurements, it is possible that some outputs are not measured on-line (no instrument is installed on the process) but require infrequent samples to be taken to the laboratory for analysis. Thus for control system design these are usually considered unmeasured outputs in the sense that the measurements are not available frequently enough for control purposes.

INTRODUCTION

14 INPUT Disturbances Unmeasured

Measured

Measured INPUT Manipulated ==~:>1 Variables

Figure 1.10.

CHEMICAL

PROCESS

The variables of a process.

Let us now illustrate these ideas with the following examples. Mbo41'\,,...l ..e....A."'

Example 1.1

THE VARIABLES OF A.s-pRRED HEATING TANK "" "'· PROCESS.

Consider the stirred heating tank process shown in Figure 1.11 below, in which it is required to regulate the temperature of the liquid in the tank (as measured by a thermocouple) in the face of fluctuations in inlet temperature T;. The flowrates in and out are constant and equaL F,T1

F,T

!

Q:Steam Flowrate

Condensate

INPUT VARIABLES: Q- steam rate in [MANIPULATED] T1- inlet stream temparature [DISTURBANCE] OUTPUT VARIABLE: T- tank temperature [MEASURABLE]

Figure 1.11.

The stirred heating tank process. .

.

In this case, clearly our main concern is with the temperature of the liquid in the tank; thus T is the output variable. It is, in fact, a measured output variable, since it is measured by a thermocouple. Observe now that the value of this variable Tis affected by changes in the values of both T; and Q. These are therefore the input variables. However, only Q can be manipulated at will. Thus, T; is a disturbance variable, while Q is the manipulated

variable.

Let us now formally consider the variables of the industrial furnace discussed in Section 1.2.

r


15

THE VARIABLES OF AN INDUSTRIAL FURNACE

Example 1.2

Referring back to the description of the process given earlier in Section 1.2, it is clear that T, the outlet temperature, is om.putput variable. Next, we note that the value of this variable is affected by a ~~'!~I o-ther variables that must be carefully considered in order to classify them prgperly.- '"" ""'-~te>. ."1>--«u.•w.n Of all the variables that can affect the value ofT, only Q,'\' the air flowrate, and Qp, the fuel flowrate, can be manipulated,!t wiLl; they are therefore the manipulated (or control) variables. ""H,......,.,oca.-1"'~·~' The other variables, F (the inlet feedrate), T; (the inlet temperature), Pp (the fuel supply pressure), and Ap (the fuel's heat content), all vary in a manner that we cannot control; hence they are all disturbance variables. This process, therefore, has one output variable, two manipulated input (i.e., control)

I I

variables, and four disturbance variables.

One final point of interest. As we shall see later on in Chapter 4, wh~JY~ A... ll-• _ r"o -. take up the issue of mathematical descriptions of process systems, it is t!ti}'Iy '~ ' o ,;--'·'''' common to represent the process variables as follows: -fo-._,J; the the the the

y u

d x ~eo....&l!'""'"'4

output variable input (control) variable disturbance variable, and state variable (whenever needed)

....,.D...,.......

The appropnate corr~sponding vector quantities, y, u, d, and x, are used whenever the variables involved in each category number more than one. We shall adopt this~~~ subsequent discussion. u

~~~

~K..

our

~Ci!IIUI.rc

,...~,~

1.4

THE CONCE!'! OF A PROCESS CONTROL SYSTEM

As earlierll1td:., the_dynamic (i.e., ever changing) nature of chemical processes makes it 0 ' era~~ that we have some means of effectively monitoring, and ·~ ~r-afiV inducing en ge m, the process variables of interest. In a typical chemical process (recall, for exam.l?_~e_J.!te fumac.I}~J §~~ t.2) the process control system is thMht\\'Y 1IUIF'rs''t:'itk~7!'d"- with~ th~ responsibility for monitoring outputs, m~'klng decisions about how best to manipulate inputs so as to obtain desired output belu!vior, and effectively implementing such decisions on the process. It is therefore convenient to break down the responsibility of the control system into the following three major tasks: 1.

2.

3.

Monitoring process output variables by measurement Making rational decisions regarding what corrective action is needed on the basis of the information about the current and desired state of the process Effectively implementing these decisions on the process · ;5o..wJU«.tff.,

J

~

When these tasks are carried out rrwi.~!!M by a human operator we have a manual control system; on the other h~cf,' control system in which these tasks are carried out in an automatic fashion by a machine is known as an

a

INTRODUCTION

16

automatic control system; in particular, when the machine involved is a computer, we have a computer control system. With the possible exception of the manual control system, all other control systems require certain hardware elements for carrying out each of the above itemized tasks. Let us now introduce these hardware elements, reserving a more detailed discussion of the principles and practice of control system implementation to Chapter 2.

1.4.1

Control System Hardware Elements

lI

The hardware elements required for the realization of the control system's tasks of !.!1f:es.~;,~pent, decision making, and corrective action impleme_l!fa~"" typically f~ll into the following categories: sensors, controllers, transmffters: l'l?;i-«2~ 1 --~ '"'-:;-, ..... , [.c. and final control elements.

I

Sensors .... ~~prt~1-!t,~c.1~2nlh

·

The first task, that of a,s~~!ng information about the status of the process output variables, is carriea" out by sensors (also called measuring devices or primary elements). In most process control applications, the sensors are usually needed for pressure, temperature, liquid level, flow, and composition measurements. Typical examples are: thermocouples (for temperature measurements), differential pressure cells (for liquid level measurements), gas/liquid chromatographs (for composition measurements), etc.

Controllers The decision maker, and hence the "heart" of the control system, is the controller; it is the hardware element with "built-in" capacity for performing the only task requiring some form of "intelligence." The controller hardware may be pnei;£matic in nature (in which case it operates on air signals), or it may be electronic (in which case, it operates on electrical signals). Electronic controllers are more common in more modem industrial process control applications. The PAr~WilJic and electronic controllers are limited to fairly simple operations "Which we shall have cause to discuss more fully later. When more complex control operations are required, the digital computer is usually used as · a controller.

Transmitters How process information acquired by the sensor gets to the controller, and the controller decision gets back to the process, is the responsibility of devices known as transmitters. Measurement and control signals may be transmitted as air pressure signals, or as electrical signals. Pneumatic transmitters are required for the former, and electrical ones for the latter.

Final Control Elements Final control elements have the task of actually implementing, on the process, the control command issued by the controller. Most final control elements are

I

I

I


17

control valves (usually pneumatic, i.e., they are air-driven), and they occur in various shapes, sizes, and have several modes of specific operation. Some other examples of final control elements include: variable speed fans, pumps, and compressors; conveyors; and relay switches.

Other Hardware Elements In transmitting information back and forth between the process and the controller, the need to convert one type of signal to another type is often unavoidable. For example, it will be necessary to convert the electrical signal from an electronic controller to a pneumatic signal needed to operate a control valve. The devices used for such signal transformations are called trpii~fiew,.,. and as will be htrther discussed in Chapter 2, various types are availa Te for .. various signal transformations. Also, for computer control applications, it is necessary to have devices known as analog-to-digital rAID\ and d~·ital-to-analov (DI A) converters. ,.~,>~~Ytlfi~'*-(r.. ) e &c.-l' 6 J.eeJ}.l; noted. Further reflection on this strategy will, however, also··· W,:~fiE a I<J:=:=:;-,

transducer

Ell transducer

'!--If--

Air pressure, 60psig

PI

l'===~l===--- Gas flow Orifice

Figure 2.10.

a!

Control sign from computer

Interfacing for the gas storage vessel.

INTRO DUCTI ON

58 ---~50%

60r----------------------

25%

0

150 Time{s)

Figure 2.11.

Dynamic behavio r of the gas storage vessel under DOC

control.

2.5.2 A Commercial DCS Network Example ation, consid er the To provid e an example of an actual commercial DCS install Figure 2.12. The in shown plastic ering engine proces s for manuf acturin g an concen trated by be must which tank hold a in n solutio feed a with process begins polym erizati on where #1 r reacto to two-st age evaporation. Then it is passed . begins. s into a flashin g From reacto r #1 the materi al goes throug h gear pump to reacto r #2 sent is it Then ed. remov are ucts section where volatile byprod is an on-line there where with its flashing unit and finally to a finishing stage stage, the final the At . quality ct produ er rheom eter for monito ring polym is a great there that Note ent. shipm for ed packag and zed pelleti materi al is rs, and reacto s, pump ate duplic deal of redun dancy in the proces s with

REACTO R

n

Figure 2.12.

DCS screen showin g process overvie w.

CHAP 2 INTRODUCTION TO CONTROL SYSTEM IMPLEMENTATION

59

pelletizing lines. This allows one of the redundant units to be shut down for cleaning while the other carries the processing load. The task of the DCS for this process is to handle approximately 100 control loops (some of them cascade) involving temperature, pressure, flow, chemical composition, and product quality. The DCS has many other detailed graphical displays of the process flowsheet showing all of the loops in whatever level of detail desired. For example, the upstream evaporation section of the plant can be seen in more detail in Figure 2.13. Note that each element is denoted by a letter A-Z and the measuring devices, transmitters, controllers, etc., are denoted by standard symbols (see Appendix A). This graphic is sometim.es called a process and instrumentation (P&I) diagram. For most measurements, the current values are displayed on the screen. For example, the feed solution level in the hold tank (top left) is controlled by a level measurement and transmitter, ~, which sends a signal to the level controller, [!9, which adjusts the valve controlling the flow of the feed solution to the hold tank. The graphic hides the fact that a multitasking DCS computer actually receives the level measurement and transmits the command to the feed solution valve. The architecture of the relatively small commercial DCS network that is controlling this plant is shown in Figure 2.14. The process has analog and ~tal signals which are sent via transmission lines to the termination units, ~, to local A/D, D/ A, and digital 1/0 slave units; there is also a field bus interface that takes processed information from sensors and actuators having their own CPU (e.g., mass flowmeters, FTIR sensors, digital valve positioners, etc.). This information goes to the lowest level data highway, the slave bus. EVAPORATOR DETAIL

Figure 2.13.

More detailed DCS graphic display of evaporation section of Figure 2.12.

INTRODUCTION

60

MASS

FLOWMETER

POSITIONER FIELD BUiS-.-j (9600 baud, 115 uo"'~·'!-----1 ON-LINE RHEOMETER

Figure 2.14.

Commercial DCS network for process in Figure 2.12.

Sitting above the slave bus are several multitask ing CPU's called multifunction processors (MFP's). For controllin g the process in Figure 212, only two MFP's are required - one for the upstream part of the process (up to reactor #2) and one for the remaining part of the process. Each of these MFP's has limited memory and uses the same CPU as found in common personal computers. However, by using multiple MFP's the computat ional load can be handled. In order to ensure against computer failures, each MFP has a duplicate which is sitting idle ready to take over in case of a computer malfuncti on. The MFP's handle all of the local controller algorithm s and report the measurem ents and controller actions to the controlway, which is the next level data highway. The data from the controlwa y is sent to a Network Processing Unit (NPU) which is a computer linking the controlwa y to the Central Communication

CHAP 2 INTRODUCTION TO CONTROL SYSTEM IMPLEMENTATION

61

Network. On this central network hang the operator's console and displays, engineering workstations, and the host (supervisory) computer. The Central Communication Network can be accessed by computers throughout the plant, and by remote connections anywhere in the world. Thus the engineer at the company headquarters on the east coast can see on his computer screen the same displays the operators see at the plant in Texas. To illustrate how this DCS works, consider the level control of the feed hold tank discussed above. The level sensor @] sends the level signal to an AID slave unit; this measurement goes on the slave bus and is picked up by MFP #1 (which is responsible for the upstream part of the plant). MFP #1 calculates the change in the feed solution valve position to be sent to the valve. This value is put on the slave bus and picked up by a D I A slave IIO device which transmits the command to the valve. Simultaneously, the MFP sends a report of measured level and control action taken to the controlway. This report is picked up by the NPM and transmitted to the Central Communication Network. The data appear on the operator console graphic display unit as a new point on a trend chart, and appear in numerical form next to the valve in the display shown in Figure 2.13. The data are also logged by the host computer and filed in archival data storage. To minimize the data traffic on each of the data highway levels, there are usually several timescales at which sensors report data and control action taken. Thus for variables that change slowly (such as temperatures in large processes), measurements and control action can be taken infrequently (e.g., every minute); for variables that change rapidly (e.g., flowrates in a transfer line), it is necessary to use a fast sampling and control action timescale (e.g., 1 s). A second way to minimize data traffic is to only report changes in variables within some band of values. Thus if the last reported value of temperature was 41.5"C and the report bounds are± O.s·c, then no changes would be reported until the temperature fell below 41.0"C or above 42.0'C.

2.6

SUMMARY

In this chapter we have presented a brief introduction to the hardware and

software elements required for control system implementation, with particular emphasis on on-line data acquisition and computer control. It is hoped that this brief overview will provide the reader with some perspective about how control system designs are currently brought to the plant and implemented. Since any control system design that is not implemented and used by the plant is economically useless, having a good grasp of these practical realities is essential to a successful control system design project. Fortunately, if the necessary sensors and actuators are available, almost any control algorithm that can be programmed into a digital computer can be implemented in hardware. This situation provides a challenge to the imagination of the control engineer as well as an incentive to the analytical chemist to develop a wider range of reliable on-line instrumentation.

INTRODUCTION

62 SUGGESTED FURTHER READING

of Industrial Control, ISA Pub!., Albert , C. L. and D. A. Cogga n (Ed.), Fundamentals (1992) na Research Triang le Park, North Caroli l, ISA Pub!., Pittsbu rgh (1978) Harris on, T. J. (Ed.), Minicomputers in Industrial Contro ations to Data Acquisition Applic with ting Compu me Retd-Ti Mellic hamp, D. A. (Ed.), (1983) and Control, Van Nostra nd Reinhold, New York , Boston (1989) (paper back Ray, W. H., Advanced Process Control, Butter worths reprin t of origina l1981 book)

1. 2. 3. 4.

REVI EW QUES TION S 1.

ter data acquis ition and contro l What is the genera l structu re of a digital compu system?

2.

What are their characteristics and What are the basic elemen ts of a digital compu ter? cost?

3.

octal to decimal numbe rs? What is the formula for converting from binary to

5.

one compu te the resolu tion of a How do AID and 0/A conver ters work? How can 20-bit AID conver ter? ter. List some analog inputs /outpu ts to a control compu

6.

List some digital inputs /outpu ts to a contro l compu

7.

chemic al compo sition on-line in a If one wished to measu re temper ature, pressu re, and s? sensor of choice good a be would what , chemic al reactor

4.

8.

9.

10. 11. 12.

13.

ter.

electri cal noise to the level Suppo se there is an electri c motor nearby adding ion this signal before sendin g condit one would How tank. a measu remen t signal from it to the compu ter? adjust ing extern al heatin g, what If one wished to contro l the tempe rature in a tank by le? availab usually rs actuato of s are the choice How do they differ in hardw are What are the two modes of compu ter contro l? require ments? l system (DCS)? What are the essential elemen ts of a distrib uted contro one of the compu ters in a DCS How does one ensure safe proces s operat ion in case fails? each new plant you see in the next What is the likelih ood of encoun tering a DCS in ? decade

PROBLEMS 2.1

Conve rt the following binary numbe rs to octal numbe (a) 100 (b) 101 (c) 111

010

000 101

001

110

111

111 110 100

101 011 011

rs:

CHAP 2 INTRODUCTION TO CONTROL SYSTEM IMPLEMENTATION 2.2

63

Convert the following octal numbers to decimal numbers: (a) 747

(b)440

(c) 100 (d) 556

2.3

A Fortran program consists of 100 lines with approximately 25 characters per line. Estimate the number of words of memory in a 16-bit machine required to store this Fortran code.

2.4

In responding to interrupts from various peripheral devices, a computer manufacturer recommends the following priority levels: Terminal-4 Real-time dock - 7 Disk drive- 7 Magnetic tape unit -7

Line printer 4 AID converter-6

Discuss why some of these peripherals are assigned higher priorities than others.

2.5

A computer has a 12-bit A/D converter with- 10 V to + 10 V span. Determine the analog input voltages and relative error for the following integers corning from the converter: (a) 570 (b) 2000

2.6

(d) 25

Provide the answers to Problem 2.5 for a 10-bit AID converter with a- 5 V to+ 5 V span and for the following integers: (a) 450

2.7

(c)- 960

(b) -200

(c) 100 (d) 25

It is desired to output the following voltages through a 12-bit D/A converter with a -10 V to+ 10 V span. Determine the integer that should be loaded into the D/ A in ' each case. (a) -1 V

(b) +2.7V (c)+8.7V (d)-6 v

2.8

Two computers are linked over a 2400-baud telephone line. The operators of one computer would like to transfer a Fortran program to the other computer over this telephone link. The Fortran program in question has about 500 lines of code with approximately 30 characters per line. Estimate how long it will take to transfer this program. An engineer suggests connecting the two computers via a 9600-baud cable link. How long would the program transfer take in this case?

part II JPJROCCES§ DYNAMKCS The main justification for a separate study of Process Dynamics as a precursor to the study of Process Control lies in the simple fact that to succeed in forcing a system to behave in a desired fashion, we must first understand the inherent, dynamic behavior of the system by itself, without assistance or interference from the controller. The task of designing effective control systems can best be carried out if it is based on a sound understanding of inherent process dynamics. Thus Part TI, consisting of Chapters 3-11, will be devoted to analyzing and characterizing the various types of dynamic behavior expected from a chemical process. The concept of employing a process model as a "surrogate" for the actual process is central to these studies. It allows us to carry out systematic analysis of process behavior most efficiently; it also provides the criteria for characterizing the behavior of an otherwise limitless number of actual processes into a relatively small number of well-defined categories.

part II JPJROCCJES§ DYNAMKCC§ CHA PTER 3.

Basic Elem ents of Dyna mic Analy sis

CHA PTER 4.

The Process Mode l

CHA PTER S.

Dyna mic Beha vior of Linea r Low- Orde r Syste ms

CHA PTER 6.

Dyna mic Beha vior of Linea r High er Orde r Syste ms

CHA PTER 7.

Inver se-Re spons e Syste ms

CHA PTER S.

Time -Dela y Syste ms

CHAP TER9 .

Frequ ency- Respo nse Analy sis

CHA PTER lO.

Nonl inear Syste ms

CHA PTER ll.

Stability

"... always form a aefinitiml ... of wnatever presents itself to tfte mitul; strip it na~ ant{ CooK_. at its essential nature, contemplating tfte wfwfe th.rougli its separate parts, ant{ tftese parts in tlieir entirety.' Marcus Aureliu s, MeditatiOIIS

CHAPTER

3 BASIC ELEMENTS OF DYNAMIC ANALYSIS There are important fundamental concepts, terminology, and mathematical tools employed in the analysis of process dynamic behavior. Familiarity with these is essential for the discussion in Part II. Thus, the objective of this chapter is to introduce these elements of dynamic analysis.

3.1

INTRODUCTION

As defined in Chapter 1, the input variables of a process are those variables capable of independently stimulating the process, and inducing changes in its internal conditions. Such changes in the internal state of the process are usually apparent from observing changes in the output variables of the process. By how much, and in what manner, the process responds to a change in the input variable clearly depends on the nature of this input change; it also depends on the intrinsic nature of the process itself. For example, the insertion of a hot piece of iron into a beaker of water will elicit a water temperature response different in magnitude and character from the response obtained by inserting the same piece of iron into a larger body of water such as a lake. For any given input change, the observed process response provides information about the intrinsic nature of the process in question. By the same token, if the intrinsic nature of the process is known, and can be properly characterized, then the response of the process to any input change can be predicted. Process Dynamics is concerned with analyzing the dynamic (i.e., timedependent) behavior of a process in response to various types of inputs. It is by such studies that we are able to characterize a wide variety and limitless number of actual processes into a relatively small number of well-defined categories. The advantages that such process characterization provides cannot be overstated, as will become clearer when we take on the issue of control system design in Part IV. To illustrate this point, consider the possible responses to a step change in 67

68

PROCESS DYNAMICS

+~~= ~t)~ ~·~ rtcJ~

y(t)

u(t)

•-i

Linear Process

~

_J

+=~-

~.,~ ~.,~ ~.,F Figure 3.1. Possible responses of a linear process.

the input u(t) for a linear dynamic process. Depending on the process, the dynamic response could take one of the forms shown in Figure 3.1. It is important for the control system designer to understand the type of process model which could produce such dynamic behavior. This will be one of the goals of our study of process dynamics. The tools required for carrying out the analysis of dynamic process behavior will be reviewed in the next section.

TOOLS OF DYNAMIC ANALYSIS

3.2

3.2.1 The Process Model and Ideal Forcing Functions The main objective of dynamic analysis is to investigate and characterize system behavior when a process is subjected to various types of input changes. To be sure, such investigations can be carried out on the actual physical processing units themselves: changing the inputs, .and recording the corresponding responses in the output variables for subsequent analysis. However, this approach suffers from two major problems that are immediately obvious: 1.

The physical process may not even exist yet. This situation arises when the issues of control systems design and analysis are being confronted before the actual construction of the plant.

2.

It is time consuming and expensive.

Most real chemical processes have such large capacities that once disturbed, a considerable amount of time must elapse before they

CHAP 3 BASIC ELEMENTS OF DYNAMIC ANALYSIS

69

finally "settle down" again. Carrying out several such investigative experiments must therefore be time consuming. Furthermore, regular process operation must often be suspended while carrying out such investigations; the financial implications of tying up profit-making processing units for extended periods of time are quite obvious. This is in addition to the cost of acquiring and analyzing data, which, in some cases, may not be trivial. There is an additional, and perhaps not so obvious, setback: certain critical theoretical information, useful in characterizing the behavior of entire classes of processes, are not easily available from the results of such strictly experimental investigations carried out on single, specific processes. Thus the dynamic analysis of process behavior is carried out with the aid of some form of a mathematical representation of the actual process. Such theoretical analyses, of course, only provide knowledge about idealized process behavior, but this has proven very useful in understanding, and in characterizing, real behavior. · The essence of dynamic analysis may therefore be stated:

Given some form of mathematical representation of the process, investigate the process response to various input changes. That is, given a process model, find y(t) in response to inputs u(t), d(t). Observe then that any successful dynamic analysis requires first and foremost

1. 2.

A process model, and Well characterized input functions (also referred to as "forcing functions" because they are used to force the observed output response).

In addition to these requirements, however, we note that the actual task of obtaining .the process response involves solving the mathematical equations that arise when the specified forcing function is incorporated into the process model. This is the central task in process dynamic analysis. As is to be expected, the successful performance of this task requires the employment of certain tools of mathematical analysis. The most commonly utilized of these tools will now be summarized.

3.2.2 Mathematical Tools As will be seen later, process models are usually in the form of differential equations: linear and nonlinear, ordinary and partial. A familiarization with such equations, especially how they are solved, will therefore be important in carrying out dynamic analysis. In this regard, one of the most important tools we will use is the Laplace transform, a technique that, among other things, converts linear, ordinary differential equations into the relatively easier-to-handle algebraic equations. It also converts partial differential equations into the ordinary type. Beyond this, however, by far the most significant contribution of the theory of Laplace transforms to dynamic analysis lies in the fact that it is responsible for the development of the transfer function model form, which is still the most widely used model form in process control studies.

70

PROCESS DYNAM ICS

requires the use of When process models are nonlinear, accurate analysis. s behavior. If proces of tion simula ter compu a out g numerical methods in carryin equati ons ear nonlin the the emphasis is on simplicity rather than accuracy, linear such from ed deduc ior behav s may be linearized and approximate proces . model s approximations of the proces form of difference Process model s used for compu ter control take the can be solved rparts, counte on equati ntial equations, which, like their differe on forms by functi r transfe lent equiva to d reduce or linear, analytically when Lapla ce the of alent equiv e the use of z-transforms - the discrete-tim transform. t variab les, i.e., Proce sses that have multip le input and outpu t, by severa l expec would we multiv ariabl e system s, are model ed, as g with such dealin in es matric of yment emplo differential equations. The intractable or s tediou ise otherw an s reduce tically drama imes equati ons somet problem to a manageable one. y consid ered most We have thus identified the mathematical tools usuall (and difference) ntial impor tant to the study of dynam ic analysis, viz., differe s. Thus a matrice and forms, z-trans equations, numerical methods, Laplace and worki ng ed requir the e provid to ed design cally specifi discussion of these tools, on of a inserti an that er, understanding, appea rs desirable. We observe, howev tute a consti must either point this at discus sion of each of these topics y to be sketch too be must or hand, at task main the from tion prolonged distrac B-D, and the reader is useful. Such discussions can be found in Appendices will introd uce only we Thus . needed as often encouraged to refer to these as - leavin g a fuller 24 er Chapt in sforms z-tran and here rms Laplace transfo discussion of each of these topics for Appendix C.

3.2.3 The Digital Com puter presen ted simpl y as On a strictly conceptual level, Process Dynamics has been s types of inputs , the analysis of the dynam ic response of processes to variou d out using carrie lly typica are es with the under standi ng that such analys tools listed l matica mathe the ng applyi and s model mathe matica l process nical aspect mecha actual, above. Beyond the conceptual, however, there is the and the ons equati bing descri the of on of dynam ic analys is: the soluti · form. cal graphi in y usuall , presentation of the results by mathe matica l Altho ugh there are classes of processes repres ented ons, a substantial equati bing descri the models that allow analytical solution of are not easily that ons equati by ed model are t interes of numbe r of processes graph ical the case, the amena ble to analyt ical soluti ons. Whate ver obtain ed are results such er (wheth se presen tation of the dynam ic respon digita l the of use The e. exercis s tediou a often is ise) analytically or otherw ical numer obtain to le possib computer in carrying out dynamic analysis makes it and icable, iriappl are ds metho ical analyt solutions to modeling equations when even when analytical relieves us of the somet imes tediou s work involv ed cs capabilities of graphi the solutions are possible. Of no less importance are ses directly in respon ic dynam the of tation the computer, facilitating the presen ic analysis dynam out carry to ary custom more re therefo is graphical form. It severa l the of one y emplo studies with the aid of the digital compu ter and to ed for design ically specif ges packa re curren tly available compu ter softwa ms progra these l, genera In . inputs s variou to se respon s simulating the proces


71

are designed to accept a process model, and a definition of the desired process input to be employed in "forcing" the response. These will then be used to produce graphical displays of the simulated process response. Appendix E has a description of computer-aided control system analysis and design software suitable for this task.

3.3

THE LAPLACE TRANSFORM

3.3.1 Definition The Laplace transform of a function j(t) is defined as:

fts) = [

e--.1 f(t) dt

(3.1)

0

The operation of {1) multiplying the function f{t) by e-51, followed by (2) integrating the result with respect to t from 0 to oo is often compactly represented by L(f(t)}, so that Eq. (3.1) may be written as:

fts)

= L(f{t)} ·

(3.2)

Remad<s

1. The function of the variable t {which, in Process Control applications is almost always time) is transformed to a function of the Laplace variable s, normally understood to be a complex variable. The Laplace transform operation is thus often thought of in terms of a mapping from the tdomain {or time domain) to the s-domain (or Laplace domain). 2. To be precise it is necessary {and instructive) to differentiate between a function and its transform by employing different 1!9menclature. Thus here we have represented the transform of j(t) by j(s); but we shall do this for this chapter only. In later chapters, for simplicity, we shall drop the overbar so that the transform off(t) will be represented asf(s) where it will be enough to retain the corresponding arguments s and t to differentiate the transform from the original function. The following examples illustrate how to obtain Laplace transforms using the definition in Eq. {3.1).

Example 3.1

LAPLACE TRANSFORM OF SIMPLE FUNCfiONS.

Find the Laplace transform of the simple function j(t);l. Solution: By definition: fts)=L{l}

f'" 0

e-st! dt

(3.3)

PROCESS DYNAMICS

72 which, upon performing the indicated integration, yields:

-

j{s)

-SI

= -e-

It=-

s

(3.4)

'" 0

to give:

jfs) =L{ I}

Example 3.2

(3.5)

s

LAPLACE TRANSFORM OF SIMPLE FUNCTIONS.

Find the Laplace transform of the functionj(t) = e-a1• Solution: In similar fashion we obtain, for the exponential function, that:

.ifs)=L(e-a1)

=

f-e-st e-at dt

(3.6)

0

which yields: .ifs)

- e-(s+a)t It=s +a t=O

(3.7)

and finally : s + a

(3.8)

Similar results for any Laplace-transformable function can, in principle, be obtained in the same manner as in the foregoing examples. However, this is hardly ever done in practice. Firstly, note that because of the integration involved, the exercise can be quite tedious for all but the very simple functions. However, more important is the fact that extensive tables of various functions and their corresponding transforms have been compiled and are commonly available (see Appendix C). It is therefore more customary to make use of such tables.

Some Properties of the Laplace Transform The following are certain properties of the Laplace transform that are of importance in Process Control applications. 1. Not all functions f(t) possess Laplace transforms. This is because the Laplace transform involves an indefinite integral whose value must be guaranteed to be finite if the Laplace transform is to exist. It is, in general, not possible to guarantee that the integral in Eq. (3.1) will be finite for all possible functions f(t).

CHAP 3 BASIC ELEMENT S OF DYNAMIC ANALYSIS

73

2. We note, however, that all the functions of interest in our study of Process Dynamics and Control are such that they possess Laplace transforms. 3. The Laplace transform J{s) contains no information about the behavior of J(t) for t < 0 since the integratio n involved is from t = 0. 1bis is, however, not a problem for our application since t usually represent s time and we are normally interested in what happens at time t>t 0 where t 0 is some initial time that can be arbitrarily set to zero. (As will be seen later, defining our process variables such that J(t) =0 for t < 0 is a common feature in Process Dynamics and Control.) 4.

The functionf( t) and its correspon ding transform J(s) are said to form a transform pair. A most importan t property of Laplace transform s is that transform pairs are unique. This implies that no two distinct functions J(t) and g(t) have the same Laplace transform. The usefulness of this property will become clearer later.

5. The Laplace transform operation is a linear one, by which we mean that for two constants c1 and c2, and two (Laplace- transform able) functions f1(t) andfit), the following is true: (3.9)

The validity of this assertion is easily proven (see Problem 3.2.)

3.3.2 Some Useful Results The utility of the Laplace transform rests on the following collection of results. (Proofs.and detailed discussions are taken up in Appendix C.)

The Inverse Transform It should not be difficult to see that for the Laplace transform to be useful, we

must have a means by which the original functionf (t) may be recovered if we are given the transform J{s). This is accomplis hed, in principle, by the following expressio n known as the Laplace inversion formula:

= -1. 21t]

f -

e'' fis)ds c

(3.10)

a complex contour integral over the path represent ed by C, called the Bromwich path (see Appendix C). However , just as was the case with the transform formula itself, this inversion formula is hardly ever used in practice. Owing to the very important fact that transform pairs are uniqu~ tables of such transform pairs can be used not only to obtain the transform J(s) given f(t), but also to obtain the inverse

PROCESS DYNAMICS

74

demo nstra te precisely how this is trans form , f(t), from a given fts). We shall done shortly.

The Transform of Derivatives fied for most of the Process Cont rol Und er cond ition s that are alwa ys satis transform of the deriv ative of a given syste ms of practical interest, the La_place know n) is give n by the follo wing func tion f(t) (who se trans form f (s) is expression: L{

41ijf } = s

fts) -fit) I,= 0

(3.11 )

l cond ition of the time func tion f(t) as Thus if we agree to desig nate the initia ) as: f(O), it is more convenient to recast Eq. (3.11 L{

~}

= s fts)-f lO)

(3.12)

w directly from here: The transforms of high er deriv ative s follo

on the first deriv ative as: If we now designate the initial cond ition tJtffi dt

I

L{

~}=

t=O

=r

(0)

(3.14)

we have that: s2 fts)- sflO) -f'(O )

(3.15)

Simi larly : L{

In general:

~}

=

SJ J(s)- s2fl0) - sf'(O )-f"( O)

(3.16)


75

or (3.18)

where J< 0 (0) denotes the value of the ith derivative of f(t) evaluated at t = 0. Let us note here that, if the initial conditions on f(t) and all its derivatives are zero, then Eq. (3.18) reduces to: L {

if'll dt" J

=

~ J(s)

(3.19)

a very important result indeed.

The Transform of Integrals If the Laplace transform of a function f(t) is known to be fts), then the Laplace transform of the integral of this function is given by:

L{f~ f(t} dt'} = ~ ](s)

(3.20)

Shift Properties of the Laplace Transform 1. Shift ins If L{f(t)} =

f(s), then: L{e"'l{t}} =

j{s

a}

(3.21}

implying that the effect of multiplying a function by e•1 is that the s variable in its transform is shifted by a units, as indicated in Eq. (3.21). For example, recall from Eq. (3.5) that we obtained the Laplace transform ofj(t)=1 as 1/s. We could now use this fact and the shift property indicated above to obtain L{e....,1} directly as being 1/(s +a) without having to carry out the integration as was done in Eqs. (3.6)-(3.8). The main utility of this particular shift property, however, lies in carrying out inversions of transforms. Thus a more useful version of Eq. (3.21) is: (3.22)

The implication is that if the transform pair [f(t), j{s)] is known, then to find the inverse of a transform involving a shift in s by a, we merely multiply f(t) by

e•'.

An example will serve to illustrate the application of this property.

PROCESS DYNAMIC S

76 Example 3.3

Given that the Laplace transform of j(t) = 1 is: l .fts)=L{l} = -;

find

ct{_l } s +a

Solution: From Eq. (3.22) we obtain immediatel y that:

L -1

{

_I- } s +a

=

e-at.I

e-at

(3.23)

2. Shift in Time If L(f(t)) = f(s), then it is possible to find the Laplace transform of f(t- a), a time-shifted version ofJ(t) (see Figure 3.2), in terms of J(s) as follows: L{j{t-a)} = e"""' j{s)

(3.24)

Thus the presence of an exponenti al function of the Laplace variable sin a transform indicates that the inverse transform will contain a "shifted" time argument . We shall encounter such transform s and the physical systems with which they are associated in Chapter 8. Situations do arise in practice in which one needs to find, from fts), only the initial, or the final value of f(t), and not the entire function itself. It is possible to do this by making use of the following results.

The Initial-Value Theorem lim [1\t)] = lim [s .f{s)]

r-.o

(3.25)

.r-too

The Final-Va lue Theorem Provided s /(s) does not become infinite for any value of s in the right half of the complex s-plane (see Appendix C), then the final value of f(t) is finite and may be obtained fromf(s) as follows: lim [fl:t)] = lim [s fts)] t-+oo

(3.26)

.r-tO

If the above stated condition s are, not satisfied, the implicatio n is that f(t) has no limit as t - t oo .


77

t

itt)

0

t--~

Figure 3.2. The time-shifted function.

3.3.3 Application to the Solution of Differential Equations Consider the _problem of finding the solution to the simple differential equation:

't"~+y

(3.27)

= Ku(t)

for the situation where 11(t) =1, and y(O) =0. Now, because of the uniqueness of transform pairs, equality of two functions implies equality of their transforms. Hence we can take Laplace transforms of both sides of Eq. (3.27) to obtain an equivalent equation in the Laplace variables, i.e.: -

-

K

-rs y(s) + y(s) = -

(3.28)

s

since u(t) =1, and therefore U{s) =1/s (recall Eq. (3.5)). Observe that Eq. (3.28) is now an algebraic equation that can be solved easily for y(s). The result is:

y(s)

K

= s(n + 1)

(3.29)

H we now know of a function y(t) whose transform is as given in Eq. (3.29), the problem will be solved. _!he next step is therefore that of obtaining y(t), the inverse transform of the y(s) function given in Eq. (3.29). We note, firstly, that Eq. (3.29) may be rewritten in the entirely equivalent form: -

K

K-r

y(s) = -;- (-rs + 1)

(3.30)

that is, the right-hand side of Eq. (3.29) has been broken into two partial fractions. By applying the linearity property of Laplace transforms we know that Eq. (3.30) is the Laplace transform of the sum of two functions, the first being K, a constant. By recalling the shift property, and noting that the second function can be expressed as:

PROCESS DYNA MICS

78

we obtai n that: or

y(t)

K- K e-tlr

y(t) == K

(I -

(3.31)

e-tt~

ral diffe rentia l equa tion in the Regardless of the complexity of the gene it is linear, and of the ordin ary as long (so t variable y as an unkn own function of dure for obtai ning the solut ion type, with cons tant coefficients), the proce ple may be gene raliz ed and illus trate d by the prece ding simp le exam summ arize d as follows: the equation. (Note that by Take the Laplace trans form of both sides of a deriv ative of any order , the defin ition of the Laplace trans form of · ded.) inclu lly atica the initial conditions are autom trans form of the ce Lapla the for tion equa raic algeb ting 2. Solve the resul unkn own function, i.e., Y{s). obtai ned for Y{s) in Step 2. 3. Obta in y(t) by inver sion of the solut ion Y{s) function into simp ler This step usual ly involves break ing up the easily recognizable) by a more are s form trans functions (whose inverse (See Appe ndix C for a proce dure know n as parti al fraction expansion. are carried out.) sions expan on fracti detai led discussion of how parti al

1.

following example. Let us illustrate the above proce dure with the Exam ple 3.4

ACE SOLUTION OF UNE AR ODE' S BY LAPL MS. TRAN SFOR

Solve the following differential equation: (3.32)

= f(t)

y"(t) + Sy'(t) + 6y(t)

by the method of Laplace transforms given the

following conditions:

f(t) = I; y(O) =I, y'(O) = 0

Solut ion:

Taking Laplace transforms of both sides of the ] [ s 2 y(s) - sy(O )- y'(O)

+ 5 [s

-

y(s)- y(O)}

=

1

+ 6 y(s) =-;

and upon introducing the initial conditions and (s 2 + 5s + 6) .Y(s)

equat ion gives:

-

rearranging, we have:

I -;+(s +5)

79


Solving this equation for y(s) gives: y(s)

=

I + s(s + 5) + 5s + 6)

(3.33)

s(s 2

By partial fraction expansion of the right-hand side, expression (3.33) becomes: (s) =

y

+ 2JL- _2}]_ lli s+2 s+3 s

(3.34)

Inverting Eq. (3.34) finally leads to the required solution: (3.35)

3.3.4

Main Process Control Application

The main application of the Laplace transform in Process Control is intimately related to the application discussed in the last subsection - solving differential equations. This is illustrated in Figure 3.3. The original problem, formulated in the time domain in terms of the time variable t, is usually easier to solve after it has been transformed to the s-domain, in terms of the Laplace variable s. Since the solution of the transformed problem is related to the solution of the original time-domain problem by Laplace transformation, the final step involves recovering the desired time-domain solution from the Laplace domain by the inverse Laplace transform operation. A significant proportion of the dynamic analysis to be carried out in Part II will be carried out, as indicated in Figure 3.3, in the Laplace domain making use of a particular process model form (the transfer function model) which will be discussed in detail in the next chapter.

INITIAL PROBLEM (In time domain)

L Laplace Transformation

TRANSFORMED PROBLEM (In s-dlmain) Manipulations ins-domain (TypicaJ1y easier thanma nipulations in time do main)

SOLUTION (in terms of t variable)

·1

L Inverse Laplace Transformation

SOLUTION (in terms of s variable)

Figure 3.3. The application of the Laplace transform to differential equation

PROCESS DYNAM ICS

80

S CHARACTERISTICS OF IDEAL FORCING FUNCTION

3.4

ior is carried out by A-c. noted earlier , the analysis of dynam ic process behav of certain wellation applic the by invest igating the process respon se forced s types of variou the of s teristic charac The ns. charac terized input functio sed in this discus be will is "forcing functions" typically used in dynam ic analys section. dynam ic analys is of In the upcom ing chapte rs, we will find that the the Laplace domain. in y entirel almost variou s types of processes is carried out of these forcin g orms transf e Laplac the re, therefo this, In antici pation of functions will also be given. The functions we shall be concerned about are: 1. 2. 3. 4. 5.

The step function The rectangular pulse function The impuls e function The ramp function The sinuso idal function

3.4.1 The Ideal Step Function Figure 3.4, is afunction The ideal step function of magni tude A, as illustr ated in A instan taneou sly at of value the on takes it until zero at s whose value remain the "starti ng time" set arbitrarily to t =0. Such a function is repres ented mathe matica lly as: u(t) = {

0; t < 0 A; t

(3.36)

> 0

Heaviside function): or in terms of the unit step function, H(t) (also called the (3.37a)

u(t) = AH(t),

where the Heavi side function is defme d as: H(t)

0; t < 0 t > 0

= { 1;

(3.37b)

A

u(t)

0 t

0

Figure 3.4. The ideal step function.


81

The Laplace transform of the step function is given by: -

A

u(s) = L{u(t)} = -

s

(3.38)

A step function may be realized in practice by implementin g a sudden change in the position of the actuator (e.g., control valve, etc.) which will correspond to a change of magnitude A in the input variable. For example, such a step change may be realized in a liquid flow process by making a sudden change in the valve position which will yield a change in liquid flowrate of A units. It should be clear that the step function, as represented here, is idealized and cannot, in general, be realized exactly in practice.

3.4.2 The Ideal Rectangular Pulse Function The ideal rectangular pulse function of magnitude A, and duration b time units is illustrated in Figure 3.5. It is a function whose value remains at zero until it takes on the value of A instantaneous ly at the "starting time" t = 0, maintains this new value for precisely b time units, and returns to its initial value thereafter. Mathematically, this function is represented as: 0; t < 0 { A;O b

recall ing the shift prope rties of Takin g Lapla ce transf orms of Eq. {3.40), and n, we imme diatel y obtain: sectio ding prece the in ssed Laplace transf orms discu

-

u(s)

A = -(I s

(3.42)

- e-bs)

funct ion illust rated in Figur e 3.5. as the transf orm o£ the ideal rectan gular pulse recta ngula r pulse funct ion is ideal the ion, funct Just as with the step ice. The proce dure for its diffic ult to realiz e perfe ctly in actua l pract taneo us chang e in actua tor instan an ng maki realiz ation , howe ver, invol ves ecifie d time perio d (b time units), positi on, main tainin g this chang e for a presp and then return ing to the initial state.

3.4.3 The Ideal Impulse Function A is a funct ion repre sente d The ideal impu lse funct ion of magn itude math emati cally as:

= A O(t)

u(t)

(3.43)

rated in Figur e 3.6, an unusu al wher e o(t) is the "Dira c delta funct ion" illust of what it does rathe r than terms in e defin to easier funct ion that is some what what it is. (if not entire ly rigoro us) to Howe ver, for our purpo ses, it is conve nient here and infinite at the point elsew define it as a "func tion" whos e value is zero r the "curv e" is 1, i.e.: nnde area total the that t = 0, and yet has the prope rty /'i..t) = {

and

J

+-

-

/'i..t) dt ==

oo; t == 0

J+£ li{t) dt -

(>Q

(3.44)

0; elsewhere

(3.45)

=

£

u(t)

Area= l

0- t-- -'- --- --- --- --- -t

0

Figure 3.6. The Dirac delta function.


83

The impulse function of area A (given by Eq. (3.43)) has the Laplace transform: u(s)

= L{u(t)} = A

(3.46)

The following are some properties of the Dirac delta function: 1.

2.

It can be obtained as the limit of the rectangular pulse function of the previous subsection. Let A= 1/b so that the area of the rectangle, i.e., A·b, is 1. In the limit as b ~0 (and therefore A ~oo) while maintaining the constant "rectangular" area of unity the result is the Dirac delta function; i.e., the Dirac delta function is a rectangular pulse function of zero width and unit area. We will make use of this property in subsequent chapters. For any functionj(t):

J

t 0 +t

t0

3.

O{t- t 0 ) f(t) d t = fl.,t0 ) -

(3.47)

E

where O(t- t0) is the general, "shifted" delta function whose value is zero everywhere except at the point t = t 0 • (Note that the function defined in Eq. (3.44) is obtained by setting t 0 = 0.) This property may be used, for example, to establish Eq. (3.46). If H(t) represents the unit step function, then:

dtd [H(t)] = ~t)

(3.48)

i.e., the Dirac delta function is the derivative of the unit step (or Heaviside) function. In terms of realization on a physical process, it is of course obvious that it is impossible to implement the impulse function exactly. However, a pulse function implemented over as short a time interval as possible will constitute a good approximation, especially for processes with slow dynamics.

3.4.4 The Ideal Ramp Function The ideal ramp function, as illustrated in Figure 3.7, is a function whose value starts at 0 (at t=O) and increases linearly with a constant slope which has been designated as A. Mathematically, this is represented as: 0;

u(t)

= { At;

t < 0 t > 0

(3.49)

Observe that this function can be obtained by integrating the step function of magnitude A. Alternatively a first derivative of the ramp function in Eq. (3.49) gives a step function of magnitude A.

84

PROCESS DYNAMICS u(t)

0 _...._ __,.

t

0

Figure 3.7. The ramp function.

The Laplace transform of this ramp function is given by:

-

u(s) = L{u(t)}

A =-a s

(3.50)

To implement a ramp function as an input to a physical system requires a systematic, graduated change in actuator position in such a way that the rate of change (increase or decrease) is constant. We must note, however, that even though the mathematical representation of this type of input indicates a function with no limiting value as t -+ oo, in real processes, physical limits are naturally imposed. For example, in implementing a ramp input in the inlet flowrate of a process flow system, we can only increase the flowrate up to the value obtained when the valve is fully- open; beyond this no flowrate increase is physically possible.

3.4.5 The Ideal Sinusoidal Function The sinusoidal input function of amplitude A and frequency w is illustrated- in Figure3.8. It is represented mathematically as:

u(t)

{

0;

t < 0 (3.51)

A sinwt; t > 0

The Laplace transform is given by:

~s) = L{u(t)} = -s-.;2...:.~o..:w=-w-.2'

(3.52)

As might be expected, it is quite difficult to implement a perfect sinusoidal input function in practice. However, the theoretical response of a process system to a sinusoidal -input provides valuable information that is useful not just for process dynamic studies, but also for designing effective controllers.

CHAP 3 BASIC ELEMEN TS OF DYNAM IC ANALYS IS

0

85

n/ro 2Jtlro

Figure 3.8. The sinusoidal function.

3.4.6 Realiz ation of Ideal Forcin g Functi ons: An Examp le We now illustrat e how these ideal forcing functions might be implem ented on real processes by considering the following example system, the stirred heating tank first introdu ced in Example 1.1 of Chapter 1 (see Figure 3.9). Here, the input variable of interest is the steam flowrate. How the various forcing functions we have discussed in this section will be implem ented on this particul ar process will now be summar ized. 1. STEP: Open the steam valve a given percent age at t = 0 such that Q changes by A units. 2. PULSE: Open the steam valve at t = 0, hold at the new value for a duratio n of b time units, and return to the old value. 3. IMPULSE: (Imposs ible to realize perfectly.) Open the steam valve (wide open) at t = 0 and instanta neously (or as soon as physica lly possible thereafter) return to the initial position. 4. RAMP: Gradua lly open the steam valve such that Q increases linearly. Ramp ends when the steam valve is fully open. 5. SINUS OID: The only practical way to realize this is to connect a sine wave generat or to the steam valve. Realizi ng higher frequen cy sinusoi dal inputs may be limited by the valve dynamics.

F,T

Q:Steam Flowrate

• Condensate

Figure 3.9. The stirred heating tank.

PROCESS DYNA MICS

86

3.5

SUMMARY

, as well as an introduction A description of Process Dynamics and its importance ic analysis, are the issues dynam for sary neces to the basic elements that are The idea of using a process that have occupied our attention in this chapter. was introduced. The role of s studie such mode l and ideal forcing functions for of the numerical and graphical the digital comp uter in the actual carrying out aspects of dynamic analysis was briefly discussed. empl oyed in dyna mic The ideal forcing functions that are typic ally introduced. In later been have tics cteris chara e analysis studies and their uniqu of information that sort the chapters we shall see how they are employed and functions. input these of each can be extracted from the responses forced by for dyna mic tools cal emati math le ensab indisp al The importance of sever of these most of ent ed treatm analysis has been stressed. Even thoug h a detail and its form trans ce Lapla the , tools was defer red until later in the book ssed. discu were ls mode ic usefulness in analyzing dynam REVIEW QUE STIO NS 1.

What is the essenc e of dynam ic analysis?

2.

Why is a proces s model a useful tool of dynam ic

3.

d in order to carry out succes sful In additi on to a proces s model , what else is neede dynam ic analys is?

4.

The following mathematical tools: • • • • •

analysis?

~entialequations

Laplace transf orms Numerical methods z-transforms Matric es

are each useful for what aspect s of proces s dynam 5.

6.

7.

In what sense was the digita l comp uter presen tool of dynam ic analysis?

ic analysis?

ted in this chapte r as an indisp ensab le

ons not being Laplace transf ormab le; What do you think is respon sible for some functi cal intere st in proces s dynam ics practi of ns functio the of most why and can you guess le? ormab transf ce and control proble ms are in fact Lapla Why is it not possible to use the final-value theore the function j{t) given that K /(s) = s-a

m in determ ining the "final value" of

inverse Laplace transforms?

8.

What is the most comm on metho d for obtain ing

9.

Why is the Lapla ce transf orm useful in solvin g linear

ODE's?


87

10. Outline the main application of the Laplace transform in process control. 11. Of all the ideal forcing functions discus~ed in Section 3.4, which do you think is most easily implementable in practice? 12. If your objective is to cause the least possible upset in your process, which of the ideal forcing functions should you employ in an experimental study?

PROBLEMS 3.1

By carrying out the indicated integration explicitly, use the definition of the Laplace transform given in Eq. (3.1) to establish that: (i)

(ii)

L {sin wtj

3.2

Use the definition given in Eq. (3.1) to establish the linearity property given in Eq. (3.9).

3.3

By expressing cos w t and sin w t respectively as: .

1

(1)

cos

(2)

sin WI

.

2(e'"" + e-Jro')

li.)f

= b<eirot- e-jrot)

use the linearity property of Laplace transforms given in Eq. (3.9) along with the fact that: s +a

(P3.l)

to reestablish the results of Problem 3.1. 3.4

By differentiating the function: j(t)

=

(P3.2)

tn

n times with respect to t, and noting that: r(n+ 1) = n(n-l)(n-2) ... l

(P3.3)

(when n is an integer) use the results on the transform of derivatives in Section 3.3.2 to establish that f(n + 1) sn +I

(P3.4)

PROCES S DYNAM ICS

88

3.5

Given that the Laplace transform of a certain process response to a unit given by: y(s)

=

K(~s

+ 1)

s(-rs

+

step input is

(P3.5)

1)

by using the initial- and final-value theorems, or otherwise, find: lim y(l), the initial value of the response, and (2) lim y(t), the ultimate, steady-state value of the response.

(I)

1-->0

,......

than the final Under what conditio n will the initial value of the response be greater 5.) Chapter in fully more studied be will value? (The process in question 3.6

Find lim y(t), for the function whose Laplace transform is given as:

,......

3(s + 2)(s- 2)

3. 7

(P3.6)

The input function u(t) shown in Figure P3.1

b

Figure P3.1. may be represented mathematically as:

u(t) =

Find u(s), its Laplace transform.

t < 0 0 { A(J t 0 ~ t < b t 2:: b

(P3.7)

CHAJP1r1EJR

4 THE PROCESS MODEL The idea of utilizing a collection of mathematical equations as a "surrogate" for a physical process is at once ingenious as well as expedient. More importantly, however, this approach has now become indispensable to proper analysis and design of process control systems. The concept of representing process behavior with purely mathematical expressions, its benefits and shortcomings, the basic principles of how such mathematical models are obtained, the various forms they can assume, as well as the interrelationships between these various model forms, are some of the issues to be discussed in this chapter. The most important model form in Process Dynamics and Control studies still remains the transfer function form. We will therefore examine transfer functions more closely towards the end of the chapter, and discuss some fundamental principles of how transfer functions are used for dynamic analysis.

4.1

THE MATHEMATICAL DESCRIPTION OF CHEMICAL PROCESSES

As stated earlier on in Chapter 3, the design of effective controllers is

significantly facilitated by a proper understanding of process dynamics. This understanding in tum has been shown to be dependent on the availability of a process model; and process models are most effectively couched in the language of mathematics. It is thus fairly accurate to say that the first step in the analysis and/ or design of a control system is the development of an appropriate mathematical model. The mathematical model is, in principle, a collection of mathematical relationships between process variables which purports to describe the behavior of a physical system. The main use of the mathematical model is therefore as a convenient "surrogate" for the physical system, making it possible to investigate system response under various input conditions both rapidly, and inexpensively, without necessarily tampering with the actual physital entity. 89

90

PROCESS DYNAMI CS

We must, however , exercise caution in distingui shing between the usually quite complex physical entity that exists in the "real world" - that we have referred to as a process - and its mathema tical description, called a process model. The latter at best captures only the salient features of the former, and should never be thought of as being its exact equivalent. This fact notwiths tanding, mathem atical descript ions in the form of in models have provided very valuable insight into the behavior of processes wide a of behavior the y efficientl rize characte to possible it general, making variety of actual processes. The extent of the usefulne ss of these mathem atical descript ions is s undersco red by the fact that it is now customa ry to classify chemical processe on. descripti tical mathema their for used according to the nature of the models the Thus a process described by linear equation s is classified as linear, while s. equation r nonlinear process is the one described by nonlinea Furtherm ore, a lumped parameter process (which may be linear or not} is not one in which, physically, its process variables change only with time but s. equation ial different ordinary by d describe with spatial position. They are hand other the on process r paramete d distribute a of s variable The process change with spatial position as well as with time. The mathematical model representing a process is comprise d of a system of equation s, and thus, strictly speaking, the term "system" refers to the process ing model rather than the process itself. Howeve r, in the jargon of engineer its as well as itself process the denote to used is " "system term the practice, both to refer will "system" term model. Thus in the discussion to follow, the the process and its modeL Let us examine all these issues a little more closely.

4.1.1 Process Characteristics and Process Models The terminology used in characterizing processes on the basis of their inherent characte ristics, and the types of mathem atical represen tation that best describe them, will now be discussed. It is importan t to note that in what follows, wheneve r the terms dependent and independent variable s are used, they should be understo od in the following context: All the process variables we identifie d in Chapter 1, be they state, output, input, or disturba nce all variable s, depend on time and/or spatial position ; they are therefore , variables te coordina spatial the and Time, consider ed as dependent variables. are the independent variables.

Linear and Nonlinear Systems A system described by linear equations (i.e., equation s containi ng only linear be functions) is said to be linear. The system which is not linear is said to further be will ation classific nonlinea r. The issues involved with this elaborated on in Chapter 10. For now, we just note that most real chemical processes exhibit nonlinea r behavior to a greater or lesser extent. It turns out, however , that some of the observed nonlinea r behavior can indeed be effectively approxim ated by linear for equation s. Such systems are then usually classified as linear, but only rigidly. too taken be not course, of should, tions classifica convenience; such

CHAP 4 THE PROCESS MODEL

91

Lumped Parameter Systems There are processes (linear or not) in which the dependent variables (i.e., all the process variables) may be considered, for all intents and purposes, as being uniform throughout the entire system, varying only with time. As will soon be shown, the theoretical models for such systems naturally occur as ordinary differential equations, with time as the only independent variable. These processes are referred to as lumped (parameter) systems, since, in a sense, the dependency of all the observed variations have been lumped into one single dependent variable, i.e., time. The order of the differential equation that describes the dynamic behavior of a lumped parameter system is used for further classification. A process described by a first-order differential equation is called a first-order system, and, in general, the nth-order system is a process described by an nth-order differential equation or by a system of n first-order differential equations.

Distributed Parameter Systems When the variables of a process vary from point to point within the system in addition to varying with time, the fundamental mathematical description must now take the form of partial differential equations in order that the additional variation with position be accounted for properly. Such processes are called distributed (parameter) systems. There are more independent variables, and the observed process variations are "distributed" among them.

Discrete-Time Systems Even though the tacit assumption is that the variables of a process do not ordinarily change in "jumps," that they normally behave as smooth continuous functions of time (and position), there are situations in which output variables are deliberately sampled - and control action implemented - only at discrete points in time. The process variables now appear to change in a piecewise constant fashion with respect to the now discretized time. Such processes are modeled by difference equations and are referred to as discrete-time systems.

4.1.2 Various Forms Of Process Models As noted earlier, the process model is a collection of mathematica l relationships between the process variables that were classified in Chapter 1 as state, input, and output variables. Those mathematical models in which the state variables occur explicitly along with the input and output variables are called state-space models. As we will soon demonstrate, when the process model is formulated from first principles, it often naturally occurs in the statespace form in the time domain. The mathematical models that, on the other hand, strictly relate only the input and output variables - entirely excluding the state variables - are called input/output models. As opposed to the state-space models which usually occur only in the time domain, input/ output models can occur in the Laplace or z-transform domain, or in the frequency domain, as well as in the time domain. In the Laplace or z-transform domain, these input/ output models

PROCESS DYNAMICS

92

usually occur in what is known as the "transform-d omain transfer function" form; in the frequency domain, they occur in the "frequency-re sponse" (or complex variable) form; and in the time domain, they occur in the "impulseresponse" (or convolution) form. In general, input/ output models occur as a result of appropriate transformatio ns of the state-space form, but they can also be obtained directly from input/ output data correlation. Thus, it is usual to cast the mathematical model for any particular process in one of four ways: 1. 2. 3. 4.

The state-space (differential or difference equation) form The transform-do main (Laplace or z-transforms) form The frequency-response (or complex variable) form The impulse-response (or convolution) form

Each of these model forms and their applications will be discussed in more detail in the sections to follow. Because these model types are obviously interrelated, it is possible to convert from one form to another. For Process Dynamics and Control applications, this is quite advantageous . Later on in Sections 4.3 through 4.6 when we examine the salient features of each model form, the importance of being able to switch from one model to the other will be better appreciated. Before examining each model form critically and discussing the interrelations hips between them, let us first consider the important issue of how process models are formulated.

4.2

FORMULA TING PROCESS MODELS

The theoretical principles of how process models used in Process Dynamics and Control studies are formulated will now be introduced.

4.2.1 The General Conservati on Principle The basis for virtually all theoretical process models is the general conservation principle which, in essence, states that

What remains accumulated within the boundaries of a system is the difference between what was added to the system and what was taken out, plus what was generated by internal production. While this may appear so straightforwa rd, almost to the point of being simplistic, it is nevertheless the basis for a countless number of mathematical models. The conservation principle written as an equation is: Accumulation

=

Input -Output + Internal Production

(4.1)

It is often more appropriate to present the conservation principle in terms of rates, in which case, we have that:


93

The rate of accumulation of a conserved quantity q within the boundaries of a system is the difference bettveen the rate at which this quantity is being added to the system and the rate at which it is being taken out plus the rate of internal production. That is: Rate of Accumulation ofq

Rate of Input ofq

Rate Rate of + of Output Production ofq ofq

(4.2)

4.2.2 Conservation of Mass, Momentum, and Energy Despite the fact that there is a wide variety of chemical process systems, the fundamental quantities that are being conserved in all cases are either mass, or momentum, or energy, or combinations thereof. Thus, the application of the conservation principle specifically to the conservation of mass, momentum, and energy provides the basic blueprint for building the mathematical models of interest. It is customary to refer to the mathematical equations obtained using Eq. (4.2) as mass, momentum, or energy balances, depending on which of these quantities q represents. Such balances could be made over the entire system, to give "overall" .or macroscopic balances, or they could be applied to portions of the system of differential size, giving "differential" or microscopic balances. We shall now give the equations to be used in carrying out mass, momentum, and energy balances over a system's "volume element,".be it microscopic or macroscopic.

Mass Balance The mass balance equation may be· with respect to the mass of individual components in a mixture (a component mass balance), or with respect to total mass. The general form of the equation is no different from Eq. (4.2), i.e.: Rate of Accumulation of mass

=

Rate of Input of mass

Rate of

+ Generation of mass (4.3)

Rate of Output of mass

Rate of Depletion of mass

If there are n components in a mixture, then n component mass balances of this form will give rise to n equations, one for each component. If one also formulates a total mass balance, it is important to remember that only n of the resulting n + 1 equations are independent. It should be stressed that the total mass balance equation will not have any generation or depletion terms; these will always be zero. The reason is that even though the mass of a component within the system may change (primarily because of chemical reaction) the total mass within the system will be constant; whatever disappears in one component appears as another component, since

PROCESS DYNAMICS

94

l process matter can never be totally created or totally destroy ed in chemica ns. conditio tivistic systems that operate under nonrela

Mome ntum Balance The general form of a momen tum balance equation is as follows: Rate of Input of momen tum

Rate of Accumulation of momen tum

+

Rate of forces acting on volume element (4.4)

Rate of Output of momen tum

tion" of where now we have paid due respect to the fact ·that any "genera this is that and , element volume the on acting forces to momen tum must be due forces. these of sum total the to equal rate a done at ization of It is importa nt to note that this equatio n is in fact just a general and mass of t produc the is force that states which Newton 's law of motion accurate to more is it , variable be could mass which in systems For tion. accelera also be express force as the rate of change of momen tum. Thus, Eq. (4.4) could , quantity vector a is force Since . balance force of n equatio conside red as an et Bird in further ed possess ing both magnitu de as well as direction, as discuss three such al. [1], a comple te force (or momen tum) balance will consist of al space. mension three-di a in ns directio three the of each for equatio ns written are very s balance tum) momen (or For our current purpose s, we note that force will not we e therefor ; systems control process g rarely the bases for modelin further. aspect this pursue

Energy Balance rise to the Modify ing Eq. (4.2) specifically for energy conserv ation gives : equation balance energy following general Rate of Accumulation of energy

Rate of Input of energy

Rate of

+ Generation of

Rate of Output of energy

energy

(4.5)

Rate of expenditure of energy via work done on surrounding

easily The importa nt point to note about this equatio n is that it is s. ynamic thermod of law first recognizable as an equival ent stateme nt of the

4.2.3 Consti tutive Equati ons system The next step in formula ting the mathematical description of a process appear in involves the introdu ction of explicit expressions for the rates that based on these balance equations. More often than not, these expressions are


95

basic physical and chemical laws, and the underlying mechanisms by which changes within the process are presumed to 'occur. Fortunately, these so-called constitutive laws are fairly well known in quite a wide variety of cases and are typically discussed in undergraduate courses in Transport Phenomena, Kinetics and Reactor Design, Thermodynamics, etc. The most widely used of these constitutive equations include: 1. Equations of Properties of Matter Basic definitions of mass, momentum, and energy in terms of physical properties such as density, specific heat capacities, temperatures, etc. 2. Transport Rate Equations: • Newton's law of viscosity (for momentum transfer) • Fourier's heat conduction law (for heat transfer) • Fick's law of diffusion (for mass transfer) 3. Chemical Kinetic Rate Expressions Based on: • The law of mass action • Arrhenius expression of temperature dependence in reaction rate constants 4. Thermodynamic Relations • Equations of state (e.g., ideal gas law, van der Waal's equation) • Equations of chemical and phase equilibria Perhaps the best way to illustrate the foregoing discussion is by presenting some examples.

4.2.4

Some Examples of Theoretical Modeling

The Stirred Heating Tank The process shown in Figure 4.1 is the stirred heating tank we encountered earlier on in Chapters 1 and 3. The purpose here is to develop a mathematical model for this process. The Process Liquid at temperature T; flowing into the tank at a volumetric flowrate F, is heated by steam flowing through the steam coil arrangement at a rate Q (mass/time). The heated fluid, now at temperature T, is withdrawn at the same volumetric rate as in the inlet stream. The tank volume is V; the liquid density and specific heat capacity are, respectively, p, and CP. The latent heat of vaporization of steam is A.

PROCESS DYNAMICS

96

F,T

t

Q:Steam Flowrate

Condensate

+steam

Figure 4.1. The stirred heating tank.

Assumptions 1.

We will assume that the content of the tank is well mixed so that the liquid temperature in .the tank and the temperature of liquid in the outlet stream are the same.

2.

The physical properties p, CP, and A. do not vary significantly with temperature.

3.

All the heat given up by the steam (through condensation ) is received by the liquid content of the tank; i.e., no heat from the steam is accumulated in the coils.

4.

Heat losses to the atmosphere are negligible.

We are now in a position to obtain a mathematical model for this process. As was discussed in the preceding subsection, we do this by carrying out mass and energy balances, and introducing appropriate constitutive equations. Overall Mass Balance (pV) Rate of mass accumulation within the tank: .!!.. dt pF Rate of mass input to the tank :

Rate of mass output from the tank: pF There is neither generation nor consumption of material in this process. Thus from Eq. (4.3) the overall mass balance equation is: d dt(pV) = pF-pF

and on the basis of the assumption of constant density p, we obtain:

(4.6)


97

dV dt = 0

(4.7)

implying that: V = constant

(4.S)

Thus the material balance equation merely tells us that we have a constant volume process. This should not come as a surprise; it is in perfect keeping with common sense that if inlet flowrate equals outlet flowrate then the volume within the system must remain unchanged. Overall Energy Balance For this system let us note that even though the total energy is a sum of the internal, potential, and kinetic energies, the rate of accumulation of energy will involve only the rate of change of internal energy. This is for the simple reason that the tank is not in motion, thus eliminating the involvement of any kinetic energy terms; and since there is also no change in the position of the tank, the rate of change of potential energy will be zero. One final point: for liquid systems, the rate of change of total enthalpy is a good and convenient approximation for the rate of change of internal energy. Let us now proceed with the energy balance. (We shall use the variable T* to represent a reference temperature at which the specific enthalpy of the liquid is taken to be zero. As we shall see, this is only for exactness; the variable cancels out in l;he long run.) Rate of accumulation of energy: pC, _dd [V (T- T* )] 1 (within. the tank) Rate of heat input: (from inlet stream)

pFC, (T;- T*)

Rate of heat input: A.Q (through steam heating) Rate of heat output pFC, (T- T*) (through outlet stream) Thus, from Eq. (4.5) the energy balance equation is: pC,

1, [V (T- T* )] = pFC, (T;- T*) + A.Q- pFC, (T- T*)

(4.9)

which simplifies to: (4.10)

since, by the mass balance, we know that V is constant. The equation may be further simplified to:

PROC ESS DYNA MICS

98

(4.11)

wher e () "" VI F is the residence time. mode l that repre sents the time Equa tion (4.11) is a differ entia l equat ion d heati ng tank as a funct ion of stirre the varia tion of the temp eratu re withi n tank resid ence time, and other inlet temp eratu re, rate of steam heati ng, phys ical param eters . The Mode l in Term s of Devi ation Varia bles ges with time withi n the syste m Unde r stead y-sta te cond itions , nothi ng chan proce ss mode ls vanis h, and the in s ative deriv any longe r. As such, time Eq. (4.11) becomes: 0

I .:t 1 -eT ,+ pVC Q_,+(JTis

(4.12)

p

to desig nate the stead y-sta te value s wher e the subs cript s has been intro duce d of the varia bles in quest ion. in terms of devia tions from the If we now defin e the follow ing varia bles, stead y-sta te value s indic ated above: x=T -T,;

u=Q -Q,;

avail able for meas urem ent direc tly, and assum ing that the tank temp eratu re is : gives then subtr actin g Eq. (4.12) from Eq. (4.11) 1 I dx - = - - x + {Ju + -ed

o

dt

with

y

=X

(4.13a)

(4.13b)

initial cond ition if T = T 5 at t =0. wher e f3 = M pVCP and x(O) =0 is the prop er l repre senta tion of the heati ng mode space statea are Note that Eqs. (4.13a,b) ut y is equa l to the state x from tank, wher e in this case the meas ured outp Eq. (4.13b). The Trans form -Dom ain Mode l the math emat ical mode l in Eq. (4.11) One of the main adva ntage s of rewri ting we now consi der the proce ss to be if that is in terms of devia tion varia bles itions on the devia tion varia bles initia lly at stead y state, then the initia l cond ce trans form ation can be done Lapla and used in Eq. {4.13) will all be zero; s that have to do with initia l with out carry ing along extra neou s term cond itions .


99

Taking Laplace transforms of Eq. (4.13) then gives:t sy(s)

1 1 = - ey(s) + fju(s) + e d(s)

This is now a transform-domain model for the dynamic response of the stirred mixing tank, an algebraic equation that may now be rearranged to give an expression for y(s) in terms of its dependence on u(s) and d(s), i.e.: (4.14)

If we now introduce the following definitions: g(s)

_1}!}_ es + 1

(4.15a)

gj..s)

1 es + 1

(4.15b)

then Eq. (4.14) becomes: y(s) = g(s)u(s) + gj..s) d(s)

(4.16)

The expressions in Eqs. (4.15a,b) are called transfer functions (see Section 4.8 for a detailed discussion) and Eq. (4.14) is a transfer function model for the stirred heating tank. Equation (4.16) is the general expression for transfer function models in the transform domain. The Frequency-Response Model If we make a variable changes= jm (where m is a frequency and j = ...f=i) in Eqs. (4.15) and (4.16), the Laplace transform, g(s), becomes a Fourier transform, g(jm), and we can convert the transform-domain model to a frequency-response model. In this case the model becomes: g(jm)

gNm) y(jm)

{3e jem + 1

jem + 1

tWO- ;me) cem) 2 + 1 1- ;me (em) 2 + 1

= g(jm) u(jm) + gNm) d(jm)

(4.17a)

(4.17b) (4.18)

Thus g(jm), gd(jm) are functions of a complex variable having both real and imaginary parts; for example, the real and imaginary parts of g(jm) are:

t Note that here and from now on in this book we have dropped the overbar on the Laplace transform for notational convenience.

PROCESS DYNAM ICS

100

of the process. If Equation (4.17) can be interpr eted as the frequency response cy range of frequen the over g of parts ary imagin and real the data one stored as sponse ncy-re freque the ute relevance, these data files, g(jm}, would constit this but model, function transfer a also is (4.18} n model for the process. Equatio type of model can this of ion discuss ed expand An . domain cy frequen the in time be found in Chapte r 9. The Impulse-Response Model der differential The process model as presen ted in Eq. (4.13} is a linear, first-or ry value arbitra any for y(t), for cally analyti equatio n which is easily solved Applyi ng the d(t}. n functio ance disturb the for and u(t}, n functio input of the to be: method s described in Appendix B, the solution is easily shown x(t)

= e-118x(0)+

r

e-(t-d)IBf3u(a)dC1+

r

e~ = ax(t) + bu(t) + '}d(t)

(4.28)

y(t) = cx(t)

n in x(t) is The analytical solution to the linear, first-order differential equatio as: easily obtaine d x(t)

=[

ea(l-d)bu(CJ')dCJ +

r

ea(I-U) yd(CJ) da

(4.72)

0

0

in the require d so that, multip lying by c, we obtain the expression for y(t) relationships The impulse-response transfer function form given in Eq. (4.21). impuls ethe and model pace state-s betwee n the variables {a,b,c,n of the be: to seen easily are g,l..t) response functions g(t) and g(t) = b

g,!..t)

(4.73a) (4.73b)

C ~I

= rc ~I

Impulse Response to Sta.te Space r function The reverse process of startin g from the impulse-response transfe (Eq. (4.21)) g(t-ci) u(ci)d a +

y(t) = [ 0

J 1

g,l..t-c i)d(d)d a

(4.21)

0

e only if the and derivin g the corresp onding state-space model is possibl aracterized well-ch in le impulse-response functions g(t) and git) are availab form. and differentiable functional

I

'


123

Whenever this is the case, Eq. (4.21} may be differentiated to obtain the equivalent differential equation. This venture is usually carried out with the aid of Leibnitz's rule for differentiating under the integral sign:

! (J

f at 0) when integrated over time gives a measure of the "reluctance" inherent in the first-order process, i.e.: J =

f-

[y*- y(t)] dt = AK

f-

J = AK

[1 - ( 1- e- 11')] dt

0

0

f-

e-t!rdt = AKr

(5.15)

0

Thus, for a unit step input, the area under the curve bounded by the "instantaneous" response, and the actual first-order response is Kr, as illustrated in Figure 5.4.

CHAP 5 DYNAMICS OF LOW-ORDER SYSTEMS

145

K

y(t)

Reluctance Area~ Kt

0

t--

Figure 5.4. Response of a first-order process to a unit step input.

Process Information Obtainable from the Step Response We are now in a position to sununarize the information that can be gathered from a graphical representation of a first-order process step response. Given y(t), the response of a first-order system to a step input of magnitude A, observe that the two characteristic parameters of the system may be extracted from this response as follows: 1.

Steady-state gain, K

K=~ A

i.e., the final, steady-state value of the response divided by the input step size (see Characteristic 1 above). 2.

Time constant, -r

Let a be the slope of the y(t) response at the origin. Characteristic 3 above:

Then from

so that:

We may also obtain -rby utilizing the fact that at t = -r the value of the output is y( -r) = 0.632 y( oo ). Let us illustrate some of the issues involved in analyzing the dynamic behavior of first-order systems with the following examples. Example 5.1

DYNAMIC 6EHAVIOR OF THE LIQUID LEVEL IN A STORAGE TANK.

A 250 liter tank used for liquid storage is configured as in Figure 5.1 so that its mathematical model and its dynamic behavior are as discussed in Section 5.1.1.

PROCESS DYNAMICS

146

uniform. The cross-sectional area of the tank is 0.25 m 2, and it may be assumed y, has a simplicit for linear be to assume shall we The outlet valve resistance, which valuec =0.1 m2/min. F; at 37 The entire system was initially at steady state with the inlet flowrate : answered be to now are s question liters/mi n (0.037 m3/min) . The following What is the initial value of the liquid level in the tank?

1.

(0.087 m 3 /min), 2. If the inlet flowrate was suddenly changed to 87 liters/m in time. with vary will tank the in level liquid the obtain an expression for how

to what final 3. When will the liquid level in the tank be at the 0.86 m mark? And settle? y value will the liquid level ultimatel Solution : 1.

From Eq. (1.13), we observe that at steady state we have:

resistance c, we If we now introduc e the given values for the flowrate and the

have the required value for the initial steady-state liquid level: hs = 0.37 m

and introduc ing 2. Recalling the transfer function for this process given in Eq. (5.7), constant are time and gain tate steady-s the that find the given numeric al values, we given by: 10 (min /m 2 ) 2.5 (min)

K= 110.1 T

= 0.25/0.1

0.05 m 3 I min. The magnitu de of the step input in the flow rate is (0.087 - 0.037) = level in response liquid in change the ting represen n expressio the (5.13) Eq. from Thus to this input change is: y(t) "' 0.5 (I - e -0.4t)

its initial Note that this is in terms of y, the deviatio n of the liquid level from by: steady-state value. The actual liquid level time behavior is represented h(t)

=

0.37 + 0.5 (1 - e- 0 · 4')

solved for t to 3. Introduc ing h = 0.86 into the expressi on shown above, it is easily value to which give the time required to attain to the 0.86 m mark as 9.78 min. The final m. the liquid level eventual ly settles is easily obtained as 0.37 + 0.5 =0.87

Example 5.2

DYNAM IC BEHAVIOR OF LIQUID LEVEL, CONTIN UED.

been suddenl y Suppose now that for the system of Example 5.1 the flowrate had 3 /min) from its initial value of 37 m (0.137 in liters/m 137 to instead changed such a change in liters/mi n (0.037 m 3 /min). An experienced plant operator says that true? If so, when the input flowrate will result in liquid spillage from the tank. Is this will the liquid begin to overflow?


147

Solution: To answer this question, we obviously need to know the height of the tank. From the given capacity (250 liters = 0.25 m3 ) assuming that the cross-sectional area is uniform, since: Volume = Area x Height it is therefore easy to deduce that the tank is 1 m high. The magnitude of the step change in this instance is 0.1 m 3 /min, and since the steady-state gain for the process has been previously found to be 10 (min/m2), the ultimate value of the change in the liquid level will then be 0.1 x 10 =1m. Thus as a result of this change in the inlet flowra te, the liquid level is scheduled to change from its initial value of 0.37 m by a total of 1m, attaining to a final value·of 1.37 m. Observe that this value is greater than the total height of the tank. The conclusion therefore is that the operator is correct: there will be spillage of liquid. Liquid will begin to overflow the instant h > 1 or, equivalently, in terms of deviation from the initial steady-state value of 0.37 m, the instant y > 0.63. The expression for the time variation of the liquid level is easily seen in this case to be given by: h(t) =

0.37 + 1.0 (I - e- 0·4')

or y(t)

=

1.0 (I - e -0.4t)

We may now use either expression to find the time when h = 1, or y = 0.63; the result is t=25min.

We therefore conclude that for this magnitude of inlet flowrate change to the liquid level system, liquid spillage will commence at about 2.5 min after implementing the flow rate change.

5.2.2

Rectangular Pulse Response

The input function in this case is as described in Section 3.4.2 in Chapter 3; 0; t < 0 { u(t) = A; 0 < t
b Since the Laplace transform of this forcing function is given by: u(s)

= A-(I s

- e-bs)

(3.42)

the rectangular pulse response of the first-order process is obtained from:

K

A

y(s) = - - - ( I 7:S

+ I s

AK

s( rs

+ I)

(I

-

e-bs) e-bs)

(5.16)

PROCESS DYNAM ICS

148 AK(l- e -bit)

y(t}

t

b

0

k-

u(t) 0 ,.-.--

t I

I

0

b

system. Figure 5.5. Rectangular pulse response of a first-order

on is made easier by The invers ion of this appare ntly compl icated functi s: follow as parts two into splitting the terms AK

y(s)

= s(-rs + 1)

AKe-b• s(-rs

+ 1)

(5.17)

side of Eq. (5.17) are We may now observe that the two terms on the right-hand as we recall, merely , which n functio e-lls the of identical except for the presence edge of the effect knowl our upon call now we If time. in tion indicates a transla easily invert ed is (5.17) Eq. rms, of the transla tion function in Laplace transfo on different takes which n functio ing follow the into the time domai n to give values over different time intervals:

y(t)

=

{

A K ( l - e-tl-r)

; t b

the change in direction This response is shown graphically in Figure 5.5. Note function at this same input the in experienced at t =b in response to the change se at this point respon peak the of value the that instant. It is easy to establish e-11/T). AK(1 is Yma>< =y(b) = Examp le 5.3

D RECfA NGUL AR PULSE RESPO NSE OF THE LIQUI LEVEL SYSTEM.

The flowrate is again to be Let us return once again to the system of Example 5.1. 2 /min), but this time we hold it at this value m (0.087 s/min 87liter to ly change d sudden s/min (0.037 m 3 /min). only for 25 min; returni ng it to its initial value of 37liter during the course of this attain will level liquid the value um maxim the is What experiment?


149

Solution: The expression for the peak value attained by a first-order system in response to a rectangular pulse input of magnitude A and duration b is given by: Ymax =y(b)

=

AK(I - e-blr)

(Note that this expression is in terms of deviations from the initial steady-state value for the liquid level.) Since in this example, A = 0.05, b = 2.5, and from Example 5.1 we recall that K = 10, T = 2.5, with an initial liquid level of 0.37 m, the peak value for the liquid level is obtained from: hmax = 0.37 + 0.5(1 - e- 1)

or 0.686 m

hmax

5.2.3 Impulse Response In this case, recall from Eqs. (3.43) and (3.46) that the input is a Dirac delta fnnction with area A, u(t) = AO(t) and u(s) = A; thus the impulse response is obtained from: y(s)

= _K_ '&S

+ I

A"

(5.19)

which is easily inverted in time to give: y(t) = A K e -1/'r 't"

The impulse response is shown in Figure 5.6.

0

u(t)

Area=A

04---L---------------------0

Figure 5.6. Impulse response of a first-order system.

(5.20)

PROCESS DYNAM ICS

150

Note:

1.

AK/-r The impulse response indicates an immed iate "jump" to a value decay. ntial expone at t = 0 followed by an

2.

impuls e If Ystep represents the step response, and Yimpulse represents the response, then: (5.21)

5.2.4 Ramp Respo nse Recalling that for the ramp input function: O·

t < 0

u(t) = { '

At;

u(s)

(3.49)

t > 0

A = 2 s

(3.50)

the ramp response is therefore obtained from: y(s)

=

Cs ~ 1) ~

(5.22)

Laplace inversion, It is left as an exercise to the reader to show that upon s: Eq. (5.22) become y(t)

= AK-r (e- 11~ + tl'r---:- 1)

(5.23)

Note: 1.

a ramp As t --? oo, y(t) --? AK(t- 'f). Thus the response is asymptotic to at origin the from units time -r by ed displac AK, slope function with

y=O. 2.

se by If the step response is represented by Ystep and the ramp respon that: h Yramp then it is easy to establis (5.24)

The ramp response is shown in Figure 5.7.


151

Slope =AK y(t)

y(t) =AK!t- •)

0

1----..... .-c::/---- - - - - - - -

t

0

u(t)

Slope =A

0

Figure 5.7. Ramp responSe of a first-order system.

5.2.5 Sinusoida l Response Recalling Eq. (3.52) for the Laplace transform of a sine wave input of amplitude A and frequency OJ, the response of a first-order system to this sinusoidal input function is obtained from: K ) y(s) = ( -rs + I

Aw s2

+

(5.25)

w2

The inversion of this to the time domain (see tables in Appendix C) yields: y(t)

= AK [

w2-r

( W'r)

+ l

e- 111 +

I

...j (w-r)2 + I

sin {rot+

tP)J

(5.26)

where

tP = tan

-I ( -w-r)

(5.27)

This response is shown in Figure 5.8. There are a few important points to note in this sinusoidal response: 1.

The first term in Eq. (526) is a transient term that decays with a time constant -r, and vanishes as t -+ oo ; but the second term persists.

2.

After the transients die away (after about four or five time constants in practice) the ultimate response will be a pure sine wave. Thus, we have, from Eq. (5.26): y(t)

I,_. ~

= --/

AK sin ( t»t + ( m-r) 2 + I

tP)

(5.28)

which we shall refer to as the Ultimate Periodic Response (UPR) because it is to this - a function which is also periodic in its own right - that the sinusoidal response ultimately settles.

152

PROCESS DYNAMICS M

y(t)

= AKI[(orcl' +

ff'

M~ 0

-M

---

- - --- 0

-----1

---

-----··

- --

.J-

-. t

I

AEllll------

.!_ ____ ,

"'"

0

~- ------c~--

-A~--·t

0

Figure 5.8. Sinusoidal response of a first-order system.

Characteristics of the Ultimate Periodic Response The following are some important, noteworthy characteristics of the ultimate periodic response (UPR) given in Eq. (5.28): 1.

The input function (u(t) =Asina>t) and the UPR are both sine waves having the same frequency ro.

2.

The UPR sine wave lags behind the input signal by an angle cp.

3.

The ratio of the UPR's amplitude to that of the input sine wave is KN (ro-r) 2 + 1, a quantity termed the "amplitude ratio" (AR) for precisely this reason.

Thus we conclude by noting that the ultimate response of a first-order system to a sine wave, the UPR, can be characterized by the following two parameters: (5.29a)

and (5.29b)

respectively known as the amplitude ratio and the phase angle. Note that both quantities are functions of ro, the frequency of the forcing sinusoidal function. Studying the behavior of AR and cp functions as they vary with ro is the main objective in frequency response analysis.

CHAP 5 DYNA MICS OF LOW -ORD ER SYST EMS

153

Frequency Response The frequency response of a dynamic syste m is, in general, a summ ariza tion of its (ultimate) responses to pure sine wave inpu ts over a spec trum of frequencies w. As indic ated above, the prim ary variables are AR, the amp litud e ratio, and ¢, the phas e angle. It is custo mary to refer to· the phas e angl e as a phase lag if the angle is negative and a phase lead if the angle is positive. The most com mon meth od for sum mari zing frequ ency resp onse infor mati on is the graphical repre senta tion sugg ested by Hend rik Bode [1]. Kno wn as the Bode diag ram (or Bode plot), it consists of two graphs: 1.

2.

log (AR) vs. log w, in conjunction with ¢ vs. log w

It is often the case that the process stead y-state gain K and time constant -r, are used as scaling factors in presenting these graphs. In such a case, the Bode diag ram will cons ist of a log-l og plot of the so-called Magnitude Ratio, MR = AR/ K, in conjunction with a semilog plot of the phas e angl e ¢, with log( Wt') as the absc issa in each plot. The main adva ntag e lies in the gene raliz ation that such scaling allows; the qual itativ e natu re of the actual frequency response characteristics are not affected in any way. The Bode diag ram for the first-order syste m (consisting of MR and phas e angle plots against log(m )) is show n in Figure 5.9. Detailed discussions about some impo rtant char acter istic s of this diag ram will be take n up later in Chap ter 9, whic h is devo ted entir ely to the stud y of frequ ency respo nse anal ysis.

5.3

PURE GAIN SYSTEMS

Cons ider the first- orde r syste m with -r = 0. (This corresponds to a physical syste m that, theoretically, is infinitely fast in resp ondi ng to inpu ts. More realistically, one migh t imagine a situa tion in which the first-order syste m is so fast in resp ondi ng that 1:is so small as to be negligible.) In this case, Eq. (5.2) becomes: y(t) = Ku(t) (5.30) and, eithe r by takin g Laplace trans form s of Eq. (5.30), or by setti ng -r = 0 in Eq. (5.4), we obtain: y(s)

=

Ku(s)

so that the transfer function for such a proc

(5.31)

ess is identified as:

g(s) = K (5.32) A process havi ng such characteristics is referred to as a pure gain process by virtu e of the fact that its transfer func tion invo lves only one characteristic para mete r: K, the process gain term. Such systems are always at stead y state, mov ing instantly from one steady state to another with no transient behavior in betw een stead y states. The transfer function for the pure gain syste m has no poles and no zeros.

PROCESS DYNAMICS

154

. --........,

~

~

0

•

~

-50

~

~

-100

Figure 5.9.

5.3.1

-

Bode diagram for a first-order system.

Physical Examples of Pure Gain System

There are a few physical processes that truly exhibit pure gain characteristics. One example is the capillary system shown in Figure 5.10. There is a flow constriction that results in a pressure drop in the incompressible fluid flowing through the capillary. The upstream and downstream pressures are measured by liquid levels in manometers. The value of the head h is observed to change whenever the liquid flowrate F changes. Thus in this case, the input variable is F, while the output is h. Owing to the fact that a capillary constitutes a laminar resistance, the headflow relationship is given by the equation: h = RF

(5.33)

where R is the resistance. In terms of deviation from an initial steady-state h(}l and Fw Eq. (5.33) becomes: y

Figure 5.10.

Ru

The capillary system.

CHAP 5 DYNAM ICS OF LOW-O RDER SYSTEM S

155

with

The import ant point to note here is that any change in F is instant aneous ly transm itted as a change in the head, h. Other physica l system s that exhibit a pure gain respon se include electrical resistors and the mechan ical spring. First-o rder or higher order systems whose dynam ics are extrem ely fast may be conven iently approx imated as pure gain processes. (Obser ve that when -r= 0, Eq. (5.4) become s approx imately the same as Eq. (5.32).) For example, a small pneum atic control valve with very rapid respon se may be approx imated as a pure gain proces s if the other parts of the proces s have much larger time constants. Perhap s the pure gain system with the most import ant applica tion in process control is the proportional controller. As we shall discuss fully later, the output of this control ler, the comma nd signal c, is depend ent on the input signal e accordi ng to: c(t)

= Kc E{t)

(5.34)

clearly indicat ing pure gain characteristics.

5.3.2 Respo nse of Pure Gain System to Vario us Input s From the mathem atical represe ntation of the pure gain process in Eq. (5.30) or (5.31), it is easy to see that: The output of a pure gain process is directl y propor tional to the input, the constant of proportionality being the process gain.

Thus the respon ses are identic al in form to the input (or forcing ) functions, differi ng only in magnit ude. The input functio n is amplif ied if K > 1, attenua ted if K < 1, and left unchan ged if K = 1. The followi ng is a catalog of the responses of a pure gain process to various input functions. Step Response Input u(t)

{ 0; t < 0 A- t > 0 '

(3.36)

Output y(t)

t < 0 A'K; t > 0

{ O·

(5.35)

PROCESS DYNAMIC S

156 Rectangul ar Pulse Response

Input

u(t)

r r

t < 0 (3.39)

A; 0 < t < b

.

O· t > b

Output

y(t)

t < 0 A'K; O 0

(5.38)

Sinusoida l Response

Input 0;

t < 0

u(t)

={

y(t)

= { AKsinwt;

(3.51)

Asinwt; t > 0

Output 0;

t

< 0 t > 0

(5.39)

We note from here that the frequency response of the pure gain process indicates an amplitude ratio AR = K, and a phase angle t/1 = 0. In other words, the UPR of the pure gain process is a sine wave whose amplitude is K times the amplitude of the input sine wave, and perfectly in phase with the input. This, of course, is in keeping with the characteristic of the pure gain process.


157

The advantage in being able to approximate the dynamic behavior of a fast dynamic process with that of a pure gain process lies in the fact that the derivation of the dynamic responses (an exercise which normally involves Laplace inversion) is now considerably simplified to an exercise involving mere multiplication by a constant.

5.4

PURE CAPA CITY SYSTEMS

Let us return to the general, first-order differential equation used to model the dynamic behavior of first-order systems:

41

a 1 dt + aoY = b u(t)

(5.1)

Recall that the characteristic first-order system parameters, the steady-state gain, and time constant, are obtainable as indicated in Eq. (5.3), provided a0 0. It is now of interest to investigate what happens when a0 = 0 in Eq. (5.1). Observe that in this case, the original equation becomes:

"*

or

~ = K*u(t)

(5.40)

K*

(5.41)

where

A process modeled by Eq. (5.40) is known as a pure capacity process. Observe that regardless of the specific nature of the input function u(t), the solution to Eq. (5.40) is easily obtained by direct integration; the result is, assuming zero initial conditions: y(t) = K*

r

u(a) da

(5.42)

0

Because the output of this process involves an integration of the input function, pure capacity processes are sometimes called pure integrators. Taking Laplace transforms in Eq. (5.40) gives: y(s) =

K* -u(s)

s

(5.43)

from which we obtain the transfer function for the pure capacity system as: K*

g(s} = -

s

(5.44)

PROCESS DYNAMICS

158

the The pure capacity system is therefore characte rized by the presence of be integrato r (or capacitan ce) element 1/s, and the paramet er K*, which may (5.42).) Eq. (see gain r regarded as an integrato From the expressio ns in Eqs. (5.3a,b) and Eq. (5.44), we may now note that the behavior of a first-orde r system approach es that of a pure capacity system the in the limit as -r~oo and K-'too while their ratio, K/ 1:, remains fixed at er first-ord a be to system capacity value K*. We may thus imagine a pure but large y extremel both are gain ate steady-st and constant time system whose for which the ratio of these paramete rs is a fixed, finite constant. The transfer function of a pure capacity system has one pole at the origin (s = 0) and no zeros.

5.4.1 Physica l Examp le of a Pure Capacit y System tank The most common example of a pure capacity system is a storage (or surge) 5.11. Figure in with an outlet pump such as shown Such a tank is typically used in the process industrie s for intermed iate the storage between two processe s. The outflow, F (usually fixed), is set by The tank. the in h, level, liquid the of ent independ therefore is pump; its value inflow Fi is usually the input variable, and its value can vary. , then If Ac is the cross-sec tional area of the storage tank (assumed uniform) a material balance on the tank yields: (5.45)

At steady state, for a fixed value of F, Eq. (5.45) becomes: (5.46)

0 = F;,-F

Subtract ing Eq. (5.46) from Eq. (5.45) and defining the deviatio n variables y = h- h5, u = F;- F;5, we obtain: (5.47) Product From Upstream Process

F;

Storage Tank

h

Feed to Downstrea m Process

I

-g

F

Pump

Figure 5.11.

The storage tank with an outlet pump.


159

from where Laplace transform ation gives: IIA, y(s) = -s- u(s)

(5.48)

which is of the same form as Eq. (5.43), with K* = 1/A,. Other example s of pure capacity processe s include the heating of wellinsulate d batch systems, the filling of tanks with no outlet, the batch preparat ion of solutions by addition of chemicals to solvent, etc.

5.4.2 Respon se of Pure Capacit y System to Variou s Inputs Once again, by combini ng the transfer function of the pure capacity system (5.44) with the Laplace transform of the input function in question , we may use the transfer function model in Eq. (5.43) to derive this process system's response to various input functions. Step Response We have, in this case: y(s)

= K*!l =7 AK* s s

which is easily inverted back to time, giving: y(t) = AK*t

(5.49)

the equation of a ramp function with slope AK". A little reflection bears out the fact that this mathema tically derived step response does in fact make physical sense. Observe that a sudden increase in the input flowrate to the example storage tank of Figure 5.11 results in a continuo us increase in the liquid level in the tank - a situation which, in theory, can continue indefinit ely but is limited in actual practice by the finite volume of the tank. (In actual practice, the liquid level in the tank increases until the tank is full and material begins to overflow.) The following is a catalog of the pure capacity process response to the other input functions . It is left as an exercise to the reader to verify these results mathema tically, and to confirm that they make physical sense. Rectangu lar Pulse Respons e

y(t)

{

~~*t; ~: ~
b

a response that starts out as a ramp function, and subseque ntly settles down to a new steady-s tate value AK*b.

PROCESS DYNAMIC S

160 Slope =AK"

AK*b y(t)

0 0

--;------~----.----------t b 0

(b) Rectangular pulse response

(a) Step response

y(t}

0

~r--------------

0

0

d) Ramp response

(c) Impulse response

Pure capacity system responses.

Figure 5.12.

Impulse Response y(t)

AK*

(5.51)

y(t)

AK*t2 2

(5.52)

a step function. Ramp Response

Sinusoida l Response AK* - - (1 - cosro t)

. y(t)

w

or y(t) =

AK* [1 + sin(wt- 90°)] ro

(5.53)

implying that the frequency response of a pure capacity process has an amplitude ratio and a phase angle given by:

AR

K*

'

= -90°

(5.54a)

(I)

(5.54b)

Diagramm atic represent ations of the step, pulse, impulse, and ramp responses are shown in Figure 5.12; the sinusoidal response is shown separately in Figure 5.13.

CHAP 5 UYNAMlCS UF LUW-Ul -r; i.e., when we have a stronger lead (or a weaker lag). The swiftness of this type of response- indicative of the dominant influence of the lead term - tends to give the impression of an impulse response of a first-order system. It is worthwhile to remark once again that the approach to the final steady state is governed solely by the lag time constant; but the nature and value of the initial response is determined by the lead-to-lag ratio. The task of deriving the lead/lag system response to the remaining ideal forcing functions is quite straightforward and is left as an exercise to the reader.

5.6

SUMMARY

This chapter has been concerned with characterizing the dynamics of processes modeled by low-order, linear differential equations. We have found that the transfer functions for these systems involve denominators having at most a first-order polynomial. The first-order system was identified as having two characteristic parameters: the steady-state gain K, a measure of the magnitude of the steadystate system response, and the time constant T, a measure of speed of system response; its transfer function has a single pole and no zeros. Several physical examples of relatively simple processes that can be categorized as first-order systems were presented. The mathematical expressions representing the response of the first-order system to other ideal forcing functions were derived and the important characteristics of each of these responses highlighted. When a 1, the coefficient of the only derivative term in the differential equation model of a first-order system vanishes, the resulting system responds instantaneously to input changes. Such a system was identified as a pure gain process. A pure gain process is therefore no more than a first-order system with

CHAP 5 DYNAM ICS OF LOW-O RDER SYSTEM S

167

zero time consta nt - one that is infinit ely fast. The transfe r functio n for this proces s was found to have no pole, and no zero: only a consta nt gain term. As such, the dynam ic respon se of pure gain proces ses are very easy to compu te since they are mere multip les of the forcing functions. Very few such system s occur in actual practic e, but first-o rder system s whose time consta nts are quite small (system s whose dynam ics are very fast, respon ding with hardly notice able transie nts) behave approx imatel y like pure gain proces ses. The disapp earanc e of a 0 , the coeffic ient of y(t), from the first-o rder differe ntial equati on model results in a uniqu e situati on in which the first deriva tive of the proces s outpu t is directl y propor tional to the input functio n (imme diately implyi ng that the output is directl y propor tional to the integr al of the input functio n). System s of this type were identif ied as pure capacity system s or pure integrators. The transfe r functio n for these proces ses has a single pole located at the origin (s = 0}, and no zero. We have noted that in the limit as the time consta nt and steady -state gain of a first-o rder system both approa ch infinit y, doing so in such a way that the ratio K./r remain s fixed, the behav ior of the pure capaci ty system is obtain ed. The storag e tank in which the outflo w is indepe ndent of the liquid level was presen ted as the typica l examp le of the pure capaci ty system . The dynam ic respon ses of this class of system s were also derive d and discussed. The final class of system s to be discus sed was the lead/l ag system . The charac teristic param eters of the lead/l ag system were identif ied as the system gain, a lead time consta nt, a lag time consta nt, and the lead-to -lag ratio. The transfe r functio n has one pole and one zero; and it is the presen ce of this zero that really disting uishes the lead/l ag system from all the other low-o rder system s. We showe d that the dynam ic behav ior of the lead/l ag system is in fact a weigh ted averag e of the dynam ic behav iors of the pure gain and the first-o rder system s. This fact was used to derive the unit step respon se of the lead/l ag system , and its somew hat unusu al characteristic s were then discus sed. What we have learne d from this partic ular respon se will becom e useful in Chapt er 16. Our study of the dynam ic behavi or of proces s system s contin ues in the next chapte r when we invest igate the behav ior of system s whose dynam ics are describ ed by model s of higher order than the ones we have encou ntered in this chapte r.

REFERENCES AND SUGGESTED FURTHER READ ING 1.

Bode, H. W., Network Analysis and Feedback Amplifier Design, Van Nostra nd, New York (1945)

REVIEW QUES TIONS 1.

What is a first-or der system and what are its charact eristic parame ters? Which of these charact eristic parame ters measur es "how much" the process will respon d to input change s, and which measur es "how fast" the process will respond ?

2.

About how long in units of "time constan ts" will it take for the step respons e of a firstorder system to attain more than 99% of its ultimat e value?

PROCESS DYNAMICS

168

3.

What does the "inherent reluctance factor" measure?

4.

How can the characteristic parameters of a first-order system be determined from its theoretical step response?

5.

What is the relationship between a process step response and impulse response?

6.

What is the relationship between a process ramp response and step response?

7.

What are the main characteristics of the sinusoidal response of a first-order system?

8.

What is frequency response in general?

9.

What is a pure gain system and what are its main characteristics?

10. What limiting type of first-order system is a pure gain system? 11. What is the main characteristic of a pure capacity system?

12. What limiting type of first-order system is a pure capacity system? 13. A lead/lag system has how many poles and how many zeros? 14. In what way does the value of the lead-to-lag ratio affect the step response of a

lead/lag system?

15. What relationship exists between the dynamic behavior of the lead/lag system and that of the pure gain system and the first-order system?

PROBLEMS 5.1

The process shown in Figure P5.1 is a constant volume electric water heater. On a particular day, as the tank water temperature reached the 80°C mark, the heater broke down and stopped supplying heat. At this time, water was withdrawn from this 100 liter capacity tank at the rate of 10 liters/min; cold water at 30°C automatically flowed into the tank at precisely the same rate. This exercise went on for exactly 5 min, and the water withdrawal was stopped. (Because of the heater design the cold water also stopped flowing in.) ColdWater

T;-3E::=~

T Hot Water Heater

Figure P5.1.

The water heater.

By developing an appropriate mathematical model for this process, show that it is reasonable to consider this a first-order process; and by solving the resulting

169


differential equation model, find the final tank water temperature at the end of this 5-minute period. State all your assumptions. 5.2

The temperature Ton a critical tray in a pyrolysis fractionator is to be controlled as indicated in the following schematic diagram, employing a feedfarward controller that manipulates the underflow reflux flowrate Ron the basis of feed flowrate measurements F. By experimental testing, the following approximate transfer function models were obtained for the response of the critical tray temperature (1) to feed rate changes, and (2) to reflux flowrate changes: (I)

(T-T*) =

~(F-F*) 30s + 1

(P5.I)

(2)

(T-T*) =

----=L 6s + I

(P5.2)

and: (R- R*)

the timescale is in minutes, the flowrates and temperature are, respectively, in Mlb/hr (thousands of pounds/hour) and °F, and the (*) indicates steady-state conditions. (a) Rewrite the model in the form: y(s) = g(s)u(s)

+ gj..s)d(s)

stating explicitly what the variables y, u, and d represent in terms of the original process variables, and clearly identifying g(s) and gd(s). (b) Obtain an expression for the dynamic response of the tray temperature to a unit step change in the underflow reflux flowrate, given that the initial temperature on the tray is 250°F and that the feedrate remained constant. To what ultimate value will the tray temperature settle after this unit change in R? (c) Obtain an expression for the dynamic response of the tray temperature, this time to a unit step change in the feed flowrate, once more given that the initial temperature on the tray is 250°F, and that this time the reflux rate has remained constant. Deduce the final steady-state tray temperature in response to this change in feedrate. PYROLYSIS FRACTIONATOR

T

Underflow Reflux

R

Feedforward Controller

Figure P5.2.

The pyrolysis fractionator.

PROCESS DYNAMICS

170

5.3

(a) Using your favorite computer simulation software, or otherwise, plot the dynamic responses obtained in Problem 5.2(b) and (c) above on the same graph, and on the same scale. Which input variable affects the tray temperature the most (in terms of absolute change in the tray temperature) , and which affects it the quickest (i.e., which response gets to steady state the quickest)? (Keep in mind that these are both unit step responses.) (b) By summing up the individual expressions for the two responses (or otherwise) obtain an expression for the tray temperature response to a simultaneou s implementat ion of unit step changes in both the feedrate and the reflux rate, and given the same initial temperature of 250°F. Plot this combined response and comment on its shape.

5.4

(a) Assuming that the model supplied for the pyrolysis fractionator in Problem 5.2 is accurate, given that an inadvertent step change of 2 Mlb/hr has occurred in the feedrate, what change would you recommend to be made in the reflux flowrate in order that at steody state no net change will be evident in the tray temperature? (b) It has been recommende d by an experienced control engineer that if the specified process model is accurate, "perfect" feed disturbance rejection can be obtained if the reflux flowrate is made to change with observed variations in the feedrate according to: (R-R*) = gFF(F-F*)

(P5.3)

where the indicated transfer function, gpp is given by: 4(6s + 1)

BFF

= (30s +

I)

(P5.4)

Using Eqs. (P5.3) and (P5.4), for the situation when the process is operating under the indicated strategy, obtain an expression for the reflux rate response to the step change of2 Mlb/hr introduced in part (a) above. Plot the response and compare the ultimate change in the reflux flowrate recommende d by this control scheme with your recommenda tion in part (a). 5.5

In Process Identification (see Chapter 13) the objective is to obtain an approximate transfer function model for the process in question by correlating input/ output data obtained directly from the process. It is known that the frequency-re sponse information derived from such plant data is richest in information content if the process input is an ideal impulse function. Since this function is, of course, impossible to implement perfectly in reality, it is customary to utilize instead the rectangular pulse input as a close approximatio n. Given a first-order process with the following transfer function: g(s)

4 =~

(P5.5)

(a) Obtain the theoretical unit impulse response; let this bey"( I). (b) Obtain the response to u1(t), a rectangular pulse input of height 1 unit and width 1 emit; let this be y 1 (t). (c) Obtain the response to uz(t), a rectangular pulse input of height 2 units and width 0.5 unit; let this be y2(t). (d) Compare these three responses by plotting y*(t), y 1(t), and y 2(t) together on the same graph and confirm that the rectangular pulse input with a narrower width provides a better approximatio n to the ideal impulse function.

CHAP 5 DYNAMICS OF LOW-ORDER SYSTEMS 5.6

171

The procedure for starting up some industrial reactors calls for the gradual introduction of reactants in several steps rather than all at once. Given that the dynamic behavior of one such reactor is reasonably well represented by the simple transfer function model: 0.32 (P5.6) y(s) = !5.5s + I u(s) where y represents, in deviation variables, the reactor temperature (with the time scale in minutes), it is required to find the response of the process to a program of reactant feed represented by the staircase function described mathematically as:

u(t)

=

r

t:S

0

I.

0 < t :s; 2

2; 3;

2 < t :s; 4 4 < t :s; 6

4·

t

(P5.7)

> 6

and illustrated in Figure P53. While it is possible to obtain first the Laplace transform of the input function u(t) and then use Eq. (PS.6) directly, this may not be very convenient. Find the process impulse-response function g(t), and recast the given process model in the impulseresponse form: y(t)

=

J'

g(a)u(t- o') da

(P5.8)

0

Now use this directly with Eq. (PS.7) to obtain the required response.

4

3

I

u(t)

2

2

4

6

Time

Figure P5.3. Reactor feed program. 5. 7

A storage tank of the type shown in Figure 5.11 in the main text is subject to periodic loading from the upstream process. Because a steady rate of product withdrawal is maintained by the constant speed pump at the tank outlet, this is a true "pure capacity" process. The tank's cross-sectional area is given as 2.5 ft 2, and that the incoming feedstream fluctuates around its nominal flowrate in essentially sinusoidal fashion with maximum deviation of 10 ft3 /hr; the frequency of the sinusoidal

172

PROCESS DYNAMICS fluctuation is different from shift to shift, but it may be considered constant on any particular shift. (a) Derive from first principles the expression for the response of the tank level (as a deviation from nominal operating conditions) assuming that the input is a perfect sinusoid with the given amplitude but a yet unspecified frequency (1). For a given frequency of 0.2 radians/hr, plot on the same graph. both the periodic input loading function and the tank level response. Compare these two functions. (b) In terms of the frequency of the periodic loading, what is the maximum deviation from nominal tank level that will be observed? (c) Given now that the tank is 10ft high and that the nominal operating level is 5 ft, what condition must the loading frequency satisfy to guarantee that during any particular shift, the tank neither overflows nor runs dry?

5.8

Consider a household pressing iron along with its thermostat as a dynamic system of constant mass m and heat capacity c,. (a) Develop a mathematical model for this system assuming that the rate of heat input through the heating element is Q (appropriate energy units/min) and that the iron loses heat to the surrounding atmosphere according to Newton's "law" of cooling; i.e., the heat loss is given by hA(T- T.), where T. is the (constant) temperature of the surrounding atmosphere, h is the heat transfer coefficient, and A is the effective area of heat transfer. (b) Given T5 and Q 5 as appropriate steady-state values for T and Q respectively, express your model in the transfer function form y(s) =g(s) u(s) where y =T- T5 and u =Q-Q5 ;show thatg(s)has the first-order form. What are K and -rin this case? (c) A specific pressing iron, with K = 5, has a thermostat set to cut off the standard energy supply (u• =10) when T attains 55°C. At t = 0, the iron was at an initial steady-state temperature of 24°C and the standard energy supply commenced; precisely 6 minutes after the commencement of this operation, the energy supply was automatically cut off. Find the effective time constant -r for this particular iron.

5.9

A certain process is composed of two first-order processes in parallel such that its transfer function is given by: (P5.9)

(a) Consolidate this transfer function and identify its steady-state gain, its poles and its zeros, and thereby establish that the steady-state gain is the sum of the steadystate gains of the contributing transfer functions; the poles are identical to the poles of the contributing transfer functions; the lead time constant (&om which the zero is obtained) is a weighted average of the two contributing lag time constants. (The dynamic behavior of such systems are investigated in Chapters'6 and 7.) (b) What conditions must the parameters of the contributing transfer functions satisfy if the composite system zero is to be positive? 5.10

Consider creating a model for the bottom part of the distillation column shown below in Figure P5.4. Here, in terms of deviation variables:

V L W8

vapor rate leaving the bottom; liquid rate entering the bottom mass in column base

mass in reboiler heat input rate bottom product withdrawal rate


173

Q

B

Figure P5.4.

The bottom part of a distillation column.

Transfer function relations are given as: WR(s)

-8

Q(s)

= J.()";'+J ;

~ Q(s)

0.01

.

= TQ;+i '

f:£!1_

1

V(s) - I OOOs + I

in addition to the differential equation in time: d [W R(t) + W 8 (c)]

dt

=

L-V- B

=

(a) Derive the transfer function W8 (s)/Q(s) assuming that B 0. (b) Obtain the mathematical expression for W8 (t) given that initially W8 (0) =0 and a step change of magnitude Q* is made in Qat t = 0.

CHAPTE R

6 DYNAMIC BEHAVIOR OF LINEAR HIGHER ORDER SYSTEMS

Having discussed the behavior of low-order systems in the last chapter, the next level of complexity involves those systems modeled by linear differential equations of higher (but still finite) order. Such systems are typically characterized by transfer functions in which the denominator polynomials are of order higher than 1. The objectives in this chapter are similar to those of the preceding chapter: to investigate and characterize the dynamic responses of this class of systems. We will begin our investigation of such systems in the most logical place: with the composite system made up of two flrst-order systems connected in series. We will flnd this to be a natural extension of our earlier discussion on first-order systems. This opening discussion of two first-order systems in series will also serve two other purposes: 1.

As a precursor to the discussion of general, second-order systems; for, as we shall see, the former is, in fact, a special case of the latter, and

2.

As a natural launching pad for subsequently generalizing to Nth order systems (with N>2).

·Our discussion in this chapter will end with a treatment of the dynamic behavior of one final class of higher order systems: those whose transfer functions have zeros in addition to the higher order denominator polynomials. There is yet another class of systems - time delay systems - qualified to be classified as "higher order." However, these systems are, in principle, of infinite order and will be discussed at a later stage in Chapter 8.

6.1

TWO FIRST-ORDER SYSTEMS IN SERIES

Consider the situation in which two liquid holding tanks are connected in series, each one being of the type identified as a first-order system in Chapter 5 175

PROCESS DYNAMICS

176

(Section 5.1.1). Such systems can be configured in one of two ways. Let us consider the first of such configurations shown in Figure 6.1 below. This indicates a situation in which the flow out of tank 1 discharges freely into the atmosphere before entering tank 2. The flowrate of this stream is therefore solely determined by the liquid level in tank 1 and is totally independent of the conditions in tank 2. Thus while the conditions in tank 1 influence the conditions in tank 2, observe carefully that the conditions in tank 2 in no way influence tank 1; i.e., tank 2 does not interact with tank 1. This is the noninteracting configuration. Contrast the arrangement in Figure 6.1 with the one shown in Figure 6.2.. In this case, the flow out of tank 1 now depends on the difference between the levels in the two tanks, with the result that variations in the conditions in tank 2 will influence the conditions in tank 1. Such an arrangement gives rise to a situation in which tank 2 does in fact interact with tank 1; it is therefore called the interacting configuration. As we might expect, the dynamic behavior of the interacting ensemble will differ from the dynamic behavior of the alternative noninteractin g arrangement. Let us look at these in tum, beginning with the noninteractin g system of Figure 6.1.

6.1.1 Two Noninterac ting Systems in Series Let us, for simplicity, assume a linear relation between liquid level and flowrate through the outlet valves in the system shown in Figure 6.1, so that outflows will be directly proportional to the liquid levels. If A 1 and A 2 represent the (uniform) cross-sectional areas of tanks 1 and 2 respectively, then the following mathematical models are obtained by carrying out material balances for each tank: • ForTank1: (6.1)

• For Tank 2 the inflow is given by F1 = c1h1; thus we have: (6.2)

Figure 6.1. Two noninteracting tanks in series.

CHAP 6 DYNAMICS OF HIGHER ORDER SYSTEMS

177

Figure 6.2. Two interacting tanks in series.

These two equations may be rearranged and presented in terms of deviation variables in the usual manner; the results are: (6.3)

(6.4)

where the deviation variables are given by y 1 = h1 - h15 ; y 2 u =F0 - F0s ; and the system parameters are given by:

= h2 -

h25 , and

and

It is important to note the following facts about this system:

1.

Each component tank is modeled by a first-order differential equation; therefore the entire process is an ensemble of two first-order systems in series.

2.

The individual transfer function representation for each of the component tanks may be obtained by taking Laplace transforms in Eqs. (6.3) and (6.4). The results are:· y 1(s)

=

y2(s)

=

KI 't"IS

u(s) + 1

K2 +

't"2S

1 YI(s)

= g 1 u(s) = 82 Y1(s)

(6.5)

(6.6)

clearly identifying each tank as a first-order system in its own right.

178

PROCESS DYNAMICS

Now, for this system, we are interested in studying the effect of changes in the input function F0 , on the level in the second tank; i.e., we wish to investigate the influence of u on y 2 • (Note that we are already familiar with the influence of u on y1; it is a first-order response problem similar to those studied in detail in Chapter 5.)

Differential Equation Model The differential equation we seek for this system is the one that represents the behavior of y 2 and its direct dependence on u, not its indirect dependence through the unimportant intermediate variable Yr Such a model can be obtained by eliminating y 1 from Eqs. (6.3) and (6.4). This may be done as follows. Differentiatin g Eq. (6.4) with respect to time and substituting Eq. (6.3) for the resulting first derivative of y1 gives:

(6.7)

By solving Eq. (6.4) for y1 and introducing the result into Eq. (6.7), we may now eliminate the last surviving y 1 term. The final expression appropriately rearranged is:

(6.8)

a second-order differential equation in y2, with u as the forcing fu..T'tction.

Transfer Function Model The transfer function representatio n for the composite system relating y2 (s) directly to u(s) is shown in Figure 6.3 and can be obtained by substituting Eq. (6.5) for y1 in Eq. (6.6); the result is:

y2(s)

u(s)

Figure 6.3.

·I

K2 'X"2S

gl(s)

+

K, u(s) -r,s + 1

yl(s)

·I

(6.9)

y2(s) g.jsl

~

Block diagramatic representation of two first-order systems connected in series.


The overall transfer function for this process is therefore now given

179

by: (6.10)

with the characteristic parameters: •

•

K = K 1K 2, the "combined" steady-state gain, Two time constants, 'rp and -r2, the individ ual time constan ts of the

contributing first-order elements.

The transfer function for this process therefore has two poles (at s = -1/7:1 and s = -1/12) and no zeros. Alternatively, we may equally well obtain the transfer functio n model for this process by taking Laplace transforms of the differe ntial equation model of Eq. (6.8), assumi ng zero initial conditions. It is an easy taskleft as an exercise for the reade r- to show that the results are identic al A very import ant point to note here is that the overall transfe r function for this process is in fact just a produc t of the two contrib uting transfer functions indicat ed in Eqs. (6.5) and (6.6). A block diagram atic represe ntation of this system shown in Figure 6.3 emphasizes this point.

6.1.2 Unit Step Respo nse of the Nonin teract ing System s in Series We may now use the transfer function of Eq. (6.9) to obtain express ions for how

this system respon ds to various inputs. When the input function is the unit step

function, for example, the system response may be obtained from:

K

1

(6.11)

since, in this case u(s) =1/s. Upon partial fraction expansion Eq. (6.11) is easily inverted (provided -r1 ~ -r2 } to give:

In the special case when -r1 =12 =1:, Eq. (6.11} becomes: (6.13)

and Laplace inversion yields:

(6.14)

180

PROCESS DYNAMICS 1.0 0.8 Y; (t)

K;

0.6 0.4 0.2

0.0 2

0

4

6

8

10

Figure 6.4. Unit step responses of the liquid level in the two-tank process.

A sketch of the two-tank system response is shown in Figure 6.4 and compared with the first-order response behavior of a single tank. The important points to note about these responses are: 1.

While the single first-order system response is instantaneous, the response of the two-tank system shows a sigmoidal behavior characterized by an initial sluggishness (at t = 0) followed by a speeding up prior to the final approach to steady state. Note the presence of an inflection point in the response curve.

2.

It is left as an exercise to the reader to show that the slope of the

two-tank response at the origin is zero, i.e.:

dh dt

I

-0

1=0 -

which is to be compared with the nonzero initial slope of the single, first-order response given in Eq. (5.14). 3.

As was obtained for the single first-order system in Eq. (5.15), the same measure of "inherent reluctance" for the two (noninteracting) first-order systems in series may be obtained by integrating, over time, the difference between instantaneous response, y• = K, and the actual unit step response given in Eq. (6.12), i.e.: }2

=

f~ [y*- y(t)] dt 0

which, when carefully evaluated, gives the interesting result: (6.15)


181

which is to be compared with the correspon ding single, first-orde r system result: 1 = K-rwhen A= 1 in Eq. (5.15). The implication of this result is that the addition of another first-order system with a time constant of -r2 increases the reluctance inherent within the system by an amount directly related to this time constant. Thus two first-order systems in series are more reluctant to respond to changes than a single, first-order system by itself. Upon careful considera tion of the physical system in Figure 6.1, a little reflection will confirm that this statement is in pe:fect keeping with what we would intuitively expect.

6.1.3 The Interacti ng System Let us now consider the interactin g system arrangem ent of Figure 6.2. The material balances for this arrangem ent are as shown below: For Tank 1:

(6.16) ForTank2 :

(6.17)

Assuming linear resistances, the flowrate out of tank 2, as before, is still given by F2 = c2h2 ; but now F1 is given by: F 1 = c 1(h 1 -h 2)

The mathemat ical models Eqs. (6.16) and (6.17) therefore become:

~dh, = c, dt A 2 dh 2

-;,;dt'=

Fo

-h, +h2 +c,

(6.18)

c1 ( 1 +c 1 ) h2 -h.c2 cz

(6.19)

As was done in the nonintera cting case (in producing Eqs. (6.3) and (6.4)), if we now introduce the same constants, and express these equations in terms of the usual deviation variables, taking Laplace transform s and rearrangi ng the resulting expression s will give rise to the following transfer function model: K

where, once again, K = K1K2.

(6.20)

182

PROCESS DYNAMICS

Observe now that the difference between the transfer function for the noninteracting case (given in Eq. (6.9)) and the one indicated in Eq. (6.20) for the interacting case lies in the presence of the term K2 -r1 in the coefficient of sin the denor.:tinator polynomial in Eq. (6.20). This is appropriately indicative of the interaction of tank 2 (via its steady-state gain K2) with tank 1 (via its time constant 1i)· The following example will assist us in appreciating the effect of interaction on the dynamic behavior of interacting first-order systems in series. Example 6.1

COMPARISON OF THE DYNAMIC BEHAVIOR OF LIQUID LEVEL FOR INTERACTING AND NONINTERACTING CONFIGURATIONS.

Consider the situation in which the system of Figure 6.1 consists of two identical tanks with identical time constants -r1 "' 1'z == 1 minute. The steady-state gains are also identical, and equal to 1; i.e., K 1 == K2 =1. 1. Obtain an expression for the response of y2 (the level in the second tank as a deviation from its initial steady-state value) to a unit step change in the inlet flowrate to tank 1. 2. If these same tanks are now arranged as in Figure 6.2, obtain the unit step response for this interacting configuration and compare with the response obtained in (1) for the noninteracting case. Solution: 1. In this case since -r1 = -r2 , the required response may be obtained by directly appealing to our earlier result in Eq. (6.14) for 'I"= 1 and K = 1; what we have is the following result:

(6.21) 2. In the interacting situation, introducing the given parameter values into Eq. (6.20), we find that the transfer function model is given by:

which, for a unit step input, becomes (after factorization of the denominator quadratic): I l Yz(s) = (2.618s + I )(0.382s + I) -;

(6.22)

Before carrying out the final step of Laplace inversion in Eq. (6.22), note first of all that this equation, when compared with Eq. (6.11), indicates that the interacting configuration of two first-order systems in series, each having a time constant of 1 minute, is equivalent to a noninteracting configuration of two first-order systems having unequal time constants 2.618 min and 0.382 min. Thus the first effect of interaction is to alter the effective time constants of the individual contributing first-order systems. Laplace inversion of Eq. (6.22) immediately yields: Y2(1) = (1- l.l71e-0.382t + 0.17le-2.618t)

(6.23)


183

1.0

0.8

0.6 0.4

0.2 0.0

0

Figure 6.5.

2

4

6

8

10

Step response of interacting and noninteractin g tanks in series.

A plot of the two responses (6.21) and (6.23) are shown in Figure 6.5. It is now clear that the main effect of interaction is to make the response more sluggish. It should be easy to see, from purely physical grounds, that this is indeed to be expected once we take cognizance of the fact that with the interacting arrangement , any increase in tank 2 level will reduce the flow from tank 1, further slowing down the rate at which this level will increase. This is not the case in the noninteractin g arrangement.

6.2

SECOND-ORDER SYSTEMS

A second-ord er system is one whose dynamic behavior is represented by the second-ord er differential equation of the type: (6.24)

where y(t) and u(t) are respectively the system's output and input variables. Once again, it is customary to rearrange such an equation to a "standard form" in which the characteristic system parameters will be more obvious. In this case, the "standard form" is:

r~

+ 2t;-r'!ft + y - Ku(t)

(6.25)

and the newly introduced parameters are given (for a0 -# 0 ) by:

~ -\J ~ ; ,-;;;

T ;;

al

2/;T = -;;;; ;

"b

K="o

Assuming, as usual, that the model in Eq. (6.25) is in terms of deviation variables, Laplace transforma tion and subsequent rearrangem ent gives the transfer function model: y(s)

so that the general transfer function for the second-ord er system is given by:

184

PROCESS DYNAMICS

(6.26)

Note that whereas the transfer function of the first-order system has a firstorder denominator polynomial, the transfer function for the second-order system has a second-order denominator polynomial (i.e., a quadratic).

Characteristic Parameters The second-order system has three characteristic parameters: 1. 2. 3.

K, the steady-state gain, t;, the damping coefficient, and 1:, the natural period (or the inverse natUral frequency)

The reason for the terminology adopted for these parameters will soon become clear. The second-order system's transfer function has no zero, but has two poles located at s = r 1 and s = r2 where r1 and r2 are the two roots of the denominator quadratic.

6.2.1

Physical Examples of Second-Order Systems

1. Two First-Order Systems in Series Let us now recall the opening discussion of this chapter involving the dynamic behavior of two first-order systems in series. We obtained the differential equation model for the noninteracting arrangement as the second-order differential equation in Eq. (6.8). Comparing this now with the standard form in Eq. (6.25), we see that:

The system consisting of a noninteracting, series arrangement of two first-order systems (with time constants 1:1 and -r2 and steady-state gains K1 and K2 ) is a second-order system with the following parameters:

and A comparison of the transfer function representation in Eq. (6.20) with Eq. (6.26) shows that the interacting configuration gives rise to a second-order system in which T and K are as given above for the noninteracting system, but with t; given by:

LHfi.t'

o

v

r 1\1/iJVllL.:l v.r-

n1unr.1\. LJ1\.LIL1\. J 1 J l LlVlJ

p

Tube of Radius R

Figure 6.6. The U-tube manometer. 2. The U-Tube Manometer

The U-tube manometer shown in Figure 6.6 is a device used for measuring pressure. The dynamic behavior of the liquid level in each leg of the manometer tube in response to pressure changes can be obtained by carrying out a force balance on this system. As shown in Refs. [1, 2], the resulting equation is: (6.27)

where h is the displacement of the liquid level from rest position, L is the total length of liquid in the manometer, R is the radius of the manometer tube; p and J1 are respectively the density and viscosity of the manometer liquid; AP is the pressure difference across the tops of the two manometer legs; and g is the acceleration due to gravity. When Eq. (6.27) is arranged in the standard form, we observe that the dynamic behavior of this system is second order with:

and 3. The Spring-Shock Absorber System Our final example is the spring-shock absorber system shown in Figure 6.7 (also known as the damped vibrator in mechanics). Performing a force balance on this system gives the following differential equation model: ' tflx dx (6.28) m dt2 = - kx- c dt + f(t) + g

PROCESS DYNAMICS

186

Figure 6.7. The spring-shock absorber system.

where m X

g j{t)

-kx dx

-edt

is the mass of the entire system is the displacement of the spring from the fully extended position is the force of gravity is some forcing function imposed downwards is the force exerted by the spring (k is the spring constant) is the force due to the shock absorber, with c as the coefficient of viscous damping

If the initial rest position for the system is x(O) = x&' with /(0) = 0 observe that the force due to the compressed spring will exactly equal the force of gravity, i.e.:

We may now rewrite Eq. (6.28) in terms of the deviation variable y result is:

mtf!::t_ dfl

= x- x0; the

= -ky-ctjz+ j{t) dt

indicating a second-order system with: u(t)=f{t);

-r=~;

K=i ;

6.2.2 State-Space Representation of Second-Order Systems We have just established that two first-order systems in series constitute an example of a second-order system. Let us now return to the original differential equation couple in Eqs. (6.3) and (6.4) from which the second-order Eq. (6.8) was obtained. Further, for consistency in notation, let us redefine the deviation variables in Eqs. (6.3) and (6.4) as x 1 = h1 - h15 , x2 = h2 - h25 ; and since the actual process output is now~~ we let y = ~- Eqs. (6.3) and (6.4) now become:


187

and

with

These state-space modeling equations are represented more compactly in vectormatrix notation as:

(6.29a)

y

(6.29b)

which is of the form: dx(t)

dt

= Ax(t) + bu(t)

y(t) = cTx(t)

(see Eq. (4.29) in Chapter 4). The general second-order system whose model is given by the standard, second-order differential equation form in Eq. (6.25) can also be represented in this state-space form. The following procedure is usually employed for this

purpose.

Define two new variables x1 and x2 as follows: (6.30a) (6.30b)

From here, we take derivatives to obtain: dxl

?!1.

dt - dt

~

dxz

Tt

=

dr

(6.3la) (6.3Ib)

188

PROCESS DYNAMICS

Now we introduce Eq. (6.30b) into Eq. (6.31a) for dyldt; solve Eq. (6.25) for d2yldP and introduce the result into the RHS of Eq. (6.31b); the result is: dxl dt

(6.32a)

(6.32b) and from Eq. (6.30a) we have the equation for the system output, i.e.: (6.32c) These three equations may now be written in the vector-matrix notation of Eq. (6.29). It is easy tp see that the vectors and matrices concerned are now given by:

cT

= [I

OJ

Thus any second-order system may be represented by the state-space differential equation form in Eq. (6.29) as an alternative to the standard, second-order differential equation form given in Eq. (6.25). Such state-space model forms are useful for obtaining real-time responses of second-order systems. A significant number of computer simulation packages utilize this model form. As an exercise, the reader should introduce the deviation variables x 1 and x 2 in Eqs. (6.18) and (6.19) and obtain the state-space form for the interacting system's differential equation modeL

6.3

RESPONSE OF SECOND-ORDER SYSTEMS TO VARIOUS INPUTS

6.3.1 Step Response The response of the second-order system to a step function of magnitude A may be obtained using Eq. (6.26). Since u(s) = A/s we have: (6.33) We now choose to rearrange this expression to read:

y(s)

(6.34)


189

where r1 and r2 are the roots of the denominator quadratic (the transfer function poles). Ordinarily, one would invert Eq. (6.34), after partial fraction expansion, and obtain the general result: (6.35)

where, as we now know, A 0 , A 1 , and A 2 are the usual constants obtained during partial fraction expansion. However, tl,e values of the roots r 1 and r1, obtained using the quadratic formula, are: (6.36)

By examining the quantity under the radical sign we may now observe that these roots can be real or complex, depending on fue value of fue parameter (. Observe furfuer that fue type of response obtained in Eq. (6.35) depends on the nature of these roots. In particular when 0 < ( < 1, Eq. (6.36) indicates complex conjugate roots, and we will expect the response under these circumstances to be different from that obtained when ( > 1, and the roots are real and distinct. When ( = 1, we have a pair of repeated roots, giving rise to yet another type of response which is expected to be different from fue oilier two. The case ( < 0 can occur only in special circumstances and signals process instability. This will be discussed in detail in Chapter 11. There are thus three different possibilities for the response in Eq. (6.35), depending on the nature of the roots rl' r 2, that are dependent on the value of the parameter (. Let us now consider each case in turn. CASE 1:

0< ( 1

(r1 and r2 Real and distinct)

With fJ as defined in Eq. (6.38a), the roots are now given by:

and the response is given by: y(t) =

A{

1 - e-,tl
0, defines this as the origin, and then suddenly releases this force at t = 0. The resulting responses from the model

194

PROCESS DYNAMICS worn shocks ( l;= 0.2)

0.05 0.04

0.03

y(t)

meters 0.02 0.01

4 regular shocks (~= 0.6)

Figure 6.9.

t(sec)

6

heavy dut,y shocks ( ~= 0.8)

Response of suspension system to step change in force of 50 newtons.

for the three shock absorbers tested are shown in Figure 6.9. Note that both new shocks give good response with the heavy duty ones giving less overshoot. The worn shocks oscillate for a long time before coming to rest, while removal of the shock absorbers (c = 0) gives sustained oscillations with no damping. Note that for this example, the characteristic time 'f, or equivalently, the natural period of oscillation ron =1/'f, is determined by the mass of the load and the spring constant k. The damping factor { depends on these factors as well as the shock absorber viscous damping coefficient c.

Underdamped behavior in chemical process systems often arises as a result of combining a feedback controller and the process itself. Since the issue of feedback control is not to be addressed until later, we will at this point settle for satisfying ourselves about this issue with the following example. Example 6.4

UNDERDAMPED BEHAVIOR ARISING FROM THE COMBINATION OF A FIRST-ORDER SYSTEM AND A PROPORTIONAL + INTEGRAL CONTROLLER.

As we know, the differential equation model for a first-order system with time constant r 1 and steady-state gain K is given as: 'fl

~ dt + y = K u(t)

(6.46)

This system is now to be operated in conjunction with a certain type of controller known as a Proportional + Integral controller. The implication of this arrangement is that the input variable, u(t), is now determined by the controller according to the control law:

(6.47)

where the constants Kc and 1/-r1 are controller parameters; Yd is the desired set-point value for y. (We should point out that the name given to this controller derives from the fact that Eq. (6.47) indicates control action proportional to the error (yd- y) as well as involving the integral of this error.) Investigate the dynamic characteristics of this combination, first in general, and then for the specific case in which the system's

CHAP 6 DYNAMICS OF HIGHER ORDER SYSTEMS steady-state gain and time constant are respectively given by: K = 2, and controller parameters are: Kc = 0.5 , -r1 = 0.25 .

195 T1

= 4; and the

Solution: By introduci ng Eq. (6.47) into Eq. (6.46), we obtain the integra-di fferential equation:

(6.48)

where now yd has become the input. To proceed from here, we consider, for simplicity, that whatever changes scheduled to occur in Yd will take place at t = 0, and are such that Yd is constant for t > 0. This is to enable the vanishing of the derivative of yd for t > 0. Thus for I> 0, differentiating Eq. (6.48) gives, upon some simple rearrangement: (6.49)

This second-or der differential equation may now be rearrange d into the "standard form" to give:

(6.50)

so that now the characteristic second-order system parameters are seen to be given

-~

7:=-\J~-c; ~=

by:

(I +KKJ 2

(6.51)

Thus, whenever:

the combined (process + feedback controller) system will exhibit underdam ped behavior. For the specific situation where K =2 , and -r1 = 4 ; and K, =0.5, and -r = 0.25, we 1 deduce from Eq. (6.51) that T= 1, and~= 0.25 and the response to a step change inyd will exhibit underdam ped behavior, since ~ < 1 in this case.

In light of this example , we now state that the dynamic behavior of many systems under feedback control is very similar to the underda mped second-o rder response . For this reason, the underda mped response takes on special significan ce in process control practice, and as a result, some special terminol ogy (which will now be defined) has been develope d to character ize this behavior .

196

PROCESS DYNAMICS

Characteristics of the Underdamped Response Using Figure 6.10 as reference, the following are the terms used to characterize the underdamped response. 1. Rise Time, t; Time to reach the ultimate value for the first time; it can be shown that, for a second-order system: (6.52)

where {3 and

~have

been defined as: =

ls 2 -1P12

(6.38a)

tP =

tan- 1(/3/0

(6.38b)

f3

2. Overshoot: The maximum amount by which the transient exceeds the ultimate value AK; expressed as a fraction of this ultimate value, i.e., the ratio atfAK in the diagram. For a second-order system, the maximum value attained by this response is given by:

(6.53)

so that the overshoot is now given by:

~

Overshoot -- AK -- exp

(-=.!!_(_) {3

(6.54)

The time to achieve this maximum value is: tmax

=-1r

(6.55)

(I)

Settling time limits y(t)

------------···j --- -

AK

' .___ '

t,

Figure 6.10.

t,

-----------

t-

Characteristics of the underdamped response.


197

3. Period of Oscillation: From Eq. (6.37) observe that the radian frequency of oscillation is: (I)

=

11 'r

and since the period (in times/ cycle) is given by 2n/ w we therefore have: T _ 2n-r

(6.56)

- f3

4. Decay Ratio: A measure of the rate at which oscillations are decaying, expressed as the ratio aja 1 in the diagram:

~

Decay Ratio=~

= exp (~) f3 =(Overshoot)2

(6.57)

5. Settling Time, t 5 : A somewhat arbitrary quantity defined as the time for the process response to settle to within some small neighborhood of the ultimate value; usually taken to be within ± 5%.

6.3.2 Impulse Response The response of the second-order system to the input function u(t) =A B(t) is obtained frotn: K

y(s)

= 1:ls 2 + 2,-rs +

IA

(6.58)

Again, depending on the value of ', there are three possible types of responses, and they are summarized below. Case 1:

'
1 (2 imaginar y roots, 2 real, distinct roots)

The constants P1 , P2 ; Q 1 , Q2 ; and R1 , R2 are obtained by partial fractions. We now point out some salient features of these responses.

2.

As t -+ "" the first two terms in each case persist; the other terms vanish by virtue of the exponential terms becomin g zero.

Thus in each case, the ultimate response is also sinusoid al and is of the form: y(t)

I, . . _=

C 1 cos rot + C2 sin rot

and now if we evaluate the constant s C 1 and C , and apply the usual 2 trigonometric identities to combine the sine and cosine terms into one sine term, we obtain the ultimate periodic response for the second-o rder system as:

(6.67) with If! given by:

(6.68)

Frequency Response A comparis on of the input sine wave, Asincot, with the ultimate output shown in Eq. (6.67) indicates that the frequenc y response of the second-o rder system has amplitud e ratio given by:

AR

K

(6.69)

and phase angle t/J given by Eq. (6.68). There are several very interesti ng characte ristics of this frequenc y response , a full-scale discussion of which is deferred till Chapter 9.

200

6.4

PROCESS DYNAMICS

NFIRST-ORDER SYSTEMS IN SERIES

Consider the situation shown in block diagramatic form in Figure 6.12, in which N noninteracting, first-order systems are connected in series. The arrangement is such that the output of the ith system is the input of the (i+1)th system, with u, the process input, as the input to the first system in the series (see Figure 6.12). The analysis of this system's dynamic behavior is actually a straightforward extension of 'the discussion in Section 6.1.1. Firstly, because each of the N systems in the ensemble is first order, the transfer function relationship for each component is given by: y 1(s) = Y2(s) =

Y;(s} =

YN-l(s} =

Kl

u(s} 'l'IS + 1 K2 'l'2S

+ 1 yl(s)

Kl

'l'iS

Y1-1 + 1

KN-1

'l'

N-1

y~s)

=

(s)

s

+ 1 YN-2 (s)

KN

'l' s N

+ 1 YN-1

(s)

Next, by a series of cascaded substitutions (an extension of the procedure by which Eq. (6.9) was obtained from Eqs. (6.5) and (6.6)) we obtain the fact that the overall transfer function for this system is simply a product of the contributing transfer functions - a generalization of our earlier result for two first-order systems in series, ie.: y~s) =

(

N

IT

f.~

K- ) ' u(s) + 1

'l';S

(6.70)

Just as two first-order systems in series turned out to be a second-order system, the ensemble of N first-order systems in series is an Nth-order system.

Characteristic Parameters The characteristic parameters for this system are easily identified. They are: N

• K = II K1 the combined steady-state gain, a product of theN i=l

individual, contributing system steady-state gains, • N time constants, 'l'; , i = 1, 2, ... , N.


Figure 6.12.

201

Block diagramatic representation of N noninteracting systems in series.

The transfer function for this process, extracted from Eq. (6.70), is: K.

N

g(S) =

(,.u;

~

T;S

1)

(6.71a)

The N factored terms in the denominator expand out to give an Nth-order polynomial ins so that Eq. (6.7la) may be expressed as: (6.71b)

where it is easy to see that the first N coefficients of the Nth-order denominator polynomial are directly related to the time constants shown in Eq. (6.71a}, and a0 = 1. Thus an ensemble of N first-order systems in series behaves like an Nth-order system. This is a generalization of our earlier statement regarding the fact that two first-order systems in series constitute a second-order (overdamped or critically damped) system. Observe that the transfer function for N first-order systems in series has N poles located at s = -1/ -r;; i = 1, 2, ...,Nand no zeros.

6.4.1 Unit Step Response The response of this system to a unit step change in the input is obtainable from . Eq. (6,70) with u(s) = 1/s, i.e:

y~s)

=

(~

)I

K; + I s

'r;S

(6.72)

which, by partial fraction expansion becomes: y~s)

= K ( A-S0

A.

N

+ ) r-'1

T;S

)

' 1

+

(6.73)

where the constants A; (obtained from the partial fraction expansion) are given by: A 0 = I; A; = lim S-+ (-1/T;)

[(T; s + l)g(s)]

Ks

(6.74)

when the poles of the process transfer function g(s) given in Eq. (6.81) are all distinct. By inverting Eq. (6.73), the unit step response is obtained as:

202

PROCESS DYNAMICS (6.75)

Concen:Ung the nature of this response, let us note the following points. 1.

We recall that the sluggish ness introduc ed into the response of a first-order system by adding a second one in series was demonst rated in Figure 6.4. It turns out that subseque nt addition of more first-ord er systems actually causes the sluggishness to become more pronoun ced. This fact is illustrate d in Figure 6.13, a plot of different unit step responses for N first-order systems in series, when N takes on various values.

2.

It can be shown that, as long as N > 1, the slope of the unit step

response at the origin will be zero, i.e.:

-dyN dt 3.

I

-0

t=

(6.76)

0 -

The "inheren t reluctan ce" factor, obtained by evaluati ng the following integral where y .. = K and YN(t), is as given in Eq. (6.75): J

N

=

f-

[y*- y(t)] dt

0

(analogous to I 2 obtained in Eq. (6.15) for two first-order systems in series), can be shown to be given in this case by: (6.77)

once more, a mere generalization of Eq. (6.15). The response of this system to various other inputs can be derived along the same lines as illustrated with the unit step response. This is left as an exercise Lo 1---::::::=--=~~=~-,

0.8 0.6 YN(t)

1{0.4 0.2

2

Figure 6.13.

4

6

8

10

Unit step response of several noninteracting first-order systems in series.

CHAP 6 DYNAMICS OF HIGHER ORDER SYSTE MS

203

to the reader. The sinus oidal response, and the subse quent frequency response deriv ed from the ultim ate, purel y periodic portio n of this response, are of special significance and will be discussed in Chap ter 9.

6.4.2 A Spec ial Limi ting Case To round up our discussion of N first-order system s in series, we wish to consider a very special case of such systems. The situat ion unde r consideration is as follows: • All theN steady-state gains involved are equal to 1 • All the time constants are identical, and • Each time const ant is equal to a/N The implication of this third point is that each time the numb er of systems in the ensemble increases, the time constant of each of these systems is decreased. The transfer function for such a sys~em is given by: gJ._s) ==

-(-~-s---'l,__+_l_J_

(6.78)

We are partic ularly intere sted in what happ ens in the limit as N--+ oo, when there are infinitely many such systems in this ensemble. (We shoul d ment ion that our intere st at this point is purel y qualitative, prima rily to set the stage for the quant itativ e discussion to be found in an upcoming chapter.) Observe that as N --+ co, each contributing "time constant" goes to zero. The immediate consequence is that we now have an infinite sequence of indiv idual systems, each of which behaves essentially like a pure gain pmcess. Let us now addre ss the issue of how this syste m will respo nd to a unit step input . From the descr iption given above, we are led to believe that a unit step change in the input will result in each comp onent syste m transferring tllis unit step insta ntane ously (for that is what a pure gain syste m does) to the imme diate neigh borin g syste m. Theoretically, this instan taneo us, stage-tostage transfer of the step function goes on throu gh the entire infinite sequence cf systems. Howe ver, we may imagine that after a sufficiently large numb er of stages, we obtai n the final outpu t from the "final " stage. Keep in mind that so far, all we know is that the (limiting) response at each stage is a step function; it is still far from obvio us what the response at the "final " stage (whic h is the overall system outpu t) will actually be. Howe ver, from Eq. {6.77) we can obtai n what the "inhe rent reluctance" factor is in this case; it is given by:

(1. 1";) N

lim J N = lim

N-too

N-+oo

r=l

(6.79)

and since all the time constants are equal, each takin g the value a/N, the sum of these time const ants is simply equal to a, regar dless of the numb er of systems in the sequence. Thus, Eq. (6.79) becomes lim JN = a

N~-

(6.80)

204

PROCESS DYNAMICS 1

y(t)

a

Figure 6.14.

TIME

An intuitive deduction of the unit step response of a special infinite sequence of first-order systems in series.

This is a very helpful result because it implies that the area under the curve between the instantaneous unit response y* = 1 and the actual unit step response of this infinite sequence of systems is equal to a. When we combine this with the fact that the response expected from the "final" stage in this infinite sequence is a step function, we are led to deduce that the overall unit step response from this system can only be as shown in Figure 6.14, a unit step function delayed by a time units. What we have deduced from such intuitive, qualitative arguments is in fact what actually obtains, but we will have to wait until Chapter 8 for a rigorous confirmation of this statement. This special infinite sequence of first-order systems behaves like a system that does not in any way alter the form of the input function: it merely introduces a delay, making the output appear as the same function as the input, only shifted by a certain amotmt of time units. Such systems are of importance in process control studies and will be discussed in Chapter 8.

6.5

THE GENERAL Nth-ORDER SYSTEM

Before proceeding to the next stage, perhaps a few statements are in order here about the general Nth-order system. Rather than starting with N first-order transfer functions and combining them to give the transfer function in Eq. (6.71a), and consequently the one in Eq. (6.71b), if we had started instead with Eq. (6.71b), we would be dealing with a general Nth-order system. And just like a general second-order system can exhibit dynamic behavior that differs from that of two first-order systems in series, the general Nth-order system can also exhibit dynamic behavior that differs from that of N first-order systems in series. Now, it is clear, following from the discussion of the general second-order system, that the behavior of the general Nth-order system will depend on the nature of theN roots of the denominator polynomial in Eq. (6.71b); i.e., theN system poles. Observe, however, that these roots are either real or they are complex; and when complex, they occur in pairs, as complex conjugates. We therefore have only two situations:

CHAP 6 DYNAMICS OF HIGHER ORDER SYSTEMS 1.

2.

205

The roots are all real (either distinct or multiple), or Some roots occur as complex conjugates while the others are real

In the first instance, the behavior will be exactly like that of N first-order systems in series, and what we have discussed in the preceding section holds. In the second instance, we only need to observe that each pair of complex conjugate roots contributes the behavior of an underdamped second-order system to the overall system behavior; the real roots contribute first-order system behavior. Thus what we have in this case is an ensemble of several first-order systemsconnected in series with as many underdamped second-order systems as we have pairs of complex conjugate roots. The final conclusion is therefore the following:

The general Nth-order system behavi(lr is composed either entirely of the behavior of N first-order systems in series, or a combination of several underdamped second-order systems included in the series of first-order systems.

Although in practice the dynamic behavior of most industrial processes tends to be higher order and overdamped, we should note that a general treatment only requires an inclusion of first-order systems plus underdamped second-order systems into the series of systems composing the higher order systems. The basic principles of decomposing the high-order system into several systems of lower order remains the same.

6.6

HIGHER ORDER SYSTEMS WITH ZEROS

A discussion of systems that exhibit higher order dynamics would be incomplete without considering that class of systems whose transfer functions involve zeros. Such systems have the general transfer function form: (6.81)

where for an underdamped response, we may consider a pair of 't'; to be complex conjugates. This model may be written more compactly as: q

K i~ (~is+ g(s)

p

1)

(6.82)

,.~(r;s+l) To facilitate our study of such systems we now introduce the following notation: We shall denote a (p,q)-order system as one whose transfer function has p poles and q zeros, or equivalently, whose transfer function has a pth-order denominator polynomial and a qth-order numerator polynomial.

206

PROCESS DYNAMICS

Thus the characterist ic parameters of the general, {p,q)-order system whose transfer function is shown in Eq. (6.81} are: • K, the steady-state gain, • p poles, located at s = -1/'r;, i = 1,2, ... p • q zeros, located at s = -1/ ~;, i =1,2, ... q

When q is strictly less than p, the model is termed strictly proper, and when q = p, it is called semiproper. For realistic process models, q will always be less

than or equal to p. The first in this class of systems is therefore seen to be the (2,1)-order system; its dynamic behavior will be investigate d first before generalizin g.

6.6.1

Unit Step Response of the (2,1)-0rde r System

The transfer function for the (2,1)-order system is given by: (6.83)

At this point we should note that the difference between this system's transfer function and that for the system composed of two first-order systems in series (see Eq. (6.10)) is the presence of the first-order lead term with the lead time constant ~1 in Eq. (6.83), contributing the only transfer function zero. Thus we may expect that any difference observed between the response of the (2,1)-order system and the pure second-ord er system will be attributable directly to the presence of this single zero. The response of the (2,1)-order system to a unit step input is obtained from: y(s)

K(~ s+I)

l

(-r1s + t)(-r2s + t)

s

1 = -:--~-:-''---

(6.84)

By partial fraction expansion, Eq. (6.84) becomes: (6.85)

where the constants are given by: -1"1 ( 1"1-

~~)

(-rl- "2)

--r2(-r2-~l)

(-r2- -rl)

(6.86)

and Laplace inversion now gives: (6.87)


207

0~~--r-~.-~-.r-~.-~~

0

Figure 6.15.

Time

10

Unit step response of the (2,1)-order system, for various values of the lead timeconstant~r

If we now compare this with the corresponding expression for the step response

of the second-order system with no zero:

we can immediately observe, very clearly, the influence of the zero. Firstly, let the lag time constants be ordered such that -r < -r • (That this 1 2 impose s no restricti on whatsoe ver is obvious from the fact that Eq. (6.87) remains absolutely unchan ged when -r1 and 12 are interchanged.) If this be the case, then it can be shown that for positive values of ~ , the respons e in 1 Eq. (6.87) can take several different forms summar ized below: Case 1: In this case it can be shown (see problems at the end of the chapter) that the response overshoots the ultimate value K. This brings out a very importa nt difference between the system with a zero (shown in Eq. (6.83)) and the one without (see Eq. (6.10)). Observe that while the system of two first-order systems in series can never oversho ot the final value, the system shown in Eq. (6.83) can exhibit oversho ot if the lead time constant is large enough.

Case 2: Here the zero cancels one of the poles, and the respons e becomes identical to that of a first-ord er system with time constan t equal to the survivin g time constant. Case 3: The respons e in this case does not show any oversho ot, and it actually resembles that of a first-order system until ~1 becomes quite small compar ed to -r1 and -r2.

208

PROCESS DYNAMICS 1.0 0.8 y(t)

K

0.6

0.4

without a zero

0.2

0.0 0

Figure 6.16.

Comparison of the unit step responses of the second-order system, with and without a zero.

These facts are demonstrated in Figure 6.15, where the unit step response of a (2,1)-order system with -r1 = 1, r 2 = 4, is shown for four different values of ~1 covering all the cases indicated above: 1. 2. 3.

=0.05, Case 3,

~1 ~1

=1 and ~1 =4, both Case 2 conditions,

~1

= 8, Case 1.

and

Figure 6.16 shows a comparison of the (nonovershooting) step response of the (2,1)-order system and the second-order system response of Eq. (6.12) to demonstrate what the presence (and the absence) of the zero does to the unit step responses for each of these systems. Let us now note the following important points about the (2,1)-order system's step response in general. 1.

Unlike the pure second-order response, the response for the system with a zero is swifter. We therefore see clearly that the effect of the lead term is to "speed up" the process response.

2.

It is possible to speed up the response to the point where overshoot

occurs. This happens when the lead time constant is larger than the larger of the two lag time constants. 3.

It can be shown that the response of the system with the zero has a nonzero slope at the origin. In particular, either directly from Eq. (6.97), or by using the initial value theorem of Laplace transforms, we obtain for this (2,1)-order system that (6.88)

which is nonzero. (Observe that the slope becomes zero when ~1 = 0, when the lead term in the system transfer function disappears.) It should also be noted that the slope is positive when the signs of K and ~ 1 are the same, and negative when K and ~ 1 have opposite signs. This fact will become useful in the next chapter.


4.

209

The "inherent reluctance" factor in this case is easily shown to be given by: J (2,tl =

J~ [y* -y(t)]dt= K ('r 1 +

-r 2

~1)

-

(6.89)

0

which provides interesting insight into the effect of zeros on the "inherent reluctance" of the process. Compared to Eq. (6.15) we see from Eq. (6.89) that the (2,1)-order system is less reluctant (more "willing") to respond than the pure second-order counterpart by virtue of the presence of the zero. In general, therefore, the tendency is for the lead term in a transfer function to reduce the inertia inherent within higher order systems. However, when ~1 becomes too large, the possibility exists that the overshoot area may overwhelm the area before the rise time, thus allowing negative values of Jc2,1,. An interesting situation arises when ~1 = -r1 + -r2• The resulting zero reluctance factor in this case is not indicative of instantaneous response; it is due to the equality of the overshoot area above the final steady-state line and the area below the line. Since these areas counterbalance each other, the net area is "computed" as zero. We will now generalize the discussion given above.

6.6.2 Unit Step Response of the General (p,q)-Order System The unit step response of the general {p,q)-order system is obtained using Eq. (6.81):

y(s)

+ 1) ... (~9s +I) ( -r 1s + 1)( -r2s + l) ... ( "tps + 1)

K ( ~ 1 s + t)(~ 2 s

s

(6.90)

Once again, by partial fraction expansion (assuming distinct poles, "ti) Eq. (6.90) becomes:

A0

y(s)

= K ( -;-

P

+ ;~1

)

A; 't';s

+

I

(6.91)

just like in Eq. (6.73), and in fact the constants A; are also given by the same expression Eq. (6.74), i.e.: (6.74) However, the actual values obtained for this set of constants will be different because the transfer function g(s) in this case is given by Eq. (6.81) (as opposed to Eq. (6.71)) and the presence of the zeros will influence the values obtained. (For example, see Eq. (6.86) for the case with p=2, and q=l.)

210

PROCESS DYNAM ICS

From Eq. (6.91} the step response is given by: (6.92) The points to note about this response are as follows: 1.

When p - q = 1 (i.e., when we have exactly one more pole than we have zeros- referred to as a pole-zero excess of 1} regardless of the actual values of p and q, as long as q ~ 1, it can be shown that the initial slope of this resp(>nse will be nonzero. In particular: q

K ~~; (6.93)

p

T1 7:; f="l

and the respons e is rapid, not exhibiti ng the typical sigmoid al characteristics of higher order sys~s. Note that Eq. (6.88) is a special case of the more general Eq. (6.93). 2.

When p - q >1 the initial slope of the response is zero, and the shape of the response curve is sigmoidal (see Figure 6.17).

3.

In general, the respons e of the system with the higher value of q (more zeros) will be swifter than the respons e of a corresp onding system with a lower value of q (fewer zeros). This is in keeping with the conclusion that lead terms tend to "speed up" process responses; and we expect that the higher the number of lead terms, the more pronounced will be the "speeding up" effect.

This fact is illustrated in Figure 6.17 for a third-order system with no zero, and a third-or der system with one zero; i.e., p 3, q = 1. Observe that because p - q = 2 (which is greater than 1}, the response for the system with the zero is sigmoidal, but it is still less sluggish than the response for the pure third-or der system with no zero. In closing, let us return once more to the (2,1}-order system. One of the most importa nt aspects of _this system's step response is that unlike the pure secondorder system's response, the slope at the origin is nonzero , and the actual value is given in Eq. (6.88) as K~1 l 'Ej 12· We did specify that for this system ~1 is positive (as such, the system' s transfer function zero, s =-1/ ~1 , is negative), and we assume d that the system gain K is also positive. As a result, this slope is positive, since K and ~1 have the same positive sign. · When the value of ~ 1 is negativ e, so that the transfer function zero is positive (with the steady- state gain still positive), observe that in this case, the slope of the step respons e at the origin will now be negativ e. The implica tion here is as follows: the system respons e starts out heading "downw ards" by virtue of the negative initial slope, turning around sometim e later to head in the directio n of the steady- state value dictated by K (remember, this is positive.)

=


Figure 6.17.

211

Effect of a single zero on the unit step response of a third-order system.

Such surprisi ng respons es are sometimes encount ered b actual practice , and in processes of some importa nce. The dynami c behavio r of such processes are so

unusual that they will be discusse d separate ly in the next chapter.

6.7

SUMM ARY

Processes whose dynami c behavio r are represen ted by higher (but finite) order models have been studied in this chapter. We have found these processe s to be somewh at more comple x in nature than the low-ord er systems treated in Chapter 5. The transfer function models for these process es have at least two poles, and may or may not have zeros. When two first-ord er systems are connect ed in series, whethe r they are connect ed in an interact ing or noninte racting format, we have found that the behavio r exhibite d is that of a second- order system. In general, N first-ord er systems connected in series constitu te an Nth-ord er system. The step response of such systems (indicat ive of typical dynami c behavio r) are in general more sluggish than the step respons e of the first-ord er system. Further more, the higher the order, the more sluggish the response. The study of the behavio r of the general second- order system confron ted us with the first major departu re from low-ord er system behavio r. We found out that unlike the first-ord er system, there are actually three differen t kinds of second- order systems: overdam ped, underda mped, and critically damped ; and that two first-ord er systems in series are of the overdam ped (or at best the criticall y damped ) type. When zeros are present in a process transfer function model, we have seen that the general tendenc y is for the respons es to be swifter than what would be observe d in the corresp onding systems with no zeros. Zeros tend to "speed up" process respons e. In particul ar we saw that for the (2,1)-order system (i.e., one with two poles and one zero) the presence of the zero can speed up the response to such an extent that oversho oting the final value is experie nced, whereas it was impossi ble for its counter part without the zero to exhibit an oversho ot. Our treatme nt of higher (but finite) order systems is not yet complete. There is a special class of higher order systems with zeros that exhibit what can only be conside red as unusual dynami c behavio r. These will be dealt with next, in Chapter 7. Also, higher order systems of infinite order have not yet been discusse d, only hinted at. As it turns out, one of the most promin ent aspects of industri al process control systems is the almost ubiquito us presenc e of time delays, and time-de lay systems are actually infinite -order systems . This fact will be establis hed in Chapte r 8 where we will underta ke a study of the dynami c behavio r of time-de lay systems .

212

PROCESS DYNAMICS

REFERENCES AND SUGGESTED FURTHER READING 1. 2.

Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, J. Wiley, New York (1960) Stephanopoulos, G., Chemical Process Control: An Introduction to Tluory and Practice, Prentice-Hall, Englewood Cliffs, NJ (1985)

REVIEW QUESTIONS 1.

An ensemble of two first-order systems in series is equivalent to a system of what order? In general, what will the resulting system order be for an ense>mble of N firstorder systems in series?

2.

What is the rationale behind the terminology adopted for { and -r, two of the characteristic parameters of the general second-order system?

3.

What range of values of the parameter{; will give rise to: (a) Overdarnped response (b) Underdamped response

(c) Critically damped response (d) Pure undamped response? 4.

Why is it impossible for two first-order systems in series to exhibit underdamped behavior?

5.

Under what condition will the step response of a second-order system with a single zero exhibit overshoot?

6.

In general, what effect does a lead term have on the theoretical process response?

7.

Is it possible to distinguish between the step response of a first-order system and that of a second-order system with a single zero, whose characteristic parameters are related according to: -r1 < ~1 < T2?

8.

If a second-order system with a single zero now has characteristic parameters related according to: ~ 1 1, and nonzero only whenp-q = 1. Also show that when the pole-zero excess is precisely 1 (i.e., p- q = 1)- all other things being equal - the initial portion of the step response becomes steeper as the values of ~i, i = 1, 2, ... , q, the lead time constants, increase; whereas, as the value of the lag time constants T;, i =1, 2, ... , p, increase, the initial response becomes less steep. 6.3

(a) Obtain the impulse response of two first-order systems in series directly by using the transfer function in Eq. (6.10), with K = K1K2• (b) Differentiate the step response given in Eq. (6.12) with respect to time and compare the result with the impulse response obtained in (a); hence establish that, at least for this system, one can obtain the impulse response by differentiating the step response. (c) Starting from the general, arbitrary linear system model: y(s) = g(s) u(s)

with no particular form specified for g(s), by introducing appropriate expressions for u(s), establish that, in general: d

d/ &step) =

(Yimpulse)

(P6.1)

where (ystep) is the step response, and (ylmpulse) is the impulse response. 6.4

A parallel arrangement of two first-order systems, each with individual transfer functions given by: gl(s)

Kl T 1s

+ I

K2

g2(s) t"2S

+ I

gives rise to an overall composite system with transfer ftu1ction given by: g(s) = g 1(s)

+ g 2(s)

(a) You are now required to show that, unlike the series arrangement discussed in the text, this g(s) has the following properties:

214

PROCESS DYNAMICS 1. 2.

3.

Its gain is K1 + K2, not K1K2; It has a single zero whose location is determined by the lead time constant, ~1 , a parameter whose value depends on all the four contributing system parameters. Give an explicit expression for the dependence of this lead time constant on the contributing system parameters. (The series arrangement, of course, gives rise to no zero.) For the situation in which both K, and K2 are positive, the lead time constant ~ 1 is related to the contributing lt.g time constants according to: -r1 < ~ 1 < '12, if -r1 < r 2, or, according to: '12 < ~1 < -r1 if the converse is the case.

Hence, or otherwise, establish that a parallel arrangement of two first-order systems, each with a posi live steady-state gain, will never exhibit overshoot in response to a step input. (b) If now K 1 is positive and K2 is negative, but with I K 1 I > I K 2 1, under what condition will the lead time constant of the system resulting from this parallel arrangement be negative? (Note that both -r1 and '12 are strictly positive.) 6.5

For the interacting system discussed in Section 6.1.3, introduce the following deviation variables:

and let y = x 2; further, introduce the parameters, r 1, -r2, Kl' and K 2, precisely as defined for the noninteracting system discussed in Section 6.1.1 and thereby combine the expressions given in Eqs. (6.18) and (6.19) into a vector-matrix differential equation model of the form:

.

x y

Ax +Bu cTx

(a) What are the elements of A, B, c, in this case? (b) Compare the A matrix for this interacting system with that given in Eq. (6.29) and

hence identify the factor responsible for the "interaction." 6.6

(a) Show that the expression for y(t) in Eq. (6.87) attains a maximum only if ~ >r1 and '12, and hence establish that a (2,1)-order system will not exhibit overshoot unless the lead time constant is greater than both lag time constants; ie., ~ >T1 and '12· (b) Under conditions guaranteeing the existence of a maximum in F.q. (6.87) derive an expression for the time tmax at which this maximum occurs; also derive an expression for the timet 1 when the ultimate value K is first attained.

6.7

Using whichever control system simulation software is available to you (e.g., CONSYD, CC, Matlab, etc.) obtain unit step response plots for systems whose transfer functions are given below: (a)

g(s)

=

2(3s + 1) (2s + l)(s + 1)

(b)

g(s) =

1.5(4s + 1) (2s+l)(s+1)

From Eq. (6.89) compute 1(2,1), the inherent reluctance factor for each system and interpret your results in light of the plots and of what /(2,1) is supposed to measure. 6.8

An industrial distillation column of an unusual design is the first part of a complex monomer /solvent recovery system used to separate the components in the product stream of a polymerization reactor. After a carefully conducted procedure for


215

process identification (see Chapter 13), the dynamic behavior of the most critical portion of the colunm was found to be reasonably well characterized by the transfer function: 6s 2 + s + I (P6.2) g(s) = (6s + i)(s 2 + 3s + I) (a) How many poles and how many zeros does this system transfer function have and where are they located? (b) Use a control system simulati'on package to obtain the system's unit step response; what do you think might be responsible for this somewhat unusual response? (c) Expand g(s) in the form: g(s)

As+ B

C

-s2~+"--c:'3-s~+- + -6-s-+-1

(P6.3)

What insight does such an expansion give in terms of an "empirical mechanism" for explaining the observed system behavior? 6. 9

In modeling the process of drug ingestion, distribution, and subsequent metabolism in an individual, a simplified mathematical model was given in Problem 4.4 at the end of

Chapter 4 as: dx 1 dt

-k 1x 1 +u

(P4.3)

klxl -k2x2

(P4.4)

x2

(P4.5)

dx2 dt

y

where, as indicated in that problem statement, x 1, .r2, and u respectively represent (as deviations from initial steady-state values) the drug mass in the Gastrointestinal tract (GIT), the drug mass in the Bloodstream (BS), and the drug ingestion rate (see Problem 4.4). For a given small mammal used in a toxicology laboratory, the values of the effective "rate constants" associated with drug distribution into the bloodstream, and with drug elimination from the bloodstream, were found respectively to be: k1 =5.63 (min)~I and k 2 =12.62 (min)~I. Using a transfer function version of the process model, or otherwise, (a) Obtain the response of the drug mass in the bloodstream to a drug ingestion program which, as shown below in Figure P6.1, calls for a 10 mg/min ingestion rate started instantaneously at t = 0, sustained for precisely 5 min, and discontinued thereafter. What is the maximum value attained by the drug mass in the bloodstream? (b) Repeat (a) for the situation in which: u(t)

= {0;

-kt

t
IK2 1, but both K1 and K2 are negative. In ethis7.2(b) case the ultimate response given by Eq. (7.4) will be negative, and the initia l slope satisfying Eqs. (7.6) and (7.7) will be positive. Thus the system will begin to respond in a positive direction and then reverse itself and ultimately reach a negative stead y state as show n in Figure 7.2(b). Systems whose step responses are characterized by such an initial inversion (starting out the process in the "wro ng" direc tion which is later rever sed so that the process response eventually heads in the "righ t'' direction) are said to exhibit inverse response. · The main point here is the following:

Inverse response occurs as the net effect of (at least two) opposing dynamic modes of different magnitudes, opera ting on different characteristic timescales. The Jaster mode, which smaller magnitude, is responsible for the initial, "wron must have the g way" response; this is eventually overwhelmed by the slower mode having the larger magnitude. Obse rve that in the specific exam ple we have used, with comp etitio n between two first-order modes, we have been able to show that inverse response is possible only when the "opposition" mode 1.

Has a lower steady-state gain than the "main " mode (recall that the condition K2 < K1 was an initial requirement), and

2.

Responds with a faster initial slope than that of the nmain" mode, for this is the implication of the condition expressed in Eq. (7.7).

Let us now combine the transfer functions of the two processes irt Figure 7.1 into one for the purpo se of illust rating the characteris tics of linear syste ms that show inverse response. The resulting transfer functi on is:

Note that this is the form of a (2,1)-order syste m described in the last chapt er:

(6.83)

CHAP 7 INVERSE-RESPONSE SYSTEMS

229

with (7.8a)

(7.8b)

To obtain the inverse response shown in Figure 7.2(a), we required both K K 11 2 positive, K 1 > K 2 , and also K 2 /-r:2 > Ktf't1 (see Eq. (7.7)). This means that in Eqs. (6.83) and (7.8}, K > 0, ~ < 0. · Alternatively, the opposite response in Figure 7.2(b) occurs if IK1 1 > IK2 1 with both K 11 K 2 negative and IK 2 / -r:2 l > IK 1 / -r:1 l. This results in K < 0 and ~ < 0 in Eqs. (6.83) and (7.8). - Thus we see that while the overall process gain can be of either sign, ~ < 0 is a requirement for inverse response of g(s) in Eq. (6.93); i.e., g(s) must have a positive (right-half plane) zero. With this as background , we will now look into some physical processes that exhibit inverse-response characteristics. We should now be in a position to understand , on purely physical grounds, why such behavior is inevitable in such processes. Once we can identify two opposing mechanisms as being responsible for the dynamic behavior of a physical process, especially if one of them exercises greater influence on the process output than the other, it should no longer come as a surprise if we find that such a process exhibits inverse response.

7.2

INVERSE RESPONSE IN PHYSICAL PROCESSES

1. The Drum Boiler The drum boiler shown schematically in Figure 7.3 is the process currently under consideration. It is typically used for producing steam as a plant utility. The boiler contains boiling liquid, vapor bubbles, and steam, and the level of nongaseou s liquid material accumulated in the drum is the key variable of interest desired to be kept under control. The observed level of material in the drum is due to both the liquid and the Steam

Feedwater Heating Figure 7.3. Schematic diagram of the drum boiler process.

230

PROCESS DYNAMICS

entrained vapor bubbles that, because of their higher specific volumes, rise through the boiling liquid and temporarily "settle" on the boiling liquid surface. This fact is central to the dynamic behavior of the level in the drum boiler. Let us now consider how the level in the drum boiler will respond to a positive step change in the input; i.e., an increase in the cold feedwater flowrate. First, we note that in the long run, the addition of more feed material into the drum boiler must ultimately result in an increase in the process inventory (the heat supply to the reboiler has not been changed). However, the same action of adding more cold feedwater sets in motion, simultaneously, an opposing phenomenon: for observe that the entrance of cold water immediately results in a drop in drum liquid temperature, causing vapor bubbles to collapse, thus reducing the observed level in the boiler drum. The overall result is that shown in Figure 7.2(a), where a step change in the cold feedwater rate initially causes a drop in the boiler level before we ultimately see the expected increase, a typical inverse response. The two opposing dynamic modes in the boiler drum process are now fairly easy to identify: 1.

The ultimate material increase in the· boiler drum as a result of material input into the process,·

2.

The initial reduction in the volume of entrained vapor bubbles as a result of the decrease in temperature brought about by the addition of colder material into the drum.

Observe carefully that the influence exerted on the drum boiler level by the first mode is by far the greater, in the long run; but this influence is exerted on a much slower timescale than the less influential second mode. However, despite the fact that the influence of collapsing bubbles on the boiler drum level is, in the long run, obviously far less significant, it is nevertheless exerted on a much swifter timescale. These factors combine to make the drum boiler process a prime candidate for inverse-response behavior as is well known (see also Shinskey [1] and Smith and Corripio [2]). 2. The Reboiler Section of a Distillation Column In operating a distillation column, it is well known that without intervention from a level controller, an increase in the reboiler steam duty will result ultimately in a decrease in the level of material in the reboiler. This is because such an action will cause more low boiling material to be boiled off from the inventory in the reboiler. It is also important to note that there will be an increase in the vapor rate through the entire column as a result. However, the immediate effect of an increase in the vapor rate is to cause increased frothing and subsequent liquid spillage from the trays immediately aBove the reboiler, sending more liquid back to the reboiler from these trays. This will result in an initial increase in the reboiler liquid level prior to the ultimate, and inevitable, reduction brought about by the more substantial influence of the vapor boil-up mechanism. Again we see two opposing mechanisms in action here: the more substantial boil up, steadily (but slowly) causing a reduction in the inventory in the


231

reboiler, pitted against the frothing and subsequent liquid spillage (caused by the same action responsible for the boil-up) responsible for the less substantial, but rapidly executed, initial increase ~ the reboiler material. The first mode exerts the more substantial influence but on a much slower timescale; the influence of the second mode is less substantial in the long run, but it is exerted on a much swifter timescale. The inevitable result of such a combination is inverse response of the form shown in Figure 7.2(b ). (See also Luyben[3].) 3. An Exothermic Tubular Catalytic Reactor When the temperature of the reactants in the feed stream to a tubular catalytic reactor is increased, the temperature of product material at the reactor exit exhibits inverse-response behavior if the reaction is exothermic. The reason for this observation is as follows. The increase in the feed temperature has the initial effect of increasing the conversion in the region near the entrance of the reactor, thereby depleting, for a short period, the amount of reactants that will reach the region around the exit. This results in a temporary reduction in the exit temperature. However, the catalyst in the bed begins to increase in temperature so that eventually the reactants reaching the exit region do so at a higher temperature, promoting an increase in the reaction rates in this region, and ultimately resulting in an increase in the exit temperature. Once more, we see the initial reduction in the amount of unreacted material reaching the exit region as the less significant, yet fast mode, standing in opposition against the increase in reactant temperature, the mode that has the more significant influence on exit temperature but whose influence, even though steady, is on a slower timescale. Thus the reactor exit temperature will respond as shown in Figure 7.2(a). Reference [2] (p. 391) reports an example of another chemical reactor that exhibits inverse-response characteristics.

7.3

DYNAMIC BEHAVIOR OF SYSTEMS WITH SINGLE, RIGHT-HALF PLANE ZEROS

Recall from Section 7.1 that for first-order processes in opposition, inverse response required that the overall transfer function g(s) have a right-half plane zero. Let us now consider the dynamic behavior of the process whose transfer function is as shown below: g(s)

K (-1]s + 1)

(7.9)

where 11 > 0. Note also that this is just the (2,1)-order system discussed in Chapter 6, with ~ = - 7]. Observe that this transfer function has two poles located at s = -1/r1 and s = -1/r2, but its only zero is located ass = + 1/11; i.e., it is a positive zero. Because of the location of the zero in the right-half of the complex s plane, such zeros are commonly referred to as right-half plane (RHP) zeros.

232

7.3.1

PROCESS DYNAMIC S

Unit Step Respons e

The unit step response for this system may be obtained using Eq. (7.9) and the usual transfer function model with, u(s) = 1/s, i.e.: (7.10)

We find it more instructive first to investigat e certain limiting behavior of this response before obtaining the entire y(t). Ultimate Response Employin g the lmal-valu e theorem of Laplace transform: lim y(t) = lim [s y(s)]

,~....

s~O

we have in this case from Eq. (7.10) that:

K

(7.11)

that is the process gain, K; i.e., the ultimate response is positive if K is positive and negative if K is negative. Initial Slope Let dy(t)/dt represent the derivative of the unit step response function. We may now employ the initial value theorem of Laplace transform s to obtain the initial value of dy(t)/dt, the value of the initial response slope at the origin. In this case, we have: lim dyd(t) = lim [sL { ~d } ]

t-+0

I

s-+-

I

=

lim [s2y(s)] s-t-

(7.12)

Using Eq. (7.10) in Eq. (7.12) now gives:

The right-han d side limit is easier to evaluate if the numerato r and denomina tor terms are divided by s 2; i.e.:


.

233

d"ft\

Il i D = = 1-> 0 dt

(7.13)

finally giving: lim

4J.ffi

..::K!1_

I-> 0

dt

't"I 't"2

(7.14)

Since TJ, -r1, -r2 are all positive, then the initial slope takes the opposite sign of the process gain. This fact combined with the results of Eq. (7.11) prove that the system Eq. (7.9) will always show an inverse response. We may now obtain the complete unit step response from Eq. (7.10) in the usual manner, yielding: (7.15)

This is similar in every respect to the solution to Eq. (6.84) but with the singular and most important exception that the substitution ,; =- TJ < 0 has been made in Eq. (6.87). Figure 7.4 shows this response for the system with K = 1, -r1 = 2 min, 't":z = 5 min, and TJ = 3 min; i.e., the system whose transfer function is given by: g(s)

= (2s

( -3s + I) + I)(Ss + I)

(7.16)

The plotted response is: y(t)

=

(I

+

~ e-t/2 - ~ e-t/5)

(7.17)

The response of this system to other input functions can be found using the same principles and is left as an exercise for the reader (also see problems at the end of the chapter). 1.00

-------------:::--=------

0.75 0.50 y(t)

0.25 0.00

Figure 7.4. Unit step response of a system with one right-half plane zero.

234

PROCESS DYNAMICS

7.3.2 Additional Characteristics of the Unit Step Response The most prominent feature of the unit step response of the inverse-response system is its initial inversion and subsequent reversal, necessitating that the response pass through a minimum at some time tm > 0. By differentiating Eq. (7.15), setting the result to zero, and solving the expression for t, we find that the response passes through a minimum at t =tm given by: (7.18)

where

-r. + 71 = _,__ ; i = 1,2 'r;

IJ.· I

(7.19)

A second differentiation of Eq. (7.15) establishes that this is truly a minimum point. Example 7.1

CHARACI'ERIZING THE UNIT STEP RESPONSE OF AN iNvERSE-RESPONSE SYSTEM.

For the system whose transfer function is given in Eq. (7.16) and whose step response is given in Eq. (7.17) show from this step-response expression that this is an inverseresponding system and find the time at which the response experiences "tum around." Solution: The first characteristic of the response obtained from systems of the type indicated in Eq. (7.6) is that the initial slope possesses a different sign from that of the ultimate value. From Eq. (7.17) we see immediately that: y(oo) =

which is positive. Differentiating Eq. (7.17) gives:

41_

dt =

_i..-112+!....-tJS 6~

IS~

(7.20)

which, upon setting t =0 gives:

kd t

I

1•0

=

-0.3

so that with the negative initial slope and a positive final value, the response is identified as an inverse response. The process response experiences "tum around" at the point where the response passes through a minimum. By setting the expression in Eq. (7.20) to zero, we observe

235

CHAP 7 INVERSE-RESPONSE SYSTEMS that this occurs when:

1 e-tn 6

_

-

_!_ e-t/S 15

which when solved for t gives the time of the minimum tm:

or tm

= 1.488

min

Alternatively, one could substitute directly into the general result Eqs. (7.18) and (7.19) to obtain tm.

7.3.3

Effect of Additional Poles and Zeros

We have shown that the system in Eq. (7.9), whose transfer function has a single RHP zero and two poles, exhibits inverse-response behavior. Let us now consider the more general system whose transfer function is given by: (7.21)

or, more compactly: q

g(s)

=

K (-7]s + 1)~( ~is+

!ft (

l) (7.22)

P

'r;S

+

1)

which is a system with p poles, q regular, negative zeros, and a single RHP zero (with p > q + 1; ie., tot.al number of zeros less than the number of poles). It can be shown readily that this system will also exhibit inverse response. The issue now is: how will the presence of the additional poles and zeros influence the dynamic behavior of such systems? In our previous study of the dynamic behavior of higher order systems (in Chapter 6), we discovered that the addition of lag terms (which contribute the poles) has the net effect of slowing down the response; on the other hand, the addition of lead terms (contributing the zeros) has the net effect of speeding up the system response. The same is true for the inverse-response system whose transfer function has a single RHP zero: additional poles will tend to slow the response down while additional zeros will tend to speed up the response. To illustrate this point, the unit step responses of higher order systems (which we shall call Systems B and C) are compared with the unit step response of the system given in Eq. (7.16), which we shall call System A, i.e.:

Ljb

PROCE SS DYNAM ICS

• System A: ( -3s + I) g(s)

= (2s + 1)(5s + I)

(7.I6)

the system with one RHP zero and two poles, whose unit step response was present ed in Eq. (7.17) and in Figure 7.4. • System B:

g(s)

= (2s

( -3s ~ I) + 1)(5s + I)(4s + I)

(7.23)

a system with one RHP zero and three poles, one more than System A. • System C:

g(s) =

( -3s + l)(s + I) (2s + l)(5s + 1)(4s + I)

(7.24)

a system with one RHP zero, one regular , negativ e zero, and three poles, one zero more than System B, and one pole and one zero more than System A Figure 7.5 shows the unit step respon ses of Systems A and B. Right away, we clearly see that the influence of the additio nal pole possess ed by System B is to slow down its respon se in compar ison to that of System A. Observ e also that as a result of the increas ed sluggis hness of System B, the inverse respon se it exhibit s is less pronou nced than that of System A Thus the effect of an additio nal pole is to slow down the overall respon se while reduci ng the strengt h of the initial inverse response. When one zero is added to the System B transfe r functio n, the result is System C. A compar ison of the unit step responses of these two systems is shown in Figure 7.6. As expected, the additio nal zero causes the System C respon se to be so.mewhat swifter than that of System B. 'J,'he effect of a simulta neous additio n of a pole and a zero to the transfe r functio n of System A, to produc e System C, is demon strated by the unit step responses shown in Figure 7.7. 1.00

0.75 0.50 y(t)

0.25 0.00 -0.25~rrrT~-.rT~-rrT~~rro

0

10

20

30

40

time

Figure 7.5.

Unit step respons es of systems A and B, demons trating the effect of an additional pole.


237

1.00 0.75 0.50 y(t)

0.25 0.00 -0.25 0

10

20

30

40

time

Figure 7.6.

Unit step responses of systems Band C, demonstrating the effect of an additional zero. 1.00 0.75 0.50 y(t)

0.25 0.00 -0.25 0

10

20

30

40

time

Figure 7.7.

Unit step responses of systems A and C, demonstrating the effect of an additional zero and pole.

Because the additional lag term, (4s + 1), contributing the pole has a stronger effect than the added lead term, (s + 1), contributing the zero, the overall effect is biased towards that due to the added pole; the system response is slowed down, and the strength of the initial inverse response is reduced.

7.4

DYNAMIC BEHAVIOR OF SYSTEMS WITH MULTIPLE, RIGHT-HALF PLANE ZEROS

The discussion in the preceding section centered around the influence of additional poles and zeros on the dynamic behavior of an inverse-response system, with the understanding that the inverse-response characteristics are due to the presence of a single RHP zero. Note carefully that the additional zeros have been regular, negative zeros, so that in the midst of all the multiple zeros contained in the system transfer function, we still find only one RHP zero. At this point, we might well ask the question: what effect would the addition of another RHP zero have on the dynamic behavior of a system already exhibiting inverse response? Or, more generally: what are the main characteristics of the dynamic behavior of a system with multiple RHP zeros? In this regard, we have the following results: The initial portion of the step response of a system with multiple RHP zeros will exhibit as many inversions as there are RHP zeros.

238

PROCESS DYNAMICS

From this general result, we may now make the following deductions: 1.

The step response of a system with one RHP zero exhibits (as we now know) one inversion. If the step response is therefore to end up in the right direction, it must first start out in the wrong direction; this, of course, is what we have seen in the examples presented thus far.

2.

The step response of the system with two RHP zeros exhibits two inversions in the initial portion, with the following consequences: for this response to end in the direction of the final steady state (as it must) after executing two inversions, it must start out heading in the right direction first, take a first inversion and temporarily head in the wrong direction, and return to the right direction upon the second and final inversion.

3.

The step response of the system with three RHP zeros exhibits three inversions in the initial portion. Consequently, we observe once again that such a response must start out in a direction opposite to that of the final steady state. The first inversion temporarily turns the response in the right direction, and the second inversion reverses the trend; the third and final inversion is what puts the response in the direction of the final steady state.

In taking such arguments as these to their logical conclusions, we may thus deduce the qualitative nature of the step responses upon consecutive additions of RHP zeros. The following is a summary of these conclusions: The system with an odd number of RHP zeros exhibits true inverse response in the sense that the initial direction of the step response will always be opposite to the direction of the final steady state, regardless of the number of inversions involved in this response. On the other hand, the initial portion of the step response of a system with an even number of RHP zeros exhibits the same even number of inversions before heading in the direction of the final steady state, but the initial direction is always the same as the direction of. the final steady state. To demonstrate these results, the unit step responses of: the following systems are plotted in Figures 7.8, 7.9, and 7.10. • System D:

g(s)

= (2s

( -3s + 1)(-s + 1)

+ 1){5s + l)(4s + 1)

a system with two RHP zeros (Figure 7.8);

(7.25)


239

• System E: g(s) = (2s

( -3s + 1)(-s + 1)(-2.5s + 1) + 1)(5s + 1)(4s + 1)(3.5s + 1)

(7.26)

a system with three RHP zeros (Figure 7.9); and finally • SystemF: g(s)

( -3s + 1)(-s + 1)(-2.5s + 1)(-6s + 1) (2s + 1)(5s + l)(4s + 1)(3.5s + 1)(1 + 7s)

a system with four RHP zeros (Figure 7.10). 1.00

------------==-~---

0.75 0.50 y(t)

0.25

0.00 -

0.25-J-..~...,...,~....-~.......,.~.....,,................,

0

10

30

20

50

40

time

Figure 7.8.

Dynamic behavior of System D with two RHP zeros. 1.00

- ---------- - · ; - . - - - - -

0.75

0.50 y(t)

0.25

0.00 -0.25 -f..-~T"T"T.....,~.-.,-~......~"T"'""~ 0 10 20 30 40 50 60

time

Figure 7.9.

Dynamic behavior of System E with three RHP zeros. 1.00

- - - - - - - - - - - - - - - - --

0.75 0.50 0.25

y(t)

0.00 -

0.25-t..-~............,.............~......~..............,

0

Figure 7.10.

10

20

30 time

40

50

60

Dynamic behavior of System F with four RHP zeros.

(7.27)

240

7.5

PROCESS DYNAMICS

SUMMARY

We have been concerned in this chapter with investigatin g the dynamic behavior of systems exhibiting inverse-resp onse behavior. We have found that such responses are possible whenever there are two (or more) opposing mechanism s (operating on two different timescales) responsible for the overall system behavior. Several examples of real processes that exhibit these unusual dynamic characterist ics were presented. It was shown that the transfer functions of inverse-resp onse systems are usually characterize d by the presence of a RHP zero; the presence of additional poles and regular, negative zeros may modify the dynamic behavior, but the inverse-res ponse characterist ics will still be in evidence. The presence of multiple RHP zeros, however, introduces some more interesting characteristics, bringing about multiple inversions in the initial portions of the step responses, a phenomeno n that is rarely observed in practice. With the material covered in this chapter, we have now completed the discussion of the dynamic behavior of higher (but finite) order systems with zeros. The dynamic behavior of time-delay systems, perhaps the most important of the infinite-ord er systems will be discussed in the next chapter. REFERENCES AND SUGGESTED FURTHER READING 1. 2.

3.

Shinskey, F.G., Process Control Systems (2nd ed.), McGraw-Hill, New York (1979) Smith, C.A. and A.B. Corripio, Principles and Practice of Automatic Process Control, J. Wiley, New York {1985) Luyben, W.L., Process Modeling, Simulation, and Control for Chemical Engineers (2nd ed.), McGraw-Hill, New York (1991)

REVIEW QUESTION S 1.

Describe the phenomenon of inverse response in a process.

2.

When a system is made up of two opposing modes of different magnitudes, each operating on different characteristic timescales, will the faster mode have to be smaller or larger in magnitude for the system to exhibit inverse response?

3.

What characterizes the transfer function of an inverse-response system?

4.

Can a series arrangement of two first-order systems exhibit inverse response?

5.

If two inverse-response systems A and Bare otherwise identical, except that the system A transfer function has an additional (left-half plane) pole, how will one system's step response differ from the other?

6.

What aspect of the transfedunct ion determines the number of inversions exhibited by a system's initial step response?

7.

Why is it that a system whose transfer function has an even number of right-half plane zeros is, strictly speaking, not really an "inverse-res ponse" system in the strictest possible sense of the term "inverse response"?


241

PROBLEMS 7.1

(a) Derive individual expressions for the response obtained when the system whose transfer function is given in Eq. (7.16) is subjected to the following inputs: 1. u(t) = lX_t); a unit impulse function

2. u(t) = {:: : : : < b;

0;

I;?;

a pulse input of magnitude A and duration b

b

(b) For the two distinct situations in which the magnitude and duration of the pulse input investigated above are given.as:

1. A= 1/8, b =8,and 2.A=l/2,b=2 obtain plots of the resulting pulse responses and compare them with a plot of the unit impulse response. (c) Compare the unit impulse response obtained in part (a) with that of the system whose transfer function is given as:

=

g(s)

7.2

(3s + l) (2s + l )(s + l)

(P7.!)

From the expression given in Eq. (7.15), show that the inherent reluctance factor for this inverse-response system- defined in the usual fashion as: J =

f-

[y*- y(t)] dt

0

(where y* = K in this case) - is given by: (P7.2)

Compare this with the expression given in Chapter 6 for J(2,1), for the corresponding system with a left-half plane zero. 7.3

While the dynamic behavior of a system with a "strong" left-half plane zero is expected to be different from that of a system with a "strong" right-half plane zero, the nature of such differences are, in themselves, quite informative. Plot the unit step responses of the two systems whose transfer functions are given as follows: (3s + 1) (-3s + 1) St(s) = (2s + l)(s + 1) ; and g2(s) = (2s + l)(s + 1)

Compare and contrast the two responses. 7.4

Two systems, each with individual transfer functions g 1(s) and g 2(s) are connected in a parallel arrangement such that the resulting composite system has a transfer function given by:

PROCESS DYNAMICS

242

In the specific case with: (-r 1s

+ 1)(-r2s + 1)

K2 Cz(s) = As

+ I

(P7.3)

(P7.4)

(a) How many poles and how many zeros will the resulting composite system possess? (b) Assuming that both K1 and K,_ are positive, what conditions must be satisfied by K 2, and A. so that g(s} will no longer be an inverse-response system? (You may find it useful to know that: A necessary and sufficient condition for the quadratic equation ax2 + bx + c = 0 to have no roots with positive real parts is that all its coefficients, a, b, and c be strictly positive.} (c) It has been proposed that the inverse response exhibited by the system in Eq. (7.16) be "compensated" by augmenting it with the following first-order system in a parallel arrangement (P7.5)

i.e., the parallel arrangement of this first-order system and the inverse-response system in Eq. (7.16) will result in a composite system that does not exhibit inverse response. Confirm or refute this proposition. 7.5

(a} A system consisting of a first-order mode in opposition to the main pure capacity mode, with respective transfer functions g 2(s) = K,_/{-r,s + 1) andg1(s) = K 1 /s, can exhibit inverse response only if K 1 < K,_/12- Establish this result (b) The level in a certain ~boiler process is known to exhibit inverse response. In identifying a dynamic model for this process, input/output data were collected in accordance with a planned experimental procedure supposedly designed so that the two opposing modes of the overall process can be identified independently. After carrying out the appropriate data analysis, the following candidate transfer ftmction model was obtained: g(s)

gl(s)- g2(s)

(P7.6)

where

c1(s)

5.3

(P7.7)

s

and g2(s) =

10.1 2.2s

+

(P7.8)

It is known that the experimental procedure is much better at identifying the ultimate behavior, characterized mostly by g 1(s), than at identifying the initial transient behavior, characterized mostly by g 2(s). In fact, an independent researcher has suggested that this model identification procedure tends to overestimate the firstorder time constant in each typical g 2(s) by as much as 20%. In light of the identified transfer hmctions given above, provide your own assessment of the identification procedure. 7.6

Show that a system composed of a critically damped second-order system in opposition to a first-order system, whose transfer hmction is given as:

243

CHAP 7 INVERSE-RESPONSE SYSTEMS g(s)

=

K2

_K_1_

(r2 s + I y

r1s + I

(P7.9)

will exhibit inverse response only if:

:: >

2C:)

(P7.10)

given that (P7.11) Compare this result with Eq. (7.7) . .7.7

In the science of immunology, it is known that the process of vaccination results in responses from two subsystems in opposition: the "disease-causing" mode promotes the growth and replication of the intruding "disease-causing" i::ells; the immune system. on the other hand, produces the appropriate antibodies required to fight off the intruding cells. The antibodies produced by the healthy immune system ultimately overwhelm and completely destroy the disease-causing cells, and then remain behind in the bloodstream as prophylactic sentries against future attacks. A dynamic systems approach to the analysis of this process shows that immunization in the human body is an inverse-response process. Let x 1 represent the "disease-causing cell count" (in total number of cells per standard sample volume); in general its "isolated" response to the amount of injected vaccine u (in cm3) is reasonably well approximated by: x 1(s)

=

K! s(r1s + I)

u(s)

(P7.12)

The corresponding "isolated" response of the "antibodies cell count," say, x 2, is reasonably well approximated by: (P7.13) Define a normalized "extent-of-immunization index" y as a weighted average of the disease-causing cell count and the antibodies cell count according to: y

= roixi

+ ·ror2

(P7.14)

with the weights given by: rol ~

-I (K2- Kl) I (K 2 -K 1)

(P7.15a) (P7.15b}

With regards to the approximate models given above, the following two facts are pertinent to the analysis:

1.

For t.'le healthy immune system: (P7.16)

2.

For effective vaccines:

244

PROCESS DYNAMICS

(P7 .17)

(a) Show that under the conditions represented in Eqs. (P7.16) and (P7.17), the transfer function relating y to u has precisely one -not two- right-half plane zeros and therefore that the response of the "extent-of-immunization index" y to the amount of injected vaccine u will exhibit true inverse response. (b) For the specific situation in which a 5 cm3 ampoule of a certain vaccine was injected into an individual for whom: K1

-r1 = 2 hr, 12

=

0.5 K2

(P7.18a)

= 3 hr. -r3 = 5 hr

(P7.18b)

first establish that this is a situation in which the individual's immune system is healthy and the vaccine is effective; then approximate this vaccination input as u(t) = SO(t) and obtain a plot of the response of the "extent-of-immuniz ation index" y. (c) Provide your own interpretation of what the "extent-of-immuniz ation index" y measures. Explain carefully what you think a value of y = 0, a value of y < 0, and a value of y =1 mean. 7.8

Revisit Problem 7.7. Establish from the general expression obtained there for the overall transfer function relating y to u that: (a) For the unhealthy immune system (with K 1 > K 2) into which a unit impulse u(t) = O(t) of an otherwise effective vaccine has been introduced, the "extent-ofimmunization index" y will settle ultimately to a value of -1 (as opposed to the desired value of +1) without exhibiting inverse response. (b) When a healthy immune system receives a unit impulse u(t) = O(t) of a vaccine that is considered less effective because: KI K2 ( Tz + TJ) < Tl•

(P7.19)

as compared to the condition indicated in Eq. (P7.17), the "extent-of-immuniz ation index" y will still settle ultimately to the desired value of +1, and still exhibit inverse response. (c) When an unhealthy immune system receives a unit impulse u(t) = li(t) of the less effective vaccine considered in part (b), the "extent-of-immuni zation index" y will still settle ultimately to the undesired value of -1, and its initial response will show two inversions. (d) Given the following three specific cases in which the parameters are given as:

2. K1 =0.5K2 ; r 1 = 4 hr,

12

= 3 hr,

1J

= 2 hr

In each case determine the "health" of the immune system and the "effectiveness" of the vaccine, and then obtain a plot of the response of the "extent-of-immuniz ation index" y to an idealized unit impulse input of vaccine, u(t) = O(t). In terms of the two components of y, provide a qualitative explanation for the observed response in each case particularly with respect to the immune system's state, and the effectiveness of the vaccine.

CHAJPTJER

8 TIME-DELAY SYSTEMS

As diverse as the various classes of systems we have studied so far have been, they do have one characteristic in common: upon the application of an input change, their outputs, even though sluggish in some cases, begin to respond without delay. In the process industries, one often finds systems in which there is a noticeable delay between the instant the input change is implemented and when the effect is observed, with the process output displaying an initial period of no response. Such systems are aptly referred to as time-delay systems, and their importance is underscored by the fact that a substantial number of processes exhibit these delay characteristics. This chapter is devoted to studying time-delay systems: how they occur, why they are classifiable as infinite-order systems, their dynamic behavior, and how they are related to other higher order systems.

8.1

AN INTRODUCTORY EXAMPLE

Consider the process shown in Figure 8.1, involving the flow of an incompressible fluid, in plug flow, at a constant velocity v, through a perfectly insulated pipe of length L. The fluid is initially at a constant temperature T 0, uniformly, throughout the entire pipe length. At time t = 0, the temperature of the fluid coming in at the pipe inlet (z = 0) is changed to Ti(t). We are now interested in investigating how the temperature at the pipe outlet (z = L) responds to such an input change.

8.1.1 Physical Considerations On purely physical grounds, it is easy to see that any temperature changes implemented at the pipe inlet will not be registered instantaneously at the outlet. This is because it will take a finite amount of time for an individual fluid element at the new temperature, to traverse the distance L from the inlet to the outlet so that its new temperature can be observed. 245

246

PROCESS DYNAMICS Transport Delay

L

a=v

0

L

Figure 8.1. Flow through an insulated pipe.

Since the fluid velocity is v, and it is assumed constant, the time for each fluid element to traverse the required distance from the pipe inlet to the outlet is L/v. Since the pipe is also assumed to be perfectly insulated, there will be no heat losses, or any other changes for that matter, experienced by each fluid element in the process. Thus, any changes implemented in the inlet will be preserved intact, to be observed at the outlet after the time requiied to traverse the entire pipe length has elapsed. From such physical arguments we have arrived at the conclusion that in response to changes in the process input (i.e., the inlet temperature) the process output (i.e., the exit temperature) will have exactly the same form as the input, only delayed by L/v time units. Let us now show, mathematically, that this is in fact what obtains.

8.1.2

Mathematical Model and Analysis

Consider an element of fluid of thickness L1z whose boundaries are arbitrarily located at the points z and z + L1z along the length of the pipe.(see Figure (8.2)). For a pipe of constant, uniform cross-sectional area A, an energy balance over such an element gives:

where p, CY, are, respectively, the fluid density, and specific heat capacity, and T* is the usual, reference temperature (see Chapter 4); the notation lz is used to indicate that the quantity in question is evaluated at z. By dividing through by Liz, and taking the limits as Az -+ 0, Eq. (8.1) becomes: dT(z,t)

ar

dT(z,t}

+v

=0

ik

z

O .,. f3 [T.,(t)- T(z,r>]

;

o< z < L

(8.34)

where {3 is a heat transfer paramete r (see Chapter 12):· In terms of the usual deviatio n variables, the process model is:

~ at +

v

~ t - y (z,t )] ik "' {3[ u ()

O has the exponential polar form: then establishing Eqs. (9.28) and (9.29) from Eqs. (9.1) and (9.3) and Figure 9.2 is straightforward, and it is left as an exercise to the reader. As we already know from Chapter 6, g{s), the overall transfer function for the composite system, is given by:

Substitutings = j(J)now gives: (9.30)

from where writing each g;(j(J)) in polar form gives:

which rearranges to: (9.31)

If we now write the overall transfer function in its polar form, i.e.:

g(j(J))

= lg(jw)l e j;

(9.32)

a direct comparison of Eqs. (9.32) and {9.31) gives the required result. It is now important to point out that taking logarithms in Eq. (9.28) gives the following alternative, but equivalent form which is particularly useful in constructing Bode diagrams: log (AR)

overall

= log AR 1 +log AR2 + ... +log ARN

(9.33)

implying that the logarithm of the overall amplitude ratio is a sum of the logarithms of the contributing system amplitude ratios. · These results will be useful in our subsequent discussions as we start to look at the frequency response of the various classes of process systems. First, let us summarize the implications of these results on the construction of Bode diagrams from complex transfer functions.

288

PROCESS DYNAM ICS

Implic ations for Construction of Bode Diagrams Any complex transfer function g(s) can be broken down into simpler individu al compon ents g1(s), g2(s), ... , gN(s), whose frequency-response parame ters, AR1, AR2, ••• , ARw and ~1' ~2 , ••• , ¢N can be easily found. As a result of Eqs. (9.28), {9.29) and (9.33), the following are the classical rules for constru cting the Bode diagram corresp onding to the comple x transfer function, using the frequencyresponse characteristics of the simpler components: 1. The steady-s tate gain of the complex transfer function is extracte d and used as a scaling factor for the overall AR plot. 2. The logarith m of the (scaled) overall AR is obtaine d by summin g up the logarith ms of the individ ual contribu ting AR's. 3. The overall phase angle is obtaine d by summin g up the individ ual contribu ting phase angles.

The emphas is on the word classical as used above is deliberate: it is meant to convey the fact that the construction of Bode diagram s is hardly ever done by hand any longer; the current practice is to employ comput er softwar e package s for this purpose . These rules are therefor e primari ly of classical interest; but they are present ed here for pedago gical reasons also, since they aid the underst anding of the frequency-response characteristics of complex systems .

9.3

LOW-ORDER SYSTEMS

9.3.1 First-O rder System As shown earlier, the frequency-response characteristics of a first-ord er system are given by: (9.34a)

and ¢ = tan

-I

(-an)

(9.34b)

Taking logarith ms in Eq. (9.34) we have: (9.35)

and for purpose s of the Bode diagram , we shall retain K as a scaling factor for AR, and use the magnitude ratio, MR = AR/K, as a scaled amplitu de ratio. We will also use m-r as the abscissa (i.e., use -r as a scaling factor for the frequency axis).

CHAP 9 FREQUENCY-RESPONSE ANALYSIS

289

Asymptotic Considerations Amplitude Ratio Concerning the AR expressions, we make the following observations about its asymptotic behavior: 1. As em~ 0, AR/K ~ 1, and log AR./K ~log 1 Thus, at very low frequencies, the log-log plot of MR tends to the horizontal line represented by:

AR/K vs.

MR=AR= I K

W'f

(9.36)

This line is called the low-frequency asymptote for this reason. 2.

As vs.

CiJ'f ~

oo, log AR./ K ~ -log

CiJ'f tends

CiJ'f and the log-log plot of MR = AR/ K to the line represented by:

log

AR K = - log w-r

(9.37)

a straight line with slope -1. Because this situation arises at very high frequencies, this line is known as the high-frequency asymptote. One final point: the high-frequency asymptote intersects the low-frequency asymptote at the point where w-r takes on the value of 1 (since the lowfrequency asymptote, for which AR/K = 1, must have this point in common with the high-frequency asymptote, for which log AR/ K =-log w-r ). The value of the frequency at this point is CiJn, known as the corner frequency or the natural frequency. Thus, since:

we have the expression for the corner frequency: (iJ

n

= -1:I

(9.38)

the reciprocal of the system time constant. This is illustrated in Figure 9.3. Phase Angle The important aspects of the asymptotic behavior of the phase angle are as follows: 1. As em~ 0, ¢~ 0, and 2. As CiJ'f ~ oo, ¢ ~ - 90° 3. At the corner frequency, em= 1, and ¢=-tan -l (1) = -45°

290

PROCESS DYNAMICS

--

~]

" .....!

0.01

. _]

" ....J

" ....J

0.1

1

10

" .....1

" .....1

" .... J

0.01

0.1 1 Frequency (radians/time)

10

Figure 9.6. Bode diagram for a pure gain system.

These facts tell us that the phase angle for the first-order system becomes asymptotic to -90° as frequency increases, never falling below this value at any time. All these points are illustrated in the Bode diagram given in Figure 9.3.

9.3.2 Pure Gain System The transfer fwlction for this process, as we know, is:

=K

(9.39a)

g(jm) = K

(9.39b)

g(s)

so that

and in terms of the form in Eq. (9.14a), we have Re = K and 1m= 0, with the immediate implication that AR = K

(9.40a)

=0

(9.40b}

and ~

Thus the Bode diagram for this system has the simple form shown in Figure 9.6. The important point to note here is how the frequency response of the pure gain system correspond s to that of the first-order system at low frequency.

9.3.3 Pure Capacity System Since the transfer fwlction in this case is:

K*

g(s) = -

s

(5.44)


291

we have: K*

g(jw) = jw

Rationalization now gives: K*"

g(jw) = - !i:l.

(9.41)

(I}

so that in this case, Re(w)= 0 and lm(w) =- K• I w; therefore: K* AR=w

(9.42a)

and giving (9.42b)

It is important to note that these results are exactly the same as earlier obtained in Eq. (5.54). Observe from Eq. (9.42a) that:

AR

log K* = -log w

(9.43)

Thus the log-log plot of the magnitude ratio vs. w for the pure capacity system will be a straight line with slope -1; and the phase angle plot will be a horizontal line fixed at~= -90°. The actual Bode diagram is shown in Figure 9.7. It is important to note that both the MR and ~ behavior correspond to the asymptotic behavior of a first-order system at high frequency.

w-:]J::EIJ 0.01

0.

1

·_] . .J. . .J .... 0.01

0.1

1

10

100

..J ......J 10

100

(I)

Figure 9.7. Bode diagram for a pure capacity system.

292

PROCESS DYNA MICS

9.3.4 First -orde r Lead Syste m The first-order lead process has the transfer function form: g(s)

= K(g+ 1)

(9.44a)

so that: g(jOJ) = K (~OJj + 1)

(9.44b)

Findin g the magn itude and argum ent of this compl ex numb er whose real and imagi nary parts are directly available witho ut ration alization is a very easy exercise; the results are: (9.45a)

and

f

= tan -J (~)

(9.45b)

Asym ptoti c Considerations Ampl itude Ratio The AR expression has the following asymptotic behav ior: 1. As me-+ 0, AR/ K-+ 1, and log AR/ K-+ log 1

Thus, at very low frequencies,. the log-log plot of MR tends to the horizo ntal line:

=AR/ K vs. ro~

Thus the low-frequency asymptote for this system is the same as that for the first-order system. 2. As me-+ oo, log AR/K -+ log me and the high-f requency asymp tote is the line: AR log K

=

log ro~

(9.46)

a straig ht line with slope +1. This is the exact oppos ite of the case with the first-order system. 3. The come r frequency in this case is: (9.47)

the reciprocal of the lead time constant.

CHAP 9 FREQUEN CY-RESPO NSE ANALYSIS

10-·

293

10-1

1

10

10'

100

•

·/

50

0

lo-"

"'

/

lQ-1

'" 1

10

10"

OJ!;

Figure 9.8. Bode diagram for a first-order lead system.

Phase Angle The phase angle asymptoti c behavior is as follows: 1. As w.;~o, ~~o 2. As ~ ~ oo, ~ ~ +90" 3. At the comer frequency,

cug =1 and ~ =tan -I {1) = + 45"

The Bode diagram for this system is shown in Figure 9.8. There are some very important points to note about this frequency response: 1. The magnitud e ratio for the first-order lead process is greater than or equal to 1; with the first-order system, MR is less than or equal to 1.

2. The phase angle is always greater than or equal to zero, with the implicatio n that the output sine wave actually leads the input sine wave. This is the basis for the term "lead system." It is interestin g to note that just as the transfer function for the first-orde r lead is like the inverse of the transfer function for a first-orde r system, the Bode diagrams for these systems are mirror images of each other; for the magnitud e ratio plots the reflection is about the line MR = 1, while for the phase angle plots, the reflection is about the line ~ = 0.

9.3.5 Lead/Lag System The lead/lag system has a Laplace domain transfer function given by: g(s) = K

100

1; (~

=2, -r =1).

It is easy to see, once again, that there are three possible types of behavior arising from the lead/lag system, depending on the relationship between the lead and lag time constants. Eliminating the trivial case when ~ = -r (a pure gain system), the two possible Bode diagrams, when ~ -r, are shown in Figures 9.9 and 9.10 respectively for the specific case when -r= 1 and~ takes on the values 0.5 and 2. Note the high-frequency behavior when the lag term dominates (Figure 9.9) and when the lead term dominates (Figure 9.10). The asymptotic value of MR = AR/ K lies either above the line MR = 1, or below depending on whether p = ~I-ris larger or smaller than 1.

9.4

HIGHER ORDER SYSTEMS

One of the main points emerging from the· discussion in Chapter 6 is the fact that, with the exception of the underdamped second-order system, higher order systems are composed of combinations of lower order systems. This being the case, the issue of obtaining the frequency-response characteristics of such systems is greatly l!implified by the application of the results (and rules) given in Section 9.2. Let us illustrate with some higher order systems that frequently arise.

9.4.1 Two First-Order Systems in Series For two first-order systems in series, the transfer function is given as: (6.10)

and forK= K 1K 2, we may apply the results of Section 9.2 to immediately obtain the following: (9.50a)

296

PROCESS DYNAMICS

and (9.50b)

Taking logarithms in Eq. (9.50), or directly from Eq. (9.33), we have: (9.51)

It is now easy to see that:

1.

The high-frequ ency asymptote of the amplitude ratio plot is, in this case, a straight line of slope -2;

2.

The low-frequ ency asymptot e is the same as for the first-orde r system;

3.

The phase angle approaches -180° at high frequency, and approaches oo at low frequency.

To illustrate how the Bode diagram is generated by combining the AR and ' of two first-order elements, we present Figure 9.11, for the system: g(s) =

(s

(9.52)

+ 1)(5s + 1)

It is importan t to emphasiz e again that, as far as the constructi on of Bode

diagrams go, employin g the methods illustrated by this example are of historical, and pedagogic al interest only; the typical computer generated plot will show only the overall AR and 'curves.

100

-2001-~~~--~~-.~nn~rrnn~ 0.0~

0.1

1

10

100

(J)

Figure 9.11.

Bode diagram for two first-Qrder systems in series.


9.4.2

297

Second-Order System

For the general second-order system, from the earlier results given in Eqs. (6.78) and (6.79) (and confirmed in Example 8.2), we have: AR

K

= --/ (1 -

oiil-) 2 + (2Cmr )2

(9.53a)

and ¢ = tan _1 ( 1-2(mr) _ w1 il

(9.53b)

Asymptotic Considerations Considering the expressions in Eq. (9.53) as functions of m'C, with ' as a parameter, the following asymptotic characteristics are worthy of note: Amplitude Ratio 1. As·mr-+ 0, AR/ K-+ 1, and log AR/ K-+ log 1 Thus, the low-frequency asymptote is the horizontal line: (9.54)

2. As ro'C-+ oo, log AR/ K -+ - 2log m'C asymptote will be the line: log

AR K

= - 2log wr

so that the high-frequency

(9.55)

a straight line with slope -2. It is important to point out that the high-frequency asymptote intersects the low-frequency asymptote when m-r = 1. However, the parameter 'f for the second-order system is defined as the natural period, or the reciprocal of the natural frequency. Thus from this definition, it is easy to see that value of the corner frequency at this point is the same as the natural frequency of oscillation

mn. Phase Angle 1.

As mr becomes much less than 1: tan¢ "' - 2(w-r

so that as mr-+ 0, ¢-+ 0.

298

PROCESS DYNAMICS

2.

When arr = 1, regardless of the value of', tan ¢ =- oo, so that¢=- 90°.

3. As arr ~

oo,

¢~

tan- 1

{~) = tan- 1 (0)

= -180°

implying that the phase angle for the second-order system becomes asymptotic to -180° as frequency increases, never falling below this value at any time. Note also that these facts imply that the various phase angle plots, obtained for different values of the parameter t; pass through the same point at ¢ =- 90° when arr = 1. The Bode diagram is shown in Figure 9.12. One peculiar characteristic of the frequency response of a second-order system is the fact that the AR plot shows a "hump" for certain values of t;. By differentiating Eq. (9.53a) with respect to ro-r, and setting to zero, we may use elementary calculus to determine that this phenomenon, known as resonance, occurs for:

t;

1

< ...[2 = 0.707

(9.56)

The frequency at the point where the AR plot passes through this maximum, called the resonance frequency, mr , is:

10 MR=AR tt---~~~~ K 0.1 0.01-t---r---r"T""TTT"T"rT---,.-....--,.....,~-rtol

0.1

1

10

1 0

Figure 9.12.

Bode diagram for the second-order system.


299

(9.57)

This resonance phenomenon leads to maximum values of MR > 1, with increasingly pronounced MR peaks as the damping coefficient (gets smaller. In the undamped case, 0, the MR peak actually becomes infinite. Note finally from Eq. (9.57) that the resonance frequency gets closer and closer to the natural frequency of oscillation as 'gets smaller.

'=

9.4.3

Other Higher Order Systems

It is now a fairly easy matter to generalize the results obtained thus far. We

have that: The Bode diagram for the Nth-order system has for the amplitude ratio high-frequency asymptote a straight line of slope -N, and for the low-frequency asymptote, the horizontal line: AR/K = 1. The phase angle plot starts at 0° at zero frequency and asymptotically approaches - (90 x N) o at very high frequencies.

The complete Bode diagram for an Nth-order system will have the properties of the lower order systems of which it is composed, and will therefore not be much different in character from the ones we have already encountered.

Systems with Zeros In ~uch the same way that the step responses of systems with zeros exhibit interesting characteristics, so also do their frequency responses. Rather than give a general treatment, we will illustrate some of the several possibilities by presenting some example Bode diagrams for the (2,1)order system:

g(s)

(6.83)

The Bode diagram in Figure 9.13 is for the system:

g(s)

= (s

(0.5s + I) + 1)(4s + 1)

Some of the important aspects to note here are:

(9.58)

300

PROCESS DYNAMI CS

---.....

1

AR

10-2

0.1

~

1

10

100

1

10

100

--....!'--.. 0.1

~

""'

Q)

Figure 9.13.

Bode diagram for the (2,1)-order system with ~1 < T1 < 'li·

1. The high-fre quency amplitud e ratio asympto te is a straight line of slope -1; and the phase angle tends asympto tically to -90° at high frequencies. It is left as an exercise to the reader to confirm that this is indeed what should be expected when two lag terms are combined with a lead term. 2. The phase angle plot shows an "undersh oot," with the value of~ falling below the ultimate value of -90° over a certain range of frequency values. It can be shown that wheneve r ~1 < -r < -r , i.e., the 1 2 lag time constant s are all larger than the lead time constant (as is the case here), this phenome non will be in evidence.

The Bode diagram for the system: 9

_

g( ) - (s

(2s + 1) + 1)(4s + 1)

(9.59)

for which -r1 < ~1 < 12 is shown in Figure 9.14. The most significa nt characteristic of this Bode diagram is that it can easily be mistaken for that of a purely first-order system. This will always be the case wheneve r the value of the lead time constant lies in between the values of the two lag time constants. When the lead time constant is larger than both lag time constants, as with the following system: 9

_

g(,) - (s

(Ss + 1) + 1)(4s + 1)

(9.60)

we recall fll"St that, as was shown in Figure 6.15 (for this same system), the step response overshoo ts the final steady-state value.

CHAP 9 FREQUENCY -RESPONSE ANALYSIS

--

AR

301

~

10-4

~

........

10

Figure 9.14.

Bode diagram for the (2,1)-order system with T1 < ~ 1 < Tz·

The Bode diagram for this system is shown in Figure 9.15. It is interesting to note that both the AR and f aspects of this Bode diagram exhibit "overshoot." The AR plot overshoots the value of 1 over a certain range of frequency values, and the phase angle is actually positive for some low-frequency values. This is, of course, due to the fact that the influence of the lead term is stronger in this case than in the earlier two cases.

1

~ .......

AR 10~

10--4 0.01

~ 0.1

1

10

100

10

100

50 0

-50 -100

O.ol

""

0.1

""' "--. ""

1 (l)

Figure 9.15.

Bode diagram for the (2,1)-order system with a stronger lead influence; '~"1 < "~"2 < ~1"

302

PROCESS DYNAMICS

---..

1

~ .......... ............

0.1 0

•

i

-100

--......

-200

10

100

10

100

"

-300 0.01

1

0.1

··"

""--1

(J)

Figure 9.16.

Bode diagram for the (2,1)-order system with ~1 < 0; i.e., an inverse-

response system.

Inverse-Response Systems When the zero present in the system transfer function is a RHP zero, ie., ~1 < 0 < -r1 < 12· we already know that the system will exhibit inverse response. h\ terms of the eHect of the RHP zero on the frequency response, the main point of interest is summarized below:

Whereas the presence of a LHP zero improves the phase angle characteristics of a system by adding +90° to the high1requency asymptotic value of lfl for every LHP zero present in the system transfer function, the effect of the RHP zero is the opposite: it worsens the phase angle characteristics by adding -90° to the high-frequency asymptotic value of lfl for every RHP zero present in the system transfer Junction. Thus, for example, the high-frequency asymptotic value of lfl for the secondorder system by itself, with no zeros, is -180"; this value is improved to -90° for the second-order system with a single LHP zero, and worsened to -270" for the corresponding system with one RHP zero. The sign of the zero has no effect in the amplitude ratio. For the purpose of illustration, the Bode diagram for the inverse-response system used as an example in Chapter 7:

g(s)

( -3s + I) l){Ss + 1)

= (2s +

(7.16)

is shown in Figure 9.16. Note the asymptotic value of the phase angle at high frequencies tends towards -270".


303

9.4.4 Time-Delay Systems For the pure time-delay system: g(s) = e-as

(8.15)

with frequency-response transfer function given by: g(jm)

= e-i=

(9.61)

which, when compared with the polar form: g(jm) = lg(jro)l eif

establishes at once the following frequency-response characteristics: lg(jro)l =AR

=1

(9.62a)

and ~

= -am

(9.62b)

once again confirming the results given earlier in Eq. (8.19). The Bode diagram is shown in Figure 9.17.

The most important features are the constant, unity AR characteristics, and the perpetually increasing nature of the phase lag. This is in contrast to each of the systems we have considered so far, where there has always been a limiting value to which the phase angle tends at very high frequency: the pure timedelay system is unique in that its phase angle decreases indefinitely as frequency increases. 10

AR

1

~ ~

... 10-4

1

"" 10

Figure 9.17 Bode diagram for the pure time-delay system.

304

PROCESS DYNAMICS

10~--~rn~n--.,.~m+~ro~nm

0.01

0.1

10

~o,~--.,,Trr~-rTOTTinr_,~~mm

0.01

Figure 9.18.

0.1

1

10

Bode diagram for a first-order-plus-time-delay system.

For any system with an associated delay, we note from Eq. (9.62a) that the logarithm of the AR contributed by the time-delay element is zero; we therefore have the following result:

If AR*

is the amplitude ratio of the undelayed portion of a time-delay system, with the corresponding phase angle of 1/1*, then the frequency response of the system including the delay element e-as is: AR

= AR*

(9.63a)

and 1/1 =

1/1*- am

(9.63b)

In other words, the AR characteristics of a system are unaffected by the

presence of a time delay, but the phase angle characteristics are considerably altered. For the purposes of illustration, Figure 9.18 shows the Bode diagram for the first-order-plus-t ime-delay system used in Example 8.1: 0.66 e-2•6• g(s) = 6.1s

+1

(8.32)

Observe that the first-order AR characteristics are still retained; but the phase angle no longer has the -90° limiting value. The fact that the phase angle no longer has a limiting value is due directly to the presence of the timedelay element.


9.5

305

FREQUENCY RESPONSE OF FEEDBACK CONTROLLERS

Listed below with their corresponding transfer functions are the four types of feedback controllers we will deal with later on in Part IV: 1. Proportional (P) Controller:

g.(s) = K,

(9.64)

2. Proportional + Integral (PI) Controller: (9.65)

3. Proportional + Derivative (PD) Controller: g.(s)

= K,

(I + 7:Ds)

(9.66)

4. Proportional + Integral + Derivative (PID) Controller: g,(s)

= K, ('

+

)r + 1:Ds)

(9.67)

Close observation will reveal that the P controller has identical characteristics with the pure gain system, while the PD controller has identical characteristics with the first-order lead system Eq. (9.42), which have all been previously discussed. It is left as an exercise to the reader to use the methods we have discussed . above to obtain the frequency-response characteristics of the PI and the PID controllers, and confirm that the Bode diagrams are as given in Figures 9.19(a) and (b). (See also problems at the end of the chapter.)

Figure 9.19.

Bode diagrams for (a) the PI controller and (b) the PID controller.

306

9.6

PROCESS DYNAMICS

SUMMARY

We have studied the frequency-response characteristics of the various classes of processes in this chapter, deriving the individual AR and ¢ expressions, and studying their respective Bode diagrams. We have also introduced the Nyquist plot as an alternative method of displaying the frequency response. Looking back, we should now be able to see why it is true that the fundamental features of these process systems are fully characterized by their frequency responses. Firstly, we have shown that the frequency-response characteristics are immediately obtained from the magnitude and argument of g(jm), the process frequency-response transfer function. We have seen that the frequency response of the process, represented by g(jm), fully characterizes a linear process. Even though we have presented several technical piece~ of information useful in the construction of Bode diagrams (especially for systems with complex transfer functions) it should be kept in mind that the standard practice is to employ computer software packages for generating such diagrams. Since Bode diagrams are just as characteristic of individual classes of linear systems as their transform-domain transfer functions, it is important that the essential aspects of these diagrams for each class of systems be well known. It is perhaps helpful to summarize these salient features here: First-Order System: The MR plot is 1 at very low frequencies and approaches a straight line with slope -1 as m-+ co, The phase angle plot starts out at 0° at very low frequencies, takes the value- 45° at the corner frequency mn =1/ 1', and approaches a limiting value of- 90° as m-+ oo, Pure Gain System: Constant MR = 1 and ¢ = 0 at all frequencies, characteristics similar to those of the first-order system as m-+ 0. Pure Capacity System: The entire MR plot is a straight line with slope -1 and ¢ is constant at -90° at all frequencies, characteristics similar to those of the first-order system as m-+ co. Second-Order System: The MR plot takes the value 1 at very low frequencies and approaches a straight line with slope -2 as ID -+ co; in between, for values of damping coefficient t; < 0.707, the MR plot exhibits resonance; for other values of t;, the MR values are less than or equal to 1. The phase angle plot starts out at 0° at very low frequencies, takes the value - 90° at the natural frequency (J)n = 1/ 1' (regardless of the value of t;), and approaches a limiting value of- 180° as ID-+ =. Nth-Order System: The MR plot takes the value 1 at very low frequencies and approaches a straight line with slope -N as m -+ co. The phase angle plot starts out at 0° at very low frequencies and approaches a limiting value of - (90 x N ) as ID-+ oo. 0

First-Order Lead System: The MR value is greater than or equal to 1 at all frequencies, and the phase angle is always positive. The MR plot is 1 at very low frequencies and approaches a straight line with slope + 1 as ID-+ =. The phase angle plot starts out at oo at very low frequencies and approaches a


307

limiting value of + 90° as w ~ oo, The Bode plot for this system looks like a mirror image of that for the first-order (lag) system. Time Delay: MR = 1 for all frequencies while ¢ has monotonically decreasing phase angle characteristics.

It is also advantageous to be familiar with the effect of system zeros on the frequency response; LHP zeros in general improve the phase angle characteristics, while these characteristics are worsened by RHP zeros. We should mention, finally, that this chapter has actually raised some issues that have very significant bearings on controller design for process systems. We will be recalling several of these issues in Chapter 15, where the application of frequency-response analysis to conventional feedback controller design will be discussed. REVIEW QUESTIONS 1.

The complete response of a linear system to a sinusoidal input consists of two parts: what are they and which part is studied in "Frequency-Response Analysis"?

2.

What is the main concern of frequency-response analysis?

3.

What is the fundamental frequency-response analysis result, and what is its main benefit?

4.

What is a Bode plot and how is it different from a Nyquist plot?

5.

How are the amplitude ratio and phase angle characteristiC'.s of the individual elements of a composite system related to the corresponding overall system characteristics?

6.

Why is the reciprocal of the first-order system time constant referred to as the "corner frequency"?

7.

Which portion of the frequency response of a first-order system is related to the frequency response of the pure capacity system? Which portion is related to the frequency response of the pure gain system?

8.

From a frequency-response perspective, why is the transfer function in Eq. (9.44) referred to as a "lead element" and the transfer function for a first-order system (with K =1) referred to as a "lag element"?

9.

What key parameter determines the shape of the le~d/lag system's Bode diagram?

10. What conditions must the damping coefficient of a second-order system satisfy in order that the phenomenon of resonance be evident in its Bode diagram? 11. A common misconception about underdamped second-order systems is that they all exhibit resonance. What do you think is responsible for this misconception? Give a range of values of the damping coefficient for which the second-order system will be underdamped without exhibiting resonance. 12. What is the difference between the effect of a left-half plane zero and that of a righthalf plane zero on the phase angle characteristics of a process?

308

PROCESS DYNAM ICS

13. How can you distinguish the frequency response of a system with a strong lead from that of a system that exhibits resonance? 14. What is unique about the frequency-response behavior of a time-dela y system? PROBLEMS

9.1

Establish that the complex number:

z

a +bj

=

may be expressed in the exponential polar form: z

=

lzfeN

(P9.1)

where lzl

(P9.2a) (P9.2b)

9.2

The derivation of the fundamental frequency-response analysis result given in the main text, from Eq. (9.5) through Eq. (9.19), was for the case when the transfer function did not involve a time delay. Consider the situation in which: g(s) = g*(s) e-as

and rederive these results for the time-delay system and show that the expressions given for AR and~ in Eq. (9.4) hold regardless of the presence of time delays. 9.3

(a) Establish Eqs. (9.56) and (9.57). (b) Plot the Nyquist diagrams for the systems whose transfer functions are given as

follows:

s 2 + 4s + I s 2 + 0.2s +

Is the effect of the resonance exhibited by System 2 obvious? 9.4

Show that the response of a pure capacity system with transfer function: g(s)

K

= -;

to a sinusoidal input u(t) =A sin rot is exactly equivalent to the unit step response of an undamped second-order system; specify the undamped system gall. 9.5

Obtain Bode plots for the system whose transfer function is given by: (P9.3)


309

for b = 1, 0.1, 0.01 and compare with the Bode plot for g(s) = 1. How can these results help you in assessing the effectiveness of approximating the ideal impulse function with a rectangular pulse function? 9.6

(a)Sketch the Bode diagram for each of the systems whose transfer functions are given below. Be sure to identify clearly on each sketch all the important distinguishing characteristics: K

as 3 + bs 2 + cs + d

(P9.4)

(P9.5)

(b) A pulse test identification experiment performed on a process gave rise to the Bode diagram shown below in Figure P9.1. Postulate a possible transfer function form that you believe is consistent with the features displayed by this Bode diagram. Support your postulate adequately.

MR

- - - - - - - Frequency (Radians/Time)

Figure P9.1. 9.7

The following differential equations were given in Problem 8.7 (Chapter 8) for a chemical reactor operating under two different feed conditions:

41_· dt = -2y(t) + 0.5 1y(t- 2) + 3u(t)

(P9.6)

1Jf = -2y(t)

(P9.7)

and + 0.5 1y(t- 2) + 3u(t- 5)

assuming the disturbance is zero. Obtain the transfer functions in each case and from these obtain the rorresponding Bode diagrams and rompare them. How does the

310

PROCESS DYNAMICS Bode diagram for the system described by the model in Eq. (P9.6) differ from the standard first-order system Bode diagram?

9.8

The transfer function for a steam-heated heat exchanger is given as:

(1 -

g(s)

o.se- 10•)

= (40s + l)(l5s + 1)

(P9.8)

Obtain the co~sponding Bode diagram. How does this differ from the Bode diagram of a typical time-delay system of the type considered in Section 9.4.4? 9.9

(a) In each case, by obtaining fi.rst,g(jm), or otherwise, derive the expressions for the AR and 'for the PI and PID controllers whose transfer functions are given in Eqs. (9.65) and (9.67). (b) Derive expressions for the asymptotic behavior of the AR and 'for these controllers and confirm that Figure 9.19 is, in fact, consistent with such results.

9.10

An anaerobic digester used in a small town's waste management facility, which is approximately modeled by the following transfer function: 0.52

(P9.9)

g(s) = (45.5s + 1)(l2.2s + 1)

is subjected to periodic loading which, for simplicity, will be approximated as a pure sine wave: u(t) =A sin OJt. The process output is the "normalized" digester biomass, and the input is the sludge flowrate into the facility, in "normalized" mass flow units; the unit of time is minutes. F"md the amplitude of the digester output when the periodic loading amplitude is 5 units, at a frequency of m = 0.2 radians/min. If the frequency were to go up by an order of magnitude to 2 radians/min., with the amplitude remaining constant, by how much will the amplitude of the digester output change? 9.11

9.12

For each of the following pairs of transfer functions, plot and compare the Bode diagrams, noting which of the two has the minimum phase angle behavior. (These results will be useful in Chapter 18.) (a)

Ct(s)

(b)

Ct(s)

(c)

Ct(s)

5 6s

+ I; 5(4s+l)

(2s + 1)(6s + 1);

=

5 2s + 1;

Repeat Problem 9.11 for Nyquist diagrams.

g2(s)

se-'ls + I

6s

g2(s)

5(-4s + I) (2s+ 1)(6s + 1)

g2(s)

_5_ 2s :... 1

CHAPTE R

10 NONLINEA R SYSTEMS The processes we have studied so far have all been represented by linear models. Because this entitles them to be classified as linear systems (and linear systems, as we know, are fully characterized by their transfer functions) we have been able to obtain fairly general results by taking full advantage of the convenience offered by the transfer function model form. However, the truth of the matter is that, to varying degrees of severity, all physical processes exhibit some nonlinear behavior. Furthermore, when a process shows strong nonlinear behavior, a linear model may be inadequate; a nonlinear model will be more realistic. Unfortunately, this improved closeness to reality is attained at a cost: the convenience and simplicity offered by the linear model is sacrificed. To be sure, many processes exhibit only mildly nonlinear dynamic behavior and therefore can be reasonably approximated by linear models. However, even for such systems, the behavior over a wide range of operating conditions will be noticeably nonlinear, and no single linear model is able to represent such behavior adequately. The issue of analyzing the dynamic behavior of processes represented by nonlinear models is the main concern in this chapter; this is motivated by the fact that situations do arise when it is undesirable to neglect the inherent nonlinearities of a process, be they mild or otherwise. The objective here is to provide an overview of how we can expect to carry out dynamic analysis when the process models are nonlinear. The nature of nonlinear problems (in all areas of analytical endeavor, not just in Process Dynamics and Control alone) is such that they are not as easily amenable to the general, complete treatment possible with linear systems. To keep nonlinear dynamic analysis on a realistic and practical level therefore usually involves trade-offs between accuracy and simplicity. Keeping this in mind will help us maintain a proper perspective throughout this chapter.

311

312

10.1

PROCESS DYNAMICS

INTRODUCTION: LINEAR AND NONLINEAR BEHAVIO R IN PROCESS DYNAMICS

There are two basic properties that characteriz e the behavior of a linear system: 1. The Principle of Superposition If the response of a process to input 11 is R 1, and the response to 1 is R , then 2 2 according to the principle of superposition, the response to (11 + 12) is (R1 + R ) if 2

the system is linear. In general, this principle states that the response of a linear system to a sum of N inputs is the same as the sum of the responses to the individual inputs. As a result of this principle, for example, the response of a linear system to a step change of magnitude A is exactly the same as A times the system's unit step response. The same is true for all other types of input functions. 2. Independen ce of Dynamic Response Character and Process Conditions

The dynamic character of the process response to an input change is independen t of the specific operating conditions at the time of implementi ng the input change, if the system is linear. In other words, identical input changes implement ed at different operating steady-stat e conditions will give rise to output changes of identical magnitude and dynamic character (see Figure 10.1). The principle of superpositi on also means that from the same initial steady-stat e conditions, the output change observed in response to a certain positive input change will be a perfect mirror image of the output observed in response to a negative input change of the same magnitude (see Figure 10.2). According to this property, the step response of a linear system, for example, is the same regardless of the actual initial value of the output or input variable, or, for that matter, the direction of the input change, the "step down" giving rise to a perfect mirror image of the "step up" response. It is because of these two characterizing properties that we have been able to use the linear transfer function representat ion for carrying out dynamic analysis in the last five chapters. A nonlinear system does not exhibit any of these properties; the response to a sum of inputs is not equal to a sum of the individual responses; the response to a step change of magnitude A is not equal to A times the unit step response; the magnitude and dynamic character of the step response are dependent on the initial steady-state operating conditions; a "step down" respo~ is not a mirror image of a "step up" response; and what is true for the step input function is true Output

r···-

Input

Figure lO.L

lc=

/

·-·

Bespooaeat

UpperSteadySt ate Response at Lower Steady State

Step response of a truly linear system at two different steady states.

CHAP 10 NONLINEAR SYSTEMS

v--

Input

313

Output

tc

"Step up" Responae

~-"Step down• Responoe

Figure 10.2. "Step down" and "step up" response of a linear system.

for all the other input functions. In particular, the ultimate response of the nonlinear system to a sinusoidal input function is not a pet;fect sinusoidal output function. The model equations that will represent the dynamic behavior of such systems adequately must necessarily contain nonlinear terms; and immediately, the convenience of being able to completely characterize the system's dynamic behavior by the linear transfer function is lost. That nonlinear dynamics are inevitable in most chemical processes is a fact that needs very little justification; one only has to call to mind, for example, the equations of chemical phase e_quilibria when Henry's Law doesn't apply; heat flux expressions for heat transfer by radiation; chemical kinetic rate expressions, especially the well-known Arrhenius expression, all of which are hardly linear. A survey discussing nonlinear processes that occur in the chemical industry may be found in Ref. [3]. To buttress this point, we will now provide a sample of some very simple, very common process systems and their process models.

10.2

SOME NONLINEAR MODELS

1. Outflow from a Tank

Regardless of tank geometry, the outlet flowrate through an orifice at the bottom of a tank is proportional to the square root of the liquid level in the tank. Thus, for the liquid level process considered in Chapter 1 (Figure 1.19), and used as the first example in Chapter 5, a more realistic model is: dh .r. Ad-= F.-c-vh t '

(10.1)

more realistic, that is, than Eq. (1.13) which was based on the simplifying approximation that outflow varied linearly with liquid level. The nonlinearity of this model is, of course, due to the presence of the square root term. 2. Isothermal Continuous Stirred Tank Reactor (CSTR)

If in the isothermal CSTR of Chapter 5 an nth-order reaction were taking place, the mathematical model would read: (10.2)

which is nonlinear when n :1: 1.

314

PROCESS DYNAMICS

3. The Nonisothermal CSTR Perhaps the most popular example of a classic nonlinear chemical process is the nonisothermal CSTR. As we had earlier shown in Chapter 4, the mathematical model for this process system is: deA dt=

-

!

~

C ()A "'0

e- (EIRT)C +! C A

()Af

(4.26)

(4.27)

Observe that the nonlinearities in this model are due to two factors: 1.

2.

The nonlinear functions of temperature - involving the exponential of the reciprocal. The product functions- involving products of cA and a function ofT.

The point here is that if such simple, common systems have nonlinear mathematical models, we can imagine the extent of nonlinearities to be encountered in the models for more complex systems, in which more complicated chemical and physical mechanisms are responsible for the observed system behavior. A good catalog of such system models is available, for example, in Refs. [1, 2].

10.3

METHODS OF DYNAMIC ANALYSIS OF NONLINEAR SYSTEMS

Coming to grips with the fact that realistic process models are almost always nonlinear, the next major question we wish to ask ourselves is the following: How do we analyze the dynamic behavior of processes when they are represented by nonlinear models? Assuming, for the moment, that we are able to answer this question satisfactorily, there is yet another important question, raised primarily because of our experience with linear systems analysis: Is there a technique for analyzing the dynamic behavior of nonlinear systems that can be generalized to cover a wide variety of such systems? The answers to these questions form the basis of the material covered in this chapter. To analyze the dynamic behavior of a system represented by nonlinear equations, there are usually four alternative strategies to adopt:


1. 2. 3. 4.

315

The rigorous analytical approach Exact linearization by variable transformation Numerical analysis Approximate linearization

We shall consider each of these in tum, with greater emphasis laid on the third and fourth strategies for reasons that will become clear as the discussion progresses.

10.3.1 Rigorous Analytical Techniques It is well known that very few nonlinear problems can be solved formally, primarily because there is no general theory for the analytical, closed-form solution of nonlinear equations. This means that, unlike linear systems where analytical techniques abound for the general solution of the modeling equations, nonlinear systems face a more restrictive situation. Nevertheless, there are a few analytical methods for analyzing the dynamic behavior of nonlinear systems; for example, classical methods due to Poincare, Krylov, and Bogolyubov (see Friedly Ref. [2]), and the techniques of Liapunov, Hop£, Golubitsky, and Arnold (see Refs. (4-7]). These methods will allow one to characterize qualitatively the dynamics of nonlinear systems. However, these approaches find limited use in process control practice; thus a detailed discussion of their salient features has been excluded from the intended scope of this textbook. Perhaps, with further research, the analytical treatment of nonlinear systems will evolve to a state where it will be more useful for process controL

10.3.2 Variable Transformation On the basis of the fact that linear systems are easier to analyze than nonlinear ones, it seems attractive to consider the following proposition:

If a system

is nonlinear in its original variables (say, x, y, and u) is it not possible that it will be linear in some transformation of these original variables?

If this were possible, the general strategy for dynamic analysis would then involve first carrying out this transformation, then performing dynamic analysis on the linear, transformed version, and finally transforming the result back to the original variables. Let us illustrate with an example. Example 10.1

VARIABLE TRANSFORMATION FOR THE ISOTHERMAL CSTR WITH A SECOND-ORDER REACTION.

The dynamic behavior of this process is given by:

(10.3)

PROCESS DYNAMICS

316

which is nonlinear because of the square term. Let us now define a new state variable: l

(10.4)

=

Since:

then, according to this definition Eq. {10.4), and from Eq. {10.3) we have: (10.5)

which simplifies to: (10.6)

H we now define another new input variable u as: (10. 7)

and recall Eq. (10.4), Eq. (10.6) becomes: dz

dt

=

I

I

-~z+ ~v

(10.8)

which is now linear in the new variables z and v defined in terms of the old variables as shown in Eqs. (10.4) and {10.7). In principle, we may now use Eq. (10.8) to investigate the dynamic behavior of this system and relate it to the original system variables via the expressions in Eqs. (10.4) and (10.7).

As promising as this idea of variable transformations may seem, there are some important questions which indicate that this strategy might be of limited practical utility:

1. How was the transformation in Eq. (10.4) found? 2. Is there a general procedure by which such transformations can always be found, for all nonlinear systems? 3. How easily can these transformations be found? There is, in fact, no general procedure for finding such transformations for all nonlinear systems; and for the restricted classes of systems for which a general procedure exists, it is not easy to apply (cf. Ref. [8]).

317

CHAP IO NONLINEAR SYSTEMS

There are other related methods based on differential geometry, but these also suffer from similar limitations (cf. Ref. [9]). A good overview of such transformation methods may be found in Ref. [10].

10.3.3 Numer ical Solutio ns and Compu ter Simula tion Let us remind ourselves that the first step in the analysis of a system's dynamic we behavio r is the solution of the modelin g equation s; and the models, as partial). or y (ordinar s equation ial different of recall, are usually in the form r For linear systems, analytica l solution s are possible , while for nonlinea a is there , however ; possible systems, analytica l solution s are usually not fallback position: we can obtain the solution in numerical form. Thus, instead of having an analytica l expressi on that represen ts the behavio r of the process output over time, we make use of a scheme that provides numeric al values for this process output at specific points in time.

General Procedure as The general procedu re for analyzing the system's behavior numerically is follows:

1. Discretize the Process Model

'!1$ involves taking the original system model, consisting of one, or possibly several, equation s of the type: dx; dt

=ft(x 1, x 2, ••• , xn, u, d, t)

(10.9)

and appropri ately discretizing it to obtain: x,{k+l)

= //(x1(k), x2(k), ..., xn(k), u(k), d(k), k, .1t)

(10.10)

A variety of teclu1iques are available for carrying out this step. 2. Recursive (or Iterative) Solution of the Discretiz ed Model

If we are now given the initial values x;(O) for each of the variables, and the for value of the forcing functions, these equations can now be solved recursively interval. time the value taken by each x;(k) in the next Owing to the recursive nature of this aspect of the problem , the digital compute r is the perfect tool for doing the job. When the model for a process system is thus solved with the aid of a computer, we are said to have obtained a compute r simulation of the process behavior. There are many specialized techniques for obtaining numerical solutions to on different ial equation models and these are typically covered in courses es techniqu These B). ix Append also (see analysis and methods numeric al from model the taking in d usually differ in the discretization strategy employe the continuous form in Eq. (10.9) to the discrete form in Eq. (10.10).

PROCESS DYNAMICS

318

In addition, computer packages are available that offer a wide variety of these numerical techniques, thereby facilitating the otherwise potentially tedious task of numerical dynamic analysis. For the purpose of illustrating the principles involved in numerically obtaining a process response, we now present the following example. Example 10.2

NUMERICAL STEP RESPONSE OF THE LIQUID LEVEL IN A TANK.

The dynamic behavior of the liquid level in a tank has a model given as: (10.1)

with a cross-sectional area A of 025 m 2 and was initially at steady state with inlet flowrate Fi at 0.3 m3 /min, while the level in the tank h was 0.36 m. It is required to obtain the response of this system when the inlet flowrate is increased to 0.4 m 3 /min. Derive an expression to be used for obtaining the numerical value of the level in the tank system at intervals of 0.1 min using the explicit Euler method for discretization. The constant c has the value 0.5 mS/2/min.

Solution: Using the given data, the model for this specific process is:

dh dt

_r

= 4F;-2"1h

(10.11)

Now, with the explicit Euler method (cf. Appendix B), a first derivative is represented in the following discrete form: y(t + At) - y(l)

~

dt =

At

or, taking time t as the kth time interval, with At as the width of each of these time intervals: ~

y(k

dt =

+ 1)- y(k) At

(10.12)

According to this scheme therefore, Eq. (10.11) becomes: h(k + 1)- h(k)

At

__ r:::

= 4F; (k)- ~ h(k)

(10.13)

or h(k + 1)

= h(k) +A{4F; (k)- 2...J h(k)]

and for Lit= 0.1 as given, the required expression is: h(k + 1) = h(k) +

o.{

J

4F; (k)- 2...JhW

(10.14)


319

We may now use this equation along with the given initial condition h(O) =0.36, and set F; (0) to 0.4 to obtain the response of the liquid level to this change in the input variable. For k =0, for example, we have: h(l)

= 0.36 + 0.1(1.6- 1.2) =0.4

implying that 0.1 minutes after the implementation of the step change in the inlet flowrate, the liquid level will be 0.4 m. A recursive application of the expression:

h(k +I)

= h(k) +

o.{

1.6- 2.V h]

(10.15)

gives the value of the liquid level at intervals of 0.1 minutes; a compilation of these values constitute the required numerical step response. Note that the recursive calculations required by this procedure are best carried out with the aid of a digital computer. There are many special subroutines available for the numerical integration of differential equations, so that details of this procedure are usually handled by the software.

Let us now give a quick recapitulation of the three techniques we have so far discussed for nonlinear dynamic analysis. The first two are clearly of limited practical application and need no further comment; the technique of numerical analysis, even though straightforward, provides limited information because it is usually impossible to infer anything of a general form from the numerical response of a process system. Numerical analysis provides specific numerical answers to specific problems; it is not possible to obtain general solutions in terms of arbitrary parameters and unspecified inputs in order to understand the process behavior in a more general fashion, as was the case with linear systems. The question now is: can we ever achieve this desirable objective of being able to analyze, in a general fashion, the behavior of an arbitrary nonlinear system, in response to unspecified inputs? The answer is yes: but at a cost. For observe that the nonlinear process model can be linearized around a particular steady-state value, and with this linearized approximation of the true nonlinear model, we can obtain fairly general results regarding the process behavior, but these will only be approximate, never completely representing true behavior accurately. When faced with the problem of analyzing the behavior of a nonlinear process, therefore, we have a choice to make: if accuracy is more important, then numerical analysis is the recommended approach; but general inferences are ruled out. When the emphasis is more on simplicity, and it is also important to be able to make some general statements about the dynamic behavior, one will then have to sacrifice accuracy• and use the method of linearization. Because the compromise involved in the latter strategy is often considered more advantageous, linearization is the most commonly utilized technique in the analysis of nonlinear systems.

320

PROCESS DYNAMICS

10.4

LINE ARIZ ATIO N

In the context of process dynamics and control, linear ization is a term used in general for the process by which a nonlinear system is appro ximat ed by a linear one; i.e., the nonlin ear process mode l is someh ow appro ximat ed by a linear process model. The most popul ar technique for obtaining these linear appro ximat ions is based on Taylor series expansions of the nonlinear aspects of the process model, as demo nstrat ed below. Any function f(x) (unde r certain conditions which are alway s satisfied by the mode ls of realistic processes) can be expressed in the following powe r series aroun d the point x = x.:

(.&.) AY) +(

ftx) = f(x,)

+

d

X

dx x=x (x-x, )

n

,

+

(R)

dx2 x=x

(x _ x ,)2

,

2!

+ ...

(x- x,)n

x=x,

+ ...

I n.

If we now ignore second- and higher order terms, Eq.

(10.16)

(10.16) becomes:

f(x) "' f(x.) + / (x,)(x - x,) (10.17) which is now linear. Obser ve that if (x - x5 ) is very small, the highe r order terms will be even small er and the linear appro ximat ion is satisfactory. The implication of this linearization process is depic ted in Figure 10.3. The appro ximat ion is exact at the point x =x , remai ns satisfactory as long 5 as xis close to x5 , and deteriorates as x moves farthe r away from x, - the rate of this deteri oratio n being directly relate d to the severi ty of the nonlin earity of the functionf(x). T'nis is easily extended to a function of two variab les f(xl' x2), expan ded in a Taylor series aroun d the point (xw x ); the result 25 is:

+ higher order terms

(10.18) This purel y mathe matic al idea will now be applie d to the probl em of nonlin ear dynam ic analysis.

x,

Figure 10.3.

Nonlinear function and linearized approximation.


321

The General Linearization Problem Consider the general nonlinear process model: dJc Jt =

R

,

(10.19a)

J\X,U)

y = h(x)

(10J9b)

where f(•, •) is an arbitrary nonlinear function of the two variables, x, the process state variable, and u the process input; h(•) is another nonlinear function relating the process output, y to the process state variable x . The linearized approximat ion of this very general nonlinear model may now be obtained by carrying out a Taylor series expansion of the nonlinear functions around the point (x5 , U5 ). 1his gives:

(K)

dt1xt = f(x,, u,) + a X

(x 1 •,)

(x-xs) +

(K) au

(u-u,)

(x1 u)

+ higher order terms

(10.20a)

(10.20b) Ignoring the higher order terms now gives the linear approximat ion: (10.2la) y = h(x,) + c(x5 ) (x- x,)

(10.2lb)

where

a(•, •) =

b(•, •)

(1?1,.) (~£ )x 1

c(•) =

u)

(~1•

It is customary to express Eqs. (10.2la,b) in terms of deviation variables:

x = x-x ii

=

5

u- u,

y = y- Y,

322

PROCESS DYNAMICS

H in addition to this, the linearization point (x5 ,u5 ) is chosen to be a steadystate operating condition, then observe from the definition of a steady state and from Eq. (10.19) that both dxj dt and f(x 5,u5 ) will be zero. Equations {10.21a,b) then become: dX dt

a;;+ bu

(10.22a)

y =ex

(10.22b)

where, for simplicity, the arguments have been dropped from a, b, and c. A transform-domain transfer function model corresponding to Eq. (10.22) may now be obtained by the usual procedure; the result is:

_ y(s) =

[c(x,) b(x,, u,)] _ s - a(x

s•

u ) s

u(s)

(10.23)

with the transfer function as indicated in the square brackets. This transfer function should provide an approximate linear model valid in a region close to (x 5 ,u5 ).

The principles involved in obtaining approximate linear models by linearization may now be summarized as follows:

1. Identify the functions responsible for the nonlinearity in the system model. 2. Expand the nonlinear function as a Taylor series around a steady state, and truncate after the first-order term. 3. Reintroduce the linearized function into the model; simplify, and express the resulting model in terms of deviation variables. The application of these principles to specific problems is usually much simpler than one would think at first, as we now demonstrate with the following examples. Example 10.3.

LINEARIZATION OF A NONLINEAR MODEL INVOLVING A NONLINEAR FUNCTION OF A SINGLE VARIABLE.

Obtain an approximate linear model for the liquid level in a tank whose nonlinear model was given as: ( 10.1)

where Fi, the inlet flowrate, is the manipulated variable. Solution; The nonlinear function in this case is f(h) =h112, a function of a single variable. A Taylor series expansion of this function around the steady state h5 gives: h 112 = h,

112

+

. 2I h,-112 (h- h,) + htgher order terms

(10.24)


323

Ignoring the hi~her order terms and introducing the linear approximation into Eq. (10.1) for h11 now gives:

lh -1/2 (h = F- [h 112 A dh + 2 S S C I dt

_h)] S

(10.25)

The deviation variable y = (h - h5 ) naturally presents itself; if we now add the variable u = (F; - F;5 ), realizin1 that under steady-state conditions (dh/dt = 0), Eq. (10.1) implies that Fis = c h511 , Eq. (10.25) reduces at once to: 'C

iJJ. dt

= Ku-y

(10.26)

where we have defined: 2h l/2

2(h) 112 T=A---

K =

c

s

c

(10.27)

From this approximate linEar model, we may now take Laplace transforms in the usual manner to obtain the following approximate transform-domain transfer function model:

K

y(s) = --u(s) u + I

(I 0.28)

a first-order transfer function with steady-state gain and a time constant given by Eq. (10.27). It is now very important to note that both K and 'Care functions of h5, so that the apparent steady-state gain and time constant exhibited by this system will be different at each operating steady-state condition. Example 10.4

LINEARIZATION OF A NONUNEAR MODEL INVOLVING A NONLINEAR FUNCTION OF TWO VARIABLES.

The dynamic behavior of the liquid level h in the conical storage tank system shown in Figure 10.4 can be shown to be represented by Eq. 10.1, where now the cross section area of the tank is given by: (10.29)

so that the tank model becomes:

Figure 10.4.

The conical tank system.

324

PROCESS DYNAMIC S (10.30) where a and f3 are parameter s defined by:

f3

=

ca

Here, F; is, again, the inlet flowrate, the manipula ted variable. Obtain an approxim ate linear model for this system. Solution: The process model has two types of nonlinear functions: F; h-2, a product of two functions, and h-312 . We shall have to lin~arize each of these functions separately around the steady state (h5, F;.). 1.

The linearizat ion off (h, F;) = F; h-2 proceeds according to Eq. (10.18) as follows:

whereupo n carrying out the indicated operation s now gives: (10.31)

if we ignore the higher order terms. 2.

The steps involved in linearizing the second nonlinear term are no different from those illustrated in the previous example; the result is: h -3/2-

h-312 =

s

l2 hs-512 (h- hs)

(10.32)

We may now introduce these expressions in place of the correspon ding nonlinear terms in Eq. (10.30). Recalling that under steady-st ate condition s aF;s = fJh/ 12 , and introduci ng the deviation variables y = (h -h5 ) and u (F; -fis), the approxim ate linear model is obtained upon further simplification as:

=

T

!!J. dt + y =

Ku

(10.26)

where the steady-sta te gain, and time constant associated with this approxim ate linear model, are given by: K

=-2a h 112 {3 s

2h/ 12 c

{10.33a)

and 2h 5/2

T

= _ s{3 _

(l 0.33b)

/


325

If we desire an approximate transform-domain transfer function model, Laplace

transformation in Eq. (10.26) gives:

K y(s) = --u(s) -rs + 1

(10.34)

Note that the approximate linear model for the conical tank has the same process gain, K, as for the cylindrical tank; but the time constant is a much stronger function of the liquid level hs.

Some important points to note about the results of these two illustrative examples are the following: 1. In each case, the approximate transfer function model is of the first-

order kind, with the time constants, and steady-state gain values dependent on the specific steady state around which the system model was linearized. 2. Using such approximate transfer function models will give approximate results which are good in a small neighborhood around the initial steady state; farther away from this steady state, the accuracy of the approximate results becomes poorer. 3. Using such approximate models, we are able to get a general (even if not 100% accurate) idea about the speed and magnitude of response to expect from any arbitrary nonlinear system. Numerical analysis can give more accurate information, but only about the specific response to a specific input, starting from a specific operating condition for a specific set of parameters. In Chapter 13, where we discuss model building from experimental data, we will introduce another approach to the problem of creating approximate linear models for nonlinear processes - through fitting approximate linear models to data. Let us conclude with a comparison of the responses of the nonlinear model and an approximate linear model for the conical tank of Example 10.4. Suppose that a= 2, fJ = 1, c =0.5 so that the nonlinear model is:

dhd = 2F. h-2- h-312 f

(10.35)

I

while the approximate linear model has the transfer function y(s)

4h/ 12 n u(s) (2h/ )s + I

(10.36)

which in the time domain is

~ dt -- _1 ( 2h/'-2) { h,- h (t) + 4 h s 1/2 [ F; (t)- F;, ]

}

(10.37)

326

PROCESS DYNAM ICS 1.000 0.900

1---:: p

0.800

,t.';/"

0.700

--

linear

tline!r

0.600

h

...... 0.600 It...... 0.400 ~ 0.300 0.200

'

-.

~

0.100

I

nonlinear

a~ nonlinear

nonlinear

I

linear 0.000 0.0 0.2 0.4 0.6 O.B 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.B 3.0 Time

Figure 10.5.

Response of nonlinear model Eq. (10.35) and linear model Eq. (10.37) for two different steps in feed flowrate up and down .df; = ± 0.05 and ±0.15. 1.000 0.900 0.800

~

0.700 0.600

h

0.500 0.400

I

i;::ra nd~' I'-. nonlinear I

0.300 0.200

I

f-DO'j'-"

/

y "'\ \ J.V ~

j-

v

\

A j

~

s-·v ~

I

~

\

A J

v

\' 'l

0.100 0.000 0.00 0.50 1.00 1.60 2.00 2.50 3.00 3.li0 4.00 4.50 6.00 Time

Figure 10.6.

Response of nonlinear model Eq. (10.35) and linear model Eq. (10.37) to sine wave in inlet flowrate with two different amplitudes A = 0.05, 0.15.

The step respons e (up and down) for these models is shown in Figure 10.5 for steps of two differen t magnitu des. The response to a sine wave input of two differen t amplitu des is shown in Figure 10.6. Note that for small magnitu de step inputs and small amplitu de sine waves, the linear model is a good approxi mation of the nonline ar model; however, for the large amplitu de inputs the linear model predicti ons deviate somewh at from the nonline ar model behavior. Also note the distinct differences in nonline ar respons e for a step up and step down.

10.5

SUMMARY

Our discussi on in this chapter has been motivat ed by the fact that it is indeed the rare process that is absolut ely linear in behavio r. Unders tandabl y, the analysis of the dynami c behavio r of nonline ar systems is not as easily, or conveniently, carried out as the analysis of linear systems; but it is importa nt to know the various options available to us when we cannot ignore the inheren t nonlinearities in a process model.


327

We have identified four techniques by which nonlinear systems may be analyzed, passing relatively quickly over two of them because they are of limited practical application, and focusing mostly on the two most viable options - numerical analysis and model linearization. It has become clear t..hat with nonlinear analysis, it is not possible to have accuracy and generality at the same time, unlike the situation with linear analysis. When accuracy is the more important objective, the numerical approach is the appropriate choice. Apart from requiring computer assistance, the other drawback associated with this approach is that it makes available only simulation results for a specific set of parameters. The model linearization technique is the most popular, primarily because it allows the use of the convenient methods of linear analysis and it is quite general. It is generally perceived that this advantage outweighs the disadvantage of potentially poor accuracy, especially when analysis is required to cover a wide range of operating conditions. All in all, when the limitations of linearization are kept in proper perspective, it can prove to be a very powerful tool for analyzing the dynamic behavior of nonlinear systems. (This last statement takes on even greater significance in the next chapter where we consider system stability.) We must now draw a most important conclusion from the proceedings in this chapter, enabling us to justify why so much time is invested on linear systems analysis when real process systems are almost always nonlinear to some extent.

The objective of process control is the design of effective control systems that will hold the process conditions close to its desired steady-state value. Even though the system is inherently nonlinear, the influence of effective regulatory control is to ensure that deviations from this steady state will be small, in which case the behavior will be essentially indistinguishable from that of a linear system. It is in this sense that the process system is primarily considered to be approximately linear, and the linear methods we have studied in considerable detail are applicable. Before leaving the main subject of Process Dynamics, there remains one final issue to be discussed, that of system stability. This is the subject of the next chapter, the last chapter of Part II.

REFERENCES AND SUGGESTED FURTHER READING

1. 2.

3. 4. 5. 6.

Luyben, W. L., Process Modeling, Simulation, and Control for Chemical Engineers (2nd ed.), McGraw-Hill, New York(1991) Friedly, J. C., Dynamic Behavior of Processes, Prentice-Hall, Englewood Cliffs, NJ (1972} Longwell, E., "The Industrial Importance of Nonlinear Control," Chemical Process Control IV (Y. Arkun and W.H. Ray, Ed.), AIChE, (1991) Iooss, G. and D. D. Joseph, Elementary Stability and Bifurcation Theory, SpringerVerlag, New York (1981) Guckenheirner, J. and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York (1983) Golubitsky, M. and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York (1985)

1C :S PROCESS D YN A/ v£ Eq ua ti. -o ra :s, dinary Differential in the Theory of Or s od eth M l ica etr om 7. Arnold, V. I., Ge w::JI'.'\S,;' w York (1988) of No nli ne ar Syste Springer-Verlag, Ne al Transformations lob "G r, ye Me G. d Su, an 8. Hu nt, L. R., R. ntrol, 28, 24 (1983) York (1989) IEEE Trans. Auto Co ringer-Verlag, New l.e -a r Control Systems, Sp ar pr oa ch to No nl i.I ne Ap nli y No etr , A. om ri, Ge l tia ren 9. Isido ffe 1) Di 99 he (1 .), AIChE d Y. Ar ku n, "T n an d W.H. Ray, Ed 10. Kravaris, C. an s Control IV (Y. Arku ces Pro l ica em Ch " Control,

328

REVIEW QUESTI

ONS

d< > tfi ey systems, an d ho w behavior of linear the e riz cte ara ch es rti 1. What tw o prope nonlinear behavior? assist in recognizing ical pr O CE ss nlinearities in chem no of es urc so on of the most co mm 2. What are so me models? r, w ha t ar e ms dynamic behavio ste sy ar ne nli no of ted with the analysis 3. When confron consideration? proaches available for ap the some of ry pr ac tic al ? tion currently no t ve ma for ns tra ble ria earization by va pr oa ch to 4. W hy is exact lin of the numerical ap s and disadvantages ge nta va ad in ma of the 5. What are some analysis? na m ic nonlinear dynamic ach to nonlinear dy del linearization appro mo the of ge nta va in ad of this approach? 6. W ha t is the ma main disadvantage ab ou t analysis? What is the ximate in fe re nc es aw general bu t appro dr dr aw to to ble ble ssi ssi po it po s it make pr oa ch makes ap ich wh d 7. Which ap pr oa ch an m, nlinear syste the behavior of a no rate inferences? cu ac t bu c cifi on ly spe tio n (o f a of a nonlinear fu nc ylor series expansion Ta a ate nc ? tru ed to sir ry de ear approximation is 8. Why is it necessa the first term if a lin th eir single variable) after in na tur e, so th at iversally no nli ne ar on s un t ati os im alm ox pr are ap s r cesse ar, wh y are linea ne nli no 9. If chemical pro s ck ay ba ed alw t fe ls are almos ecially wh en mathematical mode nu mb er of cases, esp l tia tan bs su a in te fo un d to be ad eq ua control is involved?

PROBLEMS

10.1

ons of chemical on constitutive relati mm co me so of og ple catal cess models: The following is a sam y introduce into pro nlinear functions the no the d an g rin ee engin S: CHEMICAL KINETIC

rhenius expression: rate "constant": The Ar n ctio rea of e enc end (P IO .l) (a) Temperature dep k(T )

= k0e-EIRT

ants. wi th k0, E, R, as const ATE: EQUATIONS OF ST

halpy: ture dependence of ent (b)(Empirical) Tempera T4 3 + b2T 2 + b3T + b4 H(T) = b0 + b1T

/

(PI0.2)


329

with b(}l b1, b2, b3' b4 as constants. (c) Antoine's equation: Temperature dependence of vapor pressure: pO(D

=

e!A-M;T+C)]

(Pl0.3)

with A, B, Cas constants.

PHASE EQUILIBRIUM:

(d)Vapor-Iiquid equilibrium relation for binary mixture: y(x)

=

ax I + (a - 1 )x

(Pl0.4)

with a, the relative volatility, as a constant.

HEAT TRANSFER:

(e)Radiation l1eat transfer: q(T) = t:uATI

(PI0.5)

with e, a, A as constants. Linearize each of these five expressions, expressing your results in terms of deviation variables which you should clearly define. 10.2 The differential equation model for a first-order bilinear system: dY(t) dt

=

aY(t) + nY(t)U(t) + bU(t)

(Pl0.6)

differs from the differential equation model for the first-order system: dY(t) dt

aY(t) + bU(t)

(Pl0.7)

only by the bilinear nY(t)U(t) term. (a) Linearize the bilinear system model around the steady-state values, Y3 and US' define deviation variables, y = Y- Y3 and u =U- U3 , and obtain the approximate, transfer function relating y(s) to u(s). (b) Recast Eq. (P10.7) in terms of the deviation variables of part (a) and obtain the first-order system transfer function. Compare this transfer function with the approximate transfer function obtained for the linearized bilinear system model. What effect does the presence of the bilinear term have on the steady-state gain and the time constant of a first-order system? 10.3 The dynamic behavior of the liquid level in a certain cylindrical storage tank is modeled by: dh

A -dt = F.l - ch 113

(Pl0.8)

where A is the uniform cross-sectional area (m2); F; is the inlet flowrate (m3/min); h is the liquid level (m); and cis a constant. (a) Linearize this process model around a steady-state level h5 , and obtain an approximate transfer function model relating the deviation variables y =h - h5 and u = Fi- F; 5 , where Fis is the steady-state inlet flowrate corresponding to h5• Explicitly specify what the steady-state gain and the time constant of the approximate transfer functions are. (b) The specific tank with c =0.5 (m7 /3 /min), A= 0.25 m2, and whose total height is 1 m, had a steady-state liquid level of 0.512 m when operating at an initial steady-

330

PROCESS DYNAMIC S

state flowrate of 0.32 m3 /min. It is believed that if the flowrate were to increase suddenly from this initial value to a new value of 0.52 m3 /min, and remain there, the approximate, linearized model will mislead us into concluding that the liquid level will rise to a new steady-state level within the limits of the total tank height, and the tank will not overflow; the more accurate nonlinear model, on the other hand, predicts that this change will, in fact, cause the tank to overflow. Confirm or refute this statement. 10.4

A mathematical model for the level of gasoline in a hemispheri cal storage tank is given as: dh _ l l ( (PI 0.9) dt - tr (2Rh- h 2 ) F;- ch

112)

where R is the radius of the hemispherical tank, and f; is the flowrate of gasoline into the storage tank. Show that by introducing the following transformations: (PIO.IO)

and (PIO.Il)

the original nonlinear model equation is converted to the first-order linear differential equation: dz dt=a:r.+ bv (PIO.l2) Show explicitly what the constants a and bare. 10.5

Revisit Problem 10.4. Given a specific tank with parameters R =2m, and c = 1.5 (m 5 12 jmin.), operating at initial steady-stat e conditions : f;s = 1.35 m 3 /min, h6 =0.81 m; (a) Linearize the original mathematical model around this steady state, using a firstorder Taylor series expansion; obtain an approxima te transfer function model, and use this approximate model to obtain a response of the gasoline level to a step change in f; from the initial value to 1.8 m3 /min. Plot this response. (b) Using a numerical integration software of your choice, obtain a more accurate response of the gasoline level (to the same change in f; investigate d in part (a)) by solving the original modeling equation, incorporat ing the given parameters . Superimpo se a plot of this response on the approxima te response plot obtained in part (a) and comment on the "goodness" of the linearized, approxima te transfer function modeL

10.6 In Example 10.4, an approxima te linear first-order model was derived for the conical tank by linearizing the original nonlinear model. The following problem (Olson 1992)t shows an alternative method for doing the same thing. Starting from the original material balance equation: dV

dt = F;-chl/2

(PIO.I3)

given initial steady-stat e values V5 for the tank volume, h for the tank level, and Fis 5 for the inlet flowrate, define the following new variables:

v

= VIVS

+J. H. Olson, private communications, March 1992.


331 H = h/h 3

F; = F;fFis

respectively, the relative volume in the tank, the relative tank level, and the relative inlet flowrate. Show that wilh this definition, the material balance equation becomes: T S

dV = F. - H In. dt I

(PIO.l4)

=

with -r5 VJFis' the residence time of ~e tank at the current operating stea_dy stat~. Now introduce the deviation variable V = V- V5 = V -1, with the others, H, and F; similarly defined, and show that upon linearization, Eq. (Pl0.14) becomes: (PJO.l5)

an equation that is easier to use than Eqs. (10.26) and (10.33) since no constitutive relation is needed to find the time constant and the gain. 10.7 In Problem 6.10 the model for an experimental biotechnology process was given as: d~I

d/ d~2

dt

= ai~I

-

a12~1~2 + b)l

(P6.6a)

= -a2~2 + a2I~Ig2

(P6.6b)

where: ~ 1 is the mass concentration of microorganis m 1, M (microgram/l iter) 1 ~2 is the mass concentration of microorganis m 2, M 2 (microgram/l iter) Jl is the rate of introduction of M 1 into the pond (microgram/ hr) a 1, a12, a2, a21, b, are fixed parameters Linearize this nonlinear model around the reference values g• , g• , and J.L*, 1 2 defining the following deviation variables:

U=J.L-Jl* Show that the resulting linearized model is of the form given in Eqs. (P6.7a,b ): dxl

dt =

a 11 x 1

- a 12x 2 +

bu

(P6.7a)

dx2

d/

= a2Ixl - a21x1

(P6.7b)

az on,

Provide explicit expressions for the new parameters ~ 1 , ~ 2 , 1, and bin terms of the original model parameters and the reference values around which the model was linearized. 10.8

Perhaps the most popular nonlinear chemical engineering system is the nonisotherm al CSTR in which the single, irreversible reaction A -> B is taking place. The model for this reactor is given in Chapters 4 and 12 as: (4.26) (4.27)

332

PROCESS DYNAM ICS

where, as previou sly noted: cA is the concent ration of reactan t A in the reactor while Tis reactor temperature, 8 =V/F is the mean residenc e time in the reactor, k = k0e-(E.IRT) is the reaction rate constant, f3 = (-L1H)/pCP is a heat of reaction parameter, and X is the heat removed by the tubular cooling coil: X

=

_IM_(T -T) pC V c

(12.3)

p

with h as the coil heat transfer coefficient, A the total coil heat transfer area, and Tc is the average tempera ture along the length of the coil. Lineariz e this set of equatio ns around the steady states cAs and T • 5

10.9

A detailed mathematical model describing the behavio r of a jacketed CSTR used for the free-radical polyme rization of methylm ethacry late (MMA) using azo-bisisobuty ronitrile (AIBN) as initiator, and toluene as solvent, was present ed by Congalidis et al. (1989) t and has been used in several other studies. Recently, Doyle eta/. (1992), tt by making some simplify ing assump tions, and introdu cing the publish ed kinetic and other reactor parameters, reduced this reactor model to the following set of equations;

ds.

d(=

ds2

(P10.16) 60Jl- 3.03161~2

dt ds3

-

o.OOI34I06~ 1 % +

(PI0.17 ) o.o34677

~2 -

10,;3

(Pl0.18 ) (PI0.19)

along with: I)

=

S4

s3

(PI0.20 )

Here, ~ 1 , ~2 , ~3 , and ~4 respecti vely represe nt the dimens ionless monom er concentration, initiator concentration, bulk zeroth moment, and bulk first moment; 1J is the number average molecular weight; and Jl is the volume tric flowrate of the initiator . (a) Define the deviatio n variables: X;= ~i- ~;'"; i = 1, 2, 3, 4; u = Jl- Jl*; andy= 71-71* and linearize this set of modeling equations around the steady state indicated by ~t; i = 1, 2, 3, 4; J.('; and 71*. (b) Given the following specific steady-state operatin g conditio ns: ~1 * = 5.50677, '2. =.0.132906, ~3.. = 0.001975, ~/ = 49.3818, and Jl* =0.01678,

take Laplace transfor ms of the lineariz ed equatio ns and obtain a single transfer function that relates y(s) to u(s).

t J.P. Congalidis, J. R. Richards, and W. H. Ray, "Feedfo rward and Feedback Control of a Copolym erizatio n Reactor" AIChEJ, 35-891 (1989). tt F. J. Doyle, III, B. A. Ogunnaike, and R. K. Pearson, "Nonlinear Model Predictive Control Using Second Order Volterra Models," Preprints of the Annual A.I.Ch.E. meeting, Miami Beach, Florida, Nov. 1992.

CHA PTER

11 STABIL ITY A very important aspect of the dynamic behavior of a process that so far has not received any formal attention has to do with the following issue: when a (fmite) input change is implemented on a physical system, does the resulting transient response ultimately settle to a new steady state, or does it grow indefinitely? This issue did not arise in all our previous discussions because of a deh"berate attempt on our part to defer it until all other aspects of dynamic analysis have been discussed. If it is indeed possible for a process not to settle eventually to another steady state, but have its output grow indefinitely when disturbed from an initial steady state, what characteristics of the process determine whether or not such behavior will occur? This chapter is primarily concerned with investigating those issues having to do with the stability properties of a dynamic process as represented by its model.

11.1

INTRODUCTORY CONCEPTS OF STABILITY

When a process is "disturbed" from an initial steady state, say, by implementing a change in the input forcing function, it will, in general, respond in one of three ways: • Responsel: • Response 2: • Response 3:

Proceed to a steady state and remain there, or Fail to attain to steady-state conditions because its output grows indefinitely, or Fail to attain steady-state conditions because the process oscillates indefinitely with a constant amplitude.

These possible patterns of behavior are illustrated in Figure 11.1.

333

334

COMPUTER CONTROL

t: = ~

Steady State

y(t)

I

I

Steady State

Steady State

Figure 11.1.

The three possible responses of a dynamic system to a disturban ce: (a) returns to a steady state; (b) output grows indefinitely; (c) output oscillates indefinitely with a constant amplitude.

It now seems perfectly logical to consider the process that ultimate ly settles down after having been disturbe d as being Stable" and the one that fails to settle down as being "unstable." A classic illustration of this concept is provided by the .simple pendulum, consisting of a small bob of mass m connecte d to a fixed point pivot by a rod of length R, considered to be essentially rigid and of negligible mass. It can be shown (and we will do this later on) that there are two possible steady-s tate (equilibrium) positions for the pendulu m. These are shown in Figure 11.2(a), where the pivot is above the bob, so that the pendulu m is hanging straight down, and in Figure 11.3, where the bob is directly above the pivot. The nature of the pendulum is such that when disturbed from an equilibrium position, the bob will swing in a circular arc about the pivot Its position at any time during this period is indicated by 6, the angle of deviatio n from the straight- down equilibrium position. The equilibrium position represen ted in Figure 11.2(a) therefore correspo nds to 6 = 0, while the one in Figure 11.3 corresponds to 6 = 1t Now, if in Figure 11.2(a) the bob is displaced to the left by an angle 6, as shown in Figure 11.2(b), and then released, we find that this disturbance causes the pendulu m to swing back and forth for a while and eventually settle back to the initial equilibrium position. We will then consider Figure 11.2(a) to be a stable configuration for the pendulu m system; the point 9 = 0 represen ting a stable equilibrium position or a stable steady state. In contrast, for the configuration in Figure 11.3 where the bob is above the pivot, it is easy to see that any perturba tion in the position of the bob will cause a change in the pendulu m position is such a way that the initial equilibri um position is no longer attainable. This configur ation will then 0

CHAP 11 STABILITY

335 Pivot

a.

Figure 11.2.

b.

A stable configuration for the pendulum.

be considered as being unstablei that is, 6 =1r is an unstable steady state. Note that the upward pointing configuration is indeed a. steady state because all the forces are balanced when 6 = 1r exactly. However, this steady state is in fact unattainable by the system unless some external force is used to hold the pendulum in this position.

11.1.1 Definitions of Stability Although there are many slightly different mathematical definitions of the terms stable and unstable, we shall choose to use the concept of stability in the sense of Liapunov (see Refs. [1,2]). In this section we shall introduce several rather precise definitions that will aid in understandin g the concepts of stabilityi we conclude this section with a more restricted definition of stability whiCh is used most frequently in process control. ~or a given process steady state, we have the following definitions. Definition 1:

A system steady state x5 is said to be stable if for each possible region of radius e > 0 around the steady state, there is an initial state x 0 at t = t0 falling within a radius 6 > 0 around the steady state that causes the dynamic trajectory to stay within the region (x - x) < e for all times t > t0•

I

I

This is illustrated in Figure 11.4 where we see that Definition 1 simply says that for all possible regions e (large or small), there is a corresponding region of initial conditions bounded by 6 so that the dynamic system stays within e. This, of course, implies that 6 < E and that even for a very small region e near the steady state, there must exist an initial radius 6 > 0 that keeps the system within e of x5 •

Pivot

Figure 11.3.

An unstable configuration for the pendulum.

336

COMPUTER CONTROL

...~---··-··...

. :

. .

.

.' '' '' ~

~

\··--··. ·:::·~. . . ....... --

Figure 11.4..

Regions bounded byE and 6 in the definition of stability.

Definitio n 2: A system steady state x 5 is said to be asympto tically stable if it is both stable and in addition there exists a region of initial conditions of radius 60 > 0 around x5 for which the system approaches x as t -? ""· 5

Thus Definitio n 2 means that any steady state .that is asympto tically stable will return to the steady state :x, as t -+ oo after some initial perturba tion x 0• Usually for process dynamic s and control problem s there is no practical difference between steady states that are stable and asymptotically stable. Definitio n 3: A system steady state is said to be unstable

if it is not stable.

This definition needs no explanation. Note that these precise definitions refer to the steady states of the system rather than to the system itself. If we wish to discuss the stability of the system itself we may do so in some domain of the space of state variables x.

Definitio n 4: A dynamic system is said to be stable on some domain M if for every region of radius E in the domain there is a region of radius 6 in the domain such that the system trajectories are bounded by E if the initial condition is bounded by 6.

Note that this is just the definitio n of stability (Definition 1) but limited to some domain M. If we allow the domain M to be the whole space, then the system is said to be globally stable. This concept of system stability on a domain is importan t in interpret ing the behavior of nonlinea r systems where there may be multiple steady states or domains in the state space that are unstable. The precise definitio ns above are necessar y when we consider all the possibilities arising from nonlinea r systems; however , the simpler concept of Bounded -Input/B ounded- Qutput (BIBO) stability is often sufficient for linear systems:

CHAP 11 STABILITY

337

Defittition 5 (Bounded-Input/Bounded-Output Stability):

If in

response to a bounded input, the dynamic trajectory of a system remains bounded as t --+ oo, then the system is said to be stable; othenvise, it is said to be uttstable. For the study of linear systems in process control, it is normally the concept of BIBO stability that we shall use. Based on these definitions, we can now classify the types of behavior shown in Figure 11.1. Obviously, the steady state in Figure 11.1(a) is stable and probably asymptotically stable, while the steady state in Figure ll.l(b) is clearly unstable. The sustained oscillatory trajectory in Figure 11.1(c) implies that the steady state is unstable because it violates Definition 1; however, it satisfies Definition 5 for BffiO stability of the system. Thus, in this case the steady state is unstable while the system has BIBO stability. It is for situations such as this that we have been so careful in our definitions of stability. If we analyze the pendulum shown in Figures 11.2- 11.3, we see that the downward-poin ting configuration (95 = 0) is stable (Definition 1) and the upward-pointing configuration (95 = n) is unstable. Thus the system is stable on a domain M that excludes the unstable steady state (Defmition 4). However, the system remains bounded, so that it has BIBO stability.

11.1.2 Practical Issues in Process Dynamics and Control There are several practical questions concerning system stability that arise in process dynamics and control. Some of the most important are: • • • •

When is a linear system stable? When is a nonlinear system stable? Can a system that is unstable by itself (i.e., a "naturally unstable" system) be made stable by addition of a control system? Can a "naturally stable" system be made unstable by the addition of a control system?

We will apply the stability concepts just discussed in order to answer these questions in this chapter.

11.2

STABILITY OF LINEAR SYSTEMS

For the purpose of establishing conditions for the stability of a linear system, we shall consider the response to bounded inputs such as the unit step. We shall begin by dealing with systems without time delays because these are amenable to exact analytical treatment. Methods for analyzing the stability of systems having time delays are introduced in Chapters 15 and 17. Let us only state here that the open-loopt stability of a linear transfer function g*(s) is not affected by the presence of a time delay e-as. Thus the open-loop stability character of g*(s) is the same as that of g*(s)e-as.

t Here the term open-loop refers to a system without a feedback controller.

338

COMPUTER CONTRO L

Let us represe nt the linear system (withou t time delays) by the transfor mdomain transfer function model: y(s)

= g(s) u(s)

(11.1)

where g(s) is the system transfer function that consists of a qth-ord er numera tor polynom ial and a pth-ord er denomi nator polynom ial: b

sq

+ b _ 1 sq- 1 + ... + b 0

g(s) = aPsP+ aP_ 1 sP

1

+ ... +a 0

_

M

-

A(s)

(11.2)

Recall that the polynom ials in g(s) can be factored into the form: (11.3)

so that zk, k =1, 2, ... q the roots of the numera tor polynom ial, B(s), are the zeros of g(s). Similarly, r;, i = 1, 2, ... p, the roots of the denomi nator polynom ial, A(s), are the poles of g(s). To illustrate the respons e to a bounde d input, conside r the unit step respons e given by: 1

y(s) "' g(s)-

(11.4)

s

Observe that upon partial fraction expansi on in the usual manner , Eq. (11.4) may be rewritte n as: y(s) = A -

0

s

+

A. IP [ --' ; = 1 (s- r;)

J

(11.5)

where the r;'s are the p system poles. Recall that the ri can be either real or complex number s, r; = 7i + jTt;, where 7i is the real part and 71; is the imagina ry part of the ith system pole. Laplace inversio n here immedi ately yields: (11.6)

We now note that the stabilit y charact eristics of this respons e is determi ned entirely by the p poles, r 1, r 2 , ••• rp: the respons e will remain bounde d if and only if all these poles have negative real parts (}I< 0). If just one of these poles has a positive real part, the exponen tial terms associat ed with it will grow indefini tely and the respons e become s unboun ded. The special case when some of the Yi are exactly equal to zero is a singular ity that is unlikely to occur for a real process. Thus these are only of interest as the point of transitio n from stability to instabil ity. We therefor e have the followi ng stability test:

CHAP 11 STABILI TY

Figure 11.5.

339

The complex plane showing the location of system poles (x) and system zeros (0).

A linear system is stable if and only if all its poles have negative real parts so that they lie in the left half of the complex plane; otherwis e it is unstable.

This is a very powerfu l result because it clearly states that the stability of a linear system is depend ent only on its poles; zeros do not affect system stability. Thus the issue of determi ning the stability character of a linear system is no more than that of determining the nature of its poles; it is not at all necessa ry to determine the details of the dynamic process response. Figure 11.5 shows the complex plane with some poles and zeros plotted. Note that from the stability test above, if any pole is in the right-half plane, the system is unstable . Let us illustrate this with an example. Example 11.1

STABILITY OF TWO LINEAR SYSTEMS: THE EFFECT OF POLES AND ZEROS.

Investigate the stability properties of the systems whose transfer functions are given as: 5

(11.7)

(11.8)

where the only difference between System 1 and System 2 is that the latter has a zero while the former has none; they both have identical denominator polynom ials, i.e., identical poles. Solution : The denominator polynomials for each of these systems can be factored to obtain the roots, in which case the transfer functions can be represented as: gl(s)

=

5 (s + 3)(s- 2)

340

COMPUTER CONTROL

5(s + 1) gz(s) = (s + 3)(s- 2)

Observe that both systems have identical poles: two each, located at s = -3 and s = +2. According to the stability result given above, therefore, both systems are unstable by virtue of the pole at s = +2. The zero possessed by System 2 exerts absolutely no influence as to whether it is stable or not. Let us confirm this further by obtaining each system's unit step response. For System 1, this is obtained from: _ _5___ ! (s + 3)(s - 2) s while for System 2, it is obtained from:

Y2(s)

5(s + I) 1 = (s + 3)(s- 2)-;

Partial fraction expansion of each of these expressions now gives: Y1

-516 1/3 1/2 ( ) s = -s- + s + 3 + s - 2

and -516 - - +-2/3 - - + 3/2 -s s+3 s-2

which upon inversion become:

and

It is important to observe the following regarding these responses:

• They each have the same set of exponential functions, eI. • The constant coefficients of each of the corresponding exponential functions are different for each response. Now, each of these responses grows indefinitely with time by virtue of the e21 terms, and therefore both systems are in fact unstable, as we had earlier concluded. Thus, the answer to the question "when is a linear system stable?" is: when all the transfer function poles (the roots of the denominator polynomial) have negative real parts; i.e., when they lie in the left half of the complex s-plane. If the linear system in Eq. (11.2) were expressed in terms of a pth-order differential equation model of the type: ~ tJP-Iy cf:t_ aP di' + ap-1 dt!'-1 + ... + a2 dfl +

tfJ:_ al

dt + aoy = f[u(t)]

(11.9)

CHAP 11 STABIUTY

341

where f[u(t)] is some forcing fnnction (which could involve derivatives of u(t)), then it is easy to show (d. Appendix B) that the solution of Eq. (11.9) is of the same form as Eq. (11.6): (11.10)

where the p characteristic values, m1, are found by using the coefficients of the derivatives in Eq. (11.9) to form the characteristic equation: aP m p + aP _ 1 m p-1

+ . . . + a 2 m 2 + a 1 m + a 0 -- 0

(11.11)

Thus for models in the form of Eq. (11.9) we have the following stability result:

A linear system in the form of Eq. (11.9) is stable if and only if all its characteristic values m1 have negative real parts so that they lie in the left half of the complex plane. Recall from Chapter 4 that we could also represent the transform-domai n transfer function model Eqs. (11.1) and (11.2) or the high-order equation (11.9) in the state-space: (11.12)

where x is a p vector of states defined by:

;,_I =

x,

• = -;; l ( -aP _ x, -aP _ xP _ x, 1 2 1 p

withx 1 = y (Here we consider f = Ku(t) in Eq. (11.9) for simplicity.) In this case the quantities A, b, c are given by:

A

0 0

1 0

0

0

0

0

-<Jo

-al

a,

a,

0

342

COMPUTER CONTROL

0 0 b K

1.67 the process become s unstabl e. This exampl e clearly shows that it is possible for this process to becom e unstabl e as a result of overly aggress ive control action (i.e., by having the control ler gain too large). We note, finally, that just as it is possible for an inheren tly stable system to be made unstab le by feedba ck control , it is also possibl e to stabiliz e an inheren tly unstabl e system by feedback control. We will discuss these issues further in Chapte r 17.

11.6

SUMM ARY

We have focused attentio n in this chapter on the issue of system stability, and we have provid ed method s for determ ining the stabilit y charac ter of both linear and nonline ar systems. For a linear system to be stable, we have found that the poles of the transfe r functio n must all lie in the left half of the complex plane. This means that an open-lo op-uns table system is one whose transfe r functio n possess es at least one RHP pole. Equiva lent time-do main criteria for linear system stabilit y were presen ted with the result that the eigenva lues of the system coefficient matrix must lie in the left half of the comple x plane. Shortc ut method s for determ ining the locatio n of the poles or eigenv alues of the system from the polyno mial charact eristic equatio n are presen ted in Chapte r 14. It comes as a pleasan t surpris e to find that we can establis h the stabilit y of the steady states of a nonline ar system by investi gating the behavi or of the linearized model in the vicinity of the steady state. A few exampl es were used to illustra te the proced ure for carryin g out nonline ar stabilit y analysis. We have also investi gated the dynam ic behavi or of inheren tly unstab le system s, pointin g out that they exhibit two kinds of unstab le respon ses: expone ntial (nonos cillator y) instabi lity, when the RHP poles are real, and oscillat ory instabi lity. when the RHP poles occur as comple x conjugates. The possibi lity of destabi lization of a process by feedba ck control was briefly illustra ted, awaitin g full treatme nt in Chapte rs 14 and 15. REFERENCES AND SUGGE STED FURTHER READI NG

Hahn, W., Theory and Application of Liapunov's Direct Method, Prentice -Hall, New York (1963) 2. Hahn, W., Stability of Motion, Springer-Verlag, New York (1967) 3. Iooss, G. and D. D. Joseph, Elementary Stability and Bifurcat ion Theory, SpringerVerlag, New York (1981) 4. Guckenheimer, J. and P. Holmes, Nonlinear Oscillations, Dynami cal Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York (1983) 5. Coughanowr, D. R. and L. B. Koppel, Process Systems Analysis and Control, McGrawHill, New York (1965) (2nd ed., D. R. Coughanowr, McGraw-Hill, New York (1991)) 1.

356

COMPUTER CONTROL

REVIEW QUESTIONS 1.

When a process is "'disturbed" from an initial steady state, what are the various ways in which it can respond?

2.

Why is it necessary to distinguish between the stability of a system steady state and the stability of the system itself?

3.

What is the difference between the stability definitions 1, 2, and 4 given in the main text?

4.

When is a system considered globally stable?

5.

Define "Bounded-Input-Bounde d-Output" stability. Is this definition applicable to all systems?

6.

What are some of the practical issues concerni.. 0 for which the system represented by g+(s) will be unstable. (b) Show also that for g(s) given as: 6 g(s) = (2s + I){ 4s + 1) (Pll.9) once again, there is no value of K > 0 for which the system represented by g+(s) will be unstable. ·

CHAP 11 STABILI TY

359

(c) When g(s) is given as: 6 g(s) = (2s + 1)(4s + 1)(6s + I)

(Pil.lO)

determin e the stability of g+(s) forK= 1 and K = 2. You may find it useful to know the following result:

The cubic equation: will have no roots in the right half of the complex pl11ne a 0, a!' a 2, a 3 ala2

11.9

if and only if:

> 0; and >

aaa3

(a) When an open-loop-unstable system with the transfer function:

4.5

g(s) = (3s - I)

(Pll.ll)

is under proportional-only feedback contrbl, it is easy to show (see Chapters 14, 15, and 17) that the transfer function for the resulting "closed-loop" system is given by: (Pil.12)

where Kc is the proporti onal controlle r gain. Evaluate KcL(s) in this case and determin e the conditions that Kc must satisfy so that the closed-loop system is now stable; show that this condition is entirely independent of the location of the openloop-unstable system's right-half plane pole. (b) Consider now the case that the open-loop-unstable system is given by: g () s =

4.5 (3s- l)(-r2s +I)

(P11.13)

which differs from the transfer function given in Eq. (Pll.ll) only by the presence of the addition al pole: show that under proportio nal-only control, the conditio ns for closed-loop stability in this case now depend on the location of the addition al lefthalf plane pole relative to the location of the unstable pole in the right half of the

complex plane.

part Ill MOD ELliN G AND liDJEN1rliJFliCA TliON

JPROCJE§§

Three organically related elements are essential for effective control systems analysis and design: the inputs to the process, the process model as an abstraction of the process itself, and the outputs from the process. Once any two have been specified, the third element may then in principle be derived. When the input is specified along with the model, and the resulting process output is to be derived, this is the Process Dynamics problem, a rather comprehensive treatment of which has just been completed in Part II for various processes represented by diverse models. When the desired process output is specified along with the process model, and the input required to produce such an output is to be derived, this is the Process Control problem; it will be discussed in Part IV. Deriving the process model from process input and output information (and perhaps some knowledge of the fundamental laws governing process operation) is the Process Modeling problem, quite often the most important (and most difficult) problem of all; it will be the focus here in Part mconsisting of Chapters 12 and 13. Chapter 12 is devoted to the development of theoretical process models from first principles, using material and energy balances (and fundamental physical/che mical laws) to determine the model equation structure, and estimating unknown model parameters from the process input and output data. Chapter 13 is devoted to Process Identification: the alternative approach of developing purely empirical models, in which the model form, and all the model parameters, are obtained strictly from input and output data acquired from the operating process. The relative merits of each approach will be discussed in the context of examples illustrating model development for distillation columns, chemical reactors, etc.

part II I PROCE§§ MO DEl liNG ANI D liDE NTK FKC ATli ON CHAPTER12.

Theoretical Process Modelin g

CHAPTER13.

Process Identification: Empirical Process Modelin g

tfie Science of 'Detfuctinn aruf Jllnafysis is one wft.icli can on{y 6e acquiretf 6y fong aruf patient stuay, nor is {ije fong enougli to allow any mortal to attain tft.e ft.igliest possi6k perfection in it.' K•••

Sherlock Holmes, A Study in Scarlet (Sir Arthur Conan Doyle)

'Jllristotk couU ft.ave avoitfetf tfie mistakl of tliinf;jng tft.at women ft.atf fewer tutft. tlian nun 6y tft.e simpf'e tfevice of asking Mrs Jllristotk to open fier moutft.' Bertrand Russell (1872-1970)

CH AP TE R

12 THEORETICAL PROCESS MOD ELI NG In Chapte r 4, we had our first contact with the process model primari ly as a

tool of dynami c analysis. There the -various forms of process models were present ed and the interrelationships between them were discussed. In the rest of the book, the model is used as a tool for process analysis and for the design of control systems. The question of how one obtains good process models will now be dealt with in this chapter. In this regard, there are three fundam ental issues that will occupy our attention: model formulation, parame ter estimation, and model validation. Regardi ng model formulation, the principl es involve d will be illustrat ed by several examples coverin g a spectru m of chemical processes; regardin g the other two issues, even though the scope of coverage will be limited by space, we will examine enough of the principles involved to enable the reader to obtain a flavor for what is required in estimating the unknow n parame ters that show up in a process model, and how to check the adequacy of process models before they are used for process control or other applications.

12.1

INTRODUCfiON

Inasmuch as entire books have been written on process modeling (cf. Refs. [1-3]), our objective in this chapter is not to present a compre hensive treatme nt of theoretical modeling. Further, since mathematical modelin g is recogni zed as a curious mixture of "art" and science, the subjective and the objective , neither is it our objective to present a rigid, cookbook procedu re for developing theoretical models. Instead , our aim is to present first the philoso phy and principl es involved in modelin g the behavior of process systems, and then to illustrat e the application of these principles with several examples. Process Modeli ng is not a rigid science that can be practice d in only one "right" way, but neither is it amorph ous and unstruc tured: there is some technique involve d, and the procedu re, even if not rigid, must be systema tic if the modelin g exercise is to be carried out successfully and efficiently.

363

NTIFICATION PROCESS MODELING AND IDE

364

of Dy 12.1.1 Mo del ing the Behavior

namic Processes ma y be

objective dynamic processes, the ma in In mo del ing the beh avi or of n the various wee bet ship tion of obtaining a functional rela

red uce d to tha t behavior. The lain s the obs erv ed process pro ces s var iab les wh ich exp al relationship, - but unk now n - fundament les, and our und erly ing belief is tha t a true iab var s sts bet wee n the proces dic tate d by nat ura l laws, exi . ship tion rela this towards discovering to por tray modeling exercise is directed ible oss imp is it t tha set the out al form. It is also und ers too d righ t at atic hem mat actual process behavior in del ing mo perfectly aU the details of the al ent dam fun a seeks to arrive at g elin mod cal reti theo re, most refo the The ure to application of the laws of nat The s. ces relationship thro ugh systematic pro the to determine the behavior of important phenomena assumed cal process model. end result is known as a theoreti this unk now n l modeling seeks to approximate irica emp On the oth er han d, al functions, using e (usually simple) mathematic functional relationship by som ause they dep end Bec entally from the system. information gat her ed experim app rox ima ting the , nce erie rma tion and exp ent irel y on exp erim ent al info . ed are known as empirical models mathematical functions obtain roa ch wh en the app ing del mo l tica ore the It is typ ica l to ado pt the reasonably well wh ich a process operates are by s nism cha me ing erly und retical approach s is too complicated for the theo und erst ood . When the proces al nat ure of the ent rmation about the fundam (usually because very little info eno rmo usl y are ns atio theoretical mo del equ ire subject pro ces s is available, or the ent The ice. cho e riat ch is the approp (with no complex) the empirical approa tion rma from pur ely experimental info Process as of constructing a process model wn kno is r) es for the observed behavio recourse to any theoretical bas Cha pte r 13. Identification; it is the subject of to be mutually s to process modeling appear che roa app two e Although thes be completed not can tha t theoretical modeling tha t some ind epe nde nt, we will soon see find also l wil n; in Cha pte r 13, we atio ent erim exp e som t hou wit to be successful. required for empirical modeling theoretical considerations are

nt al Process Mo del De vel op me 12.1.2 Procedure for Theoretic a process model and ultimately abstract from it stages: To star t from a physical system eral sev of is a process that consists that can be use d as its surrogate

Stage 1:

Problem Definition

e defined very make it very important to hav There are several factors tha t these are listed m we wis h to solve; some of dea rly the scope of the proble below: cess; we can nt all aspects of the physical pro 1. It is impossible to represe t to the aspects tha t are most rele van onl y hop e to cap ture those problem at hand. mathematically ic process can be interpreted 2. The behavior of a dynam ed to varying lain various phenomena exp in several different ways, the mo del s are nt ere diff l is tha t severa deg ree s of detail: the res ult

S MODELING CHAP 12 THEORETICAL PROCES

365

mp t to , all of which mig ht eve n atte possible for any given process and to les, ang ous vari from but ess, cap ture the same aspect of the proc varying degrees of complexity. obtaining ful as the tools available for 3. A process mo del is as use solved be e modeling equ atio ns can solu tion s to its equations; som hods met al eric num be solv ed by analytically, while some can only using a computer. pin g a on the actu al task of dev elo As a resu lt, before emb ark ing ch are whi ns stio que l process, a num ber of mathematical mo del for a physica ed: wer ans be first ano the r- should by no means independent of one model for? 1. What do we intend to use the the model have to be? will , plex 2. How simple, or com t and do we consider the mo st relevan 3. Which aspects of the process such a process model? therefore should be contained in the ent al prin cip les .un der lyin g dam fun the 4. To wh at ext ent are wn? ess kno operation of this aspect of the proc model? of cy qua ade the test we can 5. How eling exercise? mod the for e hav 6. How much time do we

the

e, whether will enable us decide, for exampl The ans wer to these questions h. roac or the alternative empirical app to use the theoretical approach

Stage 2:

Model Formulation

Stage 3:

Parameter Estimation

briefly retical mo del dev elo pm ent was Wh at this stag e entails for theo detail e mor in ed uss pter 4 and will be disc ation) tific introduced in Section 4.2 of Cha Iden cess Pro (or t men model develop below. This stage in empirical will be discussed in Chapter 13. retical or sical process (wh ethe r by theo In dev elop ing a model for a phy specified be st mu es ters app ear who se valu e, the empirical means), certain parame mpl exa For r. avio beh to predict process and 6) before the mo del can be use d (4.2 . the non isot her mal CSTR in Eqs theoretical mo del obtained for (4.27}: (4.21i) (4.27) ers: contains the following paramet 8, the reactor residence time constant k = k 0 e- (E/RTJ, the reaction rate ided by the up of the hea t of reaction div e mad o rati a , P= (-ti H}/ pCP y of the reactor content. density and specific hea t capacit

366

PROCESS MODEUNG AND IDENTIFIC ATION

For any process mod el a subs et of the para mete rs may be know n a priori, avail able in the liter ature , or estim ated from inde pend ent expe rime nts perfo rmed , for exam ple, on a pilot plan t, or in the laboratory; the rema ining unkn own para mete rs must be determined throu gh other means. It is customary to obta in such para mete r estimates using expe rime ntal data obtai ned directly from the physical process.

Stage 4:

Model Validation

Before proc eedin g to use the model, it is essential to evalu ate how closely it predicts the beha vior of the physical syste m it is supp osed to represent. This is usua lly accomplished by testing the mod el again st addi tiona l process data in the context of our process experience. Let us now consider, in order, the issues invo lved with Model Formulation, Para mete r Estim ation , and Mod el Vali datio n for the theor etica l mod eling appr oach ; a similar discussion for the empi rical appr oach will be prese nted in Chap ter 13.

12.2

DEVELOPMENT OF THEORETICAL PROCES S MODELS

The detai led steps in Theoretical Process Mod eling are illus trate d in Figu re 12.1. The phys ical laws gove rning the proc ess are anal ysed to prod uce mod eling equations, whic h are usually comp lex and nonlinear, and may involve parti al differential or integ ral equations. This mod el is usua lly simplified to a form that is easil y hand led, but that still retai ns the essen tial featu res of the process. At this poin t the equa tion struc ture of the mod el is fixed. Next, those mod el para mete rs that are completely unkn own or not know n precisely enou gh are estim ated using process data taken from dynamic inpu t/ outp ut experiments perfo rmed on the actua l process. As indic ated in the figure, vario us type s of errors are intro duce d at each stage of the mod el deve lopm ent 1. Model Formultztion Errors: In postu latin g the phys ical phen omen a and appl ying their laws , appr oxim ation s are mad e to the mod el equations that intro duce errors. 2. Model Simplification Errors: In simp lifyi ng the mod el to mak e it math emat icall y and num erica lly tractable, appr oxim ation s are mad e that intro duce errors. 3. Experimental Measurement Errors: In collecting the proc ess data, both systematic cahb ratio n error s and rand om experimental error s occur. In addi tion, there can be instr umen t or hum an errors resulting in erroneous or miss ing data. 4. Parameter Estimation Errors: In addi tion to the effects of expe rime ntal meas urem ent error s on estimated parameters , there are othe r sources of para mete r estim ation errors. For exam ple, inad equa te expe rime ntal desig n may prod uce data sets that intrin sical ly lack the essen tial infor mati on nece ssary for obta ining reaso nable para mete r estimates. Choo sing a para mete r estim ation algo rithm that is inap prop riate for the data set may also resul t in poor para mete r estimates.

:Mo del

: Formulation

------------------

Model

---~

Formulation

~

I

.....

N

:1 tT1

0

~ :::'!

,_ -- --

s

• Para mete r : Estim ation

'1:1

:;.:,

R

Ci

\1:)

~

0

g ~ G')

'

' '-

'

Model

Validation

'----------------------------Figu re 12.1.

-·~---------------------------

Theoretical process modeling.

----·' c... 0\

',J

368

PROCESS MODELING AND IDENTIFICATION

Primarily as a result of all these errors, the final result - a theoretical process model - will be an imperfect representat ion of the process; this provisional model must therefore be evaluated against process data to determine its adequacy in a final model validation step.

12.3

EXAMPLES OF THEORETICAL MODEL FORMULATION

It is necessary to reiterate that the objective here is not to be exhaustive, but merely to present illustrative examples we believe are representat ive of typical chemical processes.

12.3.1 Lumped Paramete r Systems In Chapter 4 several examples of lumped parameter processes whose variables are essentially uniform throughout the entire system were introduced. The stirred heating tank and the nonisothermal CSTR considered in Section 4.2.4 of Chapter 4 are typical examples. In this section we will revisit the nonisotherm al CSTR in which the exothermic reaction A-+B is taking place. We will use this process as an example of how one carries out all aspects of theoretical process modeling. The energy and material balances for the reactor (shown in Figure 12.2) were formulated in Chapter 4 and resulted in the nonlinear differential equation model of Eqs. (4.26) and (4.27): dcA. -

!

,.(E/R1)

dt- -flcA.-koe

!

(12.1)

cA.+(JcA.f

ar 1 1 dt = -6 T + f3ko e-{Eitn)cA + 6 Tr X

(12.2)

Recall that cA is the concentrati on of reactant A in the reactor, while Tis reactor temperatur e, 6 =VI F is the mean residence time in the reactor, k = kcf-IE!R1) is the reaction rate constant, f3 = (-!J.H)/ pC, is a heat of reaction parameter, and is the heat removed by the tubular cooling coil:

z

X=

hA

-

pCp V(T-Tc)

(12.3)

T

Figure 12.2.

The nonisothermal CSTR.

CHAP 12 THEORETICAL PROCESS MODEUNG

369

where h is the coil heat transfer coefficient, A the total coil heat transfer area, and: (12.4)

the average temperature along the length of the coil 0 < z < L. It is known (Aris [4]) that X depends nonlinearly on the coolant flowrate q, according to: (12.5)

where T cf is the coolant inlet temperature, and p,Cpc is the volumetric heat capacity of the coolant. In addition, the parameters U, a are given by:

U = p,CP, pC, V

a =hA -P, c,,

(12.6)

(12.7)

Observe that our model has five parameters, {J, U, lea- E, a; two of these (fJ and U) are typically available in the literature; and the other three are to be determined. In addition there are five variables cAf, 6, T1 , Tcf, q, determined by the reactor operating conditions. In subsequent sections of this chapter, we shall show how the unknown parameters are estimated and how the final CSTR model is validated against process data.

12.3.2

Stagewise Processes

There are many processes in chemical engineering that are carried out in stages: gas absorption, extraction, and distillation, for example, are carried out on several individual plates enclosed in a column, or alternately, carried out in packed columns with "'equivalent stages" equal to a specified length of packing in the column. Leaching tanks, and multistage chemical reactor trains, are also examples of stagewise processes. These processes are characterized by the occurrence of stepwise changes in the value taken by the process variables from stage to stage. We will illustrate how such processes are modeled, and show the typical form their theoretical models take, using an example of the binary distillation column - one of the most important chemical processes. Following the distillation column model proposed by Luyben [5], let us consider the schematic diagram of a typical distillation column, shown in Figure 123, where there is a total of N trays, numbered from the bottom to the top. The distillation column is used to separate a binary mixture of two components, A and B, the former being the lighter component. The feed, a saturated liquid containing Xp mole fraction of component A, is fed to the column at molar flowrate F (mole/time); the point of entry into the column is tray f, the feed tray. The overhead vapor is condensed by the indicated heat exchanger arrangement, using cooling water; the condensed liquid flows into the reflux

370

PROCESS MODEU NG AND IDENTIFICATION

Top Section

Feed Section

Bottom Section

Figure 12.3.

The binary distillation colUllUl (adapted from Ref. [5]).

receiver from where it is returned as reflux to the column, at molar flowrate R; and overhea d distillat e produc t is withdra wn at molar flowrate D. The content of the reflux receiver is assumed to have a uniform composition Xw and the liquid holdup is Hn· Note that the composition of the distillate produc t stream and the reflux stream will be the same as Xn· The vapor flowing through the column at the molar rate V (mole/time) is generated at the base of the column through the steam-heated, thermosiphon reboiler arrangement. Bottoms liquid, of composition Xs, is withdra wn as produc t at the molar flowrate B, while i:he rest is sent through the reboiler. The liquid holdup at the bottom of the column is Hs-

Vapor in

from Platen-!

Figure 12.4.

L 0 z0

Platen of a binary distillation column.

CHAP 12 THEOR ETICAL PROCESS MODEL ING

371

The activities taking place on a typical tray n are shown in Figure 12.4. At each step a vapor flowing upward is brough t into contact with the liquid at that stage and achieves thermodynamic equilibrium. The liquid from the nth stage flows downw ard to the next lowest stage in the column. Here: x. mole fraction of material A in the liquid phase y. = mole fraction of material A in the vapor phase L. liquid flowrate out of platen V n = vapor flowrate out of plate n

If the liquid hold up (total volume of accumu lated liquid) on plate n is Hn, and the vapor holdup is h., assumi ng there are no heat effects, a theoretical model for this process may now be obtaine d by taking a compo nent A mass balance on tray n as follows: Compo nent A Mass Balance Rate of materia l input with liquid coming in from plate n + I Rate of materia l input with vapor coming in from platen - I Rate of materia l output with liquid leaving plate,n . (for plate n- I) Rate of materia l output with vapor leaving plate n (for plate n + I)

v.Y.

Rate of accumulation of A in the liquid phase on plate n Rate of accumulation of A in the vapor phase on platen The materia l balance equations for the typical nth tray are therefo re given by:

dh.

dt ;: ;

V. _ 1 -V.

(12.10)

These form the basic buildin g blocks for modeli ng the entire distillation colurrm compo sed of several such stages.

372


Overall Problem Definition and Assumptions

In constructing a model for the complete multistage distillation column we must represent the dynamic behavior of the mole fraction of the lighter component A on each tray, as well as in the overhead and bottom product streams. (Frequently, it is the latter two compositions that are really of importance, but as we shall see, these depend on the compositions of the other trays.) Thus, the complete theoretical model for this column will involve overall balance equations written for each of theN trays, as well as for the condenser/receiver, and the reboiler. It is now convenient to divide the column up into distinct sections, each with its own distinguishing characteristics as follows: 1. The ccndenser/receiver, 2. The top tray (Tray N), different from all the other trays because it is the point of entry of the reflux stream), 3. An arbitrary tray n representative of all the other trays like it, 4. The feed tray (Tray f>, different from the others because it is the point of entry of the feed stream, 5. The first tray, 6. The reboiler and column base.

The following assumptions are typically used to simplify the task of formulating the balance equations for each section of the distillation column: 1. The liquid on each tray is perfectly mixed and of uniform composition xn. 2. There are negligible heat losses from the column to the atmosphere. 3. The vapor holdup hn on each tray is negligible. 4. The molal heat of vaporization of components A and B are approximately equal, with the implication that for each mole of vapor that condenses, exactly enough heat required to vaporize one mole of liquid is released (this is the equimolal overflow assumption). 5. The relative volatility a is assumed constant throughout the column; this implies that the effect of temperature on the vapor pressures of A and B is approximately the same. 6. Plate efficiencies of 100% are assumed, implying that the vapor leaving each tray is in equilibrium with the liquid accumulated on the tray. (This means that a column with lower plate efficiencies is modeled by fewer trays at 100% efficiency.) 7. The dynamics of the reboiler and condenser heat exchangers may be ignored.

Assumption 1 allows us to carry out overall balances around each tray; assumptions 2, 3, and 4 imply that the vapor rate through the column is the same from tray to tray, i.e.: V = v.; n = I, 2, ... , N and also that the temperature on each tray is determined from thermodynamic equilibrium, making an energy balance unnecessary.

CHAP 12 THEORETICAL PROCESS MODEUNG

373

Assumptions 5 and 6 provide us with one of the constitutive relations needed for this system: for each tray, the mole fraction of A in the vapor phase is related to the liquid phase mole fraction by: Yn

= I +(a- l)x.

(12.ll)

Assumption 7 allows us to treat the heat exchangers used as the condenser and the reboiler as units which perform their tasks instantaneously. This is obviously not entirely realistic, but neither is it completely unreasonable: the dynamic responses of these heat exchangers occur on a much faster timescale than the response of most industrial distillation columns. Nevertheless, we recognize that these heat exchangers are processing units in their own rights, and the dynamic models for such units will be obtained in the next section. We are now in a position to obtain the mass balance equations that constitute the theoretical model for this system, starting from the top and working our way to the bottom of the distillation column. CONDENSER AND REFLUX RECEIVER

• Component A Mass Balance: (12.12)

• Total Mass Balance: dHv

dt = V-R-D

(12.13)

TOP TRAY (TRAY N)


• Total Mass Balance: (12.15)

ARBITRARY TRAY n (n = 2, 3, ... , N -I; n ¢f)

• Component A Mass Balance:



374

FEED TRAY



FIRST TRAY



COLUMN BASE AND REBOILER • Component A Mass Balance: (12.22)

• Total Mass Balance: (12.23) Consti tutive Relati ons the mole fraction of A For each of the compo nent balance equations given above, fractio n by using mole· phase liquid the to related be can in the vapor phase Eq. (12.11). the colum n base With the exception of the holdu p in the reflux receiver and relate the rate to le (which are usuall y kept under feedback control) it is possib Francis weir the ushig tray the on of liquid flow from each tray to the holdup formula: (12.24) of the liquid above the where l is the length of the weir (ft); 6n is the height try, and the (average) geome tray the of edge knowl a weir (ft) on tray n. From

CHAP 12 THEORETICAL PROCESS MODELING

375

to relate the density of material on each tray, this expression could be used liquid holdup H,. and flowrate L... in one of the The equations may now be made more compact and rearranged l balances materia total the uting Substit 4. r Chapte of forms standa rd model s gives balance mass nent compo {12.13, 12.15, 1217, 12.19, 12.21, 12.23) into the the equations: (12.25)

(12.26)

(12.27)

n

= 2, 3, ..., N- 1; n *f (12.28)

(12.29)

(12.30)

liquid phase Note that we have (N + 2) differential equatio ns for the the liquid for ns equatio ntial differe nal additio compositions, and (N + 2) phase vapor The d. couple and ar nonline all are ns equatio the and holdups; (12.11), Eq. relation rium equilib the compositions can be eliminated by using s, H ,., by while the liquid flowrates Ln can be related to liquid holdup

Eq. (12.24).

resulting The manipulated (control) variables are reboiler vapor rate V, tively alterna (or rateR reflux the and r, reboile the to from adjusting the steam e F, and feed the distillate produc t flowrate D). The disturbances are feedrat composition of composition x1 . The measured output variables are typically the model would ar nonline the Thus • x xDt ts, produc s 8 bottom and the overhead take the form of Eq. (4.31): dx(t)

dt

= f(x(t), u(t), d(t); 9)

y(t) = h(:x(t))

s of all liquid where the state vector x is of dimension 2N + 4 and consist dimension 2, of is y vector output compositions and liquid holdups, while the e vector variabl lated manipu the and 2, ion dimens has the disturbance vector d (12.31): Eq. in ly explicit seen is This 2. ion dimens u also has

376


xa XI

x2

XI u

XN Xo x=

HB HI

H2

= [~]

d =

[~]

e = [~]

(12.31)

y = [:;]

HI HN Hv Observe that if there are 50 trays in the column, the model would consist of 104 nonlinear differential equations even though we have only two outputs and two control variables. In Chapter 13 we demonstrate that in some cases, simple empirical models often do a fine job of representing this two-input/two-o utput system with much less work. In conclusion, we note that the situation with most industrial distillation columns is more complicated: the feed mixtures are usually multicomponent; the trays are not 100% efficient; the vapor holdups on the trays are not negligible; there are multiple feed entry points and multiple sidestream draw-offs; and the condenser and reboiler heat exchanger dynamics may be important. However, the basic principles involved in developing the more complicated model (which takes all these factors into consideration) are exactly the same as those illustrated with the relatively simpler situation we have just dealt with. Such a complex theoretical distillation column model has been derived in Ref. [5].

12.3.3

Distributed Parameter Systems

As we have seen, the process variables of a lumped parameter system can be assumed to be homogeneous within the process, so that spatial variations can be considered as negligible. The resulting theoretical models therefore involve ordinary differential equations expressing time variations of these process variables. At the next level are the stagewise processes for which, even though we can no longer assume total homogeneity within the entire process, the spatial variations have been localized to finite stages; but each of these stages is considered as a lumped parameter system in its own right. The resulting theoretical models involve sequences of differential equations jointly expressing the time variation of the process variable on each stage, as well as the stage-to-stage variations.

CHAP 12 THEORETICAL PROCESS MODELING

v

Figure 12.5.

t

Condensate

I•

"'

377

T(z,t)

The steam-heated shell-and-tube heat exchanger.

We are now interested in systems in which the changes in spatial position cannot be quantized into stages as with stagewise processes; the theoretical modeling equations will therefore have to be partial differential equations, if the variations in both space and time are to be represented properly.

Shell-and-Tube Heat Exchanger The process in question, first introduced in Chapter 8, is shown in Figure 12.5. The Process A fluid with constant density p, and specific heat capacity CP is flowing

through the tube of a shell-and-tube heat exchanger with velocity v, as shown in Figure 12.5; it enters at a temperature T0 and is heated from the shell side by condensing steam. The tube has a uniform cross-sectional area A, and the surface area available for heat transfer per unit length is A 5 • It is clear that the fluid temperature will vary along the length of the heat exchanger, and also with time at any particular point. Thus fluid temperature is both a function of time as well as spatial position along the length- of the tube. Problem Definition and Assumptions We wish to develop a theoretical model that will predict the variation of the fluid temperature both with time and with position. To do this we will consider changes taking place during the infinitesimally small time interval [t, t + at], in the space contained within the infinitesimally sized elemental ring section shown in Figure 12.6, whose boundaries are at z and z + az. The following assumptions are used in conjunction with this microscopic description of the process:

Figure 12.6.

Shell balance for heat exchanger.


378 1.

The material within the element is at a uniform temperahtre T, but the temperahtre at the boundaries of the element are T I, and T I.+ Az'

2. The fluid is flowing through with a flat velocity profile: this eliminates the need to model variations in the radial direction. 3.

Physical fluid properties are assumed constant; and the cross-sectional area of the heat exchanger is assumed uniform.

4.

The dynamics of the rube and shell walls are assumed small enough to be negligible: thus any accumulation of energy within the elemental ring is due entirely to the fluid occupying the element, not the heat exchanger wall.

5. The steam on the shell side is assumed to be at a constant temperahtre T 51 and heat exchange across the tube wall is assumed to follow the Newton's "law" of cooling. Microscopic Energy Balance Amount of energy in at z : with flowing fluid, over the time interval [t, t + At] Amount of energy out at z + Az : with flowing fluid, over the time interval [t, t + At] Amount of energy in through: the heat exchanger wall over the time interval [t, t + At] (Note that the total heat transfer area on this elemental ring is A, Az.) Amount of energy accumulated: over the time interval [t, t +At] An energy balance now demands that the total amount of energy accumulated in the element, over the time interval [t, t +At], be equal to the total amount coming into the element less the total amount leaving within the same time interval. Mathematically, this means: pACPAz(TI,+At-TI, ) = pAvCPAt (T- T*)l.-pAvCPAt (T- T*)lz+. using a measuring device 1. "Mea sure" y (ie., measure T to produ ce T tion variable Ym =Tm- T,; devia the (say, a thermocouple}; determine value as y. same the have not may or the result, Ym' may d set-point value y4 to 2. Compare the meas ured value Ym' with the desire obtain the deviation as an error signal: (14.1)

se it is gener ated This is know n as the "feedback" error signal becau ss. proce the from back" "fed been from measurement that has signa l£, to be used in 3. Supp ly the controller with the feedback error flowrate} which will steam the in e chang (the u of obtaining the value be implemented on the process.

463

CHAP 14 FEEDBACK CONTROL SYSTEMS

ated value of u on 4. Imple ment u (the command to implement the calcul case, the steam this in t, the process is issued to a final control elemen valve ). 5. Measure y again, and repeat the entire procedure. ed in implementing this Let us now look more closely at the elements involv feedback control strategy.

ram 14.1.2 Elements of the Feedback Loop and Block Diag Repr esen tatio n ck controller, as shown The combination of the physical process and the feedba This feedback contro l . system l contro ck feedba a in Figure 14.1, is know n as system is seen to consist of the following elements: • • • • •

The process itself The measuring device A comparator (comparing y,. and Yd) The controller The final control element

in its own right, Observe that each of these elements is a physical system can each be they hence ts; outpu with clearl y identi fiable input s and notice how, to tant impor also is It on. functi er chara cteriz ed by a transf infonnation uced, introd just according to the feedback control strategy we have of the view a Such loop. a of form the flows from one element to the other in entati on repres m diagra block the by rced reinfo is feedback contro l system each of these elements are shown in Figure 14.2, where the-tfansfer functions of t signal s are linked outpu and input show n as indiv idual blocks whose feedback loop. the d aroun ation inform of flow the te appropriately to indica convenient, very very a is t As we shall see, the block diagra m forma consider how to now us Let s. system l contro intuit ive way of repres enting l al contro system. develop the block diagram representation of a physic r ----- ----- ----- ----- ---- _1]!~-•M•--

r---- ----- ___ /:Pro cess

'

I

II

'

ol

..,,,, II

I

Final

Error

Control

Signel

\

''

Controllor

II

Element

: : Outpu;( s)

c(s)

..: ,,,,

u(s) Measure d Output

' - - - r - - ' ,,

Manipulated Variable

I

(s)

Measurement Device

'

L---- ------ ------ ------ ------ ------ ------ 1 Figure 14.2.

Equivalent block diagra m for the process in Figure 14.1.

464

SINGLE LOOP CONTROL

14.2

BLOCK DIAGRAM DEVELOPMENT

Block diagram representations of control systems are developed by:

1. Identifying the individual elements of the control system (as was done in the previous section for the process in Figure 14.1), 2.

Identifying the input and output for each element,

3.

Representing the individual input/ output transfer function relationship for each element in the block diagram form introduced in Chapter 4,

4.

Finally, combining the individual block diagrams for each element to obtain the overall block diagram.

We will now continue with our example process in Figure 14.1 and show how the block diagram in Figure 14.2 was developed.

14.2.1 Individual Elements and Transfer Functions The individual elements for the process as identified in Section 14.1 have the following characteristics. The Process

Inputs: Output:

u, d, the process input and process disturbance

y

As obtained earlier in Eq. (4.16), the transform-domain transfer function model for the process is: y(s) = g(s) u(s)

+ gj..s) d(s)

(14.2)

As we will recall from the discussion in Chapter 4, this model form is, of course, generally applicable to linear single-input, single-output (SISO) systems, not just specific to the stirred heating tank system that we have used merely as a backdrop. The block diagramatic representation of this transfer function model is shown in Figure 14.3. d(s) ~-------------------------------~

u(s)

y(s)

Figure 14.3.

The process.


465

The Measuring Device (or Sensor)

Input Output

y Ym

In the specific case of the stirred heating tank system, using a thermocouple as a measuring device, y represents the temperature, and Ym the voltage output of the thermocouple. Other types of sensors were mentioned in Section 1.4 of Chapter 1.

~ Figure 14.4.

The measuring device.

Designating the transfer function for the measuring device in general as h(s), we have, in terms of the Laplace transforms of the input and output to this system: Ym(s)

= h(s)y(s)

(14.3)

The Comparator (Part of the Controller Mechanism)

Inputs: Output

Ym' Ytl E

related according to: E(s)

= y,f.s)-ym(s)

T

Figure 14.5.

(14.1)

The comparator.

The Controller (The NHeart" of the Feedback Control System)

Input Output

e c

The transfer function for the controller is usually designated gc; thus, we have: c(s)

= Bc(s) E(s)

(14.4)

As we might expect, gc(s) can take various forms; and the form it takes obviously determines the nature of c(s). We shall examine the different forms of gc(s) below.

.. ,·I · "' Figure 14.6.

1 ..., ·

The controller.

SINGLE LOOP CONTROL 466

The Final Control Element

er c, command signal from the controll variable t inpu ess proc the of e u, actual valu 1.4 for othe r most often a valve (see Section Since the final control element is

Inp ut Output:

Figu re 14.7.

The final control element.

transfer function is usually designa types of final control elements), its ther efor e have: Cv(s) c(s) u(s)

=

ted g..,; we (14.5)

Typically, g., is of the form: (14.6)

ess itseU. It pared to the dynamics of the proc and -rv is usually quite small com be ignored, can e valv that the dynamics of the is therefore customary to assume ively, it is rnat Alte . ctly dire er, of the controll and u(s) is taken to be the outp ut process; the of cs of the valve along with that also possible to lum p the dynami gral par t of the process. i.e., the valve is treated as an inte

Block 14.2.2 Th e Closed-Loop Sys tem

Dia gra m

feedback out of each of the elements of the The flow of information into and Figures in s ram diag k bloc al vidu indi by the control loop is aptl y represented each of ing link ks, bloc bine these indi vidu al 14.3 thro ugh 14.7. If we now com wn in sho as ram diag k bloc with the overall trol the indi cate d signals, we end up con een each component in this feedback of ysis Figure 14.2. The relationship betw anal ent sequ as we might suspect, sub loop is now easier to visualize, and facilitated. the ove rall syst em is also greatly systems, of all stan dard feedback control cal typi is ram diag k This bloc con trol l fina or or, ess, con trol ler type, sens ns·on rega rdle ss of the specific proc atio vari ated plic com e Even thou gh mor the elem ent in any part icul ar case. ing erly possible, the basic principles und ents the sam e them e are sometimes pon com the een the relationships betw dev elop men t of suc h diagrams and are alw ays the same. the simple to be for a closed-loop system for · Such a block diag ram is said with the ted nec con "closed": the controller is reas on that the control loop is and also ler, trol con the s to the process from process so that information flow en (hence rok unb one in ly, ous tinu con troller, from the process back to the con . loop ) sed" "clo ysis of s are particularly useful for the anal These closed-loop block diagram , we lved invo taking a closer look at the issues closed-loop systems; but before . take ers forms that feedback controll wish to consider first the various

467

EMS CHAP 14 FEEDBACK CONTROL SYST

14.3

CLASSICAL FEEDBACK CONTROLL

ERS

loop r is that com pone nt of the feedback As note d above, the feedback controlle ent) elem rol cont final the (via ess proc that issues cont rol com man ds to the oint set-p red desi its ess mea sure men t from based on e, the deviation of the proc any for e issu to al sign d man s wha t com value. How the controller calculate her. anot from r rolle mnt one given value of e is wha t differentiates s controller, along with the equa tion The nam e of each classical feedback n. give be now to c(t), will that describe how they each relate e(t)

14.3.1 The Pro por tion al Con trol ler the controller (a pres sure signal in the Let p(t) be the actual outp ut signal from l sign al in the case of an electronic trica case of a pneu mati c controller; an elec the t) valu e whe n e(t) is zero. Then stan (con its be Ps let controller), and to: g rdin acco ates proportional feedback controller oper ·

p(t) :: Kc e(t)

+P,

(14.7)

al, r" value p5 is the control com man d sign The deviation of p from its "zer o erro i.e.: (14.8) K0 e(t) c(t) = p(t) - Ps controller is sche dule d to be dire ctly Thu s the com man d sign al from this e its name. proportional to the observed error, henc leads immediately to: ) (14.8 Laplace transformation of Eq.

=

(14.9)

the prop ortio nal controller as: identifying the transfer function for (14.10)

nal cont rolle r K, is called the proportio The characteristic para met er of this in the er met para the e, war hard r rolle gain. Sometimes in commercial cont as: as the proportional band, PB, defined prop ortio nal controller is repr esen ted ut) PB _ IOO·(max range of controller outp ble) - K,·(m ax range of measured varia Thus: PB

cc

-

1

Kc

(PI) Con trol ler 14.3.2 The Pro por tion al + Inte gra l In this case, p(t) is given by: p(t)

= K{ e(t)

+if~ e(t) dt] + P,

(14.11)

468

SINGLE LOOP CONTRO L

which contains a proportional portion as well as an integral of the error over time. Upon taking the Laplace transform, we obtain: c(s) =

Kc (1 + -1:11- ) E(s) s

(14.12)

so that the transfer function for the PI controller is: (14.13)

The addition al paramet er 1:1 is called the integral time, or the reset time. Often the reciprocal, 1/7:1 is the preferred parameter; this is known as the reset

rate.

14.3.3 The Propor tional + Integral + Derivative (PID) Contro ller Here the operatin g equation is: (14.14)

and Laplace transformation gives: (14.15)

The transfer function for the PID controller is therefore given by: (14.16)

The paramet er -r0 is called the derivative time constant.

14.3.4 The Propor tional+ Derivative (PD) Contro ller This is the controller obtained by leaving out the integral action mode from the PID controller, and as such its transfer function is given by: (14.17)

14.4

CLOSED-LOOP TRANSFER FUNCTIONS

In terms of the outer box in Figure 14.2, observe that the overall closed-lo op system in reality just has two inputs, y4 and d, and one output y. This indicates that the individu al transfer functions for the separate compone nts of this feedback control system may be consolidated to give composite, overall closed-


469 d(s)

/Outer /

I Yis> 1jf{s)

Figure 14.8.

Box

Vis)

.1

y(s)

The concept of the closed-loop transfer ftmction.

loop transfer functions: one relating y to yd, and another relating y to d, as illustrated in Figure 14.8. Let us now consolidate the transfer functions for the closed-loop system by analyzing Figure 14.2 and its componen t parts. Even though the process by itself is given by Eq. (14.2), the input u is now obtained according to Eq. (14.5); i.e., u(s) = g v(s)c(s). However, c itself is obtained according to Eq. (14.4); so that (14.18)

Using Eq. (14.1) for eand Eq. (14.3) for Ym' Eq. (14.18) becomes: U

= 8v8c{Yd-

hy)

(14.19)

,,~

Introduci ng Eq. (14.19) into the process equation for u now completes the consolida tion exercise, having eliminate d every other variable except y, yd, and d; the result is: (14.20)

which, when solved for y, now gives: BBvBc gd d Y = 1 + ggv gch Yd + 1 + gg g h v c

(14.21)

and by defining: lfl(s) =

(14.22)

and (14.23)

we obtain: y(s)

l/l(s) yjs)

+ ytJ..s) d(s)

(14.24)


; IJI{s) and 'I'Js) function repr esen tati on we seek This is the closed-loop transfer tions. func sfer tran oop are the closed-l defi ned in Eqs. (14.22) and (14.23) ram from diag k bloc the and ns ctio fun sfer Reg ardi ng these closed-loop tran erve that: whi ch the y wer e obtained, obs ominators. 1. 'If and 'I'd hav e identical den tran sfer gg,$c' is the pro duc t of all the 2. The num era tor of 'If, i.e., or of lfld erat num the gd, le whi Yd; to y een functions in the direct pat h from betw h pat ct dire sfer functions in the 11 is also the "pr odu ct of the tran ). tion func sfer tran h suc y and d (in this case, there is only one re loop. the transfer functions in the enti 3. gg,$)1 is the pro duc t of all These results may now be general

ized as follows:

(such as that in Figure 14.2) the For single-loop feedback systems TF) are given by the general (CL closed-loop transfer functions expression:

(14.25)

whe re

functions in the '7- = pro duc t of all the transfer (yd or d).

direct pat h betw een the

out put y and the inp ut

nL

= pro duc t of all the transfer functions in the entire

loop.

foll owi ng tion of thes e resu lts wit h the Let us illu stra te the app lica exa mpl es. FUN CI'I ON FRO M SIN GLE CLOSED-LOOP TRANSFER M. L SYSTEM BLOCK DIA GRA LOOP, FEEDBACK CON TRO in Figure 14.9, first who se block diag ram is sho wn For the feedback control syst em tion relating find the closed-loop transfer func and d2, y (c) and , d (a) y and Yd• (b) y and 1 ress ion rela ting the ed-loop tran sfer func tion exp and then obta in the over all clos outp ut to all the inputs.

Exa mpl e 14.1

Figu re 14.9.

. Block diag ram for Example 14.1

471

EMS CHAP 14 FEEDBACK CONTROL SYST Solution:

y is g2 g1 tions in the direct path between y and 4 (a) The prod uct of the transfer func hr gchz g g isg 11 1 loop 2 e tions in the entir g11 g,; and the prod uct of the transfer func relation between y and Yd is: tion Thus the closed-loop transfer func (14.26)

d is g gd; tions in the direct path betw een y and 1 2 (b) The prod uct of the transfer func n by: give be will tities quan two relating these thus the closed-loop transfer function g,_ ld

(14.27)

of transfer d , we have the following prod uct (c) In the dire ct path betw een y and 2 therefore be given will case this in tion func fer trans functions g2 Kr The closed-loop by: (14.28) function we now obta in the overall transfer Combining all three expressions, as: ut control system to the outp relating all the inputs of this feedback

y

=

B2B J8v lc l+g 2g1 gvg ch2 h1Y d

+

d B28 d 1 2h1 g.,h ,g., l+g ,_g

(14.29)

cont rol is poss ible to hav e a SISO feed back As we will see in Cha pter t( it such In s. loop me invo lves mul tiple con trol syst em for whi ch the over all sche is to tion func sfer tran oop the over all clos ed-l case s, the stra tegy for find ing sing le to s loop r inne the cing redu ram by first simp lify the mul ti.lo op bloc k diag whi ch is the mul tiple loop s to a sing le loop ces redu cise exer an h Suc ks. bloc d. niqu e we hav e disc usse then easi ly han dled by the tech ept with an exam ple. conc this trate illus It is best to Example 14.2

CI'IO N FROM CLOSED-LOOP TRANSFER FUN L SYSTEM BLOCK TRO CON CK DBA FEE OP MULTILO DIAGRAM.

back cont rol fer function for the multiloop feed Find the over all closed-loop trans see later in will we (As n in Figure 14.10. syst em who se bloc k diag ram is show ) tion. igura conf system Chapter 16, this is a crrscade control

Solution:

gnated r loop by itself. It has two inputs, desi Let us start by considering the inne relating thes e tions func fer trans op ed-lo clos m2 and d2, and one outp ut m1• The ple 14.1; the g the appr oach illus trate d in Exam variables are easily obta ined usin result is: (14.30)

472

SINGLE LOOP CONTROL

:

i !

!Inner

!Loop

~.........................................................................................................l

~------------_, hl ~-----------------~

Figur e lUO. Block diagra m for Example 14.2. If we repres ent these as: ml

= Ba~ + gbd2

(14.31)

then the multil oop block diagra m in Figure 14.10 is simplified to the single-loop one shown in Figure 14.11. It is now easy from here to obtain the overal l transf er function expres sion for the complete system; it is given by:

(14.32)

If necessary, the newly introd uced transfer functi ons g8 andgb may be replac ed by their explicit expressions obtainable from Eqs. (14.30 ) and (14.31).

14.5

CLOSED-LOOP TRANSIENT RESPONSE

14.5.1 Preli mina ry Con sider ation s In Part II, we studie d the dynamic behavior of processes (in the open loop) using their trans fer function representations to investigate their responses to various forcing functions. Obse rve now that for the closed-loop system, we have obtai ned the overall closed-loop transfer function representation as: y(s)

= IJ(s) yj.s) + 'lfJ.s) d(s)

(14.33)

y

~------------~ hl ~---------------~

Figur e 14.11.

Single-loop equivalent of the multiloop block diagra

m of Figure 14.10.


473

The input s for the closed-loop system are y , the set-po int, and d the disturbance. In precisely the same manner as with 4 open-loop dynamic analysis, we may now use Eq. (14.33) to investigate the dynam ic response of closed-loop systems to changes in the set-point (the servo, or set-point tracking problem), or the response to input disturbances (the regulatory problem). To be sure, the transfer functions involved in such closed-loop dynam ic analyses are more complicated than those used for open-loop analysis, but the principles are exactly the same in each case. Let us now combine all the blocks in the direc t path betwe en y and y d (excepting the controller, gc) in a general feedback control system block diagr am and refer to this produ ct as simply g; and let gd represent the produ ct of all the transfer functions in the direct path between y and d; furthe r let the produ ct of all the blocks in the feedback path be referred to as h, giving rise to the generic closed-loop system show n in Figure 14.12. (It is important g and h stand for here; we have introduced these "com to keep in mind what bined" transfer functions as a short hand notat ion only; by no means are we restricting the following discussion to only those systems having one g and one h block.) For this generic system, the closed-loop transfer functions are given by: ggc 'I' = 1

(14.34a)

+ ggch

and gd 'I'd=

1

(14.34b)

+ ggch

and the transfer fu.'lction repre~~tation for the closed

-loop system is: (14.35)

These will now be used to investigate the transi ent behav ior of closed-loop systems in response to input changes. It is of prime importance to note that both closed-loop trans fer functions, lJI and 'lfd, have ident ical denom inato rs, i.e.: 1 +gg~. '

y

Ym

h

Figure 14.12. A generic closed-loop system.

SINGLE LOO P CON TRO L

474

t-Order Sys tem und er 14.5.2 Transient Res pon se of a Firs trol Con Proportional Feedback rise er feedback control (Figure 14.1) gave The stirr ed heating tank system und t sien tran the te . Let us now investiga to the block diag ram in Figure 14.2 nce urba dist the and oint in the set-p resp onse of this syst em to changes d"). "loa d (sometimes calle ted ns for the transfer functions represen To do this we need explicit expressio its that 4 pter Cha from ll itself, we reca in the block diagram. For the process transfer functions are given by: g(s)

=

{39

(4.15a)

-::-'-~-

9s + 1

gj.s) =

es +

(4.15b)

1

to time, the ratio of the tank volu me V whe re f3 = }.j pVC and 9 is the residence and gain tate dy-s stea with em syst er Thus this is a first-ord the thro ugh put by: n give tant cons time

l

(14.36a)

K = {38

and

'1:

=

(14.36b)

e

that valve dynamics are negligible and Let us assume, for simplicity, that and ect perf are e oupl moc from the ther tem pera ture mea sure men ts obtained instantaneous, implying: gv = 1; h

=1

rolle r with prop ortio nal gain K,, the cont Und er prop ortio nal only control, transfer function is given by: (14.37)

ion to investigate the We will now use this process informat inputs:

response to two

o responsei with yd = 1 and d = 0) 1. A unit step change in set-point (serv (regulatory response; with d = 1 and 2. A unit step change in disturbance Yd = 0)

point Response to Unit Step Change in Setrelationship in this case is: From Eq. (14.35) the transfer function (14.38)

475


matio which, upon introducing the provided infor

y(s)

n becomes:

) ( KKc es + 1

!

(14.39)

KK ) s 1 + ( es : 1

ssion simplifies to: (with K as given in Eq. (14.36a)). This expre 1

KK y(s) =

6s + 1

~

KK

c

s

: which we now put in the more revealing form y(s) =

! K"' + 1 s

e•s

(14.40)

where the "closed-loop gain /' K* is: (14.41a)

and the "closed-loop time cons tant/ '

fJ+

is: (l4.4 1b)

nse of Eq. (14.40) we observe that it In orde r to solve for the transient respo order system with steady-state gain looks like the unit step response of a firstof this fact therefore, we may on gniti K .. and time cons tant 8*. In reco ient response: immediately write dow n the required trans y(t) = K*

(I - e-vs•)

(14.42)

control syste m (the heating tank , This equa tion represents how the entire to a unit step change in the setnds respo ) valve, and prop ortio nal controller nse to a change in yd from 0 to 1. In point; i.e., how y changes with time in respo temp eratu re will respo nd whe n physical terms, it show s how the actual tank cont rolle r trying to meet this the with 1° its desir ed valu e is chan ged by demand. Let us note the following from Eq. (14.42): y ~ K* .

1.

As t ~

2.

is exponential. The appr oach of y to the ultimate value K*

3.

1a) we see that K• will How ever , from the definition in Eq. (14.4 K,. of es valu finite all always be less than 1, for

oo,

476

SINGLE LOOP CONTROL l.O

- - - - &fs~t

- - - - - - - - - - - - - - - - -

y(t)

Increasing K ,

0

Figure 14.13. Response of a first-order system under proportional feedback control to a unit step change in the set-point.

4.

Because the final value attained by y is not the same as the desired value yd, there is said to be a steady-state offset defined by: offset

= yd- y(oo)

(14.43)

which, in this specific case, is given by: offset = 1 -

KKC = --=-1 + KKC 1 + KKC

(14.44)

The inability to eliminate offset is one of the limitations of proportional-o nly controllers. 5.

Note that as the controller gain, Kc

~

oo, the offset tends to zero.

A plot of this transient behavior is shown in Figure 14.13.

Response to Unit Step Change in Disturbance For this situation, the transfer function relationship is: y(s)

gd = -1 --d(s) + ggc

(14.45)

and from the expressions for g, gd, gc, we have:

y(s)

1+ (

KK (Js :

(14.46)

) s 1

which again may be rearranged to give the familiar form: y(s) =

Kd* 1 8*s + 1 s

(14.47)


477

1.0

No Control, Kc • 0 y(t)

_1_

l+KKc Offset

Figure 14.14. Response of a first-order system under proportional feedback control to a unit step in disturbance.

where the closed-loop disturbance gain is:

K *I d - I+ KKC

(I4.48)

and(}* is as given earlier in Eq. (14.41b). Once again the transient response is easily obtained as: (14.49)

representing how the tank temperature changes in response to a unit step increase in the inlet stream temperature as the controller tries to handle this disturbance. Once again we note from Eq~(14.49) that:

1.

As t -7 oo, y -7 Kl and that the approach is exponential.

2.

The controller is supposed to keep y at zero in the presence of the disturbance when Yd =0; however, from Eq. (14.48) we see that this is not the case, since:

K/' = 1 + KKC '# 0 The offset in this case (from Eq. (14.43)) is: 1 offset= 0-I + KKc 3.

I I+ KKc

(I4.50)

In the absence of control, i.e., with Kc = 0, Kd,. will be 1, 8,. will be 8 and the response to this disturbance will be given by: y(t) =

(I - e-118)

(14.51)

Figure 14.14 shows the closed-loop response, Eq. (14.49), along with the open-loop response, Eq. (14.51). Note that using the proportional controller reduces the offset, and that the amount of offset decreases as the value of Kc increases.

SINGLE LOOP CONTROL

478

Syst ems Effect of Proportional Control on First-Order al feedback control on a general Let us now summ arize the effect of propo rtion first-order system. fer function: The first-order system with the open-loop trans y(s) =

(-rs~ 1)

(14.52)

u(s)

form ed to a syste m with the closedunde r propo rtion al feedback control is trans loop transfer function representation: (14.53)

(assuminggv

= 1,

h

= 1) where: (14.54a)

and

(14.54b)

the dyna mic beha vior of the Thus prop ortio nal feedback contr ol modifies fying the value s of the modi but r orde first- order syste m, retai ning the chara cteris tic param eters . t of propo rtion al control on the It is parti cular ly impo rtant to note the effec so long as KK, > 0, then r* 1.67, two tabl uns and de litu amp ng easi incr that the resp ons e is oscillatory with ry llato osci s em' soid al port ion of a syst • Since the frequency of the sinu tic eris ract cha the of t ima gina ry par resp ons e is dete rmi ned by the er ons e take s on high er and high resp ry llato osci the s, equ atio n root frequency as K, increases. e as K, s real and becomes mor e neg ativ • Since the thir d root, r3, stay in Eq. e'i A term 3 the from ing e aris increases, the tran sien t resp ons s. (14.82) becomes faster asK , increase Let us now generalize the results

from this example.

t Locus 14.7.2 Ge ner al Features of a Roo

Dia gra m

techniques ineers wer e required to mas ter the Earlier generations of con trol eng no long er is this r, eve locus diagrams. How for efficiently con stru ctin g roo t grea t dea l a with s ram diag such that generate necessary. Com pute r pro gram s ent or stud available, ther eby allo win g the thes e of facility are now rou tine ly t rpre inte to how the task of lear ning prac ticin g eng inee r to focus on . diagrams, not how to draw them roo t locus general features exhibited by the the are e efor ther ng owi The foll ng, and acti extr of ch will facilitate the task diag ram , a kno wle dge of whi . ram diag the in tion contained prop erly interpreting, the informa . w as man y roots as the poles of gg)r 1. The root locus diag ram will sho is sim ply a the fact that the root locus diag ram This is easily esta blis hed from c equation: plot of the roots of the characteristi

(14.83)

499


there are no time delays with the controller gain Kc as a parameter. When of polynomials: ratio ing follow the as d sente involved, gg)z can be repre (14.108)

N(s) is a polynomial of order r, where KP is a combination of all the loop gains, tic closed-loop systems, r ~ n. realis and D(s) is a polynomial of order n; and for Thus gg)z has r zeros and n poles. The characteristic equation is therefore given by:

or D(s) + K,Kc N(s)

=0

(14.109)

r::;; n) and will therefore have which will be an nth-order polynomial (because gg,!z. as many roots as the poles of process, ggch will take the In case there is a time delay a in the open-loop form: gg)l

=

K K N'(s) e-as P c D'(s)

(14.110)

te numb er of roots (recall Since e-as is periodic, it will introd uce an infini harmonics are not of interest Example 14.10). However, all o.f--the higher order appro xima te (say, mth-order), so that if we represent cas by a high-order Pade we woul d have: NJs)

cas

= DJs)

(14.111)

s. Then if we let: wher e both Na(s), Da(s) are mth-order polynomial

= N'(s) NJs)

(14.112)

D(s) = D'(s) DJs)

(14.113)

N(s)

case of time delays. It can be we may use the argum ents above even in the ional highe r harm onic loci addit uce introd show n that even thoug h delay s close to the origin need be (branches) on the root loctis diagram, only those considered. trajectory of each root of Since the loci of the root locus diagr am show the that: ore theref s follow it ion, the characteristic equat as the poles of The root locus diagram has as many branches (or loci) gg.ft.

500

SINGLE LOOP CONTROL

2. The root locus diagram is symmetrical about the real axis. This is simply because of the fact that "branching" off the real axis occurs when there are complex roots, and such roots only occur in mirror image pairs, as complex conjugates. 3. The root loci commence at the poles of gg)r and terminate at its zeros. The key to establishing this is the representation of the characteristic equation given in Eq. (14.109). The root loci commence when K, =0, and terminate when Kc = oo. Setting K, = 0 in Eq. (14.109) leaves: D(s) = 0

(14.114)

which, of course, gives rise to roots identical to the poles of gg,!z in Eq. (14.108). To obtain the temrinating point of the root loci, we divide through in Eq. (14.109) by Kc first, to obtain:

M K

+KPN(s)

0

(14.115)

c

and now, asK,~ oo, Eq. (14.115) becomes: K,N(s)

=0

(14.116)

which gives rise to roots identical to the zeros of gg,!z in Eq. (14.108). It is in this sense therefore that we may ·consider the root locus diagram as the ''history of the migration of the poles of gg,!z to its zeros" as a function of the controller gain. This raises an interesting question, however: what happens in the situation where gg,!z has more poles than zeros? The answer is that the remaining roots go to infinity. We therefore have the result: The root locus has n branches, r of which terminate at the zeros of gg,h, while the remaining (n - r) branches go to infinity as K, increases.

Note that the system with the root locus diagram shown in Figure 14.16 has no zeros so that all three branches go to infinity. 4. The magnitude of gg)t is 1 for all values of s. By rearranging the characteristic Eq. (14.83) we see that:

gg,!z = - 1 and taking the absolute value of each side, we obtain: (14.117)

The zeros and poles of gg,!z are easily identified on the root locus diagram, but not the process gain. This fourth property may be used to evaluate the total


501

process gain K 11 from the root locus diagram. Observe that from Eq. (14.108) that this property implies: (14.118)

from which K, may be obtained. The following three properties have to do with extracting transient response information from the root locus diagrru;n, and, on the basis of our knowledge of the relationshi p betweer characteris tic equation roots and transient response, they are self-evident. .> S. The closed-loop system response is nonoscillatory for points on the loci that

lie completely on the real axis.

6. When the loci branch away from the real axis, the closed-loop system response becomes oscillatory. 7. The closed-loop system is on the verge of instability at the point where one

of the branches first intersects the imaginary axis in going from the left half to the right half of the complex plane.

It should be quite obvious by now that gg,ft plays a very important role in issues concerning the root locus and in the determinati on of the characteristics of the closed-loop system in generaL This quantity is referred to as the openloop transfer function of the closed-loop system. This is because if the feedback loop is "opened" right at the point where the feedback information Ym is passed on to the comparator (see Fi~ 14.12), the transfer function relation between this output and the set-point y 4 is: (14.119)

This is the transfer function we obtain by "opening" the otherwise closed feedback loop. Let us now use two examples to illustrate the general features of the root locus which we have just presented. It should be mentioned that the notational standard is to represent the poles of the open-loop transfer function by the symbol x while the zeros are represented on the root locus diagram by o . Example 14.11 ANALYSIS OF CLOSED-LO OP SYSTEM BEHAVIOR FROM A ROOT LOCUS DIAGRAM.

The root locus diagram shown in Figure 14.17 is for the process with: 2Kc(s + 1)(- s + 1)

ggjl = (O.Ss + 1)(2s + 1)(4s + I)

(14.120)

(a) Confirm that the poles and zeros shown in the root locus plot correspond to those in the transfer function Eq. (14.120) (b) What will the nature of the transient response be when Kc =17 (c) At what Kc value is the system on the verge of instability?

/

/

\


m a g i n

1

a r

y

0

A X

i -1

8

-2

-1

2

0

3

Real Axis

Fig ure 14.17.

Root locus diag ram for process mod

eled by Eq. (14.120).

Sol utio n:

at s = -2, - 0.5 0,- 0.25) and indicates three pole s (located (a) The roo t locu s diag ram mod el. Obs erv e that the s is consistent wit h the orig inal two zeros at s -1, + 1. Thi s and one of whi ch goes zero em s, two of whi ch go to syst roo t locu s has three branche to pos itiv e infinity. t two of the roo ts hav e ram , whe n K, 1, we see tha (b) Pro m the roo t locu s diag ion is therefore that the the complex plane. The conclus alre ady bran che d out wel l into ped oscillatory. tran sien t response will be dam the bran che s are on the s diag ram sho ws that two of locu t roo the , 3.21 = Ke en (c) Wh is therefore that this ion clus con The . cross into the RHP ima gina ry axis and abo ut to instability. of e whi ch the syst em is on the verg is the controller gain valu e for

=

=

BEH AV IOR FRO M OF CLO SED -LO OP SYSTEM ARE TIM E Exa mpl e 14.12 ANALYSIS RE THE WH EN A RO OT LOCUS DIA GR AM DELAYS. s add itio nal bran che s to the pres enc e of a time dela y add As was indi cate d abo ve, the of the firs t-or der pro cess ram 8 sho ws the roo t locu s diag roo t locu s plot . Fig ure 14.1 side red earl ier in Exa mpl e por tion al-o nly controller con . wit h a time dela y and a pro 14.10: K (14.104) ggrft = (lO s: 1) ~~-Ss izon tal harm onic bran che s infinite num ber of alm ost hor The tim e dela y intr odu ces an sho w onl y thr~ of these). (we axis ry dow n the ima gina at reg ular inte rval s up and real axis is of imp1>rtance to c:oming in fro m- oo on the s How eve r, onl y the bra nch le pol e at- 0.1 and bran che sing the roo t aris ing from are cs ami dyn cess stability. Thi s roo t mee ts the pro the nt 0.446. Thu s bey ond this poi from the real axis whe n Kc = ler gain s K, > 3.81 the ram pred icts that for con trol diag s locu t roo oscillatory. The ulat ions of the clos edsim by se resu lts can be veri fied The e. tabl uns be will cess pro ure 14.19. loop beh avio r sho wn in Fig

e

1-

503

CHAP 14 FEEDBACK CONTROL SYSTEMS 2 I I I I I I

1

Kc=3.81

: Kc:=2.0t 1 0.25 in Figure 15.2) when an approximate model of the form Eq. (15.30) is known. Table 15.2 provides their guidelines. Smith and Corripio [6] point out that these tuning rules are

very sensitive to the time delay/time constant ratio; they recommend these rules be limited to models with 0.1 < (a/-r) < 1.0.

Cohen-Coon Tuning Rules Cohen and Coon [5] also developed tuning rules based on the concept of achieving a quarter decay ratio. However, because this criterion does not produce a unique set of tuning parameters, the Cohen-Coon rules given in Table 15.3 are somewhat different from those of Ziegler-Nichols. Again, these should be applied only for a limited range 0.1 < (a/-r} < 1.0. Table15.2 Ziegler-Nichols Approximate Model PID Tuning Rules [3) (as reinterpreted by Smith and Corripio [6])

K,

"r

"v

p

t(:)

-

-

PI

0:(:)

Controller Type

3.33a

!

PID

~2(:)

2.0a

O.Sa

CHAP 15 CONVENTION AL FEEDBACK CONTROLLER DESIGN

537

Table15.3

Cohen-Coon Approximate Model PID Tuning Rules [5] Controller Type

Kc

j ( ~ )]

p

k(: )[

PI

k(:)[o. 9 +112(~ )]

1+

a

PD

7:1

"v

-

-

[~J 9+20(~)

t(:)[~+~(~)]

-

-

{~J~) 22 + 3(

PID

t (: )[~ +~ (~ )]

[~J +s( ~) ill+~d

a 13

Time-Integral Tuning Rules Smith, Murrill and coworkers [7-9] have developed correlations for PID tuning parameters which minimize integral-time objectives such as IAE, ISE, ITAE, and ITSE. Their correlations distinguish between set-point changes and disturbance-rejection responses. A rather complete set of results are summarized in Ref. [6]; however, we will choose to present only the ITAE results in Table 15.4 because these are often the best solution in a practical'sens e (e.g., see Figure 15.4). Again, these should only be applied in the approximate range 0.1 < (a/7:) < 1.0. It is particularly interesting to note now how these different tuning rules based on somewhat different criteria all depend on the time-constant -to-timedelay ratio. It is therefore easy to see why many practical industrial control engineers consider this ratio to be a very crucial process parameter when it comes to tuning feedback controllers.

SINGLE-LOOP CONTROL

538

Tab le15 .4 es [7-9] e Mo del Controller Tun ing Rul mat roxi App E Min imu m ITA

Kc

Controller Type of Response Type

p .

Disturb.

o:9 (~Jos4

PI

Set-point

o.;r(~J916

PI

Disturb.

O-~r(~J977

PID

Set-point o.i65

PID

Disturb.

't'z

't'o

-

't

[ 1.03-0.165 (~]

(~J680 -"0.674 't

(;Jsss

T

[ 0.796-0.147

(~)

(~J738 -"0.842 't

1.~7(;J"947

-

- 929 0.308-r (aJT

- 99S 0.381T (aJ. T

del Tuning Ru les Direct Synthesis Approximate Mo in Eq. (15.30), it is en an app rox ima te mod el as As sho wn in Cha pter 19, giv n requiring a ed s for PID con trol lers bas upo pos sibl e to dev elop tuning rule wit h a tim e lay e-de -tim plus ch is firs t-or derclos ed- loo p traj ecto ry q(s) whi des ired speed of ame ter -r,.. This dete rmi nes the con stan t ind icat ed by the par resp ond mor e will for sma ll -r,., the con trol ler the clos ed-l oop resp ons e, so that . .15.5 le Tab in hming rule s are giv en quic kly than for larger -r,.. The Tab lelS .S Cha pter 19) del Con trol ler Tun ing Rul es (cf. Mo te Dir ect Syn thes is App rox ima

Controller Type

Kc

1'

PI

K('t'r +a)

PID

2-r+ a 2K (-r, +a )

1'1

't'D

1'

a

H2

_!( !_ 2-r+ a

. Add itio nal First-Order Filt er Parameter 9•

a-rr 2(1'r +a )

,. 539

CHAP 15 CONVENTIONAL FEEDBACK CONTROLLER DESIGN

IMC Approximate Model Tuning Rules d tuning As noted in Section 15.3, Rivera and Morari et al. [4, 10] have propose l (IMC) Contro Model l Interna the on based lers control rules for PID select a must r designe the method ology. In additio n to the model parame ters, t constan time oop closed-l desired the to s "filter parame ter," .l., which amount more respond will er controll the small, .l. for Thus e. respons er for a first-ord quickly than if it is larger. The tuning rules are given in Table 15.6. Table15 .6 IMC Approx imate Model Controller Tuning Rules [4,10]

Control ler Type

Kc

1'

PI "Improv ed" PI

PID

.l.K

't'D

Tr

Recommended Choice of.l. (it > 0.2 1' Always )

1'

a

2-r+ a 2-l.K

1'+2

2-r+a 2K (it +a)

1'+2

a

-

!>1.7

-

l ->1.7

_!!!_

2-r+ a

a

a

it ->0.25

a

those based Note the similari ty betwee n the direct synthes is tuning rules and 19. r Chapte in this about say to more have shall We on IMC. but the A number of other approxi mate model tuning rules are availabl e, us now Let choice. wide a provide to nt sufficie y ones present ed here are certainl . example an with rules tuning various the of ance perform compar e the

Example 15.6.

CONTROLLER TUNING USING AN APPROXIMATE PROCESS MODEL

PI and PID Consider again the three-tank process of Example 15.1 and now design the process controllers based on the approximate model Eq. (15.33) determined from using change t set-poin a to s response the e Compar 15.5. reaction curve in Example different tuning guidelines. Solution : ers Based on all of the tuning rules discussed above, the controller tuning paramet for the recall, we (where table g followin the in given are control PID for PI and approximate model K = 6, 1'= 15, a= 3).

SINGLE-LOOP CONTROL

540 PI

Tuning Rules

Kc

PID

.!..

Kc

!.

'!"I

TD

TI

Ziegler-Nichols ITable 15.2)

0.75

0.100

1.00

0.167

1.50

Cohen-Coon ITable 15.3)

0.764

0.142

1.15

0.147

1.05

ITAE ITable (15.4)

0.427

0.066

0.637

0.051

1.04

Direct Synthesis (Table 15.5) -r,.=3forPI -r,.=2forPID

0.417

0.067

0.55

0.061

1.36; (6 ..=0.6)

IMC (Table 15.6) l=6forPI l=2forPID

0.417

0.067

0.55

0.061

1.36

0.458

0.061

(hnpro ~PI)

0

20

40

60

80

100

Figure 15.11. Closed-loop set-point change responses for the three-tank process using a PI controller designed with various controller tuning rules (the IMC and direct synthesis results are identical).

CHAP 15 CONVE NTIONA L FEEDBACK CONTROLLER DESIGN

0

20

40

t

60

541

80

100

Figure 15.12. Oosed- loop set-point change responses for the three-tank process using a PID controller designed with various tuning rules (the IMC and direct synthesis results are identical). The set-point responses of the true third-order tank process for PI and PID controller designs using each tuning rule are shown in Figures 15.11 and 15.12 respectively. Note that for PI control, both Ziegler-Nichols and Cohen-Coon are much too dose to the stability limit to provide good responses. The IMC, improved IMC, Direct Synthesis, and ITAE designs have nearly identical settings and provide nearly identical responses. (In fact the Direct Synthesis, and the IMC settings are, for both PI and PID, identical except for the additional filter time constant of 0.6 in the PID case which is so small as to be negligible when compared with the effective process time constant of 15.) The PID responses, shown in Figure 15.12, have similar trends. The ZieglerNichols and Cohen-Coon settings are somewh at overly aggress ive, but the ITAE, Direct Synthesis and IMC tunings are quite good. Based on this example and other general experience, the ITAE and IMC/Direct synthesis settings appear to be the best choice with the approximate model approach.

15.5

CONTROLLER TUNI NG USIN G FREQUENCYRESPONSE MODELS

As we saw in Chapte r 13, process identification sometimes produc es frequencyresponse models in the form (cf. Chapte r 4); y(j{J))

= g(j{J))u(j{J)) + gd (j{J))d(j{J))

(4.18)

where g(j{J)) is a complex function having a real and a comple x part: g(j{J))

= Re( {J)) +jIm( {J))

(4.41)

These transfer functio ns are often represe nted in the form of Bode or Nyquis t plots (see Chapte r 9). Thus it is useful to consid er how to tune controllers

ROL SINGLE-LOOP CONT 54 2

the closed loop. Figure 15.13. Opening

ecialized

There are several sp ll this class of models. of sis ba the on However, he re we wi effectively cy-domain models.

re d on frequen we will discuss mo h.ming methods base ; in a later chapter ds tho me in rg ma focus on stability ds. mo de ls ar e advanced tuning metho m eth od s for fre qu en cy -re sp on se .3.3. The basic St ab ili ty m ar gi n cussed in Section 15 wi ll bring the dis se tho to l ca nti er parameters tha t philosophically ide op system termining the controll ., when the closed-lo principle involves de (i.e y ilit tab ins of rge the ve cking off on the closed-loop system to tability) an d then ba ins d an ity bil sta n the Ziegleree is at the border betw ovide some stability margin. Therefore ed in this pr pli to ap n also be tuning parameters rules of Table 15.1 ca ing tun in rg ma ity Nichols stabil of determining case. ds, we need a means tho me in rg ma ity bil models. Consider In order to apply sta m frequency-response fro y ilit tab ins op -lo feedback loop were to the point of closed n in Figure 15.13. If the ow sh m response ste sy ol ntr ted in the diagram, the the feedback co ica ind as e, vic de ing asur be opened after the me ll be given by: wi Yd in s ge an ch of Ym to (15.34)

or (15.35)

wh er e

this closed-loop transfer function for op -lo en op the as to d ou tp ut variables Here, gl. is referred op system's in pu t an -lo sed en clo the s ate rel -loop response is giv system because it recall tha t the closed we w No d. ene op is when the loop by:

) may be rewrit which, from Eq. (15.35

ten as: (15.36)

the d from the roots of stability is determine m ste sy p loo dse so that the clo given by: aracteristic equation closed-loop system ch ) (15.37

CONTROLLER DESIGN CHAP 15 CONVENTIONAL FEEDBACK

543

rmining the stability of a closed-loop Thus, gL is seen to play a key role in dete system. n by: In the frequency domain, gL(jro} is give (153 8)

stability of the syst em by analyzing the Let us now dete rmin e the closed-loop domain using Eq. (15.38). There are characteristic Eq. (15.37) in the frequency can be readily appl ied - one base d on two equivalent graphical criteria which a Bode plot. a Nyq uist plot, and the other based on

15.5.1 Nyq uist Stability Criterion plot of any process is simp ly a plot of Recall from Cha pter 9 that the Nyq uist plane. Thu s the open -loo p transfer the tran sfer function in the complex be written two way s (d. App endi x B) in function, gu defined by Eq. (15.38) can cartesian form: (15.39) gL(jro) = Re(ro) + j Im(ro) or in pola r form: (15.40)

whe re the amp litud e ratio, AR, and phas

e angle, ,, are given by (15.41)

tKro)

o) J =LgL(jro) = tan- r( lm(r Re(ro)

(15.42)

back to Cha pter 9 and Appendix B for If necessary, the read er may wish to refer more discussion of these relationships. be represented in the complex plan e The relations (15.39) - (15.42) can then . by a Nyq uist plot as show n in Figure 15.14 Im

Re

Figure 15.14.

A Nyquist plot of gL(jOJ).

544

SINGLE-LOOP CONTROL

Note that the curve traced by KLVW) sweeps out a path as frequency w is varied. As we recall from Chapter 9, depending on the order of the system, the curve rotates through 90° (first order), 180° (second order), 270° (third order), etc., or rotates an infinite number of times if there is a time delay. Note also that the AR (the distance of the c..trve from the origin) tends to decrease with increasing frequency. Also note that the AR is proportional to the controller gain because of the definitions Eqs. (15.38) and (15.41). For example, by doubling the controller gain we cause the Nyquist plot to have twice the value of the AR at every frequency.

The Nyquist Map of the Cartesian Complex Plane To use the Nyquist plot in determining the closed-loop system stability, we must first carefully translate our current understanding of stability into the "new language" of the Nyquist diagram. Our current understanding of stability involves investigating where in the Cartesian complex plane the roots of the characteristic equation are located; we must now reconsider what this means in terms of the Nyquist diagram. Let us therefore consider all the possible locations of the roots, r 1 = a1 + j /31 of the characteristic Eq. (15.37) in the complex plane. As shown in Figure 15.15, we recall that roots that cause the process to be unstable will be in the righthalf plane while those which allow stable process behavior will be in the left-half plane. Let us now translate this into the new setting of the Nyquist diagram. For this purpose, we consider the open-loop transfer function gL(s) as a function that takes values of the complex variable s (say, s1) that can have locations in Figure 15.15 and produces another complex number, (say,f1) given by:

I; = gL(si) which can have locations on the Nyquist plot, Figure 15.14. Mathematically, we then say that gL(s) maps the chosen value of s into the resulting value of f. For example, by this "mapping," all the values s = jw (for all values of w) - corresponding to the upper portion of the imaginary axis in the complex plane - give rise to:

f =

gL(jw)

stable

t 10

Re

l Figure 15.15. Plot of possible locations of the roots, r 1 = a 1 + j {31, of characteristic Eq. (15.37).

CHAP 15 CONVE NTIONA L FEEDBACK CONTROLLER DESIGN

545

But recall that the Nyquist diagram is nothing more than a plot of gL(jOJ) in the complex plane. Observe therefore that this mappin g actuall y corresponds to the very curve traced out by the Nyquist diagram. Thus we say that the upper portion of the i'fiUlginary axis is mapped onto the curve gL(jOJ) of the

Nyquist plot.

It is easy to show that the lower portion of the imagin ary axis, corresponding to the values s =- jOJ (for all OJ), gives rise to:

where gL*(jOJ) is the complex conjugate of &CiOJ). Thus, the lower portion of the i'fiUlginary axis is 11Ulpped onto the mirror image of the curve gL(jOJ) of the Nyquist plot. It is now possible to show that all the values s = a+ j fJ for which a> 0, i.e., all the values in the right half of the complex plane, are mappe d by gL(s) into the shaded region ;nside the curve gL(jOJ) in the Nyquist plot of Figure 15.14. Thus we may now conclude:

1. The imagin ary axis in Figure 15.15, the stabilit y bound ary in the Cartes ian coordi nate repres entatio n of the comple x variab le s, translates to the curve gL(jOJ)- and its mirror imag e- in the Nyquis t plot. 2. The location of all the potentially unstable roots of 'the charact eristic equatio n (the entire right-half plane (RHP) in the Cartesian coordin ate representation) translates into the shaded portion in Figure 15.14, the interior of the region swept out by gL(jOJ) as OJ goes from 0 to -. It is now a relatively simple matter to use the Nyquis t plot to investigate stability. For any value of s to be a root of the characteristic equatio n: (15.37)

it must obviously satisfy the condition that gL(s) = -1; i.e., the complex numbe r gL(s) has a real part= -1, and an imaginary part= 0. Observ e therefore that the shaded region in Figure 15.14 must include (or encompass) the point (-1,0) if there are any roots of the characteristic equatio n in the right half of the complex plane. By the same token, if there are no values of s in the RHP that satisfy the characteristic equatio n (indicating that the charact eristic equatio n has no roots in the RHP) the shaded region will exclude the point (-1,0). This leads us to the Nyquist Stability Criterion:

In the Nyquist plot, if N is the net number of times that the curve gL(s) encircles the point (- 1,0) in the clockwise direction, and P is the number of unstable open-loop poles of gL(s) (i.e., P > 0 implies an open-lo op unstable process), then the number of unstable roots of the closed-l oop characteristic Eq. (15.37) (i.e., roots that lie in the right-half plane) is U=N+ P.

There are some immediate implications of this result.

=

1. If the open-loop process gL(s) is stable, then P 0 and the closed-loop process is unstable only if there are net clockwise encirclements of (-1,0).

SINGLE-LOOP CONTROL 546

.,

'1,.

e mu st be at least unstable, then P > 0 and ther controller 2. If the open-loop process is of (-1 ,0) (i.e., N ==- P) for the P counterclockwise encirclements . to stabilize the uns tabl e process ure 15.14, all of controls the size of AR in Fig 3. Since the controller gain, K,, obviously the d (an K, of ice cho ically on the these conclusions dep end crit therefore be y well). The Nyq uist plo t ma oth er controller par ame ters as det erm ine to n erio crit Nyq uist stab ility use d in con jun ctio n wit h the es. valu ter ler par ame stability limits on the control

Detennining Stability Limits

trol ler par am ete r whi ch stab ility lim its on con The re are sev era l way s by roa ch is out line d app One . the Nyq uist diagram from ned rmi dete be may es valu below:

question.

in 1. Obtain gL(s) for the process iabl es; inse rt wil l occ ur as unk now n var ters ame par ler The con trol . K,, the pro por tion al gain pro visi ona l valu es for all but g the value of gL(s) resulting from Step 1, usin 2. Obtain the Nyquist plot for K,= 1. uist plot. The r to as the "ga in neu tral " Nyq This gives wh at we ma y refe the AR at K..,, arb itra ry val ue of K,, say, pre mis e is tha t for any oth er the "ga in on ue val the cise ly K,a tim es eac h freq uen cy wil l be pre erat ing gen for le ilab ava are ms gra uter pro neu tral " Nyq uist plot. (Co mp suc h plots.)

Nyquist plot t at which the "gain neutral" 3. Determine the critical poin intersects the real axis.

(-y,O). Let this poi nt be den ote d as

limits, from the ultimate gain K,u and stability 4. Determine the value of 3.

Step the critical point determined in mat e gai n erio n, we kno w tha t the ulti crit ility stab uist the Nyq

Fro m inte rsec t the d to cau se the Nyq uist plo t to K,u is the valu e of K, req uire in neu tral " "ga us vio pre the wh en K, = Kcu, real axis at (-1,0). How eve r, tha t the so val ues mu ltip lied by K,u, Ny qui st plo t wil l hav e AR axis at l rea the t val ue wil l now inte rsec is the n Ny qui st plo t for this gai n 0) (-1, h wit cide coin this to mak e (-Kcur, 0). The valu e req uire d eas ily dete rmi ned : (15.43a) l K =c•

r

of the real w) wit h onl y one intersection For sim ple Nyq uist cur ves , gL(j e for: axis, the sys tem is the n uns tabl (15.43b) K > lr c

of the real r, for mu ltip le inte rsec tion s Ho wev er as we sha ll see late plex. axis, the situ atio n is mo re com wit h an example: Let us illustrate this pro ced ure

547

ER DESIGN CHAP 15 CONVENTIONAL FEEDBACK CONTROLL Exam ple 15.7

NYQU IST STABILITY TEST OF THRE E-TA UNDE R PID CONT ROL.

NK SYST EM

earlie r, now opera ting with a PID Consi der the three- tank system invest igated a perfectly respo nding contro l valve controller, a perfect measu ring device (h = 1), and ngL(s) will be given by: functio er transf oop open-l the (g11 =1). In this case (15.44)

Determine the range of stable controller gains when: 1

-r0 =0,-= 0

(Pcon trol)

(b)

1 0.0S -r0 =0,-=

{PI control)

(c)

1 -r0 = 10,-= 1.33 (PID control)

{a)

1'1

'~'I

1'1

Soluti on:

·-.-;_

1 (obtai ned using the progr am The "gain neutra l" Nyqu ist plots forK. = P contro l (Figure 15.16{a)), the for that CONSYD) are given in Figure 15.16. Note cts the real axis at (- 0.60, 0), interse curve st Nyqui the and 1 = Kc for system is stable 1/0.60 = 1.667, the system is > Kc gains ller contro 0.6. Thus for all indica ting that by once; the same token the system unstab le becau se the curve will then encircle {-1,()) 1.667. is stable for all Kc < 1/0.60 = l.. Nyqui st plot (for Kc = 1) begins For PI contro l (F'tgure 15.16{b)) the "gain neutra ingL{s)) and finally interse cts factor 1/s the of se (becau axis at-- on the imagi nary le for all contro ller gains unstab is system the real axis at (- 0.732,0). Thus the Kc > 1/0.73 2 = 1.37. ex. As indica ted in Figure For the PID controller, the situati on is a bit more compl oo on the imagi nary axis, but ats begin curve st Nyqui l" neutra "gain the 15.16{c), (-1.48,0) for Kc = 1. Note howev er interse cts the real axis at two places: (-19.8,0) and point (-1,0) is never encircled for the se that the net numb er of encirclements is 0 becau

r=

Kc=l. Obser ve now that: will still not cause the result ing 1. As Kc is increa sed beyon d 1, the amplification process remains stable. the and Nyqui st curve to encircle the point (-1,0), to the right, and the first shifts curve ing result the 2. Howe ver, in decreasing Kc encirc lemen t of (-1,0) will occur when: 1

Kc < 1.48 = 0.674 for Kc values betwe en 0.674 and Subse quent ly, as we contin ue to decrea se Kc, d this point, for: (1/19.8) = 0.051, the system remains unstable. Beyon 1

Kc < 19 . 8 = 0.051 the entire curve shifts beyond the point (-1,0) and

the system becomes stable again.

SINGLE-LOOP CONTROL

548

{a)

lm

(-1,0)

('

-2

( -0.80,0)

I

~

lJ

-6 2

-2

(b)

Im

c:· (-1,0)

-6

~

~

J II -2

-4

(c)

lm

Re (-19.8,0) ...,....--

\ a

lm

"4 2

~

(-1,0) 1\e

(-1!8,0~ 2

4 -4

-2

0

2

4

Figure 15.16. Nyquist plots of the three-tank system under (a) P control, (b) PI control (-r1 = 20), (c) PID control (t1 = 0.75, To= 10).

CHAP 15 CONVENT IONAL FEEDBACK CONTROLLER DESIGN

549

2 I

I

I

~

I I I I I

1

I

A

G I N A

: K0 =0.674

--f~-·~1

0

R y

A

X I

s

: K.,=0.674

1

I I I I

I

- 2:-lr---~--.----r---r~--~--~l~--r---,1 -0.6

-0.2

0

0.2

REAL AXIS

Figure 15.17. Root locus diagram for the three-tank system under PID control (l/T1 = 1.33, TD = 10). Compare with Nyquist plot (Figure 15.16(c)) and Bode plot (Figure 15.20). Thus with the PID controller, the system is "conditionally stable," i.e., forK,< 0.051 and K, > 0.674 the system is stable, but in the range 0.051 < K, < 0.674, the system is unstable. The explanatio n for this type of behavior may be made more explicit by the Root Locus diagram of Figure 15.17. There it is seen that a pair of closed-loop poles cross into the RHP when K, = 0.051, but tum around and cross back into the LHP when K,=0.674.

15.5.2 Bode Stability Criterio n An alternate form of frequency domain represent ation is the Bode plot, discussed in some detail in Chapter 9. Because it is closely related to the Nyquist diagram, it can also be used for determining closed-loop stability from open-loop frequency-response data. Putting the Nyquist stability criterion in the context of the Bode diagram, we observe first that the critical point (-1,0} correspon ds to a situation in which AR = 1 for a phase angle ; =-180. The frequency at which this occurs (shown explicitly on the Bode diagram but not on the Nyquist plot} is known as the crossover frequency, m,. Thus the encirclement condition of the Nyquist criterion may be reinterpre ted in the context of the Bode diagram leading to the Bode Stability Criterion:

If

the AR of the open-loop transfer function, gL(s), is greater than 1 when ifJ = - 180°, the closed-loop system is unstable.

Determining Stability Limits Stability limits for controller paramete rs may now be determine d from the Bode diagram, using the Bode stability criterion. The following is a procedure for doing this:

SINGLE-LOOP CONTROL

550

s) and the n obt ain ler par am ete rs but K, in gL( 1. Specify all oth er con trol K, = 1. a "ga in neu tral " Bode plo t for (using a com put er program) ue of m for wh ich ver fre que ncy m, the val sso cro the ine term De 2. t{l=-180°. tral " am plit ude rati o val ue AR, of the "ga in neu 3. Det erm ine the critical at this crossover frequency. wil l pro duc e an ry val ue of K,, say , Kw 4. Since any oth er arb itra cy, the ulti ma te uen freq , at this cro sso ver am plit ude rati o of K•• AR at m = m,. K,u is ue for wh ich K,, .AR , = 1 val ue K,u is the n tha t val therefore obt ain ed as: (15.45)

st has ten to add the le in Figure 15.16(c), we mu As illu stra ted by the examp to sys tem s tha t cross ility crit erio n onl y app lies restriction tha t the Bode stab st criterion. qui Ny le crossings one mu st use the tfJ = - 180° once. For mu ltip criterion wit h an ility stab e Bod the lica tion of Let us now illu stra te the app example.

THREE-TANK PROCESS BO DE STABILITY TEST OF UN DE R PID CO NT RO L. usse d in Example 15.7, to the thre e-ta nk process disc For the PID controller app lied ility criterion. stab e Bod the g controller gains usin determine the rang e of stable

Exa mpl e 15.8

Sol utio n: K, =1) under: for the control system (when The "gain neutral" Bode plots

a) b) . c)

P control PI control (1/ T1 =0.05) PID control (1 /'1"1 = 1.33, T0

=10) -----r------~

A

1~~-------r-------r--

~ L I

T

u

AR,

D

--r----~~~----~

--+ E 1~•-r-------r--

p H A

0

-200

---------

'' I

--~

.,......':'

G L E -300

to-• Fig ure 15.18.

to•

I

~

~ -100

~

tO

t~·

to-•

t0-1

-----

10

10•

FREQUENCY (RADIANS/l'IME)

Open-loop Bode plo t for thre

e-tank process with P control

; K, = 1.

551

CONTROLLER DESIGN CHAP 15 CONVENTIONAL FEEDBACK A

M p L I

10"

T

-

~~.

u

D E

t-

;I I

to-•

--------

I I

R

I I

A lQ-10

T

-........._ 1111

""10

I 1111

"" 10- 1

""to-•

I 0

1'---.

l.O'

.------r------,

--~-

0~-----,------.-

p H A

~

~

-100

"" :

~-:~+----+----1

~

- 200-4----_-_--_-_-_-l----+--"-_--_.....

G L

\;

E

I

:

~~nrr~_,~m*~,Tn~

-300~-rTTrnmr-rT~~

to-•

10-•

1 FREQUENCY (RADIANS/riME) lQ-1

to•

10

ol; for three -tank proc ess with PI contr Figu re 15.19. Open -loop Bode plot K, = 1, 1/ r1 = 0.05. that the same proc edur e is appl ied in each are show n in Figures 15.1 8- 15.20. Note . fi~ 1.

2. 3.

e ~ =- 180". Determine the crossover frequency, w,, wher itude ratio, AR, , at the crossover Determine the value of the critical ampl frequency we < 1/AR,. Calculate the stability limit on K, from K, A

M p L I T

104

u

D E

R A T I

to-•

0 p H A

I

to-•

lo-•

I I I I

-150

G

102

I I

-50

~ -100

~

10

lo-1:

.,.

L E -200

to-s

______

~

I I I I

{'\

; .. "'--~j "" "" ""

-----~':!"'

to-•

to-t

1

"" 10

102

FREQUENCY (RADIANSfl'IME)

rol; for three -tank proc ess with PID cont Figure 15.20. Open -loop Bode plot K, =1, 1/ r1 =1.33, r 0 = 10.

552

SINGLE-LOOP CONTROL

A little thought will show that this is exactly the same procedure as followed for Nyquist stability analysis and obviously the results are the same: (a) For P control, the crossover frequency is w, = 0.5 and AR, = 0.60; thus the closed-loop system is stable for K, < 1.667. (b) For PI control, the crossover frequency is w, = 0.457 and AR, = 0.732 so that the closed-loop system is stable for K, < 1.37. (c) For PID control, note that there are two crossover frequencies, w,, =0.172, = 0.338, and two critical amplitude ratios AR,, = 19.8, AR,2 = 1.48, and thus the Bode stability criterion does not apply. If one were to apply it in this case, the system would be declared unst;;.ble for K, = 1 which is clearly incorrect.

w,,

15.5.3 Stability Margin Tuning Methods in the Frequency Domain Now that we have stability criteria which we can apply to frequency domain models, let us illustrate some stability margin tuning strategies that have found widespread use in practice.

Gain and Phase Margin Tuning Consider the "gain neutral" Bode diagram shown in Figure 15.21 for open-loop transfer function gL(s) (i.e., with K, = 1). Every "gain neutral" Bode diagram of this type has the following important characteristics: 1. Two critical points, one on each plot: the phase plot has m,, the frequency for which tfl = -180°, and the AR plot has the point where

AR=l.

2. The corresponding values of AR and tfl at each of the noted critical points: AR, is the value of AR at the crossover frequency m,; 1/11 is the value of tfl at the point when AR = 1, the value of the frequency here being m1.

AR

-180 ------------

Figure 15.21. Bode diagram for the open-loop transfer function, gL(s).


553

According to the .Bode stability criterion, if AR, > 1 then the system is unstable; conversely, the system is stable if AR., < 1. Furthermore, we observe that an intuitively appealing measure of how nearly unstable a system is may be defined as how far AR, is from the critical value of 1; or how far ~1 is from -180°. We therefore define a gain margin, GM, as: 1 GM=ARc

(15.46)

and a phase margin, PM, as: PM =

~~-

(-180")

(15.47)

These are quantitative measures of stability margins - measures of how far the system is from being on the verge of instability. (Note that for a stable system, GM > 1 and PM> 0.) Obviously these values depend on the actual value chosen for K, and other controller parameters. Conceptually, we could now use this notion of gain and phase margins for controller design. Observe that we could conceive of a scheme by which a controller g, is chosen for a process, the Bode diagram for the resulting openloop transfer function KL obtained, and the critical points of the Bode diagram investigated first for absolute stability, then for stability margins if the system is stable. Regarding stability margins, it is typical for design purposes to choose the controller parameters to give a GM of about 1.7, or a PM of about 30°. If the chosen g, fails to give a control system that meets the.noted stability margin specifications , another choice is made, and the resulting control system reinvestigated until a g, is found for which the specifications are met. Strictly speaking, this concept of controller design using gain and phase margin specifications is most useful for selecting K, values for proportional-only controllers. For selecting the additional parameters required for other controller types, a time-consuming, trial-and-error procedure must be followed. A more useful, and more popular method for obtaining controller settings from Bode diagrams is now discussed.

Ziegler-Nichols Stability Margin Tuning From previous discussions in this section, we know that two critical pieces of information concerning closed-loop system stability can be extracted from the Bode diagram: the value of K, at which the system will be on the verge of instability, and the frequency at this point; the crossover frequency In Section 15.3.3 we discussed a method due to Ziegler and Nichols [3] by which appropriate controller parameter values can be derived from these two parameters. The following is the procedure for obtaining Ziegler-Nicho ls controller settings from Bode diagrams:

m,.

1. Choosing g, as a proportional- only controller, with K, = 1, obtain the "gain neutral" Bode diagram for the open-loop transfer functiongL of the given system. 2. From this plot, obtain the critical value of AR (i.e., AR,) at the crossover frequency m, (see Figure 15.22}.

SINGLE-LOOP CONTROL

554

(1)• -18 0°

Fig ure 15.22.

(j)-

------------I I

~

e diagram. Controller tuning from a Bod

tion 15.5.2, the re tha t as explained in Sec Observe from her e therefo 1) is given by: = , AR K,u ke ma l (which wil ultimate controller gain K, 11 (15.48)

bility criterion, , according to the Bode sta The imm edi ate implication wh en K, = Kcu. ility tab l be on the verge of ins is now tha t the system wil verge of instability wh en the system is on the ual ly as an 3. It can be sho wn . tha t te out put will oscilla per pet oscillation, (wh en K, = K,u), the sys tem this of iod per The uency m,. und am ped sin uso id wit h freq P , is given by: kno wn as the ultimate period 11 (15.49)

s determined, the ultimate gain and per iod thu 4. With the values of the controller types nt ere diff for rgi n settings Ziegler-Nichols stability ma le 15.1 in Section 15.3.3. ma y be determined from Tab knowledge of the s which dep end only on a Clearly, any oth er tun ing rule ll Alternatively, the re per iod could be use d as we ultimate gain and ultimate uency response; these on oth er aspects of the freq are tun ing rules tha t dep end chapter. will be discussed in a late r

G FRO M A BILITY MA RG IN TU NIN ZIEGLER-NICHOLS STA BODE PLOT. 8. the Bode plo t of Figure 15.1 tem wit h P control hav ing Con side r the thre e-ta nk sys troL con settings for P, PI, PID Determine the Ziegler-Nichols n: Sol utio h pro por tion al con trol . tha t ru, = 0.5, K, 11 = 1.667 wit Fro m Fig ure 15.18, we see P, hols PI, PID settings are as P =12.56 and the Ziegler-Nic Thus the ulti mat e per iod is 11 given in Example 15.4.

Exa mp le 15.9

ROLL CHAP 15 CONV ENTI ONAL FEEDBACK CONT

15.6

555

ER DESIGN

CONTROLLER TUNING WITHOUT A MOD

EL

le loops, it may not be necessary to In many pract ical situations invol ving simp in order to tune the controller. In obtai n a mode l (fund amen tal or approximate) preve nt the deve lopm ent of any t migh s other cases, time or other limitation one. Unde r these circumstances, it form of mode l, even if there was the need for e experiments, in conjunction with is possi ble to use simple; unob trusiv e on-lin to provi de good initial controller 15.1, Table in tunin g rules such as those found expe rime ntal appr oach es for para mete r value s. Let us cons ider two impl emen ting this strategy.

oop 15.6.1 Experimental Dete rmin ation of Closed-L Bou ndar ies

Stab ility

contr oller and expe rimen tally This appr oach invol ves using a prop ortio nal p syste m begin s to oscillate at d-loo close the turni ng up the controller gain until is an expe rimen tal deter mina tion of cons tant ampl itude with perio d Pu· This KCU' P" for a propo rtion al controller. guide lines such as the ZieglerWith these value s deter mine d, the tunin g Table 15.1) may then be used to Nich ols stabi lity marg in rules (show n in rapid ly respo nding loops (such as confi gure a PI or PID controller. For simple, is easy and quick to impl emen t ach appro this local flow or temp eratu re loops), d-loop response, this appro ach is Obvi ously , for large processes with slow close one is bringing the entire process se much less convenient and may be unsafe becau appro ach must be used with this Thus vior. to the poin t of unsta ble beha cauti on.

to Dete rmin e 15.6.2 11Auto Tun ing" with Relay Con troll er Stab ility Boundaries may be used to ampl itude deter mine the ultim ate gain and ultim d in the foun is as such oller contr f on-of le expe rimen ts.. The controller is a simp relay The . 15.23 e is show n in Figur home furnace. The syste m block diagram 6. It zone dead l smal a have may contr oller has a speci fied ampl itude h and with oller contr nal ortio prop gain high very can be creat ed easil y by using a exam ple of the resul ting close dcons train ed contr oller actio n - h S u S h. An and Hagg lund have show n that m loop respo nse is show n in Figure 15.24. Astro close d-loo p respo nse is the the of d perio to a very good appro xima tion, the al contr oller gain is given by: ultim ate perio d Pu' and the ultim ate propo rtion

le relay controller Astrom and Hagg lund [11] show how a simp ate perio d from smal l

(15.50)

resul ting outp ut ampl itude . wher e h is the contr ol ampl itude and A is the s one to deter mine the essen tial allow g," This appro ach, terme d "auto tunin m with only one expe rime nt stabi lity chara cter of the close d-loo p syste (with magn itude h), and outp uts invol ving relati vely low-a mplit ude input s, l distu rbanc es into the norm al smal only s (with magn itude A). This intro duce entire process to be brou ght to the proce ss opera tion and does not requi re the

556

SINGLE-LOOP CONTROL (a)

Yd

set point -

(b)

h------~--._--~----~

Controller output, u

---4--------- --

0

.

-h~----~--~---+

-li

li 0 Error Signal, E

Figure 15.23. Feedback control with relay controller: (a) block diagram, (b) relay controller with amplitude h and dead zone o.

verge of instability. Thus this approach is much more convenient and safer than the first approach. The disadvantage is that we must have available (in hardware or software) a relay controller with which to perform the tuning. In industrial practice, some vendors provide auto tuning cards for electronic controllers, while for distributed control systems, these relay controllers are provided as software. 0.2

(a)

-0.1 -0.2+-~-r~~r-~--.-~-.--~..,

0 0.2

20

40

t

60

80

100 (b)

-0.2 +---.---.--..--.-~.-~--..--~-, 20 0 80 100

Figure 15.24. Closed-loop response with the relay controller: (a) e= y4 -y; (b) u.


557

Figure 15.24 illustrates the performan ce of this auto tuning approach applied to the three-tank process. Here the relay controller is a high-gain proportio nal controller (e.g., Kc = 100) with constraine d small amplitude control action (e.g., h == 0.1). The closed-loop response of this relay controller to an initial disturban ce leads to period Pu = 12.6, and the output amplitude is A = 0.076. Thus the ultimate controller gain, calculated from Eq. (15.50), is Kcu =1.67. Semina [12] shows how these ideas can be applied for tuning unstable and nonlinear processes.

15.7

SUMMARY

We have taken the first step in what is to be an extensive study of controller design for chemical processes by undertaki ng in this chapter a rather comprehe nsive treatment of the design of conventional, single-loop, feedback controllers. We have introduce d the basic principles of single-loop feedback controller design, identifying the main issues as: the controller type decision, and choosing controller parameters (also known as controller tuning). Various concepts, methods, criteria, and techniques typically used for making rational decisions about what controller type to use, and what parameters to choose for these controllers have been introduced. The nature of a substantial number of industrial processes is such that they can be adequatel y controlled by straightfo rward, conventio nal, single-loop, feedback controller s. This chapter is therefore directly relevant for such systems. However, important situations arise in which the conventional singleloop feedback control scheme proves inadequat e. The remaining chapters in Part IVA deal with the issue of controller design (for single-inp ut, singleoutput systems) under circumsta nces in which more is required than the conventional scheme can offer. The issue of controller design for multivariable systems is discussed later, in Part IVB.

REFERENCES AND SUGGESTED FURTIIER READING 1.

2.

3.

4. 5. 6. 7. 8. 9.

Considine, D. M., Process Instruments and Controls Handbook, McGraw-H ill, New York (1974) Smith, R. E. and W. H. Ray, CONSYD EX- An Erpert System for Computer-Aided Control System Design (in press) Ziegler, J. G. and N. B. Nichols, "Optimum Settings for Automatic Controllen;," Trans. ASME,64, 759 (1942) Rivera, D. E., M. Morari, and S. Skogestad, "Internal Model Control 4. PID Controller Design," I&C Process Design Development, 25, 252 (1986) Cohen, G. H. and G. A. Coon, "Theoretic al Considerat ions of Retarded Control," Trans. ASME, 75, 827 (1953) Smith, C. A. and A. B. Corripio, Principles and Practia of Automatic Process Control, J. Wiley, New York (1985) · Murrill, P. W., Automatic Control of Processes, Internation al Textbook Co., Scranton, PA (1967) Lopez, A. M., P. W. Murrill, and C. L Smith, "Controller Tuning Relationships Based on Integral Performanc e Criteria," Instr. Tech., 14, 11, 57 (1967) Rovira, Alberto H., PhD. Thesis, Louisiana State Univen;ity (1981)

SINGLE-LOOP CONTROL

558

d Cliffs, NJ 10. Morari, M. and E. Zafirou, Robust Process Control, Prentice-Hall, Englewoo (1989) s with 11. Astrom, K. J. and T. Hagglund , "Automa tic Tuning of Simple Regulator Specifications on Phase and Amplitude Margins," Automntica, 20, 645 (1984) ," 12. Semina, D., "Automa tic Tuning of PID Controllers for Unstable Processes (1994) Japan Kyoto, '94, Proceedings ADCHEM

REVIEW QUESTI ONS 1.

Apart from the design of the controller itself, what other considerations are important in control systems design?

2.

What are the most important points about sensors and transmitters?

3.

What is the difference between an "air-to-open" and an "air-to-close" valve?

4.

What are the three most common control valve flow characteristics?

5.

Why is the "equal" percentage valve so called?

6.

Apart from the control valve, what are some other final control elements?

7.

action Why is it necessary for a controller to be equipped with a direct/rev erse selection feature?

8.

What is "proportional band" and how is it related to the proportional gain?

9.

What is the main essence of the controller design problem?

ce may 10. What are the three types of criteria by which closed-loop system performan assessed in general?

be

11. What are the most salient characteristics of: (a) a P controller (b) a PI controller (c) a PO controller (d) a PID controller? 12. Under what conditions should the lise of a P controller be avoided? of the PI 13. Why is it that a large number of feedback controllers in a typical plant are type? ? When is 14. When is it advisable to use a PID controller as opposed to a PI controller le? this inadvisab advantage 15. Why do you think a PD controller is seldom used? When it is used what accrues? 16. What is "controller tuning"? al 17. What are some of the strategies available for controller tuning when a fundament available? is model


559

18. What is the fundamental principle behind the use of stability margins for controller design when process models are available?

19. Explain the philosophical difference between the models developed via the principles of process identification discussed in Chapter 13, and the models constructed for the sole purpose of feedback controller tuning. 20. For feedback controller tuning purposes, what is the most commonly employed approximate process model form? 21. What is the process reaction curve, and how is it used for feedback controller tuning? 22. What is the Nyquist stability criterion, and how is it used for controller tuning? 23. What is the Bode stability criterion, and how is it used for controller tuning? 24. Within the context of controller design using frequency-response models, what is a "gain margin" and how is it different from a "phase margin"? 25. What strategies are available for controller design without the use of a "formal" model? 26. What is the fundamental principle underlying the "auto tuning" technique using relays?

PROBLEMS 15.1

The dynamic behavior of a pilot scale absorption process has been approximated by the transfer function model: -0.025 (PIS.!) y(s) = l.Ss + 1 u(s) where y is the deviation of 502 concentration from its nominal operating value of 100 ppm, measured at the absorber's outlet; and u is the absorber water flowrate, in deviation from its nominal operating value of 250 gpm; time is in minutes. (a) It is desired to design a PI controller for this process (assuming for now that the dynamics of the sensor as well as those of the valve are negligible) by assigning the dosed-loop system poles to the same location as the roots of the quadratic equation: (P15.2) Find the values of Kc and 1/ -r1 required to achieve this pole assignment, and identify the location of the resultant closed-loop zero which this technique is unable to place arbitrarily. [Hint: Pay attention to the sign which Kc must have for the controller's

action to be reasonable.] (b) Implement this PI controller on the process and obtain a simulation of the closed-

loop response to a step change of -10 ppm in the set-point for the absorber's outlet 502 concentration. Use the g(s) given in Eq. (PlS.l) above for the process model. (c) To investigate the effect of plant/model mismatch and unmode!ed dynamics on control system performance, consider now that 1. the "true" process model has the same first-order structure, but with a gain of -0.03, and a time constant of 1.0 min; 2. the unmodeled dynamics ignored in part (a) are given by:

560

SINGLE-LOOP CONTROL

Composition Analyzer Dynamics: e-o.2s

(Pl5.3a)

= 0.2s + I

h(s)

Water Line Valve Dynamics: (P15.3b)

0.25s + 1

g.(s)

Introduce this information into a control system simulation package, implement the controller of part (a) on this as the "true" process, and compare the closed-loop response to the same -10 ppm set-point change investigated in part (b). Comment briefly on whether or not the difference in performance is significant enough to warrant a redesign of the controller. 15.2

Revisit Problem 15.1. Reduce the K, value obtained in part (a) by a factor of 10, and implement this new controller first on the process as indicated in part (b), and then implement it on the process as modified in part (c). Compare these two closed-loop

responses. By evaluating this set of responses in light of the corresponding set of closed-loop responses obtained in Problem 15.1, comment briefly on what effect (if any) the use of a smaller controller gain has on the control system's "sensitivity" to the effect of plant/model mismatch and unmodeled dynamics. 15.3

A conventional controller is to be used in the scheme shown below in Figure P15.1 for controlling the exit temperature of the industrial heat exchanger first introduced in Problem 14.2 Assuming that T 8, the brine temperature (in "C), remains constant, the transfer function relations for the prooess have been previously given as:

-(t -

o.se- 10•)

(T -T*)("C)

= (40s + I )(1 Ss + I) (FB - FB•) (legis)

(T -T*)(0 C)

=

o.se-1o.r (T;- T;•)

("C)

(Pl4.3a) (PI4.3b)

Assume that the valve d}"namics are neglig~ble. (a) Use the provided model in a control system simulation package to obtain a

process re~~ction curve; use this process reaction curve to design a PI controller for this prooess based on: 1. The Ziegler-Nichols approximate model tuning rules 2. The Cohen-Coon tuning rules 3. The IMC approximate model tuning rules (Improved PI) ChUled

Brine

Process

Heat Excbanaer

Feed

Stream

Cbllled Brine Out

Figure Pl5.1.


561

(b) Using each of the PI controllers designed in part (a) in turn, obtain the overall closed-loop response of the heat exchanger system (using the actual transfer fl.Ulction models given in Eq. (P14.3)) to:

1. a + 1"C exit temperature set-point change 2. a + 2•c change in the inlet temperature In each case briefly evaluate the performance of each of the three PI controllers. If you had to recommend one controller for use on the actual process, which one would you recommend? 15.4 The process shown in Figure Pl5.2 consists of an insulated hollow pipe of uniform cross-sectional area A m 2, and length L (m), through which liquid is flowing at a constant velocity, v (m/s). It is desired to maintain the temperature of the liquid stream flowing through the pipe at some reference value, T,OC, and a control system is to be designed to achieve this objective. A temperature controller (TC) has been set up to receive T, a temperature measurement, and use this information to adjust the quantity of heat, Q (kW.hr) supplied to the inlet stream by the electrical heater. The electric heater is so efficient that the temperature at the pipe inlet T; responds instantaneously to changes in Q; however, the only thermocouple available for temperature measurement has been installed right at the pipe outlet (see diagram). Thus, if we define the following deviation variables:

assuming the thermocouple has negligible dynamics, and that it is equipped with built in calibration so that its output is identical to the measured temperatUre, in •c, then the equations describing the behavior of this system are: x(s)

=

Ku(s)

(PIS.4)

where K is the heater constant (units • •ctkW.hr), and: y(s)

= e-as x(s)

(PIS.S)

(a) Draw a block diagram for the entire feedback control system; be careful to label

every signal and identify every block.

(b) Consider the specific situation in which a pure proportional controller is used on a pipe whose length is 1 m, and whose cross-sectional area is 0.05 ull; liquid is flowing through this pipe at a volumetric rate of 0.2 m 3 /min. If the heater constant for this system is 0.4 "C/kW.hr, what is the ultimate gain, K1,? T

Figure P1S.2.

SINGLE-LOOP CONTROL

562

(c) Use the Ziegler-Nichols stability margin tuning rules to design a PID controller for the system descnbed in part (b). Implement this controller and obtain the closedloop system response to a step change of +2oC in the fluid temperatur e set-point. 15.5 The liquid level in a drum boiler used in the utilities section of a research laboratory complex is controlled by the feedwater flowrate as indicated in Figure P15.3. An approximat e transfer function model representing this process is given by:

=

y(s)

0.1(1 - 2s) s(Ss + 1) u(s)

(P15.6)

where, in terms of deviation variabies, y is the drum level (in inches) and u is the feedwater flowrate (in lb/hr), time is in minutes. Steam

l

LC

I;! I" ....,

~

•

J Feedwater

Heating

Figure P15.3. It is known that there are no significant sensor dynamics, but that the valve

dynamics are represented by:

1.2 g.(s) = (0.8s + 1)

(PJ5.7)

(a) Design a proportion al controller for this process by generating an appropriat e Bode diagram and using the Bode stability criterion in conjunction with a gain margin of 1.7. Redesign the proportiona l controller using a phase margin of 30". Which controller is more conservative? (b) Using a control system simulation package, implement the two controllers obtained in (a) and obtain a simulation of the closed-loop system response to a step change of 5 inches in the drum level set-point. Plot both they and u response in each case. Briefly evaluate the performance of each controller. 15.6

For a pilot scale, binary {Ethanol/W ater) distillation column, the process reaction curve, representin g the experiment ally recorded response of the overhead mole · fraction of ethanol to a step change in the overhead reflux flowrate, has been · approximat ely modeled by the transfer function: g(s) =

0.66e-2.6s 6.1s + 1

{P15.8)

(a) From an appropriat e Bode diagram for this process, obtain the parameters necessary for designing a PI controller using the Ziegler-Nichols stability margin tuning rules (fable 15.1). (b) It has been suggested by Fuentes (1989),t that given a process reaction curve represented by the transfer function: g(s)

=

Ke - - - - - ' shut

0

100%

Controller output signal ('ILl

Figure 16.16. A nonisothermal batch reactor under split-range control; (a) controller configuration; (b) split-range schedule.

16.3.1 Split-Range Control Split-range control is the usual solution when one requires multiple manipulated variables in order to span the range of possible set-points_ The concepts may be explained with a simple, but common example. Let us suppose that we have a nonisothermal batch reactor with a certain specified temperature program required to produce the product. However, the temperature program covers the range from 15"C at the beginning of the batch to too·c at the end. In the plant the cooling water is at s·c and the lowpressure steam is at tso·c. Obviously we will require both cooling water and steam in order to span the temperature range of interest. Split-range control allows both cooling water and steam to be used as shown in Figure 16.16. Note that the schedule allows the appropriate fraction of cooling water and steam to be used for each value of the controller output signal and reactor jacket inlet temperature required. Usually the design calls for overlap in the range of each valve for more precise control. Split-range control is often employed for broad span temperature control of small plant or pilot scale processes, and for year-round heating and cooling of office buildings.

16.3.2 Multiple Inputs for Improved Dynamics Sometimes multiple inputs are employed in order to speed up the dynamic response of a process during serious upsets or transitions between set-points (i.e., to improve servo behavior). To illustrate the basic principles, consider the stirred heating tank of Figure 16.3, but now assume that there is an additional auxiliary electrical heater and cooler that can be used to control the tank temperature as shown in Figure 16.17. The heating tank has several

SINGLE LOOP CONTROL

584

temperature set-points depending on the product being made in a downstream reactor. The normal regulatory operation at each of these set-points can be readily handled with the process steam going to the tank heating coil; however, it is important for the downstream reactor that set-point changes be made rapidly. Unfortunately, the stirred heating tank temperature responds rather slowly even with the steam valve full open (to go to a higher temperature) or full shut (to go to a lower temperature). Thus for an increase in temperature set-point the auxiliary heater helps speed the move to the new set-point. Similarly when the temperature set-point change is down, the auxiliary cooler helps the tank quickly achieve the new set-point. The drawback is that this auxiliary heater and cooler require expensive electrical energy while the steam to the coil is very cheap because it is produced as a byproduct from another process unit. Thus energy efficiency requires that the auxiliary heater and cooler be used oniy during set-point changes. This could be accomplished with three controllers as shown in Figure 16.17 with a schedule defining when each controller is active. For example, one could choose the following schedule to achieve the control objective.

CONTROLLER SCHEDULE Deviation from Set-point ("0

Controllers Active

AT!< 1----flrt+ Water

Figure P16.7.

SINGLE LOOP CONTROL

596

Produc tion Rate

Production Rate

Estima tor

Controller

Figur e P16.8. use of a Vanadium catalyst and an alumi num 16.10 A polymerization process requires the the catalyst flowrate is used to control the ver, Howe cocatalyst in a fixed ratio. tly as shown in Figure P16.8. produ ction rate and therefore varies independen ring the catalyst and cocat alyst measu for ble availa are itters Two flow transm about desig ning a ratio control go would you flowrates. Briefly describe how am is Figure P16.8 to show your diagr ss scheme for this process. Modify the proce e. ratio control schem of a parallel control struct ure to obtain 16.11 Figur e P16.9 show s a typical application nger (cf. Balchen and Mum me 1988t). excha heat a in l contro better exit temperature heat exchanger is divide d into two; the by heated The stream of cold liquid feed to be throu gh the heat exchanger; the other the main portion, of fixed flowrate, passes ses the heat exchanger, and is later bypas ate, flowr table portio n, with an adjus Figure P16.9. It is desire d to control in ted mixed with the heated stream, as indica both the steam control valve on the the final, mixed stream, tempe rature T, with the heat exchanger bypas s stream as well as nger, excha prima ry side of the heat control valve, available for manipulation. on T is on a slower timescale than the Because the effect of steam flowrate changes el control schem e indica ted in the parall the ate, flowr effect of the bypas s stream TCl, a proportional-only controller, ller figure is to be used. The temperature contro

T Mixed Steam Temperature

Conde nsate

Figur e P16.9.

s Control: Structures t J. G. Balchen and K. I. Mumme, Proces . Nostr and Reinhold (1988), pp. 33-35

and Applications, Van

CHAP 16 DESIGN OF MORE COMPLEX CONTROL STRUCTURES

597

adjusts the bypass stream valve on the basis of the mixed stream temperature measurements. However, the output of this controller is fed to the second temperature controller, TC2, a PI controller, which uses this signal to adjust the steam valve. (a) Draw a block diagram for this process, assuming that the transfer functions representing the effect of the bypass stream flow on Tis g1(s), and that representing the effect of steam flowrate on Tis gis). (b) Provide a qualitative discussion of how such a control scheme will respond to a disturbance of a unit step decrease in the cold liquid feed temperature, assuming the bypass valve was initially 50% open just before this event. To what value will the bypass valve opening ultimately settle? Contrast this with a "standard" feedback scheme in which TC2 alone is used to control T. Which scheme do you think will be more effective, and why?

16.12 An opportunity for investigating the "parallel" control structure arises with the following temperature control problem for an exothermic reactor subject to feed temperature disturbances. The reactor is traditionally cooled by jacket cooling water, and an approximate mathematical model for the process (in the vicinity of the normal operating conditions) has been obtained as: -4.0e-O.ls 1.5e-o.b y(s) = 15.0s + I "•(s) + 2.5s + I d(s)

(P16.9)

withy as the reactor temperature (0 F), u1 as the jacket cooling water flowrate (gpm{, d as the reactant feed temperature (0 F), all in terms of deviation variables. The e-o. 5 is the approximate dynamics introduced by the temperature sensor. In order to increase the cooling capacity, a condenser was installed as shown in Figure P16.10, with the objective of using condenser cooling water flowrate as an

~Condenser Cooling Water

I

@

Feed

--~--'-+-----f)!-:--'-7~---

0~----------------TimeNel!"tive

Positive

Steady State

Steady State

Gain

Gain

Time ..... (b)

(a)

Figure 17.1.

Step responses for processes with "normal" open-loop dynamics.

The control system structures discussed up to this point do an adequate job of controlling processes with "normal" dynamics as we have seen from many previous examples. By contrast, conventional control systems often do a poor job of controlling the three classes of processes with difficult dynamics. Thus we need to analyze the key features of these difficult processes and provide new control system designs for improved closed-loop performance of these processes.

17.1.1 Characteristics of Difficult Process Dynamics From our studies in Part II, we know that the presence of any of these difficult characteristics in the dynamic behavior of a process will be identified in the process model as indicated below: • Time delay • Inverse response • Open-loop instability

transfer function has the e-as term transfer function has a RHP zero transfer function has a RHP pole

Processes that exhibit "normal" behavior do not have any of these terms in their transfer function models. As we recall from Chapter 8, when the response of a process is as depicted in Figure 17.2(a), i.e., it exhibits an initial, distinct period of no response, we have a time-delay (or dead-time) process. A process with time delay thus violates the first c~ndition noted above for normal dynamic behavior: it does not respond instantaneously to input changes. Unfortunately, a substantial number of chemical processes exhibit time-delay (or time-delay-like) behavior.

tctclL Time._

(a)

Figure 17.2.

Time._

(b)

Time._

(c)

Step responses for processes with difficult dynamics: (a) time delay, (b) inverse response, (c) open-loop unstable.

CHAP 17 CONTROUERS FOR PROCESSES WITH DIFFICULT DYNAMICS

601

A process with inverse response violates the second condition for normal dynamic behavior in that, even though its step response eventually ends up heading in the direction of the new steady state, it starts out initially heading in the opposite direction, away from the new steady state, changing direction somewhere during the course of time. Such a response is depicted in Figure 17.2(b). Examples of physical processes that exhibit this unusual dynamic behavior were given in Chapter 7. As we saw in Chapter 11, a process for which the step response is unbounded, i.e., the output increases (or decreases) indefinitely with time, as depicted in Figure 17.2(c) (or, also as in Figure ll.l(b)), is said to be open-loop unstable for obvious reasons. An open-loop unstable process thus violates the third condition noted above; its output fails to settle to a new steady-state value in response to a step change in the input. The most well-known chemical processes that exhibits open-loop instability is the exothermic CSTR. Before we analyze these three classes of processes in more mathematical detail, it is useful to consider physical reasons why they would cause difficulty for a conventional feedback controller: 1.

Processes with time delays have a significant delay before they respond to control action (cf. Figure 17.2(a)) so that controllers with aggressive action (high controller gain) will tend to overcompensate and become unstable. Thus there is a limit on the controller gain that can be used for a process with time delay.

2. Processes with inverse response will initially move in the wrong direction as they respond to control action (cf. Figure 17.2{b}). Thus if the controller is tuned too tightly (high controller gain) it will attempt to correct for the movement in the wrong direction and overcompensate. Again there is a limit on the controller gain that can be used for a process having inverse response. · 3. Processes that are open-loop unstable will "run away" without controL so most of the controller tuning procedures of Chapter 15 cannot be applied. In addition, open-loop unstable processes can be unstable for various reasons so that simple PI control may not be enough to stabilize them. These difficult dynamics translate into unusual phase behavior as we see next.

17.1.2 Nonminimum Phase (NMP) systems Processes with time delays and with inverse response are sometimes collectively referred to as nonminimum pluzse (NMP) systems, a term first introduced by Bode [1]. Let us explain the meaning of this terminology. A process with a general, "normal" transfer function, g 1 (s) has identical amplitude ratio (AR) characteristics as the process whose transfer function is g2 (s) =g 1 (s) e-as, but the phase angle. characteristics are very different: that much we know from Chapter 9. H g 1(s) has n poles and m zeros, as we may recall, its phase angle 4'1(s), approaches (n - m) x (-90') asymptotically at high frequencies. The phase angle of g2(s), on the other hand, is given by:

412(s) = 41 1(s)- am

(17.1)

SINGLE-LOOP CONTROL 602

delay term. ly due to the influence of the sho w ide nti cal AR and decreases monotonical t tha tra nsf er fun ctio ns Th us of all pos sib le be the one hav ing the ses sin g a time delay cannot characteristics, the one pos characteristics. mi nim um possible pha se functions: processes hav ing transfer two In a similar vein, the

(17.2)

(17.3)

and

c limit of the pha se ang le teristics but the asymptoti sho w identical AR charac g {s). Thus, the process the corresponding value for 1 s of all processes of ess exc in 0° -18 is ) for gis teristic allest possible pha se charac g 2(s) doe s not hav e the sm s: stic teri rac cha ntic al AR wit h wh ich it sha res ide

, gn(s)} having s of processes {g1(s), gis ), ... Thus in general, within a clas identical (i.e., all , A~} {AR 1, AR2, ••• , s stic teri rac cha o rati ude amplit , t/1 } of which the and phase angles {1/Jp , 2, ••• 11 AR 1 = AR2 = ... = AR11 ), e-delay element or a in' if gi contains a tim minimum is designated 41m RHP zero, then tP; :;t. tPmin.

to as non mi nim um pha se suc h sys tem s are referred It is in this sen se tha t

se for time-delay

m pha reserve the term nonminimu It has bee n cus tom ary to . Wh at is not wid ely [2]) le mp exa for e (se y onl s tem sys e ons esp e-r le sys tem s also and invers fact tha t ope n-l oop uns tab led ged , how eve r, is the

systems.

ack now se characteristics. exhibit ncn mi nim um pha t the first-order process: Observe, for example, tha

(17.4)

as . has identical AR beh avi or

cess: the open-loop unstable pro (17.5)

cess does not ngless since the unstable pro (alt hou gh the AR is meani t: tha r, it can be sho wn out put amplitude); howeve

hav e finite (17.6)

m phase. le process is also non min imu so tha t the open-loop unstab act ual ly the RH P pole s wa it [1], de Bo by ted sen pre ally gin ori as In fact, - and not the sys tem s in , of course, the time delay and m zero P RH the and erred to as the non min imu the y app ear , tha t we re ref se pha um nim mi wh ose tran sfe r functions non any wh ich therefore contains pha se elements. A sys tem a NMP system. P zero, or a time delay, is RH a e, pol P RH a it be nt, me ele to focus atte ntio n on in systems we have decided Observe therefore tha t the non mi nim um pha se tain feature: the y all con n mo com a re sha r pte thi s cha

DIFFICULT DYNA MICS CHAP 17 CONT ROUE RS FOR PROCESSES WITH

603

ss system's transfer function elements. The presence of NMP elements in a proce ult" dynam ic behavior; it is is thus identified as being responsible for its "diffic in control systems design. lty difficu of nt amou also the source of a considerable ics is prese nted in Anoth er aspect of processes with "difficult" dynam when, in order to arise which Chapt er 19; there we show the serious problems e of the process invers an in conta must ller contro achieve high performance, the mode l l problems presented by Let us now consider in more detail the special contro available for handl ing iques techn n desig ller these systems, and the contro them.

17.2

TIME-DELAY SYSTEMS

17.2.1 Cont rol Prob lems l loop, When a time-delay element is present in a feedback contro

the following

obvious control problems arise:

action will be based on 1. With a delay in the measuring device, contro l that is usual ly not ation delay ed, hence obsolete, proce ss inform s. proces the within ion situat t representative of the curren of a transp ort delay or 2. H the process itself has an input delay (because l action will not be contro the of effect the then other process feature), the probl em even imme diatel y felt by the process, comp oundi ng furthe r. instability. It is easy to see how such a situation could provoke ilization can be cause d destab much how To provi de a quantitative sense of function, recall Figure er transf ss proce a in nt eleme by the presence of a delay e that the phase observ ; 9.3, the Bode diagra m of a "normal" first-order system Thus from our -90". of value ng limiti angle asymp totica lly approaches a unstab le under go never can system al" "norm this 15, er discussion in Chapt attain the never can angle propo rtiona l feedb ack contro l since the phase critical value of -180". loop transfer function By contrast, consider the Bode diagra m for the openNow the phase 9.18. e Figur of a first-order system with time delay given in once was a there where : ormed transf lly radica angle characteristics have been decreases angle ·the phase limiting value of -90", there is now in fact no limit; value of ng limiti a now is mono tonica lly with frequency. Thus there and the -180° s crosse angle phase the which at propo rtiona l controller gain delay , time the of value the system becomes unstable. Furthermore, the highe r es. becom gain ller the smaller this limiting value of contro

· 17.2.2 Conv entio nal Feed back Cont rolle r Desi gn Let us consider the general model for a process with

time delay as: (17.7)

SINGLE-LOOP CONTRO L

604

where g•(s) has "normal " dynamics. Under conventional feedback control as shown in Figure 17.3, the closed-loop system has the characteristic equation: 1 + gJI•(s) e-as

=0

(17.8)

As discussed above, the increased phase lag of the delay term requires, for closed-loop stability, a reduction in the allowable value of the controller gain. The closed-loop system will therefore have to be more sluggish than the correspo nding system without delay. Therefore even though conventi onal controllers can be used for time-delay systems, we usually have to sacrifice speed of response in order to have closed-loop stability.

Classical Techniques y The techniques for designing conventional feedback controllers for time-dela 15. Chapter in d discusse already have we what systems are no different from rs As evidence d by the performance of the conventional feedback controlle system: y designed in Chapter 15 for the example time-dela 6e-3'

g(s)

(17.9)

= 15s + 1

(see Figures 15.11-15.12) acceptable results can be obtained using these classical techniqu es provided the time delay is not too large. In this case, closed-loop stability, and acceptable overall closed-loop performance can be obtained without sacrificing too much of the speed of response. The major problem with conventional feedback controller design for timedelay systems lies in the fact that for systems with large time delays, consider ably smaller controller gains are recomm ended yielding sluggish response. Under these circumstances, it is customary to employ a scheme that explicitly takes the presence of the time delay into consideration and actively y compensates for it. Such an approach is referred to, in general, as time-dela

compensation.

17.2.3 Time-Delay Compensation To implement time-delay compensation, we can alter the control scheme shown in Figure 17.3 to the new configuration of Figure 17.4 where a minor feedback loop has been introduc ed around the conventional controller, as shown. The subscript m denotes a model, while unsubscripted elements are the real process. Thus:

(17.10)

g*(s)e-as

Figure 17.3.

.. y

Block diagram of a time-delay system under conventional feedback control.

605

CHAP 17 CONTROLLERS FOR PROCESSES WITH DIFFICULT DYNAMICS u(s)

Figure 17.4.

1------..-•Y.

Block diagram incorporating the Smith predictor.

represents the model of the real process. Let us define new variables y*(s),

Ym *(s) as follows: (17.11)

y*(s) = g*(s) u(s)

Ym *(s) = gm *(s) u(s)

Since the actual process output y is given by: y(s)

= g*(s)e-as u(s)

(17.12)

then y*(s) is the output of the "undelayed" version of the process output y(s).

Let us proceed to analyze the time-delay compensator assuming for the moment that there are no model errors (so that gm(s) =g(s) and am= a). Observe that the signal reaching the controller, designated as ec in the diagram, is a "corrected" error signal given by:

ec

(17.13)

= y d- y(s)- fy*(s) - y(s)]

or (17.14)

implying, as a result, that the block diagram of Figure 17.4 is apparently equivalent to that shown in Figure 17.5. If this is so (and we will soon show that it is) then the net result of the introduction of the minor loop is therefore to eliminate the time-delay factor from the feedback loop - where it causes stability problems - and "move" it outside of the loop, where it has no effect on closed-loop system stability. The characteristic equation of the equivalent system in Figure 17.5 is: (17.15)

1 + gc8*(s) = 0

which no longer contains the time-delay element and therefore allows the use of higher controller gains without placing the closed-loop stability in jeopardy. y

y*

Figure 17.5.

Equivalent block diagram for system incorporating the Smith predictor.

'l

SINGLE-LOOP CO NTR OL 606

p of Figure 17.4 Smith [3, 4], and the minor loo This control scheme is due to l be exp lain ed wil t tha s son ith predictor for rea Sm the as wn kno lly era gen is 17.4 tha t the sho rtly . the block dia gra m in Figure Let us establish directly from in, ass um ing Aga 15). (17. Eq. in ation is really as aro und the p closed-loop characteristic equ loo or rs, we can consolidate the min u as: and tha t the re are no model erro £ n wee bet -function relationship controller to obtain the transfer

(17. 16)

wh ere

(17. 17)

The overall closed-loop transfer Y

function is now given by:

=

g*(s) e-asg, * 1 + g*( s) e-a •g, * YJ

(17.18)

17) for g/, we obtain: If we now introduce Eq. (17. (17.19)

so that: 1 + g*g , -:: :-: -- -" --g*g , e-rn 1 + g*(s) e-asg,* = -g,g* 1+

(17.20)

n in Eq. (17. 18) by the closed-loop transfer functio Simplifying the expression for 20) gives: combining Eqs. (17.19) and (17. (17.21)

establishing two things:

t to tha t in Figure dia gra m is ind eed equ iva len 1. The Figure 17.5 block function for the sfer is the closed-loop tran 17.4 , because Eq. (17. 21) Figure 17.5 system.

in Eq. (17 .15) , by n for the system is as given atio equ tic eris ract cha The 2. Eq. (17.21}. inspection of the denominator wn in general we have just considered is kno The specialized control scheme min or loop (which ause, as we hav e seen, the as time-delay compensation bec ced to compensate odu ventional controller) was intr differentiates it from the con often referred to re efo ther is p y. The minor loo dela e tim the of ce sen pre the for as a compensator.

ULT DYNAMICS CHAP 17 CONTROLLERS FOR PROCESSES WITH DIFFIC

607

this time-d elay It is impor tant to note the follow ing points about compensation scheme: the signal y* to the 1. The effective action of the compensator is to feed y. output s proces actual the of controller instead observe that: 2. By the definition in Eqs. (17.11) and (17.12), we may y*(s) = ea-; y(s)

so that: y*(t)

= y(t + a)

(17.22)

y(t) exactly a and it becomes perfectly clear that y"(t) is a prediction of genera lly tor" predic h "Smit name the time units ahead , hence e. schem the with associated

proces s model is 3. The schem e will work perfec tly as long as the its performance. affect perfectly known; modeling errors will obviously techni que is its tor predic Smith the of The most significant criticism errors. ing model sensitivity to delays are due to 4. In those chemical processes for which the time pipes, the time long h throug energy or and/ al materi of transp ort fluid flowrate; the with vary delays observed for such processes will vice-versa. and , delays time lower to rise giving te an increase in flowra delays and time nt consta for ed design is e schem tor predic Smith The delays which may therefore not perform as well for systems with time time. over vary significantly

lJesign Procedure predic tor for timeThe following is the design proced ure for using the Smith delay compensation: 1. Design the Smith Predictor (the minor loop) the controller Recall that the implication of this minor loop is to cause errore ; i.e., actual the of to utilize a "corrected error" signal e, instead the control law is: (17.23)

where

e, = e- (y* -

y)

(17.24)

by which y* The design of the minor loop involves setting up a means simula tion, by lly typica , model s proces the from and y are produ ced s model , proces the using a digital computer; y is obtained directly from . model s proces the of n versio yed and y* is obtained from the undela

608

SINGLE-LOOP CONTROL

2.

17.3

Design g, Since, according to the Smith predictor scheme, the controller "thinks" it is controlling the process g•(s) for which there is no time delay (see Figure 17.5}, we now design the controller on this basis, using any of the previously discussed techniques; i.e., design Kc for the undelayed version of the time-delay system. Observe that the absence of the time delay from the apparent process permits the use of much higher controller gains than would otherwise be allowable. However, since, for perfect delay compensation, the Smith predictor requires a perfect model, and real models are never perfect, we must be cautious in choosing the controller parameters for g,. In practice, one would choose controller parameters large enough to achieve much better performance than feedback control alone, but not so large as to cause serious deterioration in performance resulting from inevitable plant/model mismatch.

INVERSE-RESPONSE SYSTEMS

17.3.1 Control Problems The major problem created by the inverse-response system is the confusing scenario it presents to the automatic controller: observe that having taken the proper action that will eventually yield the desired result, the controller is first given the impression that it, in fact, took the wrong action. The "uninformed" controller, in reacting to this illogical state of affairs, is liable to compound the problem further. It is therefore not difficult to see that such a system's closed-loop stability stands in real jeopardy. One can obtain a sense of the destabilizing effect caused by the presence of a RHP zero in a system's transfer function by recalling Figure 9.16, the Bode diagram for the process whose open-loop transfer function is:

s _ (1- 3s) g() - (2s + l)(Ss + 1)

(17.25)

Without the RHP zero in the numerator, the transfer function will be second order, and from our results in Chapter 9, we know that the phase angle will asymptotically approach a limiting value of -180° - implying that the closed-loop system will always be stable, since, theoretically, the phase angle only attains the critical value of -180° at infinite frequency. ' With the RHP zero present, observe from Figure 9.16 that the limiting value for the phase angle has been altered to -2700 and as such, there is now a finite (crossover} frequency at which ~ = -180°, and hence there is a limiting value of K, above which the system will be unstable. Typically, inverse-response systems are controlled either using conventional PID control or an inverse-response compensator, similar in nature to the time-delay compensator discussed in the previous section. Let us look at the details of these controller designs more closely.

CHAP 17 CONTROLLERS FOR PROCESSES WITH DIFFICULT DYNAMICS

609

17.3.2 Conventional Feedback Controller Design Because the derivative mode of the PID controller endows it with "anticipator!" character, this controller, of all the conventional controllers, is able to cope somewhat with controlling the inverse-response system; it does this by anticipating the "wrong way behavior" and appropriately accommodating it. It is not difficult to visualize how the PID controller achieves this: at the initial stage of the process response, rather than the error reducing in response to the correct controller action, it actually increases because of the inversion. However, the derivative of the response is negative during this period, and when this information is incorporated into the controller equation (as is the case with the PID controller), the result is a net reduction in the magnitude of the control action. On the other hand, immediately after the inversion is over, and the response begins heading in the right direction, the derivative is positive and usually quite large; with the PID controller, this translates to a net increase in control action. Thus, at the initial stage, when the initial, temporary increase in the error would mislead other conventional controllers into increasing the control action (a potentially dangerous situation), the PID controller, with its access to the derivative of the error, actually cuts back on the applied control action; when the initial period of inversion is over, and it is now advantageous to step up the magnitude of control action in order to make up for lost ground, the PID controller is again able to do precisely this, because of the positive derivative of the error in this region.

Classical Techniques: Ziegler-Nichols Designs As demonstrated by Waller and Nygardas [5], the Ziegler-Nichols settings for PID controllers yield acceptable control of inverse-response systems. (Later in Chapter 19, we will uncover another justification for why, of all conventional feedback controllers, only the PID type can be used to control the inverseresponse system with any degree of success.) Let us illustrate with an example. Example 17.1

DESIGN OF A CONVENTIONAL PID CONTROLLER FOR AN INVERSE-RESPONSE SYSTEM BY THE ZIEGLERNICHOLS METHOD.

Design a PID controller for the inverse-response system of Eq. (17.25) using the frequency-response Ziegler-Nichols technique. Solution:

Figure 17.6 shows the Bode diagram for the system under proportional-only control: i.e., the open-loop transfer function is given by:

g(s)gc(s)

= (2s +

l)(5s

+ 1)

(17.26)

SINGLE-LOOP CONTROL

610

--- - --==---:-. I

1

0.6

I I I I

~ ............

-.........

,.

~

-............ ........_

-100 -200

it-- - (l)co

10

1

I

I

--X

= 0.55----..:

'-

"'

10 frequency (radians/time)

Figur e 17.6.

Bode diagra m for Example 17.1.

critical pieces of inform From this diagra m, we obtain the follow ing the Ziegler-Nichols design; wco = 0.55 radia ns/tim e • The crossover frequency: MR = ARIKK.c = 0.5 point this at ratio • The magn itude From here, it i5 easy to see that the ultima te gain 2 and

ation requir ed for

and period are given, respectively, by; (17.27a) (l7.27 b)

11.43

The Ziegler-Nichols recom mend ed value s obtain ed as: -r1 = 5.7;

for the PID contro ller param eters are now

We will evalu ate the perfor mance of this contro

ller shortl y.

gns Classical Techniques: Cohen and Coon Desi process reaction curve technique for It is also possible to use the Cohen and Coon se-resp()nse syste ms; again the obtai ning PID contr oller desig ns for inver procedure is the same as before: 1. Obtain the process reaction curve (PRC), te gain, an effective time 2. Char acter ize the PRC with a stead y-sta , delay time tive effec an constant, and the Cohe n and Coon PID 3. Use the chara cteri zing param eters in 15. ter Chap of ulas controller param eter form approximately as a first-order Of all these steps, characterizing the PRC difficult from a practical most the ally typic syste m with a time delay is decid ing on wher e to draw the stand point . Natu rally , the difficulty lies in tive time delay and the effective effec tange nt line in order to estimate both the ique for taking the guesswork techn ing follow the time constant. We recommend out of this step.

ICS CHAP 17 CONTROLLERS FOR PROCESSES WITH DIFFICULT DYNAM

611

for Recall that at the end of Chapter 8 we introdu ced Pade approximations a ces introdu mation approxi ar time-delay systems. We see that this particul function transfer the into pole LHP image mirror a as well RHP zero element as as an approxi mation for the time-delay element. can This suggests that, by reversing the Pade approximation, a RHP zero LHP zero; image mirror a and element lay time-de a by mated approxi be also i.e., if: 1- ~s 2 e-os ... ---=-1+

~s 2

then, by the same token: (17.28)

1 - 1jS ,. (1 + 7jS) e-2 11•

. · Let us illustrate how to apply this procedure with the following example INVERSEExample 17.2 APPROXIMATE TIME-DELAY MODEL FOR AN RESPON SE SYSTEM USING A REVERSE PAD~ APPROX IMATIO N. of Eq. (17.25) Obtain an approxim ate time-delay model for the inverse-response system s of both the using the reverse Pad~ approxim ation and compare the unit step response ation. approxim y time-dela its and inverse-response system

Solution : which 71 = 3, Since the original system transfer-function model has a RHP zero for replacin g requires ation approxim Pad~ reverse the using .28) (17 to~ g accordin then by: given therefore is model y time-dela ate approxim The . this with: (1 + 3s) e .''·' .''.'!

g(.r)

(1 + 3.s)e-6• = (2s + l)(Ss + 1)

(17.29) system of

Figure 17.7 shows the unit step response of the original inverse-r esponse le Eq. (17.25) and the time-delay approxim ation Eq. (17.29), indicatin g quite reasonab agreement. p=--

1.00

. / .

0.75

I

0.50

! I!

2

... 0.25

0.00

-0.25

~ 0

Figure 17.7.

10

20 Time

30

and its Compari son of the unit step responses of an inverse-response system ation. approxim Pad~ reverse a using ation time-delay approxim

SINGLE-LOOP CONTROl.

612

the Cohen and Let us now demonstrate the design of a PID controller by e. exampl ng followi the with method Coon process reaction OLLER FOR Exampl e 17.3 DESIGN OF A CONVE NTION AL PID CONTR AN INVERS E-RESP ONSE SYSTEM BY THE COHEN AND COON METHO D. of Exampl e 17.1 (whose Design a PID controll er for the inverse- respons e system curve method to reaction process the using transfer function is given in Eq. {17.25)) ns. Compa re the obtain the Cohen and Coon controll er parame ter recomm endatio in the set-poin t under behavio r of the closed system in respons e to a unit step change Ziegler- Nichols PID the under r behavio this controll er with the corresp onding er. controll Solutio n: mate time-delay Recall that the unit step respons e for this process and the approxi and steady-s tate gain respons e is shown in Figure 17.7. From the figure the time delay by taking the slope are directly available; the effective time constan t is easily obtained taken by these values The origin. the at e of the approxi mate time-de lay respons are: ers paramet

K = 1;

a= 6;

t"=

4

The Cohen and Coon PID formula from Chapter 15 now recomm

ends the following

paramet er values:

I 11 (as opposed to 1 =71) has some advantage in case of plant-model mismatch. However, one must be careful not to chose 1 too large because then g"'(s) will have a much faster response than go(s) (see discussion in Chapter 6) and the control loop will be sluggish. It can be shown that the choice: 1

=

27]

(17.40)

is optimal, providing minimum mean square deviation of plant output from desired set-point. ·

.CHAP 17 CONT ROllE RS FOR PROCESSES WITH DIFFICULT DYNAMICS

615

Let us demonstrate the design of the inverse-response compensator and the influence on closed-loop system stability with the follow ing examples. Examp le 17.4 DESIG N OF AN INVERSE-RESPON SE COMP ENSA TOR. Design an inverse-response compensator for the inverse

-response system of Eq. (17.25).

Solution: The design of the inverse-response compe nsator boils down to obtain ing the approp riate transfe r functio n g'(s} to use in the minor feedback loop in the block diagram shown in Figure 17.10. For the process given by Eq. (17.25), g"(s) is given by: g•(s)

=

(2s + 1:(Ss + 1}

(17.41}

g '(s)

= (2s + 1}(Ss + 1)

(17.42)

and therefore:

For this system 11 = 3; thus according to Eq. (17.40} a good value of .il to be used in

Eq. (17.42) is .il= 6 so that

g'(s}

6s

= (2s+ l}(5s +l)

(17.43 )

is the transfer function to use in the inverse-response compensator loop. Using this transfer function in the inverse-response compe nsator loop produc es the effect of interfa cing the contro ller with the appare nt proces s whose transfe r function is given by: g•(s}

= .-:--'('-"3= ,. -s,..:,+,.,l:.,j}:.._,,.,. (2s + l)(Ss + 1)

(17.44)

which does not have a RHP zero.

The influence that the inverse-response compensato

r has on the closed-loop ated in the following

system stability characteristics can be dramatic, as illustr

example.

Examp le 17.5 CLOS ED-LO OP STAB IUTY OF AN INVERSE-RESPONSE SYSTEM UNDE R CONV ENTIO NAL FEEDBACK CONT ROL WITH AND WITH OUT THE COMP ENSA TOR. Invest igate the

closed -loop stabili ty proper ties of the invers e-resp onse system of Eq. (17.25) under proportional-only feedback control, first withou and then with the inverse-response compensator design

t any compensation, ed in Example 17.4.

Solution: Under conven tional, propor tional feedback contro l, the characteristic equati on for the closed-~ system is: Kc(l - 3s} 1 + ( 2 .r + l)(S.r + 1 )

=0

(17.45 )

616

SINGLE-LOOP CONTROL

which rearranges to: (17.46)

From here, we observe that the condition for stability is: (17.47)

any value of K, higher than this will make the system unstable. With the inverse-response compensator, let us first obtain the overall closed-loop transfer function from the block diagram of Figure 17.10; we will then deduce the characteristic equation from here and use it to investigate the stability properties. Consolidating the block diagram, dealing first with the minor loop, gives: u = 8/£

(17 .48)

where

8/ =

__ 8_,_

1 + 8,g'

(17.49)

The overall closed-loop transfer function is now given by: (17.50)

and the characteristic equation is: (17.51)

1+88/=0

Upon introducing into Eq. (17.51) all the appropriate expressions for the indicated transfer-function elements, and rearranging, the characteristic equation becomes: 10r+7s+(I +

KJ

= o

(17.52)

which is stable for all positive values of K,; clearly demonstrating the restrictive effect of the RHP zero on closed-loop system stability and the advantages of using the inverse-response compensator. To illustrate the performance of the inverse-response compensation compared to conventional PID control, the response to a unit set-point change with the inverseresponse compensator combined with a PI controller (K, = 10, 1/ -r1 = 0.167) is shown in Figure 17.8 together with two responses of conventional feedback controllers using PID control. Note that the inverse-response compensator produces the smallest negative deviation and a rapid response without overshoot.

Design Procedure As with the time-delay compensator, the following is the design procedure for designing an inverse-response compensation scheme: 1.

Design the inverse-response compensator loop As demonstrated in Example 17.4, this simply involves obtaining the appropriate transfer function g'(s) to use in the minor feedback loop in the block diagram shown in Figure 17.10; it specifically entails finding A which satisfies the conditions in Eq. (17.39) and using this in the expression for g'(s) in Eq. (17.31).

CHAP

2.

17 CONTROU.EJ

rut

Design 8c Once the compensator has been designed, its effect is to make the controller "think" it is interfaced to the process g ..(s) which has no RHP zero. Again, as with the time-delay compensator, we now simply design the controller for g""(s); and the absence of the RHP zero permits the use of higher controller gains than would otherwise be possible. However, because of process/model mismatch, one must be careful not to increase the controller gains too aggressively.

17.4

OPEN-LOOP UNSTABLE SYSTEMS

The transfer function that describes the dynamic behavior of an open-loop unstable system has at least one RHP pole. The obvious problem with such a system is that, in its natural state, it is unstable, so that any upset in any direction will result in unstable response. This creates one of the most significant difficulties in control system design; ie., one cannot use the standard process model identification procedures, but must keep the process under control while carrying out the modeling experiments. Hone wishes to obtain only the critical controller gains, we can use the "auto tuning relay controller" of Chapter 15 to determine the upper bound on controller gain and then decrease the gain in closed-loop experiments until the lower bound is determined. H one wishes to obtain an explicit model, one can identify the model under feedback control with known controller parameters using a sequence of set-point changes and disturbances. The open-loop model can then be determined. This difficult model identification problem is balanced by the fact that a process which is unstable in the open loop can usually be controlled quite well by conventional' feedback control if the controller parameters are carefully chosen. Let us illustrate with some examples. Example 17.6 STABIUZATION OF A FIRST-ORDER OPEN-LOOP UNSTABLE SYSTEM BY PROPORTIONAL FEEDBACK CONTROL. Obtain the range of Kc values required to ensure that the closed-loop system involving the first-order system: g(s)

K = -n- 1

(17.53)

and a proportional controller is stable.

Solution: The characteristic equation for this closed-loop system is:

KKC

1+-- = 0 TS - 1

(17.54)

which rearranges to give: (17.55)

,,jl

618

SINGLE-LO OP CONTROL

an equation with only one root located at: I -KKc s = --T

(17.56)

Thus, observe that this root will be negative (and the closed-loop system will be stable) as long as: K

c

> !_

K

(17.57)

with the immediate implication that any Kc value which satisfies Eq. (17.57) will stabilize the otherwise open-loop unstable system.

Beyond merely stabilizing the system, observe that we may go one step further and obtain the values of controller paramete rs that will not merely stabilize the closed-loop system, but which will place the closed-loop system poles in prespecified locations in the LHP. Let us demonstrate this with the following example. Example 17.7 STABILIZATION OF AN OPEN-LOOP UNSTABLE SYSTEM BY CLOSED-LOOP POLE PLACEMENT Design a PI controller for the system of Example 17.6 that will stabilize the closedloop system with the closed-loop system poles lore ted at s = -2, and at s = - 4. Solution: The closed-loop characteris tic equation in this case becomes, upon some elementary rearrangement (17.58)

The roots of this quadratic are related to its coefficients by: (17.59)

and (17.60)

Introducin g the process parameters , K = 2, 't =5 and solving these two equations simultaneously gives the required controller parameter values: Kc

= 15.5;

T1

=

0. 775

(17.61)

It should be mentione d that not all open-loop unstable processes can be stabilized by P control or PI control. It is left as an exercise to the reader (also one of the problems at the end of the chapter) to establish that, using a P or a PI controller, it is impossible to stabilize the process whose open-loop transfer function is given as: 2 (17.62) g(s) = (2s- 1)(5s + 1)

CHAP 17 CONTROLLERS fOR PROCESSES WITH DIFFICULT DYNAMICS

619

Here eithe r a PD or a PID controller is requi red for stabilization. This illustrates the fact that open-loop unstable system s can require special types of controllers to stabilize them. Open -loop unsta ble syste ms can also have condi tiona l close d-loo p stability, where there is only a range of controller gains, K, < K, < K, , whic h stabilize the process. This occurs whenever there is an uppe~ limit in controller gain (e.g., when the open-loop phase angle is less than -180° for some frequency) combined with the lower threshold level of controller gain necessary for closed-loop stability of the open-loop unsta ble system. Let us illust rate with an example. Exam ple 17.8

COND ITION AL CLOSED-LOOP STABILITY OF AN OPEN LOOP UNSTABLE SYSTEM.

Consider the open-loop unstable process given by: g(s)

= (4s

2(-s + I) - l)(2s + I)

(17.63 )

under propo rtiona l feedback control. Determine the range of controller gains for which the closed-loop system is stable. Soluti on: The characteristic equation for the closed-loop system

is given by:

(17.64 )

and by inspection, the stable range of controller gain is exhibits conditional stability.

17.5

t < K, < 1. Thus the system

SUM MAR Y

We have focused attention in this chapter on those processes that exhibit such dynamic behav ior in the open loop as to make them quite difficult to control, for the most part, by conventional means. We have seen that there are three classes of such systems: time-delay systems, inverse-response systems, and open-loop unsta ble systems; we have also seen that they have some thing in common: they each have nonminimum phase eleme nts. The presence of these nonm inimu m phase elements (the exponential term for the time-delay system; the RHP zero for the inverse-response system; and the RHP pole for the open loop unsta ble system) constitutes the main sourc e of difficulty in desig ning effective controllers for these systems. Parti cular ly for the time- delay and the inver se-re spons e syste ms, when ever the influence of the nonm inimu m phase element is not too stron g, conve ntion al feedback controllers can give acceptable result s. Howe ver, compensators introduced as minor loops aroun d the controllers are recommended when the nonm inimu m phase elements exert subst antial influence; i.e., when the time delay is very large, and for the inver se-response system, when the negative lead time constant is large. Aside from their sensitivity to mode ling errors, these compensators can be quite useful in that they enable the use of

620

SINGLE-{.OOP CONTROL

higher controller gains; thus improving the overall closed-loop performance. Explicit procedures for designing control systems that incorporate these compensators have been given. The open-loop unstable system was shown to be a comparatively less demanding system to control; it seldom requires more than closed-loop stabilization by conventional feedback control However, the design procedure can be difficult because the identification of the open-loop model is difficult and the conditional stability properties make controller tuning more complex. REFERENCES AND SUGGESTED FURTIIER READING 1. 2.

Bode, H. W., Network Analysis and Feedback Amplifier Design, Von Nostrand, Princeton (1985) Morari, M. and E. Zafiriou, Robust Process Control, Prentice-Hall, Englewood Cliffs,

NJ (1989) 3. 4.

5. 6.

Smith, 0. J. M., "Close Control of Loops with Dead Time/' Chern. Eng. Prog., 53, (5), 217 (1951) Smith, 0. J. M., "A Controller to Overcome Deadtime," ISA Journal, 6, (2), 28 (1959) Waller, K. and C. Nygardas, "On Inverse Response in Process Control," I&EC Fund., 14, 221 (1975) Iinoya, K. and R. J. Altpeter, "Inverse Response in Process Control," I&EC, 54, (1), 39

(1962)

REVIEW QUESTIONS 1.

What are the three classes of processes with difficult dynamics discussed in this chapter, and how does the process transfer-function model indicate the presence of such difficult dynamic characteristics?

2.

What is a noruninimum phase system?

3.

Why are the time delay, the RHP zero, and the RHP pole considered nonminimum phase elements?

4.

In physical terms, how can the presence of a time delay cause problems in conventional feedback control?

5.

In terms of phase angle characteristics, how does the presence of a time delay cause problems in conventional feedback control?

6.

In applying conventional feedback control for time-delay systems, what is usually sacrificed, and why? ·

1.

When does it become inadvisable to apply conventional feedback control for a timedelay system?

8.

What is the Smith predictor, and why is it so called?

9.

What is the procedure for controller design incorporating a Smith predictor for a timedelay system?

10. How is the minor loop of the Smith predictor scheme usually implemented?

CHAP 17 CONTROLLERS FOR PROCESSES WITH DIFFICULT DYNAMICS

621

11. Why do you think the performance of the Smith predictor scheme will be sensitive to modeling errors? 12. What is the main problem created by an inverse-response system under conventional feedback control? 13. Of all the conventional feedback controllers, which kind has the potential of coping with the problems created by inverse response? Why is this controller type able to copewith this problem?

14. How is a reverse Pade approximatio n useful in conventional feedback controller design for inverse-response systems? 15. What is the main objective in the design of an inverse-response compensator? 16. How is the inverse-response compensator similar to the Smith predictor? 17. What is the procedure for controller design incorporatin g an inverse-resp onse compensator for an inverse-response system? 18. What poses the more significant challenge in dealing with an open-loop unstable system: model identification, or controller design? 19. Why is it not possible to use standard process identification procedures for open-loop unstable processes? 20. Can any process that is unstable in the open loop be stabilized by any conventional feedback controller?

PROBLEMS 17.1

For the time-delay process with the following transfer-function model:

s.oe-10.0s y(s) = 15.0s + I u(s)

(Pl7.1)

(a) Use the Cohen-Coon tuning rules to design a PI, and a PID, controller; implement each of these controllers and obtain the closed-loop system response to a unit step change in the set-point. Compare the perfonnance of these two controllers. (b) Repeat part (a) using the appropriate time integral tuning rules to design the PI and PID controllers. Compare this set of closed-loop responses with those obtained in part (a). 17.2

(a) Design a Smith predictor for the time-delay process given in Problem 17.1. by specifying the transfer function to be implemented in the minor loop around the controller yet to designed. (b) Assuming the process model given in Eq. (P17.1) is perfect, implement the Smith predictor, using for g,, a PI controller with gain 0.3 and integral time constant 15.0. Obtain the overall response to a unit step change in the set-point. (c) Investigate the effect of process gain errors on the performance of the Smith predictor by repeating part (b), retaining the same PI controller, but this time using as the "true" process model: 5.5e-10.0s (Pl7.2) y(s) = 15.0s + I u(s)

622

SINGLE-LOOP CONTR OL while using the model in Eq. (P17.1) in the Smith predicto r's minor loop. Compar e this respons e with the "perfect " Smith predicto r response. (d) Now investig ate the effect of time-de lay errors on the perform ance of the Smith predicto r by repeatin g part (c), but this time using as the "true" process model: y(s) =

5.oe-11 .0s 15.0s + I u(s)

(Pl7.3)

Compar e this respons e with the "perfect " Smith predicto r respons e, and with the respons e obtaine d in part (c) with the process gain errors. (e) Finally, investig ate the effect of time constan t errors on the perform ance of the Smith predicto r by repeatin g part (c), but this time using as the "true" process model: y(s) =

s.oe-IO.Os !6.5s + 1 u(s)

(Pl7 .4)

·Compa re this respons e with the "perfect " Smith predicto r respons e, and with the other respons es obtained in parts (c) and (d). From this specific example , comme nt on which error causes the most signific ant perform ance degradation. 17.3

It is required to control a certain inverse- respons e system whose transfer -functio n model is given as: 2(1 - 4s) g(s) = (2s + l)(5s + I)

(PI7.5)

and several control strategie s are to be investigated. (a) Under pure proport ional control, what is the range of values that K, can take for the closed-l oop system to be stable? (Assum e there are no other element s in the feedbac k loop, just the process and the feedback controller.) (b) When the system operate s under pure proport ional control with K, = 0.5, in respons e to a unit step change in set-poin t, what will the steady-s tate offset in the process output be? (c) Using an inverse -respon se compen sator in a minor loop around a pure proportional controller, if g'(s), the transfer function in this minor loop, is chosen to be:

'( )

g s

O=s.,---= (2s+.-=:-11)(5s+ I)

(PI7.6)

now find the range of values which K, can take for the overall system to be stable. (d) If the pure proport ional controll er and the inverse- respons e compen sator in the minor loop around it are combin ed into one "appare nt controll er" block referred to as g,•, show that in the limit asK,--+ oo, this apparen t controll er, gt, in fact tends to a PID controller. What are the values of K,, -r , and -r for this controller? 1 0 (e) Implem ent the limiting PID controll er derived in part (d) on the process model given in Eq. (P17.5) and obtain the closed-loop system respons e to a unit step change set-point. 17.4

The dynami c behavio r of a biological process which consists of two subproc esses in opposit ion is represen ted by the following transfer function: 3

g(s)

= (7 s +

3 l) - (s + l)(s + 3)

(Pl7.7)

(a) Assume that the process is to be under proport ional feedbac k control and obtain a root locus diagram for the closed-l oop system, and from it determi ne the range of K,

CHAP 17 CONTROLLERS FOR PROCESSE S WITH DIFFICULT DYNAMICS

623 value s requi red for closed-loop stability. Wha t is the frequency of oscillation when the dosed -loop syste m is on the verge of instability? (b) Use this infor matio n to desig n a PI contr oller for the process. Imple ment this controller and obtai n the closed-loop respo nse to a unit step change in the set-p oint. Briefly comm ent on the control system perfo rmance vis-ll-vis the natur e of the proce ss transf er function. (c) For Extra Credit Only. Derive an inver s~response comp ensat or for the class of systems with two right-half plane zeros, whos e transf er functions are of the form: g(s)

17.5

(Pl7 .8)

(a) Find the range of Kc value s for whic h the inver se-re spons e syste m with the following trans fer funct ion will remai n stable unde r propo rtiona l feedback contr ol: y(s)

5(1 - 0.5s)

= (2s + 1)(0.5 s + I) u(s)

(Pl7 .9)

(b) Cons ider now the situat ion in which the following unort hodo x contr oller is utilized:

(Pl7. 10)

a propo rtiona l contr oller with a lead-lag elem ent If now -rz is chose n to be 0.5, find the -rL value requi red to obtai n a stability range of 0 < Kc < 4 for the propo rtion al gain. (c) Deter mine whet her or not the controller of Eq. (P17.10) will leave a stead y-sta te offset. 17.6

A certai n proce ss whos e transf er function g(.s)

is given as:

2.0 = (5s+ 1)(2 s-l)

(Pl7. 11)

has prove d partic ularly diffic ult to contr ol. After careful analysis, the follow ing was disco vered abou t the process:

If either P control or PI control is used on this process, then there are no values of Kc or fdor which it can ever be stabilized.

1.

2. This process can only be stabilized with a PD or a PID controller.

3. Furthermore, fpr the PID controller, the parameters must satisfy the following three conditions: (a) (b)

(c)

Confirm or refute these statements.

624 17.7

SINGLE-LOOP CONTROL (a) Use the Ziegler-Nichols stability margin rules to design a PI, and a PID, controller for the process whose transfer function is given as:

2.0

g(s) "' (2s + l )(Ss - 1)

(P17.12)

Implement these two controllers and obtain the closed -loop responses to a unit step change in the set-point. Which controller appears to be more effective? (b) If the "true" process transfer function is now given as: g(s) = (3.0s

2.5 + l)(4.5 s- 1)

(Pl7.1 3)

instea d of Eq. (P17.12), invest igate the effect of plant/ model misma tch by implementing the controllers designed in (a) on this "true" process. Comment on the effectiveness of each of the two controllers in the face of the modeling errors. 17.8

Prove that when A.= 217, the inverse-response compensator is exactly the same as the Smith predictor with a first-order Pade approximatio n.

CHAPTJEJR

18 CONTROLLER DESIGN FOR NO NL IN EA R SYSTEMS It was in Chap ter 10, after havi ng devo ted five chap ters to the analysis of vario us class es of linea r syste ms, that we first state d that virtu ally all phys ical syste ms are nonl inea r in char acter . The impl icati ons of this statement of fact are just as serious now, after having devoted four chapters to various aspects of linear control systems desig n. Traditionally, chemical process control has focused, almo st entirely, on the analysis and cont rol of linear systems; and therefore most existing cont rol systems desig n and analysis techniques are suitable only for linear systems. It is poss ible to justi fy this state of affai rs by reco gniz ing the fortu nate circumstance that man y processes are in fact only mildly nonlinear; and that all nonlinear processes take on approximately linear beha vior as they appr oach steady state (see Chap ter 10). Thus for such mildly nonlinear processes, and for close regulatory control of arbitrary nonl inear processes near stead y state, it is entirely reasonable to use linear control system desig n methods. Natu re has been kind in that the majority of real proc ess control problems fall into these categories. Nevertheless, for those nonlinear processes whose nonlinearities are strong, and for sufficiently large excursions from steady state, linear controller desig n techniques often prov e inadequate, and more effective alter nativ es mus t be considered. The objective in this chapter is to examine the main aspects of the rapid ly deve lopin g subject of nonl inear process control, and to prov ide a summ ary of the various alternative techn iques available for desig ning control systems, inclu ding those that explicitly recognize the nonl inear ities of the process.

18.1

NONLINEAR CONTROLLER DESIGN

PHILOSOPHIES

Controller desig n may be carried out for nonlinear systems alon g the lines indicated by any of the following four schem es:

625

626

SINGLE-LOOP CONTROL

1.

Local Linearization This involves linearizing the modeling equations around a steady-state operating condition and applying linear control systems design results. It is obvious that the controller performance will deteriorate as the process moves further away from the steady state around which the model was linearized, but quite often, adequate controllers can be designed this way.

2.

Local Linearization with Adaptation This scheme seeks to improve on Scheme 1 by recognizing the presence of nonlinearities and consequently providing the controller a means for systematically adapting whenever the adequacy of the linear approximation becomes questionable.

3. Exact Linearization by Variable Transformations This is a scheme by which a system model that is nonlinear in its original variables is converted into one that is exactly linear in a different set of variables by the use of appropriate nonlinear transformations; controller design may then be carried out for the transformed system with greater facility since it is now linear. 4.

"Special Purpose" Procedures These are usually custom made for specific processes, or specific types of nonlinearities.

Some of the factors that typically determine which of these approach philosophies one adopts are now summarized below: • The specific nature of the nonlinear control problem at hand; the nature of the process, and the control system performance objectives, • The amount of time available for carrying out the design, • The type of hardware available for implementing the controller, • The availability and quality of the process modeL We will now discuss these approaches in more detaiL

18.2

LINEARIZATION AND THE CLASSICAL APPROACH

It is always possible to obtain an approximate linear model for any nonlinear system, either by linearizing the nonlinear modeling equations (as discussed in Chapter 10), or. by correlating input/ output data, on the assumption that within the range covered by the experimental data, the process is approximately linear. Either way, the result is an approximate linear model, usually in the transfer function form. The classical approach to controller design for nonlinear systems involves using the approximate linear model as the basis for the design and applying standard linear control systems design strategies. The following example illustrates how this may be done when a nonlinear process model is available.

CHAP 18 CONTRO LLER DESIGN FOR NONLIN EAR SYSTEM S

Example 18.1

627

PI CONTROLLER DESIGN FOR THE UQUID LEVEL PROCES S OF EXAMPLE 10.2.

Design a PI controller for the liquid level system of Example 10.2 using an approxim ate linear model. Choose the controller paramete rs K, and -r such that the closed-lo op 1 poles for the approxim ate linear system are located at rl' r = -2 ± 2j. 2 Solution : A linear approxim ation for this system's nonlinear model was obtained earlier on in Example 10.3; it is the first-order transfer function model given below: y(s)

(18.1)

Making use of the specific process parameters given in Example 10.2, the approxim ate model, for the indicated steady-state conditions, has an apparen t steady-s tate gain K =2.4 nr2 min, and apparent time constant -r =0.6 min. Using the results given in Chapter 15 on controlle r design by closed-lo op pole specification (in particula r, Eq. (1528) for first-order systems under PI control), we obtain the following equation s to be solved for the PI controlle r paramete rs required for placing the approximate system's closed-loop poles in the indicated positions: (18.2)

(18.3)

which when solved simultaneously yield: K,

=

0.58; -r1

= 0.29

When a nonline ar process model is not availab le, the identifi cation method s of Chapter 13 may be applied. The magnitu de of the test inputs used should span the range of expected process operatio n so that the approxi mate linear model is valid in the operatin g regime of interest. The classica l approac h of lineariz ation and subsequ ent linear control systems design may therefore be used with success when: • The system is only mildly nonlinear, • The perform ance objectives are not too stringen t, so that very tight control is not required, • Large deviatio ns from nomina l steady-s tate operatin g conditio ns (e.g., for regulato ry control) are not expected. With strong linearities, the classical scheme should not be used for carrying out large set-poin t changes , which, by definition, require moving the nonline ar process from one steady state to another, quite different, steady state.

628

18.3

SINGLE-WOP CONTROL

ADAPTIVE CONTROL PRINCIPLES

A controller desig ned for the system in Example 18.1 usin g the appr oxim ate transfer function mod el given in Eq. (18.1 ) will function acceptably as long as the level, h, is main taine d at, or close to, the steady-state value h,. Und er thes e cond ition s (h - h,) will be smal l enou gh to mak e the linea r appr oxim ation adequate. However, if the level is requ ired to change over a wide rang e, the farth er away from h 5 the level gets, the poor er the linea r appr oxim ation . Observe also from the approximate trans fer function model in Eq. (18.1) that the appa rent stead y-sta te gain and time constant are depe nden t on the stead y state arou nd whic h the linearizat ion was carri ed out. This implies that in goin g from one stead y state to the other, the approximate process para mete rs will change. The main prob lem with appl ying the classical appr oach unde r these circu msta nces is that it igno res the fact that the char acter istic s of the appr oxim ate mod el must change as the process moves away from h,, if the appr oxim ate mod el is to rema in reaso nabl y accu rate. The imm edia te impl icati on is that any cont rolle r desig ned on the basis of this chan ging appr oxim ate mod el mus t also have its para meters adju sted if it is to rema in effec tive. In the adap tive control scheme, the contr oller para mete rs are adju sted (in an auto mati c fash ion) to keep up with the chan ges in the proc ess characteristics. We know intuitively that, if prop erly designed, this sche me will be a significant improvement over the classical scheme. There are vario us type s of adap tive cont rol schemes, differing main ly in the way the cont rolle r para mete rs are adjusted. The three mos t popu lar schemes are: scheduled adap tive control, model reference adaptive control, and self-tuning controllers. We shall discuss each of these in the following sections.

18.3.1 Sch edu led Ada ptiv e Control A scheduled adaptive control scheme is one in which, as a resul t of a priori knowledge and easy quantification of wha t is responsible for the changes in the proc ess characteristics, the com men surat e changes requ ired in the controller para mete rs are prog ramm ed (or sche dule d) ahea d of time. This type of adap tive control, sometimes referred to as gain scheduling, is illustrated by the block diag ram in Figure 18.1. As the inpu t and outp ut variables of the process

r-1

Updated Controller

I(

-

I1

Output y

u

Parame~

Set-point + .11.

Controller Parameter A4justment

I I Controller II

I

u

I

p......,...

II

Output

y

l

Figure 18.1.

Scheduled adaptive control scheme.

c f

CHAP 18 CONTROLLER DES IGN FOR NONLINEAR SYSTEM S

629 change significantly, this information is sen t to the controller, and its parameters are adjusted acc ording to the preprogrammed adjustment schedule. In practice this often reduce s to a table Let us illustrate scheduled ada look-up procedure. ptive contro] with an examp le. Exa mp le 18.2

PRO GR AM ME D AD APT IVE CO NT RO L OF A CA TALYTIC RE AC TO R.

Let us con side r the situ atio n in whi ch the effect of cata lyst dec ay on the dyn ami beh avi or of a catalytic reac c tor control sys tem is to cha nge the effective stea dy- stat gai n of the ent ire process (inc e luding the mea sur ing device, and final control element) acc ord ing to the equation: K(t)

= K0 ft.t)

(18.4) wh ere Ko is the process gain wit h fresh catalyst, t is the age of the catalyst in day s, f(t) is som e unk now n catalyst and decay function. If K, is the pro por tion al gain of the feedback con trol ler atta che d to this process, the n the ove rall gai n in the con trol loop wil l be KKc. If we now wis h to keep this ove rall gai n con stan t at its initial val ue of KoKco (sin ce, for exa mp le, dos ed- loo stab ility analysis was carr ied p out on this basis), the n obs erv e tha t this wou ld req uire ~ to change according to:

(18 .5) The actual process gain, K(t) , can be determined at any time from measurements of u(t), y(t) as sho wn in Fig ure 18.1. Thu s the con trol ler par ame ter adj ustm ent wo uld inv olv e calculating K(t) and usin g Eq. (18.5) to adju st K,(t ).

An oth er com mo n typ e of sch edu led ada pti ve con tro l inv olv es predetermining a set of con troller tuning parameters, and model, for each region of ope pos sibly a process rating space (e.g., as indicated by u and y) and then applying this schedule of app ropriate controller paramete rs for each par t of pha se space. Such prepro grammed adaptive control schemes can be very effective when there is a larg e amount of prior knowledge about the process; however, in other situations in which the process is not so well known as to permit the use of scheduled adaptive control, there are more self-adaptive schemes, in which the measu red value of the process inputs and outputs are used to deduce the required control ler mechanism. These wil l be disc parameter adaptation by using some learning ussed nex t

18.3.2 Mo de l Reference Ad aptive Contr

ol

The first of these self-adapt ive schemes is called the mo del controller (sometimes abb reviated MRAC) and is illu reference adaptive strated by the block dia gra m of Figure 18.2. The key component of the MR AC scheme is the reference mo del tha t consist s of a reasonable dosed-loo p model of how the process sho uld respond to a set-point change. This cou ld be as simple as a reference trajectory, or it cou ld be a more detailed closed -loop modeL The reference model out put is com pared wit h the actual proces s out put and the observed error E,n is used to driv e some adaptation scheme to cause the controller parameters to be adjusted so as to reduce E,n to zero. The adaptation scheme

630

SINGLE-LOOP CONTROL

_I

I

Reference I Model 1

Model Output

+ fm

Updated Controller parameters

Set point

~

+I(

· : Controller

-

Figure 18.2.

~

+

y

-

Controller Parameter A p•. Also establish from the nonlinear model in Eq. (Pl8.9) the stead y-state relationship betw een the nont rivial equilibrium points ran d p•. Wha t does this indicate abou t the process steady-sta te gain? (b) Linearize this mod el arou nd an arbitrarily specified equilibrium poin t indicated by ;• and p•, define the devi ation variables x = ~- ;• and u =11- p•, obta in an appr oxim ate transfer function relating x to u, and show that the steady-state gain indic ated by this approximate transfer function is in fact entirely inde pend ent of the actua l equi libri um poin t arou nd whic h the mod el was linearized. Wha t is the indicated appa rent time constant? (c) For a parti cular syste m with chara cteristic cons tant a= 2.5, and an initia l popu latio n variable ~0 = 2, obtain the response to a change in the source term from its initial valu e to J1 = 5.0 using the nonl inear model. Obta in the com men surat e respo nse using the appr oxim ate trans fer function. Plot and comp are these two responses. (d) Given that the popu latio n variable ~ can only be meas ured with a device that introduces a time delay of 5 units, i.e.: 71 =

~(t-

5)

(Pl8 .ll) intro duce the devi ation varia ble, y = 71 - 71*, and obta in the trans fer func tion relat ing y to u. With the aid of this trans fer function, desig n a PID controller for the process using the time integral tuning criteria (for set-point changes). Implemen t this cont rolle r on the nonl inear process and evalu ate its perfo rman ce for a unit step chan ge in the desir ed set-point of the meas ured valu e of; from its initial value of 2. Plot this response and comment on the controller performance.

CHAP1rJER

19 MODEL-BASED CONTROL

So far, feedback controller design has been approached from the classical viewpoint in which the controller structure choice is restricted to the PID form, and design guidelines are used merely to choose appropriate parameters for this prespecified class of controllers. There is an alternative approach in which what is specified a priori is the dynamic behavior desired of the control system- not the controller structure; the design procedure is then invoked to derive the controller (the structure as well as the parameters) that will achieve the prespecified desired objectives. This completely different controller design paradigm involves the explicit use of a process model directly; it is therefore often referred to as model-based control. The objective of this chapter is to introduce the concepts of modelbased control, and then to discuss some of the most common model-based control techniques.

19.1

INTRODUCTION

In introducing the subject matter of Part III, it was stated that the three constituent elements of process analysis are: the inputs to the process, the process model (as an abstraction of the process itself), and the outputs from the process. By specifying any two of these elements, and requiring that the unspecified third be determined from the supplied information, we obtain the three fundamental problems of systems analysis illustrated in Figure 19.1: 1.

The "Process Dynamics" problem: in which the input u is specified along with the model, say, in the form of a transfer function g and the resulting output is to be determined. This was the subject of Part II; it is the most straightforward problem.

2.

The "Process Modeling/Identification" problem: in which the process input u and output y are provided (sometimes along with some knowledge of the fundamental laws governing the process operation), 645

646

SINGLE-LOOP CONTROL

and it is required to determine the process model g. This was the subject of Part Ill; it is the most difficult problem. 3.

The "Process Control" problem: in which the desired output y"' is specified along with the model g, and the input u required to produce such an output is to be determined. This is the subject of Part N; it is the most important problem.

19.1.1 Solving the Process Control Problem In solving the "process control" problem in practice, there is the important issue

of what physical device will be used to generate the process input u, and also how this device is to be configured in relation to the physical process. When constrained to use specific hardware elements to implement the controller, this necessarily imposes limitations on our freedom to schedule input adjustments. At the same time, by insisting on specifying the desired output behavior explicitly and arbitrarily, the resulting controller requiremen t may be impossible to realize because of hardware limitations. Observe therefore that there are two, apparently mutually exclusive, approaches to solving the posed process control problem: one in which the primary basis is the hardware element available for implement ing the controller, with the desired output behavior only of secondary concern; the other in which the primary basis is the desired output behavior, with the hardware element for controller implementation of secondary concern. The former is the situation with conventional feedback control: with the prespecifie d hardware element (the PID controller), the desired output behavior must be defined somewhat more loosely, so that feasible solutions can be found within the constraints imposed by the prespecified controller structure.

----~u--•~~~---g--~~--?~-..~ (a)

The Procesa Dynamics Problem.

____u __

(b)

_.~~~---?--~~--~Y--•~~

The Procesa Modellng/Identifll:ation Problem.

--~?--~·~~---g--~---~r--~~~ (c)

Figure 19.1.

The Process Control Problem.

The three fundamental problems of systems analysis.

CHAP 19 MODEL-BASED CONTROl..

647 The PID controller param eters thus determined, the resulti ng outpu t behavior must be evalua ted (typically by simulation) for acceptability . The latter is the situati on with model-based control: the desire d outpu t behav ior is precisely, and explicitly, specified, and, using the process model explicitly, the contro ller requir ed to obtain the prespe cified behav ior is derive d. In most cases, a microprocessor-based hardw are eleme nt may be requir ed to implem ent the resulting controller, but as we shall soon see, the resulting controllers sometimes take the familiar PID form.

19.1.2 Model-Based Appr oache s There are two approaches to model-based control system design: 1.

The "Direc t Synth esis" Appro ach: in which the desire d outpu t behav ior is specified in the form of a trajectory, and the process model is used directly to synthesize the controller requir ed to cause the proces s outpu t to follow this trajectory exactly. Some of the particu lar techniques falling under this category, and to be discussed in this chapte r, are: • Direct Synthesis Control • Internal Model Control [1) • Generic Model Control [2)

2.

The "'Opti mizat ion" Appro ach: in which the desire d outpu t behav ior is specified in the form of an objective functio n {which may or may not involve an outpu t trajectory explicitly), and the process model is used to derive the contro ller requir ed to minimize (or maximize) this objective. It is also possible to includ e some known operat ing constr aints in the optim ization objective. Apart from "Optim al Contro l" which we will review briefly in this chapte r, most of the other techniques falling under this catego ry deal with discre te, single -varia ble or multiv ariabl e system s; we will conse quentl y discus s them at the appro priate places in later chapte rs.

It should be noted that we have alread y seen some types of model -based controllers in earlier chapters: the feedfo rward contro ller of Chapt er 16, the Smith predic tor, and the inverse-response compe nsator of Chapt er 17 are all model-based controllers. They all use the process model explicitly in derivi ng the resulti ng controllers; and witho ut makin g the issue of the hardw are required for implem entatio n the foremost concern, the prima ry basis for each technique is a clearly define d outpu t behav ior objecti ve: For feedfo rward control it is "perfect" disturb ance rejection, while for the Smith predic tor and the inverse-response compe nsator it is elimination of the time-delay term, and the right-half plane zero, respectively, from the closed-loop transfe r function. Let us now begin our discussion of the other model-based control techniques by startin g with the one closest in form to the feedback controllers with which we are alread y familiar.

SINGLE-LOOP CONTROL

19.2

CONTROLLER DESIG N BY DIRECT SYNTHESIS

19.2.1 The Contro ller Synthe sis Problem ated Consider the block diagram of Figure 19.2, where g represen ts the consolid element. control final + device ent measurem + process the transfer function for In this case the closed-loop transfer function is given by: 88c

y(s) = -1 --yjs ) + ggc

(19.1)

Suppose now that we require the closed-loop behavior to be: y(s) = q(s) yjs)

(19.2)

The where q(s) is a specific, predeter mined transfer function of our choice. op closed-lo of type the on depends choice of the reference trajector y q(s) Some process. the from s response possible the on and response desired Note particula r choices for the reference trajectory are shown in Figure 19.3. es trajectori reference , response inverse or delays time no that for systems with a of type (a) or (b) would be appropri ate. Howeve r, if the process has because required is (c) type of trajectory reference a significant time delay, then . no controlle r can overcom e the initial delay in the closed-io op response trajectory reference the then , response Similarly, if the process has an inverse must allow inverse response. Having defined an appropri ate form for the reference trajector y, the controller synthesis problem is now posed as follows:

What is the form of the controller Kc required to produce in the process the closed-loop behavior represented by the reference trajectory q(s)? closedIn other words, given q and g, find the Kc required to make the process behavior op closed-lo the that Noting q. to equal loop transfer function exactly and (as represen ted in Eq. (19.1)) will be as we desire it to be in Eq. (19.2) if, only if: ggc (19.3) q{s) = 1 + ggc then the controller synthesis problem reduces to solving Eq. (19.3) for 8c to yield: g

Figure 19.2.

-

c -

l(_!i_) g 1- q

Block diagram of a feedback control system.

(19.4)

CHAP 19 MODEL-BASED CONTROL

649 (a)

lr-----~------------------~ q(s)=-1-

trs+l

(b)

c

(d)

q(s)=

(1-11,•) (t,,S+ 1Xt,, + 1)

0

Figure 19.3.

t

Some possible reference trajectories for model-based control.

This is the controller synthesis formula from which, given the process model g, we may derive the controller g, required for obtaining any desired closed-loop behavior represented by q. Observe that there are no restrictions on the form this controller can take; for this reason, we may also expect that there will be no guarantees that the controller will be implementable in all cases. However, we shall see that if q is chosen appropriately, the synthesized controller g, will take on practical forms. In particular, we will show that in some important cases, the synthesized controller actually takes the familiar form of conventional PID controllers.

Specifying The Desired Closed-Loop Behavior The type of controller synthesized using Eq. (19.4) clearly depends on the nature of the transfer function form chosen for q, the desired closed-loop response. To guide us in our choice, let us recall the earlier discussion of Figure 19.3 and supplement this by specifying some of the most important features we would normally expect to find in acceptable closed-loop reference trajectories:

650

SINGLE-LOOP CONTROL 1. 2. 3. 4.

No steady-state offset, Quick, stable response, having little or no overshoot, A dynamic response which the process is capable of achieving_ and A mathematical form which is as simple as possible.

Let us consider first a simple form that is adequate for many processes and that leads to simple controller structures; i.e., the first-order response: "'s) - ___L_ 'r,S + 1

'1\

(19.5)

For this choice of q, the controller synthesic; formula Eq. (19.4} simplifies to: g

c

1 1 ='C,S g

(19.6)

Before investigating the various types of controllers prescribed by Eq. (19.6} for various types of process models, it is important to note that this direct synthesis controller involves the inverse of the process model. This feature, common to all model-based controllers, will have significant implications later on, particularly when we consider the issue of synthesis for nonminimum phase systems. Also, it is easy to show (see Problem 19.1} that even when the actual plant dynamics are not matched exactly by the model transfer function g, used in Eq. (19.6), provided the overall closed-loop system is stable, this direct synthesis controller does not result in steady-state ?ffset.

19.2.2 Synthesis for Low-Order Systems Pure Gain Processes

For these processes, with g(s} =K, Eq. (19.6) gives rise to:

1

gc

= KT,s

(19.7)

a pure integral controller with integral time constant -c1 =K-c,.. Pure Capacity Processes Since in this case g(s}

=K/s, Eq. (19.6} gives rise to: gc

1

= K-rr

(19.8)

a pure proportional controller, with gain Kc =1/K 'Zj. First-Order Processes In this case, with the process transfer function: g(s)

K

= -rs +

1

(19.9)


651

Eq. (19.6) gives the required synthesized controller as: Kc

=

TS

+ 1

(19.10)

K-r,s

which may be rearrang ed to take the more familiar form: g

c

= _!.... (1 K-r:,

+

_1) Ts

(19.11)

immediately recognizable as a PI controller with Kc = T/ K 1:, and 1: = 1:. 1 We may summar ize the most importan t implications of the foregoing as follows: 1. Once the desired speed of the closed-loop response is specified (i.e., a value is chosen for -r,), for any of the simple low-orde r processes we have consider ed, both the controller type and the controller parameters required to achieve the closed-loop objectives specified by Eq. (19.5) are immedia tely given in the derived controll er transfer function s; furtherm ore, these controller paramet ers are given in terms of the process model parameters and -r,.. 2. For these simple low-orde r processe s, we observe that the direct synthesi s approac h prescrib es nothing outside of the familiar collection of classical feedback controllers; the advantag e with this approach is that it automatically prescribes the controller paramete rs to go with each controller type. Let us now consider an illustrative example. Example 19.1 DIRECT SYNTHESIS CONTRO LLER FOR A FIRST-ORDER SYSTEM. Design a controller for the following first-order system: g

0.66 ( s) = 6. 1s + 1

(19.12)

using the direct synthesis approach, and given that the desired closed-loop behavior is as in Eq. (19.5), with -r,= 5. Compare this controller with that resulting from choosing

-r,.,;,l.

Solution: Let us observe first that this is the system used in Problem 15.6 without the time delay. Either from first principles , or directly from Eq. (19.11), we obtain that the required direct synthesis controller (for -r,= 5) is:

(19.13)

SINGLE-LOOP CONT ROL

652

a PI contro ller with Kc = 2.03 and Tr = 6.7. 1 (the faster closed -loop trajectory) is: The contro ller obtain ed with the choice -r, = (19.14 )

llers is that the latter has a propo rtiona l and the only differ ence betwe en these contro r; the integr al times are identicaL As we forme the of that gain value which is five times value is requir ed to provi de the faster woul d expec t, a contro ller with a highe r gain chose n for -r,. Further, it shoul d be value er small the by ted indica nse closed -loop respo of 5 result ed in a fivefold increa se factor a by r, of noted that a reduc tion in the value in the value of K,.

19.2.3 Synt hesi s for High er Ord er Systems For the second-order system:

(19.15)

p response to be first order with q(s) it is still possible to require the closed-loo controller is: given by Eq. (19.5). In this case the required

iar form, gives: which, upon rearrangement into a more famil

lli(

-r

I Kc = K'r, 1 + 2{-rs + 2{s

)

(19.16)

with the indicated parameters. immediately recognizable as a PID controller Exam ple 19.2

A SECO ND· DIRE CT SYNT HESI S CONT ROLL ER FOR ORDE R SYSTEM.

Desig n a contro ller for the following secon d-ord

er system:

g(s) = (2s + 1)(5s + 1)

(19.17)

that the desire d closed -loop behav ior is as using the direct synth esis appro ach, given this contro ller with that result ing from are in Eq. (19.5), with 'IT= 5. Also comp choos ing 't;. = 1. Solut ion: ple system used in Sectio n 17.3 of Once more we obser ve that this is the exam zero. RHP the ut witho .25)) Chap ter 17 (see Eq. (17 Eq. (19.16) direct ly (being carefu l to Eithe r from first princi ples, or by apply ing l param eters correctly), we obtain mode given the '!"from 'and eters extrac t the param 5) is: 'IT= (for ller contro that the requir ed direct synth esis (19.18)

653

CHAP 19 MODEL-BASED CONTROL a PID controller with I-~ -21J+T.

v--,~.,

~

b. Ideal PID Controller with first-order filter: Tl + T2 'E',t); 'tj = 'ft +

K.,= K (21J +

-r,

12;

1'1 T2 To= 1'1 + T2;

TJ'rrt Trt

-t> = 2TJ +

664

SINGLE-LOOP CONTROL

Table 19.3 Some Formulas for Direct Synthesis Tuning of Unstable Processes

Process Model: K 't5- 1

Reference Trajectory: (1 -71,S) (-r,.,s + 1) (-r,.2s + 1)

--

-r,. -r,. PI Controller with 11,. = -r, + -r,. +-'-•: 1 2

't'

't'

11,.

K ,- K-r- -r ·, 't'r= 11r 't rz

Process Model: K (-r1s- 1) (-r2s + 1)

Reference Trajectory: (1- 'II,S) (-r,,s + 1) (-r,2 + 1)

. -r,., ..,. PID Controller w1th 11,. = -r, + -r, + - - : I 2 't'l (12 + 'lfr) 't'2 11,. . 't=ti+71 . t; = - Kc r ' D -r2 + 1], K-r,, -.,.2 ' l

19.2.5 Summary of Direct Synthesis Formulas Although the discussion above would allow the reader to derive the direct synthesis tuning parameters for any class of models and choice of reference trajectories, it is useful to provide a summary of some common cases. Tables 19.1 - 19.3 provide formulas for PI and PID controller tuning based on the most common classes of linear models and for appropriate choices of reference trajectories. These formulas provide an excellent set of tuning parameters for a wide range of problems when an adequate linear model is available for the process.

19.2.6 Direct Synthesis for Nonlinear Systems Since by its very nature the method of direct synthesis is based on deriving the controller required to make the process output achieve some prespecified objective, it is, in principle, not restricted to linear systems. However, since nonlinear systems are usually not represented in the transform domain by transfer function models, applications to nonlinear systems will have to be carried out in the time domain. A technique particularly suited to this, Generic Model Control, will be discussed in Section 19.4.


19.3

665

INTERNAL MODEL CONTROL

19.3.1 Motivation Consider the process whose dynamic behavior is represented by: y(s) = g(s) u(s)

+d

(19.51)

with the block diagram shown in Figure 19.4 Here d represents the collective effect of unmeasured disturbances on the process output y. If it is desired to have "perfect" control, in which the output tracks the desired set-point Yd perfectly, the control action required to achieve this objective is easily obtained by substituting y =Yd in Eq. (19.51): yjs)

= g(s) u(s)

(19.52)

+ d

and then solving for u(s) to obtain: (19.53)

The implication is that if both d and g(s) are known, then for any given yd, Eq. (19.53} provides the controller that will achieve perfect control. Since in reality, dis unmeasured, and hence unknown, and since g may not always be known perfectly, but only modeled approximately by, say, g(s), we may now adopt the following strategy for implementing Eq. (19.53): 1.

Assuming that g is our best estimate of the plant dynamics g, then our best estimate of d is obtained by subtracting the model prediction, g(s)u(s), from the actual plant output y to yield the estimate: d"

2.

= y- g(s) u(s)

(19.54)

Let us choose the notation: c(s)

=

_I_

(19.55)

g(s) d

+

Figure 19.4.

A process under feedback control with unmeasured disturbances d.

666

SINGLE-LOOP CONTROL

and rewrite Eq. (19.53) as: II

u(s) = c(s)[y tJ{s)- d(s)]

(19.56)

where d" is the estimate of d given by Eq. (19.54). A block diagram matic represen tation of Eqs. (19.54) and (19.56) takes the form shown in Figure 19.5. This struchlre, first suggeste d by Frank in 1974 [3], is now known as the "Internal Model Control" struchlre [1]. If we wish to extend consider ation to both measure d disturba nces d , 1 and unmeasu red disturbances d2 , we can let:

in Figure 19.5 and in Eqs. (19.54) and (19.56) so that the "perfect" controlle r for both set-point changes and measure d dishtrbance rejection is: (19.57) Observe now that the addition al term in Eq. (19.57) is no more than what has already been derived in Chapter 16 for the feedforw ard controlle r used for cancelling out the effect of measure d disrurbances.

19.3.2 Basic Features and Properties of IMC By rearrang ing the block diagram of Figure 19.5 as indicated in Figure 19.6(a) and consolid ating the inner loop into the single block indicate d in Figure 19.6(b), it is possible to compare the IMC structure with an equivale nt conventi onal feedback control struchtre . The following relations hips between the conventi onal feedback controlle r, g,(s}, and the internal model controlle r, c(s), are easy to establish: g,(s) = --"'c(u.s)_ _ (19.58a)

I - c(s) g(s) c(s)

= _ _g,(s) .:....___ 1 + gc(s) g(s)

(19.58b)

These will be useful in the discussion to follow. d Yd

+

+ 1---+o{) (}-.,---- y

d

Figure 19.5.

The internal model control structure.

CHAP 19 MODE L-BAS ED CONTROL

667 d

Yd

+

+

+

I-_.,+-<X:h .----Y

(a)

Alternate IMC Structu re d

Yd

+

Be=_£ _ l-eg

f---"0---i!~

+

g(s)

1----+-{X)-----l~Y

Equivalent Conventional Control Structu re

(b)

Figur e 19.6.

The conventional control struct ure equiva lent to

IMC.

Closed-Loop Transfer Function Rela tions and Controller Properties From Figur e 19.5, it is a straightforward matte r to deriv e the following closedloop transfer function relations: Relation of u to y4 and d: c

u = I

+ c(g- g)

(yrd )

(19.59)

Relation of y to y 4 and d: y = d +

gc t + c(g-

or y

=

g)

gc

1 + c (g -

g)

Yd

From these transfer function relations, three basic prope rties of the IMC scheme:

(yd-d )

+ w~

(19.60a)

l-g c

I + c (g -

g)

d

(19.60b)

may now estab lish the following

1~ Nom inal Close d-Loo p Stabi lity In the nomi nal case, wher e the mode l is perfe ct so that (19.60a) becom e, respectively:

g = g, Eqs. (19.59) and (19.61)

SINGLE-WOP CONTROL

668 and y

= gc(yrd)

+ d

(19.62)

From Eq. (19.61) we see that the control u will be bowtded if c(s) is stable, and the output y(s) is bowtded if g(s)c(s) is stable. Thus for the nominal case in which the model is assumed to be perfect, the IMC structure guarantees that the closed-loop system will be stable whenever the process g(s) is open-loop stable, and the controller c(s) is also stable. The requirement that g(s) also be stable is not 5o stringent, but to require c(s) =1/g(s) to be stable is often difficult to satisfy in practice. 2. "Perfect " Control

Hg=g, and c =g-1, then from Eq. (19.60), y =y4 for all time and for all d. Thus the IMC structure provides "perfect" controL However, in practice, one has limited controller power so that requiring c(s) = g-1 can seldom be satisfied in reality. 3. 110ffset-free " Control So long as c(s) is chosen such that lim c(s) :::; - 18(0)

s-> 0

then regardless of the fact that g *" g, Eq. (19.60) shows that y:::; y4 at steady state. Thus so long as the steady-state gain of the controller is the same as the reciprocal of the gain of the process model, the IMC structure guarantees offsetfree controL The IMC structure is an idealization that can seldom be realized in practice; however, it is a useful ideal for purposes of discussion, and by adding some approximations and additional constraints, IMC can be implemented in some cases.

19.3.3 Design and Implementation of Internal Model Controllers In implementing the IMC scheme, the following practical issues must be taken

into consideration: 1.

g is never equal tog in practice.

2.

Implementing c(s) as g -l is rarely feasible because the transfer fwtction is "improper," and there may be time delays or right-half plane zeros in the model.

3.

Even in the absence of such "noninvertible" dynamics, observe from Eq. (19.58a) that implementing c(s) as g -1 will often require controllers with infinitely large control action, which is both inadvisable and impracticaL

As a result, the IMC design procedure has to be modified as follows:


1.

669

Process model factorization: The process model is separated into two parts: (19.63)

where g+ contains all the noninvertible aspects (time delays, righthalf plane zeros), with a steady-state gain of 1, with g_ as the remaining, invertible part. 2.

Controller Specification and Filter Design: The controller is specified as: c(s)

= -1

J(s)

(19.64)

'jj_

where f(s) is a filter usually of the form: j(s)

= (A.s + l)n

(19.65)

with parameters il and n chosen to ensure that c(s) is proper (i.e., the numerator order is less than, or at most equal to, the denominator order). 3.

Equivalent Conventional Controller form: If necessary, the IMC controller, c(s), may be converted to the conventional form, Kc(s), for implementation. This is accomplished by using Eq. (19.58a).

Example 19.6

IMC DESIGN FOR A FIRST-ORDER PROCESS.

Design a controller for the first-order process whose transfer function is given by: 5.0 g(s) '" Ss

+ 1

(19.66)

using the IMC strategy. Convert this controller to the conventional feedback form. Solution: Observing that the transfer function in Eq. (19.66) is completely invertible, we obtain: 1 Ss + 1 g(s) =

--s:o-

which requires only a first-order filter (n from Eq. (19.64), we have, in this case: c(s) = g(ls)j{s) = 510

=1) in order that c(s) =fig be proper.

(8s ++ 1I )

· .ts

Thus,

(19.67)

SINGLE-LOOP CONTROL

670

which can be implemented using a lead/lag element Introducing Eqs. (19.67) and (19.66) into Eq. (19.58a), we obtain: g (s)

=

c

j_(t _1_) 5l

+

8s

(19.68)

as the equivalent con\•entional feedback form, a PI controller whose gain depends on the filter parameter A.

If the resulting PI controller in Eq. (19.68) appears reminiscent of the direct synthesis controller result of Section 19.2.2, it is because the IMC approach and the direct synthesis approach produce identical controllers under certain conditions. Consider the situation for which g is invertible so that c(s) = JJj)_ g(s)

then under these conditions, Eq. (19.58a) becomes: 1 ) _

g,..s -

or

fl:s)l i(s)

1 - J(s)

[....fuj_]

1 gc(s) = g(s) 1 - f(s)

(19.69)

which is identical to Eq. (19.4) with f(s) = q(s). Thus for ~um phase systen-..s, the filter of the IMC scheme and the reference trajectory of the direct synthesis scheme have the same interpretation. It is easy to show that even when g is not invertible, the IMC scheme and the direct synthesis scheme still give rise to identical controllers, only this time, f(s) and q(s) take on different forms. It can also be shown that the equivalence between the IMC controller and the direct synthesis controller is not limited to the nominal case; even in the presence of modeling errors, the direct synthesis controller and the IMC controller also give rise to identical closed-loop responses. Example 19.7

IMC DESIGN FOR A FIRST-ORDER PROCESS WITH TIME DELAY.

Design a controller for the first-order-plus-time-delay proCess whose transfer function is given by: g(s)

=

s.ae-3•

8s + 1

(19.70)

using the IMC strategy.

Solution: Observing that the presence of the time delay makes the transfer function not completely invertible in this case, we proceed to factor the transfer function as follows: Let


671

so that: g_(s)

= __2jL_ Ss + I

The desired controller is then easily determined from Eq. (19.64) as:

I

I (8s+1)

c(s) = - ()j{s) = -5 0 - g_ s . .ls + 1

(19.71)

where we have chosen a first-order filter. Note that this controller is precisely the same as the one obtained in Example 19.6 for the first-order system without the delay. It is left as an exercise to the reader, however, to show that the corresponding conventional feedback forms are in fact not identical. In this regard, the reader may want to keep in mind that in this case, c(s) is based on a "truncated" form of the model, so that the denominator of Eq. (19.58a) will be more complicated.

Problem (19.9) illustrates how the IMC approach is used to obtain PID tuning rules for processes whose dynamics are represented by first-order-plus-timedelay transfer functions. These results were presented in Chapter 15, in Table 15.6. For additional details about the IMC scheme, particularly, regarding how model uncertainty information is used to design a filter that will ensure robustness, the interested reader is referred, for example, to Morari and Zafiriou [4].

19.4 . GENERIC MODEL CONTROL Consider the situation in which the dynamics of a process are represented by the first-order differential equation:

~ = j(y, u,d)

(19.72)

where y is the process output, u is the input, d is the disturbance, and f( •, •, •) can be a nonlinear function. Further, consider that we wish for the process output to approach its desired value of y d along a reference trajectory, y r' determined by the equation: (19_73)

This can be related to q(s) in the Laplace domain by taking the Laplace transforms of Eq. (19.73) and letting y = y" to yield:

or y, =q(s)yd where:

SINGLE-LOOP CONTROL

672

The process model Eq. (19.72) may be used directly to obtain the controller required to cause the process to follow this trajectory by equating Eqs. (19.72) and (19.73) to obtain: f(y,u,d) = K 1 (yd-y) + K 2

r

(Y&-y)dr

(19.74)

0

and solving for u. In general, since it is impossible to model the process perfectly, our best approximatio n, ](y, u, d), is used instead, and the resulting implicit, nonlinear equation: (19.75)

is the control law to be solved for u. Although it may not be obvious, it should be pointed out that solving Eq. (19.75) for u involves an implicit inversion of the process model. This technique, known as "Generic Model Control," was first presented by Lee and Sullivan [2]. It has the following properties: 1.

2. 3.

Any available (possibly nonlinear) process is employed directly in the controller. The integral term provides additional compensation for inevitable modeling errors. The control law has only two parameters, K 1 and K 2, whose values determine the nature of the desired reference trajectory in a wellcharacterized form.

Relationshi p between GMC and Other Model-Based Schemes Lee and Sullivan [2] have discussed the similarities between this control scheme and IMC. We will now show that for K2 = 0, when the process model is linear, first-order, the closed-form solution of the resulting control law yields a PI controller identical to the direct synthesis controller. For the first-order system whose transfer function is given in Eq. (19.9), the differential equation model is: 1:

41: dt

= -y + Ku(t)

(19.76)

Setting K2 = 0 in Eq. (19.73) implies a reference trajectory described by:

or (19.77)


673

where e(t) is the usual feedback error term. We may now integrate Eq. (19.77) directly to obtain: (19.78)

The control action required to cause the process modeled by Eq. (19.76) to follow the trajectory described by Eqs. (19.77) and (19.78) may now be obtained straightforwardly by substituting the desired trajectory expressions into the model equation and solving for u(t); the result is: u(t)

1:K1 [ E(t) =K

+ IT

f' ] 0

e(t)dt

(19.79)

which, of course, is a PI controller. But beyond this fact, observe that this is precisely the same PI controller obtained for the direct synthesis controller in Eq. (19.11) with K 1 = 1/ 'fr . Note that if the process output y is set equal to the reference trajectory Yr in Eq. (19.73), a Laplace transform of the resulting expression gives:

and only the substitution K 1 = 1/7:r is required to establish that this desired trajectory is precisely the one indicated by q(s) given in Eq. (19.5) and used by the direct synthesis controller.

GMC for Higher Order Systems For processes modeled by the general, nth-order, nonlinear differential equation:

.~.d!L . I ( Y' dt ' dil '

. d~ . . ) · · · • dt' • u • d - 0

(19.80)

applying the GMC scheme requires no more than obtaining the required higher order derivatives of the desired trajectory, introducing these into the model in Eq. (19.80) and solving for u. Thus, control action will be obtained by solving: (19.81)

foru. For the purpose of illustration, consider the design of a GMC controller for the second-order process modeled by:

Ku

(19.82)

,, 674

SINGLE-LOOP CONTROL

and suppose that the desired trajectory is obtained by setting K2 = 0 in Eq. (19.73). Then as obtained previously, the first derivative of the desired trajectory is as given in Eq. (19.77), from which we easily obtain the second derivative as: (19.83)

And now, by introducing Eqs. (19.77), (19.78), and (19.83) into the process model in Eq. (19.82) and solving for u, we obtain: u(t) = 2 ''fK K 1

{

e(t)

+ _l 2 (;1:

J' 0

e(t}d t

s [A.llil._J} dt

+ ..!_ 2

(19.84)

the same PID controller obtained in Eq. (19.16) using the direct synthesis approach. For additional information about the GMC technique, including discussions about its applications on several experimental case studies, the reader is referred to Lee and coworkers [2, 5, 6].

19.5

OPTIMIZATION APPROACHES

Let us return, once again, to the process whose dynamics are represented by the differential equation:

~

= j(y, u,d}

(19.72)

which was used to introduce the GMC technique in Section 19.4; and instead of requiring that the process output approach its desired value of y4 along a specific reference trajectory, y,, we now wish to approach the control problem differently. We start by setting up an objective functional of the general form: (19.85)

where y(9 is the value taken by the process output at the final time t =t1 ; and GO and F(·, ·,·)are functions chosen by the control system designer to reflect the aspect of the process behavior to be optimized. The idea is now to find the control policy u(t) that minimizes (or maximizes) cP, perhaps even subject to such operating constraints as: g (y,u) :::; 0

(19.86a)

h (y,u) = 0

(19.86b)

The philosophical differences between this technique known as "Optimal Control," and GMC are as follows:


1.

2.

3.

675 Whe n control action is dete rmin ed acco rding to the optim al cont rol strat egy, the trajectory followed by y is not prespecified; rather, the "bes t" possible trajectory, consistent with the process mod el and the objective emb edde d in Eqs. (19.85) and (19.86), will be found, and this will be followed by the process outp ut. The prespecified trajectory of the GMC tech niqu e (or any othe r appr oach not base d on optimization) is arbitrary and may quite possibly be unattainable. The optim al control strategy is automati cally equi pped to cons ider process oper ating constraints; the othe r strategies mus t inco rpor ate additional considerations whe n process constraints are involved.

The general optimal control problem as pose d above cann ot be solv ed in closed form, or even implicitly, as with the GMC in Eq. (19.72); quite often, intensive computation is requ ired.

Opti mal cont rol is not limited to the first-order nonl inea r case pres ente d above; in fact, the orig inal deve lopm ent was for nonl inea r mult ivar iable systems of arbit rary orde r. The resulting optim al cont rol prob lem, how ever , simplifies cons idera bly whe n specific special cases of the gene ral form are considered. These and othe r aspects of controller desi gn by the optim izati on appr oach are disc usse d in Ray [7]. The appl icati on of these opti miza tion tech niqu es to the desi gn of disc retetime mult ivar iable cont rolle rs for industrial processes has led to the evol ution of a class of control schemes know n as Model Predictive Control; this will be discu ssed in grea ter deta il in Cha pter 27.

19.6

SUMMARY

The appr oach to controller design that we have intro duce d in this chap ter is radically diffe rent from anyt hing we have disc usse d up until now . Thes e model-based appr oach es allow the cont rol syst em desi gner to spec ify the closed-loop beha vior desi red of the process, and from this, to deriv e the controller requ ired to obtain the specified closed-loop beha vior directly, usin g the proc ess mod el. Even thou gh the two classes of mod el-ba sed cont rol techniques were identified, our discussion focussed more on the direc t synthesis approaches (Direct Synt hesi s Control, Inter nal Mod el Con trol, and Generic Model Control) whil e pass ing more light ly over the optim izati on appr oach es. This latte r class of techniques are muc h more naturally suite d to mult ivari able systems and will therefore be discussed more fully later whe n it will be mor e appropriate to do so. We show ed that model-based controlle rs all share in common the concept of using the process mod el inverse - in one form or anot her - as part of the controller; we also show ed that even thou gh philo soph icall y diffe rent from conventional feedback control, model-ba sed control strategies often give rise to controllers that may be rearr ange d to take on the fami liar form s of thes e classical controllers. The special cons idera tions requ ired for deal ing with those processes that exhibit more com plex dynamics were carefully discussed , and we foun d that the deriv ed controlle rs then take on special forms, some of

SINGLE-LOOP CONTROL

676

which we had encountered earlier. Upon some simplification even these special forms reduce to classical feedback controllers. The principles underlying model-based control make it applicable for the rational design of controllers for nonlinear systeJr.s. The Generic Model Control scheme was introduced as a strategy most suited for such applications, particularly when a nonlinear differential equation process model is available. With this, we conclude our discussion of controller design for single-input, single-output systems. Next, in Part IVB, we will devote our attention to the issue of control system analysis and design when the process has multiple input and output variables.

REFERENCES AND SUGGESTED FURlHER READING 1.

2. 3. 4. 5. 6.

7.

Garcia, C. E. and M. Morari, "Internal Model Control: 1- A Unifying Review and Some New Results," Ind. Eng. Chern. Proc. Res. Dev., 21, 308 (1982) Lee, P. L. and G. R. Sullivan, "Generic Model Control (GMC);" Computers and Chemical Engineering, 12, (6), 573 (1988) Frank, P. M., Entwurf von Regelkreisen mit vorgeschriebenem Verhalten, G. Braun, Karlsruhe (1974) Morari, M. and E. Zafiriou, Robust Process Control, Prentice-Hall, Englewood Cliffs, NJ (1989) Lee, P. L., G. R. Sullivan, and W. Zhou, "Process/Model Mismatch Compensation for Model-Based Controllers," Chern. Eng. Comm., 80,33 {1989) Newell, R. B. and P. L. Lee, Applied Process Control: A Case Study, Prentice-Hall, Englewood Cliffs, NJ (1989) Ray, W. H., Advanced Process Control, Butterworths, Boston (1989)

REVIEW QUESTIONS 1.

What are the three fundamental problems of systems analysis?

2.

What is model-based control, and what differentiates it from conventional feedback control?

3.

What are the two approaches to model-based control system design?

4.

What are some of the model-based controllers encountered in earlier chapters?

5.

How is the controller design problem posed in the direct synthesis approach?

6.

What are some of the most important features of an acceptable closed-loop reference trajectory?

7.

What are some model forms and closed-loop reference trajectories that give rise to direct synthesis controllers in the form of the classical feedback controllers?

8.

What is the advantage of using the direct synthesis approach to design feedback controllers for low-order processes?

9.

What restrictions are placed on admissible reference trajectories for direct synthesis controller design when time-delay and inverse-response elements are present in process models? Why are these restrictions necessary?

10. What additional approximations are needed in order to obtain classical feedback controllers from a direct synthesis controller designed for a first-order-plus-time-delay system?


677

11. The direct synthesis controller designed for a first-order open-loop unstable system suffers from what problem if a first-order closed-loop reference trajectory is used? How is the problem remedied? 12. Why is it rarely feasible to use the model inverse directly as the controller in the IMC strategy? 13. What is the typical procedure for the IMC design? 14. What is the purpose of the IMC filter?

15. What is the relationship between the IMC filter and the transfer function for the direct synthesis reference trajectory? 16. What are some of the main properties of Generic Model Control? 17. What is the fundamental philosophy of the optimization approaches? 18. What are some of the philosophical differences between optimal control and GMC?

PROBLEMS 19.1 Suppose that the direct synthesis controller given in Eq. (19.2) is implemented as in Figure 19.2, but the true process transfer function is Kp rather than g. Obtain the closed-loop transfer function between Yd andy, and show that, despite the fact that gP g, so long as the overall system is stable, this direct synthesis controller guarantees that at steady state, y = yd, and there will be no offset.

*

19.2 Obtain the direct synthesis controller for the process whose transfer function is given as: K (~s + 1) g(s)

if the closed-loop reference trajectory in Eq. (19.5) is desired. Show that this controller may be arranged in two ways: first, as:

(Pl9.1)

which is a standard PID controller with an additional term; and then as:

which is the commercial PID controller form. 19.3

Establish the result presented in Eq. (19.50) for the open-loop unstable system modeled by the transfer function in Eq. (19.49).

19.4

The following problems have to do with the theoretical robustness stability properties of the direct synthesis controller. (a) Consider a first-order process whose true transfer function:

678

SINGLE-WOP CONTROL K g (s) = ____5L_ P -rps + l

(Pl9.2)

is approximated by the model transfer function: g(s)

= -K-

(Pl9.3)

-rs + I

and suppose that this approximate transfer function is used to design a direct synthesis controller to achieve a first-order desired closed-loop trajectory in Eq. (19.5). Show that when the resulting controller is implemented on the "real" process in Eq. (P19.2), the overall closed-loop system will always be stable regardless of the value of the true process parameters, Kp and -rp, and for all values of -r,, the reference trajectory time constant, provided Kp and its estimate, K, have the

same sign. (b) Suppose now that the true process transfer function is second order, i.e.:

(P19.4)

containing an additional time constant, Tn, which has been neglected by the model transfer function in Eq. (P19.3). Furthermore, suppose that the direct synthesis controller has been designed on the basis of the first-order desired trajectory, using the Eq. (P19.3) model. Establish that in implementing such a controller on the real second-order process, closed-loop stability will be guaranteed provided:

(PI9.5)

where Pp is the multiplicative uncertainty in the process gain estimate, defined by: (PI9.6) (c) To illustrate how relatively weak the restriction indicated in Eq. (P19.5) is, consider the situation in which the true process transfer function is: 4 Cp(s) = (lOs+ 1)(6s +I)

(P19.7)

and that the model not only ignores the smaller time constant, but in addition grossly underestimates the dominant time COIIStant as 3.0, and the gain as 2.0; i.e.: 2.0 g(s) = 3.0s +

(P19.8)

is the model used to design the direct synthesis controller. First show that, in this case, guaranteeing closed-loop stability merely requires that -r7 > 1.5. Next obtain a PI controller for this process by direct synthesis, using the model in Eq. (P19.8) and a first-order desired reference trajectory with T7 = 2.5; simulate and plot the response of the closed-loop system to a unit step change in the set-point. 19.5

The following second-order transfer function was obtained via an identification experiment performed in a small neighborhood of a particular steady-state operating condition of a nonlinear process:

CHAP 19 MODEL-BASED CONTROL g(s)

=

679

K (3.0s + I )(5.0s + I)

(P19.9)

While the time constants (in minutes) are relatively easy to estimate, the gain is very difficult to estimate; it is therefore left indeterminate as a free design parameter. (a) Use this model along with a first-or der desired trajecto ry, with -rr =4.0, to design a controller for this process. Leave your controller in terms of the parameter K. (b) Suppose now that the true process has a transfer functio n of the form: 8p(s) "" (3.0s + 1)(5.0s + I)

(Pl9.10 )

where, as a consequence of the nonlinearity, the process sometim es passes through a region of operati on in which it exhibits inverse-response behavi or not capture d by the approximate linear model in Eq. (P19.9). The time constan ts are assume d to be essentially perfectly known. If the control ler designe d in part (a) is now to be implem ented on this process (assuming for simplicity that the parameters in Eq. (P19.10 ), even though unknow n, are fixed), and if it is known that in the worst case KP = 2K, (i.e., in the worst case the process gain is underestimated by a factor of 2), find the maxim um value of the unknow n right-h alf plane zero parame ter ~ which the closed-loop system can tolerate withou t going unstable. (c) If it is now known that for the process in part (b),~ takes the value of 3.5 in the worst case, and that the worst case for the gain estimate remain s as given in part (b), what value of -r, is required for guaran teeing closed-loop system stability in the worst case? 19.6 Revisit Problem 19.5 for the specific situatio n in which the true process transfe r function is given as: 3.0(-l. 5s + I) 8p(s) = (3.0s + 1)(5.0s + I) but the model transfer function is as in Eq. (P19.9) with K chosen as 2.0. (a) Use the model transfer function along with a first-order desired trajectory, with -r, =4.0, to design a direct synthesis controller; implem ent the control ler on the true process in Eq. (P19.10) and plot the closed-loop system respons e to a unit step change in set-point. (b) Repeat part (a) for the choice of Tr .= 6.0. Compa re the perform ance of this controller with that obtained in part (a) and comme nt on the effect of larger values of Tr on closed-loop performance in the face of plant/m odel mismatch. 19.7 The nominal transfe r function for an open-loop unstabl e process was obtaine d via closed-loop identification experimentation as: - 0.55

g(s)

= (- 5.2s +

1)

(Pl9.11 )

and the true process gain and time constant are known to -0.45 - 4.9

~

KP

~

- 0.65

~ Tp ~ -

5.5

lie in the following limits: (P19.12 a)

(Pl9.12 b) Choose an approp riate desired closed-loop transfe r functio n and design a direct synthesis controller for this process. Implement your control ler on a process whose transfer function is given as:

680

SINGLE-LOOP CONTROL

(s)

Cp

= -0.6(-1.5s+

I)

(PI9.13)

(-5.0s+l)

and plot the closed-loop system response to a unit step change in set-point. 19.8

The following sixth-order transfer function was obtained by linearizing the original set of ordinary differential equations used to model an industrial chemical reactor: 10.3 (Pl9.14) g(s) = ( 1.5s + 1)5(15s + I) The indicated time constants are in minutes. For the purposes of designing a practical controller, this model was further simplified to: gl(s) =

10.3e-7 ·5' 15s + I

(PI9.15)

(a) Design a controller for this process by direct synthesis, using the transfer function g1(s), for '~'r = 5.0 min. Introduce a first-order Pade approximation for the delay element in the controller. What further approximation will be required in order to reduce the result to a standard PID controller? Can you justify such an approximation? (b) Implement the PID controller obtained in part (a) on the sixth-order process and obtain the closed-loop response to a step change of 0.5 in the output set-point. (c) Repeat part (a) but this time introduce the Pade approximation for the delay element in the model before carrying out the controller design, and then design a direct synthesis controller for the resulting inverse-response system, again using '~'r = 5.0 mins. Compare the resulting controller with the one obtained in part (a). 19.9

Use the IMC approach to design a classical controller for the process whose dynamics are represented by the general first-order-plus-time-delay transfer function: K e-as y(s) 'I'S

+ I

by following this procedure: 1. 2. 3.

4.

Introduce a first-order Pade approximation for the delay. Factor the resulting transfer function into g_(s) g+(s) as required. Choose a first-order filter with time constant .it and obtain the IMC controller. Convert from the IMC form to the standard feedback form. The result should be a PID controller.

What values does this IMC procedure recommend for Kc,

'l'p

and T0 ?

19.10 Revisit Problem 19.8 and design a PID controller for the sixth-order process in Eq. (P19.14), based on the transfer function in Eq. (P19.15) and the IMC procedure of Problem 19.9, with the specific choice of .it= 5.0 min. Compare this controller to that obtained in part (a) of Problem 19.8. What does this example illustrate about the

relationship between the IMC strategy and the direct synthesis strategy for PID controller tuning for systems with time delays? 19.11 [This problem is open ended]

Via frequency response identification, the following model was obtained for a heat exchanger:


681 (P19.16)

with parameter values estimated as: K

Kd r1

r2

a {3

2.07 ± 0.8; (°F/Ib/hr) 0.478 ± 0.1; ("F/0 F) 7.60 ± 0.5; (s)

30.40 ± 1.2; (s) 7.6 ± 1.3; (s) 25.3 ± 5.5; (s)

It may be assumed that the bounds given above for the parameter values cover the entire range of possible values taken by the true process parameters. In terms of deviation from steady-state values, y is the exit temperature (°F); u is the steam rate (lb/hr); dis the inlet temperature of the process stream. (a) Assume that d(s) is measured and design an IMC controller for this process using the supplied model. Justify all the design choices made. (b) Implement the controller designed in part (a) on a "process" represented by: I .3e-8s 2ls gp(s) = (8s + 1 )(29s + 1} + 0·55 e-

(Pl9.17)

and plot the control system response to a change of 2.0°F in the inlet temperature of the process stream. Evaluate the performance of your controller. If it is necessary to redesign your controller, explain what needs to be done, do it, and then implement the redesigned controller for the same distwbance. 19.12 The following model has been given for a CSTR in which an isothermal second-order reaction is taking place:

(Pl9.18) Here F, the volumetric flowrate of reactant A into the reactor, is the manipulated variable; CAO• the inlet concentration of pure reactant in the feed, is the disturbance variable; and CA' the reactor concentration of A, is the output variable to be controlled; V is the reactor volume, assumed constant. The nominal process operating conditions are given as:

w-

3 m3ts F = 7.0 X V 7.0 m3/s k 1.5 x 1o-3 m3/kg mole-s CAo 2.5 kg mole/m 3 CA 1.0 kg mole/m 3

(a) Use this information to design a GMC controller for the reactor, leaving as unspecified the controller parameters K1 and K2 • Describe how such a controller would be implemented. (b) Assume now that the gas chromatograph measurement of CAO made available to the controller is permanently biased such that the true measurement is always CAo + 0.25 kg mole/m3. Under these circumstances, implement the GMC controller with K1 =1.0 x 1o-3 and K2 =2.0 x 10-3 and plot the control system response to a change in the actual inlet concentration from2.5 to 2.0 kgmole/m3 . Assume all the other model parameters are as given.

part IVB MUL TIV ARIA BLE PRO CES S CON TRO L

CHAPTER20.

Introduction to Multivariable Systems

CHAPTER21.

Interaction Analysis and Multiple Single Loop Designs

CHAPTER22.

Design of Multivariable Controllers

'... fww tliey liave increasetf tliat trou6fe mt! many are tliey tliat rise up agairut ""' Psalm3:1

'It is quite a tfiree pipe pro6ftm.

ant! I 6eg tliat you won't speaf(to'lll for fifty minutes.' Sherlock Holmes, The Red-Headed Leilgut_ . (Sir Arthur Conan Doyle)

682

CHAPTJER

20 IN TR OD UC TIO N TO MU LT IVA RIA BL E SY ST EM S When a process has only one input variable to be used in controlling one outp ut variable, the controller design problem can be hand led, fairly conveniently, by the meth ods discussed in Part IVA. In sever al impo rtant situations, howe ver, the syste m unde r consi derat ion has mult iple inpu ts and mult iple outp uts, maki ng it multivariable in nature. In actua l fact, the most impo rtant chem ical proce sses are often mult ivari able in natur e. Man y nontr ivial issues, not enco unter ed hithe rto in our discussion of singl e-inp ut, singl e-out put (SISO) syste ms, are now raise d when contr oller s are to be desig ned for these multivariable systems. Our study of multi vatia ble control systems analysis and desig n begins in this chap ter with an intro ducti on to mult ivari able syste ms and the uniq ue control problems they prese nt This is where we lay the foundation required for a thoro ugh treat ment of the controller desig n issue s to be foun d in later chap ters.

20.1

THE NATURE OF MULTIVARIABLE SYS TEMS

A multi varia ble proce ss is one with mult iple inpu ts, ul' u 2, u 3, ••• , um, and multiple outpu ts, y1, y2, y31 ••• , Yn' where m is not necessarily equal ton; it could be a single process, such as the stirred mixin g tank show n in Figure 20.1, or it could be an aggregate of many process units constituting part of an entire plant , or it could be the entire plant itself.

Inpu t/Ou tput Pairing Up until now, becau se we have had to deal with only SISO systems, the question of what inpu t variable to use in contr olling what outp ut variable did

683

MULTIVARIABLE PROCESS CONTROL

684

GoldStream

Fc,Tc

Figure 20.1.

Stirred mixing tank requiring level and temperature controL

not arise, since there was just one of each. But now, with m inputs and n outputs, we are faced with a new problem:

Which input variable should be used in controlling which output variable?

This is referred to as the input/output pairing problem. One of the consequences of having several input and output variables is that such a control system can be configured several different ways depending on which input variable is paired with which output variable. Each U; to yi "pairing" constitutes a control configuration, and for a two-input, two-output system (often abbreviated as a 2 x 2 system) we have two such configurations:

Configuration 2

Configuration 1

ut-Y2

ut-Yt

and

What this means in a physical sense is illustrated by the following example. Example 20.1

CONTROL CONFIGURATIONS FOR THE STIRRED MIXING TANK.

The stirred mixing tank shown in Figure 20.1 has two input variables, the cold stream flowrate, and the hot stream flowrate, to be used in controlling two output variables, the temperature of the liquid in the tank, and the liquid level (see Ref. [1]). Enumerate the different ways the control system can be configured. Solution: 1.

2.

There are two possible configurations: Use hot stream flowrate to control liquid level, and use cold stream to control liquid temperature (this is the configuration shown in Figure 20.1). Use cold stream flowrate to control liquid level, and use hot stream to control liquid temperature.

CHAP 20 INTRO TO MULTIVARIABLE SYSTEMS

685

Note that these two configurations are mutually exclusive, i.e., only one of them can be used at any particular time. .

It can be shown that for a 3 x 3 system, there are six such configurations; for a 4 x 4 system there are twenty-four; and in general, for an n x n system, there are n! possible input-output pairing configurations. We know, of course, that our controllers can be set up according to only one of these configurations. Furthermore, we would intuitively expect one of the configurations to yield "better overall control system performance" than the others: how, then, to choose among these possibilities? At the simplest level, therefore, the first problem in the analysis and design of multivariable control systems is that of deciding on what input variable to pair with what output variable; and the problem is by no means trivial.

Notation Because we will be engaged with this issue of input/ output pairing quite a bit, we now find it necessary to introduce the following notation, to avoid the confusion that usually attends this exercise: We shall refer to the input (or manipulated) variables of a process in their free i.e., "unpaired" form as m1, m 2 , m3 , ••• , mm; the output variables will still be Y1' Y2• Y3' .. · , Yn· After the input variables have been paired with the output variables, we shall reserve the notation ui for the input variable paired with the jth output, Yr

Thus, for example, if the third input variable m3 is paired with the first output variable yl' then m3 becomes ul' In this sense, the subscripts on them's are merely for counting purposes; the subscripts on the u's actually indicate control loop assignment. Let us illustrate again with the stirred mixing tank. The input variables are:

1. The cold steam flowrate 2. The hot stream flowrate We may then refer to these respectively as m 1 and m2, in their "unpaired state." The output variables are: 1. Liquid level . 2. Liquid temperature and let these be y1 and y2 respectively. Now, as configured in Figure 20.1, the hot stream m2 is paired with the liquid level y1; and the cold stream m1 is paired with the liquid temperature y2• Thus, in terms of keeping track of how many input variables we have, the cold stream is m1, and the hot stream is m2; but in terms of which output they have been paired with, the hot stream is u 1 (because of its association with the "first" output variable, y1), and the cold stream is u 2 .

686

MULTIV ARIABL E PROCESS CONTR OL

If the configuration changes, so will the assignm ent of the u's; this will ensure that u1 is always paired with Yt The advant age of this notatio nal convenience will become particularly import ant in Chapte r 21.

Interactions Another import ant consideration that did not feature in discuss ions involving SISO systems but that is of prime importance with multivariable systems is the following: after somehow arriving at the conclusion that input, m1, is best used to control output yi' we must now answer the very pertinent questio n: In addition to affecting yj' its assigned output variable, will m 1 affect any other output variable? That is, can m control y in isolatio 1 n, or will 1 the control effort of m 1 influence other outputs apart from Y/

Unfort unately , in virtual ly all cases, a particu lar m will influence several 1 outputs. For example, even though the hot stream flowrate is paired with the level in the stirred mixing process, change s in this input variab le will obviou sly affect the liquid temper ature, in addition to affectin g its assigned output variable, the liquid level. This is the interaction problem, recognized as the other main proble m with ·the control of multivariable systems. There is one final set of related issues that do not come into consideration with 5150 system s but that are someti mes import ant with multiv ariable systems; these will now be introduced briefly.

Controllability and Obseruability To illustra te these concep ts we introdu ce anothe r examp le multiv ariable system: the two--tank networ k shown in Figure 20.2. For this process, liquid streams flow into Tanks 1 and 2 at respective volumetric rates Fl' and F2; the outflow from each tank is assume d (for simplicity) to be propor tional to the respective liquid levels h 1 and h 2 in each tank. The liquid leaving Tank 2 is split into two with a fraction F, exiting, and the remain der R pumpe d back to the first tank (see Ref. [11). Thus this is a two--input, two-ou tput (or 2 x 2) system, with the flowrates of the two inlet streams as the two inputs, and the liquid level in each tank as the two output variables. We now wish to investigate (for now only in a qualitative sense) how this process will operate under the following conditions: 1.

As shown in Figure 20.2, with both inlet streams, and the recycle pump in operation;

2. The recycle pump is turned off, and only stream F is operati onal; 1 stream F2 is set at constant flow which cannot be changed; 3. The recycle pump is turned off, but this time stream F 1 is now set at constant flow that cannot be changed; only stream f is operati onal. 2

CHAP 20 INTRO TO MULTIVARIABLE SYST EMS

Figur e 20.2.

687

The two-tank network.

In particular, we are interested in answ ering the following question:

With what ever inpu t variables we have available, can we simu ltan eous ly influence the liquid level in both tanks? Or, put more formally, can we "drive" the liquid level in both tanks from a given initial value to any other arbitrarily set desired value, in finite time? Upo n some reflection, we can answ er this ques tion for each of the three situations note d above as follows: 1. Und er Cond ition 1, since both inlet strea ms are available, we can clearly influence the level in both tanks simu ltaneously.

2. Und er Condition 2, observe that with strea m F2 no longer available for control, we have lost our source of direct influence on h2; however, we still have indirect influence on it throu gh the outfl ow from Tank 1. Thus , even thou gh we do not have as muc h contr ol over the syste m variables (particularly h2) as we woul d unde r Cond ition 1, we have not lost total control; h2 can still be influence d by the only available inpu t variable, the strea m F1 flowrate. 3. Und er Cond ition 3, it is clear that we have absolutely no control over h1; the strea m F1 flowrate cannot be chan ged, and the recycle strea m is no long er in operation; thus we can only contr ol h2• (Note that if the recycle strea m were still in oper ation , thro ugh it we wou ld have indirect influence on the level in Tank 1, but this is not the case.) Und er Cond ition s 1 and 2, the syste m is said to be controllable beca use, loosely spea king , we can influence each of the two proc ess outp ut variables with what ever inpu t variables we have available; unde r Cond ition 3, the

MULT IVAR IABLE PROCESS CONT ROL

688

that we have no control over one syste m is said to be uncontrollable in the sense bles. of the process outp ut varia shou ld be clear that not all From this simp le illusl rativ e exam ple, it le, and that controllability is ollab contr mult ivari able systems are necessarily in desig ning effective controllers an essen tial factor to take into consideration for multi varia ble systems. vability, let us imag ine that To illust rate the closely relate d issue of obser we can now only meas ure the with the same two-tank syste m of Figure 20.2, el meas urem ent restriction, e-lev liqui d level in one tank; along with this singl tion. opera of the recycle loop is also taken out is , obser ve that since its value If the only meas urem ent available is h 2 ct extra iple, princ in can, we ow, influenced by h 1 throu gh the Tank 1 outfl h , and a fairly accurate proce ss infor matio n abou t the value h1 given only 2 mode l. der the situation in which the only If we turn the situation aroun d, and consi the physics of the process that from clear is it , h avail able meas urem ent is 1 absolutely no way we can infer without the recycle loop in operation, there is mation abou t hl' infor only anyth ing abou t the value of h2 given to be observable, while it is said is ss proce the In the form er case, unobservable in the latter case. the intrin sic prop erty of the Thus while controllability has to do with ol all the proce ss state or outp ut syste m whic h make s it possi ble to contr observability has to do with the variables with the available inpu t variables, available measurements. from bles varia state ability to infer all the process observability, and quan titati ve Form al defin itions of controllability and observability of a process are and meth ods for deter minin g the controllability [1]. available, for example, in Ref. it is clear, therefore, that in On the basis of the foregoing discussions, than SISO systems, the need so more deali ng with multivariable systems, much mpha sized . The main overe be ot cann sis analy of for syste matic meth ods e of whic h have been (som ms chara cteris tic probl ems of multi varia ble syste icatio ns on contr ol impl t fican signi very highl ighte d above) typically have e, these prob lems are such that syste m performance; and by their very natur matic analysis. syste they cann ot be hand led any other way but by math emat ical models, block (e.g., sis analy ms The same basic tools of syste systems, but because of increased diagr ams, etc.) are still used for multivariable to be intro duce d. The rest of dime nsion ality, addit ional considerations have k for multivariable control ewor fram the g this chap ter is devo ted to deve lopin exten ding perti nent ideas we are syste ms analy sis and desig n, most ly by with singl e-var iable systems, and alrea dy familiar with from our enco unter to hand le prob lems pecu liar to n arise have h intro ducin g new ones whic mult ivari able systems.

20.2

MULTIVARIABLE PROCESS MODELS

of a multivariable process occur Mod els used to represent the dynamic beha vior or in the transfer function form. most comm only eithe r in the state-space form, mode ling equa tions they are such of e Owin g to the multi varia ble natur more conveniently written in vector-matrix form.

CHAP 20 INTRO TO MULTIV ARIABLE SYSTEMS

689

State-Space Model Form As was briefly introduced in Section 4.3 in Chapter 4, the linear, multivariable system is represented in U\e state space as: x(t)

Ax(t) + Bu(t) + rd(t)

(20.1)

y(t)

Cx(t)

(20.2)

where the dot ( •) represents d tfferentiation with respect to time, and x u

y d

- [-dimensional vector of state variables m-dimensional vector of input (control) variables - n-dimensional vector of output variables - k-dimensional vector of disturbance variables

=

r

as appropriately dimensioned system matrices of with A, B, C, and conformable order with the respective multiplying vectors. Nonlinear multivariable systems are typically represented by several nonlinear differential equations, which might be represented in general as given earlier in Eq. (4.31}, i.e.: dx(t)

dt

= f(x, u, d)

(20.3)

y(t) = h(x(t))

where f(x, u, d), and h(x(t)) are usually vectors of nonlinear functions.

Transfer Function Model Form Analogous to ilie situation with SISO systems, the multivariable transfer function model form relates the Laplace transform of ilie vector of input variables to that of t.."te output variables, as given earlier in Eq. (4.38); i.e.: y(s) = G(s) u(s)

+ Gd(s) d(s)

(20.4)

where, for an m-input, n-output system, G(s) is an m x n transfer function matrix with elements: gu(s) g 12(s) ··· 81m(s)]

G(s)

[ 821(s) 822(s) ··· 82m(s)

...

...

8nt(s) Cn2(s)

...

(20.5)

Cnm(s)

and each typical giis) transfer function element is exactly like the familiar SISO transfer functions. Gis) is ann x k transfer function matrix consisting of similar transfer function elements.

690

MULTIVARI ABLE PROCESS CONTROL

Interrelat ionships between Model Forms As with SISO systems, it is possible to convert the state-space model representat ion to the transfer function form by taking Laplace transforms and rearranging . Taking Laplace transforms in Eqs. (20.1) and (20.2), assuming that the matrices involved are constant, and that we have zero initial conditions, we obtain: sx(s)

y(s)

Ax(s) + Bu(s) + rd(s)

(20.6)

Cx(s)

(20.7)

which can be rearranged (paying due respect to the rules of matrix algebra) to give:

r] d(s)

y(s) = [C(sl- A)- 1B] u(s) + [ C(sl- At1

(20.8}

Hence, by comparison with Eq. (20.4}, we obtain the relationship between the transfer function matrices and the state-space matrices: G(s)

[C(sl- At 1B]

(20.9)

and

r]

Gd(s) = [ C(sl- A)- 1

(20.10)

The reverse problem of deducing the equivalent state-space model from the transfer function model- typically known as the realization problem - is not by any means trivial. For one thing, no realization is unique, in the sense that there are several equivalent sets of differential equations that, upon Laplace transforma tion, will yield the same transfer function matrix. Further, there are several different ways of obtaining these different, but equivalent, realizations. As a result it is often necessary to utilize a method that provides the so-called "minimal realization. " We consider a discussion of how to obtain minimal realizations as being outside the intended scope of this book; the interested reader is referred to Ref. [1] for the important details. A simple procedure for obtaining state-space realizations will be illustrated with a problem at the end of the chapter. It is important to note that most computer-a ided control systems design packages with multivariab le control capabilities (cf. Appendix E) have routines for obtaining time-domai n realizations ; this step is necessary when carrying out process simulation using transfer function matrix models. Let us consider some examples of multivariable process models. Example 20.2

MATHEMATICAL MODEL FOR THE TWO-TANK SYSTEM OF FIGURE 20.2.

Assuming uniform cross-sectional areas A 1, and A 2, respectively for Tanks 1 and 2, material balances over each of the tanks yield the following modeling equations: dhl Aldt

= -alht+f3h2+ Ft

(20.11)

CHAP 20 INTRO TO MULTIVARIABLE SYSTEMS

691 (20.12 )

and if we let a2 ={3 +

r and define the following deviation variables: x 1 = h1 - h 15 ; u 1 = F1 -F15 ;

x2 = h2 -h25 u2 = F2 -F25

then these modeling equations become:

(20.13 )

(20.14)

If we introd uce the variables y 1 and y to repres ent the measurements of the liquid 2 levels (as deviations from steady-state values) we have, in the situation where both are measured: (20.15a) (20.15 b)

Observe now that these equations, (20.13), (20.14), and (20.15), are in the form given in Eqs. (20.1) and (20.2); the state-5pace vectors" and matrice s are easily seen to be:

withC = l,and r=o.

Observe that, in this case, this is a 2 x 2 system.

As the next example demonstrates, the stirred mixin g tank of Figure 20.1 is a nonlin ear multiv ariabl e system; it is mode led by two differential equati ons consisting of nonlinear functions. Examp le 20.3

MATHEMATICAL MODEL FOR THE STIRRED MIXING TANK OF FIGURE 20.1.

Assum ing unifor m cross-sectional area Ac for the tank, constant liquid physical proper ties p and CP' and F = K..Jh, we obtain the follow ing from material and energy balances over the tank: dh _ F A 0 u1 =1 but u2 = 0). 20.3

The complex theoretical model for a certain novel bioreactor has been linearized around a proposed steady-stat e operating point, resulting in the following approximat e linear state-space model:

i =

with

A=[45

-!]·

-2

'

Ax + Bu

B-[1 OJ· 0 2 •

X

=

[::l [:J U

=

For simplicity we shall assume that the two state variables are measured directly so that: y = X

CHAP 20 INTRO TO MULTIVAR IABLE SYSTEMS

719

· (a) Show that the bioreactor is open-loop unstable at this operating condition. (b) Obtain the corresponding transfer function matrix G(s) for the bioreactor at this operating point; find the system poles and zeros. (c) Certain economic considerations dictate that the bioreactor is most profitably operated at this unstable steady state. Under proportional-only feedback control, using two single-loop controllers with gains Kc, and Kc{ with: (P20.5)

where

given that Kct =2, find the range of values which Kc, must take for the closed-loop system to be stable. 20.4

A 2 x 2 system is modeled approximately by:

[

s :

3

--=L s+1

s

~I 1

___!Q__ s+4

l[m']

(P20.6)

mz

(a) Find the poles and zeros of the transfer function matrix.

(b) Obtain explicit expressions for the system outputs y (t), y (t) in response to 1 2 inputs m1(t) =mz(t) =1 at t > 0. Plot these responses.

20.5

Luyben and Vinante [4] studied a 24-tray distillation column used for separating methanol and water. Considering the temperatur es on trays 17 and 4 as the most important output variables, they studied their responses to changes in the reflux flowrate, and steam flowrate, and obtained the following transfer function matrix

model:

[:J

=

3 6 [ ;:5~ :-'1 ;:~~:-: ~ ] [ml] -2.75e-1. 8• 8.2s + I

4.28e-0· 35• 9.0s + 1

m2

(P20.7)

where, in terms of deviations from their respective steady states, the input and output variables are: y 1 = Tray #17 temperature (0 C) y2 = Tray #4 temperature ("C) m1 = Reflux flowrate (kg/hr) m2 = Steam flowrate (kg/hr)

(a) By defining four new "state" variables x 1, x2, x3, and x as follows: 4

x 2 (s)

( -2.16e-•) 8.5s + 1 m,(s)

(P20.8)

26e- 0 · 3' ( 7I..05s + I

(P20.9)

)

m2(s)

720

MULTIVARIABLE PROCESS CONTROL (

-2.75e-I.Ss) 8.2s + I ml(s)

(P20.10)

(P20.11)

write the differential equations to be solved in time to simulate the dynamic behavior of this process. Such a set of equations are called a realization of the original transfer function model. (b) Using your favorite simulation package (or otherwise) obtain the response of y1 (the tray ##17 temperature) to unit step changes implemented simultaneously in both the reflux Oowrate and the steam flowrate. By examining the given transfer function model carefully, explain the nature of the response you have obtained. 20.6

Obtain the equivalent impulse response matrix model for the Luyben and Vinante column of Problem 20.5. [Hint: The impulse response matrix G(t) has as its elements gip), the inverse Laplace

transform of gi/s), the individual elements of the transfer function matrix. Do not forget the presence of the time delays.] 20.7

For the multivariable system whose block diagram is shown below in Figure P20.1, find the closed-loop transfer function matrices relating y to both yd and d. What is the characteristic equation?

20.8

The transfer function of a 2 x 2 system is given as:

G(s) =

[

_I 5s + I

_!QQ_ 3s + I

0.0001 0.2s + I

l

(P20.12)

0.5s +

(a) Sets= 0 and obtain the steady-state gain matrix K = G(O). (b) First find the eigenvalues of K, then find its singular values. Obtain the ratio of the larger to the smaller singular values (the "condition number''; see Appendix D) and compare it to the corresponding ratio of the eigenvalues. From your knowledge of what the condition number of a matrix indicates, what does the value you have obtained imply about the "conditioning" of this process? (c) By examining the steady-state gain matrix for this system carefully, comment on the relative effectiveness of the two process input variables in affecting the two output variables.

Figure P20.1.

721

CHAP 20 INTRO TO MULTIVARIABLE SYSTEMS y

Figure P20.2. 20.9

The block diagram shown in Figure P20.2 (where all the indicated blocks contain transfer function matrices) is for a certain multivariable control system to be discussed in Chapter 22. (a) Find the closed-loop transfer function between y and Yd· (b) What is the characteristic equation for this system? (c) Given that the following relationship exists between G, the process transfer function matrix, and the other transfer function matrices V and W: G

=

WI, yT

where I, is a diagonal matrix, and the matrices V and W are unitary matrices (see Appendix D), show that the characteristic equation for this system reduces to: (P20.13) where G~ is the indicated (diagonal) matrix of controller transfer functions and I is the identity matrix. [Hint You may find the following determinant relations useful: IABCI =I AIIBIICI =I BllciiAI =I CIIAIIBI, etc., for nonsingular, square matrices A, B, and C of identical order. 2. lA I =1/IA-1 1.] 1.

20.10 A two-input/two-output system whose transfer function matrix model is as shown below:

[:J

=[

s: I

3s ; I

s + I

2s + 1

l[::J

(P20.14)

is under feedback control, using the Ycm 1 /y2-m 2 pairing, so that the process transfer function matrix, G(s), is as indicated above; the controller, Gc has been chosen as two pure proportional controllers, i.e.:

=[

G c

Kc,

0

0

K

J

c2

and it is desired to use the following specific controller parameters: Kc =1, K, = 5. (a) First show that with the given controller parameters, the closed-lobp systim will be unstable; then (b) Show that with the same controller parameters, if the loop pairing is switched (i.e., changing to a y1 - m2 /y2 - m 1 configuration) the system will be stable. {Hint:

Find the appropriate transfer Junction matrix first, since the transfer Junction matrix in this case is different from the one used in part (a).]

CCHAJPTJER

21 INTERACTION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS In design ing controllers for multivariable systems, a typical starting point is the use of multiple, independent single-loop controllers, with each contro ller using one input variable to control a preassigned output variabl e. Since such a control system will consist of several individual, single-loop feedbac k system s of the type we studied in Part IVA, it has the obvious advantage of building on familiar materiaL However, as shown in Chapte r 20, primarily because of the interac tions among the process variables, multiv ariable system s cannot, in general, be treated like multiple, indepe ndent, single-loop system s. Thus, a multiple single-loop control strategy for multivariable systems must take these interactions into consideration, and should be applied with care. This chapte r is concerned with discussing how interaction analysis is carried out and how the results of such analysis are used for multiple singleloop controller designs. The main questions regard ing when to use this control strateg y and how to design the multip le single- loop contro llers will be answer ed in this chapter; what to do to obtain improv ed perform ance when this strateg y proves ineffective will be discussed in the next chapter.

21.1

INTRODUCI'ION

As introdu ced in Chapte r 20, a multivariable system has multip le inputs and multiple outputs. When confronted with the task of using the multiple inputs to control the multiple output variables of such processes, it seems natural to start out by consid ering the possibility of pairing the input and output variables and assigning one feedback controller to link each input/ output variabl e pair, resulti ng in several feedback contro l loops, as illustra ted in Figure 20.3(c).

723

724

MULTWARl ABLE PROCESS CONTROL

With this approach, the issues to be resolved are twofold: • How to pair the input and output variables • How to design the individual single-loop controllers Consider, as an example, the transfer function model for a 2 x 2 system, which in its expanded form is given by the following equations: (21.1) (21.2)

where we clearly see how each input variable is capable of influencing each output variable. Here, we have a situation in which if u 1 has been assigned to y 1 (and, by default, u2 to y2), each of these control loops will experience interactions from the other loop; Loop 1 (the u1 - y1 loop) will experience interaction s coming from Loop 2 (the u 2 - y2 loop) via the g12 element, since it is by this that y is 1 influenced by u 2• In the same manner, Loop 2 experiences interactions coming from Loop 1 via the g21 element. This situation is clearly illustrated, once again, in Figure 20.3(c). If the input/ output pairing were switched - giving a second, alternative configurati on - each loop will still be subject to interactions from the other loop, but this time, the interactions will come about via the g11 and g22 elements in Eqs. (21.1) and (21.2). . It seems reasonable that, as long as we intend to use two single-loop controllers for this 2 x 2 system, we should pair the input and output variables such that the resulting interaction is "minimum" ; in other words, since there are two possible input/ output pairing configurations for this particular system, it would be to our advantage to choose that configuration that results in smaller net interaction among the variables. This strategy works well for the great majority of processes; however, one should be aware that there are more complex control system designs that try to use the interaction s rather than eliminate them. To proceed logically to determine which input/ output pairing configuration to use for the multiple, single-loop control strategy, we must have a means of discriminat ing between the N! different configurations ~ble with an N x N system. This means we must be able to quantify the interactions experienced by the process under each configuration.

21.2

PRELIMINARY CONSIDERATIONS OF INTERACTION ANALYSIS AND LOOP PAIRING

21.2.1 A Measure of Control Loop Interactions Consider an example 2 x 2 system (see Figure 21.1) with two output variables,

y 1 and y 2, and two input variables that we will now refer to as m 1 and m , 2

because we do not yet know which one will be paired with which output variable.

-~·

CHAP 21

INTERACTION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

725

Loop1

___ ... I

-----, - ______ .,...______

Figure 21.1.

. I

T

Loop interactions for a 2 x 2 system.

Note that each loop can be opened or closed. When both loops are open, then m 1 and m2 can be manipulated independently and the effect of each of these inputs on each of the outputs is represented by the following transfer function model: (21.3) (21.4)

where each transfer function element consists of an unspecified dynamic portion, and a steady-state gain term K;r Let us now consider m1 as a candidate input variable to pair with Yr In order to evaluate this choice against the alternative of using m2 instead, we will perform two "experiments" on the system. • Experiment 1: Unit step change in m1 with all loops open. When a unit step change is made in the input variable ml' with all the loops open, the output variable y1 will change, and so will y2, but for now we are primarily interested in the response in Yr After steady state has been achieved, let the change observed in y1 as a result of this change in m1 be referred to as lly1m; this represents the main effect of m1 on y1• Observe from Eq. (21.3) that: (21.5)

Keep in mind that no other input variable apart from m1 has been changed, and all the control loops are opened so that there is no feedback control involved.

726

MULTWARIABLE PROCESS CONTROL

• Experiment 2: Unit step clumge in nit With Loop 2 closed .

In this "experiment," the same unit step change imple mente d in m1 in the first "experiment" will be implemented; howe ver, this time, Loop 2 is closed, and the contro ller Kcz is charg ed with the responsibil ity of using the other input variable m2 to ward off any upset in y occur ring as a result of this step 2 chang e in mr Note that Figure 21.1 shows that m 1 has both a direct influence on y1 and an indire ct influence with the path given by the dotted line. In partic ular the following things happe n to the process due to the chang e in m1: 1. Obviously y1 changes because of Kw but becau se of interactions via the g21 element, so does y2 •

2. Unde r feedback control, Loop 2 wards off this intera ction effect on y2 by manip ulatin g m2 until ]h is restor ed to its initia l state befor e the occurrence of the "disturbance." 3. The changes in m2 now return to affect y via the g12 transf er function 1 eleme nt

Thus, the changes observed in y 1 are from two differ ent sources: (1) the direct influence of m1 on y 1 (which we alread y know from "experiment" 1 to be llylm), and (2) the additi onal, indirect influe nce, precip itated by the retali atory action from the Loop 2 controller in wardi ng off the interaction effect of m1 on y 2 , say, lly 1,. Note that this quant ity, lly ,, is indica 1 tive of how much intera ction m1 is able to provoke from the other contro l loop in its attem pt to controlyzAfter the dynamic transients die away and steady state has been achieved, there will be a net change ob~ed in y ,1ly • given by: 1 1

a sum of the main effect of m1 on y1 and that of the interactive effect provo ked by m1 interacting with the other loop. As we will show below, the quant ity lly • will be giv~ by, say: 1

(21.6) Observe now that a good measure of how well the proce ss can be controlled if m1 is used to control y1 is: · ~Ylm

All

= ~y,*

Au

= ~Ylm + ~Ylr

or ~Ylm

(21.7)

,. CHAP 21 INTERACTION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

72 7

This quantity is a measure of the main effect of m1 on y 1 compared to the total effect including the effect it provokes from the other controller, since it cannot control y1 without upsetting y2 • Thus the quantity, ~1 , provides a measure of the extent of interaction in using m1 to control y1 while using m2 to control y2• Observe that we can now perform a similar set of "experiments" that, this time, investigate the candidacy of m2 as the input variable to use in controlling Yr By properly interpreting the measures calculated from Eq. (21.7), we can quantify the degree of steady-state interaction involved with each control configuration, and thus determine which configuration minimizes steady-state interaction.

21.2.2 Loop Pairing on the Basis of Interaction Analysis Let us return to the results of the "experiments " performed on the example 2 x 2 system of the previous subsection and now interpret the quantity ~ 1 as a measure of loop interaction. We will do this by evaluating the consequences of having various values ~1' L The case when ~1

=1

This can happen only if Ay1r is zero; in physic 0.5, the main effect contributes more to the total effect than the interaction effect, and as the main effect contribution increases, Au becomes closer to 1. For Au < 0.5, the contribution from the interaction effect dominates, and as this contribution increases, ~ 1 moves closer to zero. For ~ 1 = 0.5, the contributions from the main effect and the interactive effect are exactly equal

728

MULTIVARIABLB PROCESS CONTROL

4. The case when ~1 > 1 This corresponds to the condition where L1y , (the interac tion effect) is opposite 1 in sign to L1y1m (the main effect), but smaller in absolute value. In this case the total effect, L1y 1 * , is less than the main effect, L1y 1 m, and thus a larger controller action m1 is required to achieve a given chang e in y1 in the closed loop than in the open loop. Obviously for A very large and positive, the 11 interactive effect almost cancels the main effect and closed -loop control of y1 by m1 will be very difficult to achieve. 5. The case when ~1 < 0 This case arises when L1y1, (the interaction effect) is not only opposite in sign to L1y1m (the main effect), but is larger in absolute value. The pairing of m1 with y1 in this case is not very desirable because the direction of the effect of m1 on y 1 in the open loop is oppos ite to the direction in the closed loop. The conse quenc es of using such a pairin g could be catast rophic as we shall demonstrate below. With these prelim inary ideas in mind, it is possible to genera lize the interaction analysis to multivariable systems of arbit:Tary dimension.

21.3

THE RELATIVE GAIN ARRAY (RGA)

The quant ity A11 introduced in the last section is known as the relative gain betwe en outpu t y1 and input m1, and as we have seen, it provides a measure of the extent of the influence of process interactions when m is used to control Yr Even thoug h we introduced this quantity in reference to a 1 2 x 2 system, it can be generalized to any other multivariable system of arbitra ry dimension. Let us define Aii' the relative gain betwe en outpu t variab le Yi and input variable mi' as the ratio of two steady-state gains:

1

Aij

=

iJyi) (am.

J all loops

open ~av~--~_u~~~~~----

(U::;)

(21.8)

all loops closed excepl for the m; loop

open-loop gain ) . =( closedloop gain for loop 1 under the control of m;

When the relative gain is calculated for all the input /outpu t combinations of a multivariable system, and the results are presen ted in an array of the form shown below: ·

All A12 ··· A1n] Azl i\.22 ··· Azn [ A=

... ...

...

Ani An2

Ann

(21.9)

CHAP 21 INTERACTION ANALYSIS AND MULTIPL.E SINGLE-LOOP DESIGNS

729

the result is what is known as the relative gain array (RGA) or the Bristol array. The RGA was first introduced by Bristol [1] and has become the most widely used measure of interaction.

21.3.1 Propertie s of the RGA The following are some of the most important properties of the RGA; they provide the bases for its utility in studying and quantifying interactions among the control loops of a multivariable system. 1. The elements of the RGA across any row, or down any colunm, sum up to 1, i.e.: (21.10)

2. A;i is dimensionless; therefore, neither the units, nor the absolute values actually taken by the variables mi, or Yi' affect it. 3. The value of A-if is a measure of the steady-state interaction expected in the ith loop of the multivariable system if its output Yi is paired with mji in particular, Aij == 1 implies that mi affects Yi without interacting with, and/ or eliciting interaction from, the other control loops. By the same token, Aij = 0 implies that mi has absolutely no effect on Y;· 4. Let K;/' represent the loop i steady-state gain when all the other loops - excepting this one - are closed, (i.e., a generalization of Eq. (21.6) for the example 2 x 2 system), whereas Kii represents the normal, openloop gain. By the definition of A.ii' observe that 1

Kq* == :;tKii

(21.11)

ij

with the following. very important implication: 1/ A.iJ tells us by what factor the open-loop gain between output Yi and input mi will be altered when the other loops are dosed. 5. When A.ij is negative, it is indicative of a situation in which loop i, with all loops open, will produce a change in Y;in response to a change in mi totally opposite in direction to that when the other loops are closed. Such input/ output pairings are potentially unstable and should be avoided. Before discussing the general principles of how the RGA is actually used for loop pairing, it seems appropriate first to discuss the methods by which it is computed.

21.3.2 Calculati ng the RGA There are two basic methods used for calculating the RGA for a square, linear system, whose transfer function matrix G (s) is available: (1) the "first principles" method, and (2) the matrix method.

730

MULTIVARIABL E PROCESS CONTRO L

Calculating RGA 's from First Principles We will illustrat e this procedu re with the 2 x 2 system of Eqs. (21.3) and (21.4). First, we observe that, by definition, the RGA is concern ed with steadystate conditions, thus we only need the steady- state form of this model, which is: (21.12)

(21.13)

To obtain 'il.11 from the definiti on in Eq. (21.8), we need to evaluat e both of the indicat ed partial derivati ves from Eqs. (21.12) and (21.13) as follows: Eqs. (21.12) and (21.13) represe nt steady-state, open-lo op conditions; therefore, the numera tor partial derivati ve in Eq. (21.8) is obtaine d from Eq. (21.12) by straigh tforwar d differen tiation:

(ay.) -

dm1

-K

all loopJ open -

II

(21.14)

The second partial derivati ve calls for Loop 2 to be closed, so that in respons e to changes in m1, the second control variable m,_ can be used ultimate ly to restore y 2 to its initial value of 0. Thus, to obtain the second partial derivati ve, we first find, from Eq. (21.13), the value m must take to keep y2 = 0 2 in the face of changes in m1; what effect this will have on y is deduce d by 1 substitu ting this value for m2 in Eq. (21.12). Setting y2 = 0 and solving for m,. in Eq. (21.13) gives: (21.15)

and substitu ting this into Eq. (21.12) gives: (21.16)

Having thus "elimin ated" m2 we may then differentiate, so that now: (21.17)

and we obtain, finally, that: (21.18)

where (21.19)

CHAP 21


731

As an exercise, the reader is encouraged to follow the illustrated procedure and confirm that the other relative gains, A,2, ~1' and A.n, are given by: (21.20)

and (21.21)

Thus the RGA for the 2 x 2 system is given by:

A= [

1 1-~

-=....{_ 1-~

-=....{_] 1-~ _1_

(21.22)

1-~

Note carefully that the elements add up to unity across any row as well as down any column. It is useful to observe that if we define: (21.23)

for this 2 x 2 system, then the RGA may be written:

A- [

.t 1-.t

(21.24)

The immediate consequence is that the single element A. is sufficient to determine the RGA for a 2 x 2 system. Thus for 2 x 2 systems, A. is often called the relative gain parameter. If we realistically assess this first-principles method of obtaining the RGA, it will not be difficult to come to the conclusion that it will be too tedious to apply in cases where the system is of higher dimension than 2 x 2. The problem is not with the numerator partial derivatives - these are always easy to obtain. The problem is with the denominator partial derivatives: the procedure for their evaluation in higher dimensional systems will involve solving several systems of linear algebraic equations. The fact that linear algebraic equations are quite conveniently solved by matrix methods should then alert us to the possibility of using matrix methods to simplify the tedium involved in calculating RGA's, especially for higher dimensional systems.

The Matrix Method for Calculating RGA's Let K be the matrix of steady-state gains of the transfer function matrix G(s), i.e.: lim G(s) s--.0

K

(21.25)

732

MUL TWARIA BLE PROCESS CONTROL

who se elem ents are Kg; further, let R be the trans pose of the inverse of this stead y-sta te gain matr1x, i.e.:

with elements r;i" Then, it is possible to show that the elements of the RGA can be obta ined from the elements of these two matrices according to: (21.26)

It is impo rtant to note that the above equa tion indicates an element-by-element mult iplic ation of the corre spon ding elem ents of the two matrices K and R; it has noth ing to do with the stand ard matr ix prod uct.

Let us illustrate this proc edur e with the following examples. Exam ple 21.1

RGA FOR A GENERAL 2 x 2 SYSTEM USIN G MATRIX MET HOD S.

Find the RGA for the 2 x 2 syste m whos e transfer function matrix mode l is given in Eqs. (21.3) and (21.4) and comp are the resul ts with those obtai ned earlie r in Eqs. (21.18) throu gh (21.22). Solut ion: For this system, the steady-state gain matri

x is given by:

(21.2 7)

and from the definition of the inverse of a

matrix, we have that

wher e I K I, the determinant of K, is given by:

IKI = K,,K22- K,2K21 and takin g the transpose of the matrix given

R

I [ = (K-1 ) T = -, I K

(21.2 8)

in Eq. (21.27), we obtain:

K22 -K 12

(21.29)

If we now carry out a term-by-term multi plication of the elements of the matrices K (in Eq. (21.26)) and R (in Eq. (21.29)) we obtai n:

(21.3 0)

j

~l

I

CHAP 21 INTERACTION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

733

or (21.31)

*

which, for K11K22 0 simplifies to:

.:til= - 1-

(21.32)

1 - '

with (21.33) exactly as we earlier obtained in Eqs. (21.18) and (21.19). By considering each element of the K and R matrices, we can generate an expression for each of the other-\·

The following are examples of how to calculate the RGA for some specific systems. Example 2L2

RGA FOR THE WOOD AND BERRY DISTILLATION COLUMN OF EXAMPLE 20.6 USING MATRIX METHODS.

Find the RGA for the Wood and Berry binary distillation column whose lransfer function matrix is as given in Eq. (20.26). Solution:

For this system, the steady-state gain matrix is easily extracted from the transfer function matrix by settings =0, giving: K

= G(O) = [

12.8

-18.9

6.6

-19.4

J

(21.34)

and its inverse is obtained, in the usual manner, as: K-1 = [

0.157

-0.153

0.053

-0.104

J

(21.35)

from where, upon taking the transpose of this matrix, we obtain: R - (K- 1)

T

[

0.157

0.053

-0.153

-0.104

-

J

(21.36)

and a term-by-term multiplication of Kij and rij now gives the RGA as: A

=[

2.0 -1.0 -1.0 2.0

J

(21.37)

The following example is computationally more burdensome because it involves a 3 x 3 system, but the basic principles are the same.

734

MULTIVARIABLE PROCESS CONTROL RGA FOR A BINA RY DIST ILLA TION COL UMN WIT H A SIDE STRE AM DRA W-O FF USIN G MAT RIX MET HOD S.

Exam ple 21.3

In Ref. [2], the trans fer function mode l for a pilot scale, binar y clistillation colum n USed to sepa rate ethan ol and wate r was given as in Eq. (21.38) below , wher e the proce ss varia bles are (in terms of devia tions from their respe ctive stead y-sta te value s):

y1

= overh ead mole fraction ethan ol

= mole fraction of ethanol in the side strea m = temperature on Tray #19 u1 = overh ead reflux flowr ate

y2 y3

u2 = side strea m draw -off rate

u3

= reboiler steam press ure 0.66e -2·6' 6.1s + I

- 0.6le - 3 ·5' 8.64 s + 1

-0.00 49e- • 9.06 s + 1

l.lle- 6.Ss 3.25 s + 1

-2.36 e-3' 5s + 1

-0.012e-1. 2s 7.09 s + 1

-33.6 8e-9 ·2s 8.15 s + 1

46.2e -9 ·4' 10.9 s + 1

0.87 (11.6 ls + l)e-s (3.89 s + l)(l8 .8s + I)

Find the RGA for this 3 x 3 system. Solu tion: The stead y-sta te gain matri x in this case

K

[

= G(O)

is given by:

0.66

-0.61

-0.00 49 ]

1.11

-2.36

-0.01 2

-33.6 8

46.2

0.87

(21.39)

The proce dure for obtai ning the inver se of this matri x is long and tedio us: first the deter mina nt is found , by using any of the meth ods indicated in Appe ndix D (the value is -0.5037), and then all the nine cofac tors for this matr ix are foun d; since the adjoi nt is the trans pose of the matr ix of cofac tors, the inver se is easil y foun d there after . The resul t is:

K-1

[

0.0083]

2.955

-0.6

1066

-0.79 7

-0.00 5

61.157

18.405

1.748

2.955

1.066

-06

-0.79 7

0.0083

-0.00 5

(21.40)

so that:

R

= (K-l) T = [

61.157 ] 18.405

(21.41)

1.748

From here, a term- by-te rm multi plica tion of the elem ents in Eqs. (21.39) and (21.41), as indic ated in Eq. (21.26), now gives the comp lete RGA for thls system as:

CHAP 21 INTERACI'ION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

1.95 -0.65 [ A = -0.66 1.88,

-0.22

-0.29

1.52

-0.2J

735

-0.3] (21.42)

Observe carefully that the elements sum to unity across any row, and down any mlmnn.

21.4

LOOP PAIRING USING THE RGA

Having established the RGA as a reasonable means of quantifying control loop interactions at steady state, and having illustrated how it is usually obtained, we will now consider how it may be used as a guide for pairing input and output variables in order to obtain the control configuration with minimal loop interaction.

21.4.1 Interpreting the RGA Elements It is important to know how to interpret the RGA elements properly before we

can use the RGA for loop pairing. For the purpose of properly interpreting the RGA element lii' it is convenient to classify all the possible values it can take into five categories, and in each case, investigate the implications for loop interaction, and what such implications suggest regarding input/ output pairing. 1. A;; = 1, indicating that the open-loop gain between Y; and mi is identical to the closed-loop gain:

• Implication for loop interactions: loop i will not be subject to retaliatory action from other control loops when they are closed, therefore mi can control Y; without interference from the other control loops. If, however, any of the gik elements in the transfer function matrix are nonzero, then the ith loop will experience some disturbances from control actions taken in the other loops, but these are not provoked by control actions taken in the ith loop. • Pairing recommendation: pairing Y; with mi will therefore be ideal. 2. lij

= 0, indicating that the open-loop gain between Y; and mi is zero :

• Implication for loop interactions: mj has no direct influence on Y; (even though it may affect other output variables). • Pairing recommendation: do not pair Y; with mi pairing m; with some other output will however be advantageous, smce we are ·sure that at least Y; will be immune to interaction from this loop. 3. 0 < l;; < 1, indicating that the open-loop gain between Y; and m; is smaller than the closed-loop gain:

736


• Implication for loop interactions: since the closed-loop gain is the sum of the open-loop gain and the retaliatory effect from the other loops: (a) The loops are definitely interacting, but (b) They do so in such a way that the retaliatory effect from the other loops is in the same direction as the main effect of miony1• Thus the loop interactions "assist" m1 in controlling y1• The extent of the "assistance" from other loops is indicated by how close A.ij is to 0.5. When A.11 = 0.5, the main effect of m1 on y1 is exactly identical to the retaliatory (but complementary) effect it provokes from other loops; when A.1i > 0.5 (but still less than 1) this retaliatory effect from the other interacting loops is lower than the main effect of m on y ; if, 1 1 however , 0 1, indicating that the open-loop gain between Y; and m is lilrger 1 than the closed-loop gain:

• Implica tion for loop interactions: the loops interact, and the retaliatory effect from the other loops acts in opposition to the main effect of mi on y1 (thus reducing the loop gain when the other loops are closed}; but the main effect is still dominan t, otherwis e A. t will 1 be negative. For large ~/ values, the controller gain for loop z will have to be chosen much larger than when all the other loops are open. This could cause loop i to become unstable when the other loops are open. • Pairing recommendation: The higher the value of A.q- the greater the oppositi on m1 experiences from the other control loops in trying to control y1; therefore, where possible, do n~t pair m with y if A. . 1 1 11 takes a very high value. 5. A.11 < 0 ; indicating that the open-loop and cl~ed-loop gains between y1 and m1have opposite signs:

• Implica tion for loop interactions: the loops interact, and the retaliatory effect from the other loops is not only in opposition to the main effect of m1on y1, it is also the more dominan t of the two effects. This is a potentia lly dangero us situation because opening the other loops will likely cause loop i to become unstable. • Pairing recommendation: because the retaliato ry effect that m provoke s from the other loops acts in oppositi on to, and in fact1 dominates, its main effect on y1, avoid pairing m with y • 1 1


737

21.4.2 Basic Loop Pairing Rules In light of the foregoing discussion, we are led first to the conclusion that the

ideal situation occurs when the RGA takes the form: I A=

0

0 0

0

0

0

0

0

0

0 0

0

in which each row, and each column, contains one and only one nonzero element whose value is unity. Such an RGA is obtained when the process transfer function matrix is diagonal, or triangular; the first indicates no interaction among the loops, while the latter indicates one-way interaction (see example below and problems at the end of the chapter). It is easy to pair the input and output variables under these circumstances. The case of one-way interaction requires some further discussion. Consider the 2 x 2 system with the following lower triangular transfer function: _1 G(s) =

[

s + 1 _ 3_ 3s + 1

. ~J

(21.43)

Calculation of the RGA yields:

A=[~~]

(21.44)

Note that because the element gu(s) is zero, the input m2 has no effect on output

y1 and thus any control action in Loop 2 has no effect at all on Loop 1 (cf. Figure 21.1). However, the input m1 does influence the output y2 through the nonzero g21 (s) element. Thus upsets in Loop 1 requiring control action by m1 would appear as a disturbance, g21 (s) m 1(s), which would have to be handled by the

feedback controller of Loop 2. Hence, even though the RGA of Eq. (21.44) is ideal, Loop 2 would be at a disadvantage. Thus in deciding on loop pairing one should distinguish between loop pairings having ideal RGA's resulting from diagonal transfer functions and those arising from triangular transfer functions. The latter case will always have poorer performance under feedback control. In the more common nonideal cases, the following pairing rule originally published by Bristol [1] should be used: • Rulel

Pair input and output variables that have positive RGA elements that are closest to 1.0.

738

.MUL TNARIA BLE PROCESS CONTROL

Thus, for a 2 x 2 system with outp ut variables y1 and y2, to be paire d with inpu t variables m1 and m2, if the RGA is:

then it is recom mend ed pair to pair y with m1, and y 2 with m 2, quite often 1 refer red to as the 1-1/2-2 pairing. On the other hand , for the 2 x 2 syste m whos e RGA is: A = [ 0.3 0.7 0.7 0.3

J

the recom mend ed pairi ng is y1 with m and y2 with m1, or the 1-2/2-1 pairing, 2 while the following RGA: A = [ 1.5 -0.5 -0.5 1.5

J

recom mend s the 1-1/2-2 pairi ng in orde r to avoid pairi ng on a nega tive RGA element. (Despite the fact that pairi ng on RGA value s great er than 1 is not very desirable, to pair on a negative RGA value is worse still.) Thus, for the 2 x 2 Wood and Berry distillation colum..1 whos e RGA was calculated in Example 21.2 as: A = [ 2.0

-1.0

-1.0 2.0

J

the 1-1/2 -2 pairi ng is recommended; in term s of the actual process variables, (see Example 20.6) this trans lates to using the over head reflux flowrate to control overh ead mole fraction of meth anol (y - m1), and the botto ms steam flowrate to control the bottoms mole fraction 1 meth anol (y2 - m2). For the 3 x 3 distillation column of Example 21.3 with the RGA:

A

=

1.95 -0.65 -0.3 ] [ -0.66 1.88 -0.22 -0.29 -0.23

1.52

the 1-1/2 -2/3- 3 pairi ng is recommended. In physical terins, this gives rise to a control configuration which uses the overh ead reflux flowrate to control y1, the overh ead mole fraction ethanol; the side strea m draw -off rate to control y2, the mole fraction of ethanol in the side stream, and the reboiler steam press ure to control y3, the temperature on Tray #19.

Niederlinski Index Even thou gh pairi ng rule 1 is usua lly suffi cient in most cases, it is often necessary (especially with 3 x 3 and highe r dime nsion al systems) to use this rule in conjunction with stability cons idera tions prov ided by the following

CHAP 21 INTERACTION ANALYSIS AND MULTIPLE SINGLE -LOOP DESIGNS

739

theore m origina lly due to Nieder linski (Ref. [3]), and later modifi ed by Grosdi dier et al. (Ref. [4]):

Consider the n x n multivariable system wltose input and output variables have been paired as follows: y 1 - u1 , y - u , ••• , Ynun, resulting 2 2 in a transfer function model of the form: y(s)

= G(s) u(s)

Further let each element of G(s), g;/s), be 1. Rational, and 2. Open-loop stable,

and let n individual feedback controllers (which have integra l action) be designed for each loop so that each one of the resulting n feedbac k control loops is stable when all the other n - 1 loops are open. Under closed-loop conditions in all n loops, the multiloop system will be unstable for all possible values of controller parameters (i.e., it will be "structurally monotonic unstable") if the Niederlinski index N defined below is negative, i.e.:

Ne

(21.45)

The following import ant points help us utilize this result proper ly: 1. This result is both necess ary and sufficie nt only for 2 x 2 systems; for higher dimens ional system s, it provid es only sufficient conditi ons: i.e., if Eq. (21.45) holds then the system is definitely unstable; howev er, if Eq. (21.45) does not hold, the system may, or may not be unstabl e: the stabili ty, will, in this case, depen d on the values taken by the control ler parame ters. 2. For 2 x 2 system s the Nieder linski index becomes:

'>

where 'is given by Eq. (21.19). Thus for a 2 x 2 with a negativ e relativ e gain, 1, the Nieder linski index is always negativ e; hence 2 x 2 system s paired with negative relative gains are always structurally unstable. 3. The theore m is for system s with rationa l transfe r functio n elemen ts, thereb y technically exclud ing time-d elay system s. Howev er, since Eq. (21.45) depend s only on steady- state gains (which are indepe ndent of time delays), the results of this theorem provid e useful inform ation about time-d elay system s also, but the analysi s is no longer rigorou s (see Ref. [4].) Thus Eq. (21.45) should therefore be applied with caution when time delays are involved.

740

MULTWARIABLE PROCESS CONTROL

This leads to loop pairing rule 2:

• Rule2 Any loop pairing is unacceptable if it leads to a control system configuration for which the Niederlinski index is negative.

Thus, the strategy for using the RGA for loop pairing may be summarized as follows: 1. Given the transfer function matrix G(s), obtain the steady-state gain matrix K = G (0), and from this, obtain the RGA, A; also obtain the determinant of K, and the product of the elements on its main diagonal. 2. Use Rule #1 to obtain tentative loop pairing suggestions from the RGA, by pairing on positive elements which are closest to 1.0. 3. Use Niederlinski's condition of Eq. (21.45) to verify the stability status of the control configur ation resulting from 2; if the pairing is unacceptable, select another.

21.4.3 Some Applications of the Loop Pairing Rules As we have shown above, for a 2 x 2 system, the Niederli nski stability condition is exactly equivalent to requiring that we pair only on positive RGA elements. Thus as long as we are dealing with 2 x 2 systems, Rule #1 is sufficient for obtaining appropriate loop pairing. This is, however, not the case for 3 x 3 and higher dimensional systems; it is possible to pair on positive RGA elements and still have a structurally unstable system, as the following example illustrates. Example 21.4

LOOP PAIRING FOR A 3 x 3 SYSTEM.

Calculate the RGA for the system whose steady-sta te gain matrix is given below, and investigate the loop pairing suggested upon applying Rule #1 (see Koppel, Ref. [5]): I

1

(21.46)

3

Solution: By taking the inverse, and calculating the RGA in the usual manner as illustrated in Example 21.3 we obtain the following RGA:

A= [

-~~5 ~· 5 -4.5

4.5

-4.5] 4.5 I

(21.47)


741

from which, by Rule #1, the 1-1/2-2/3-3 pairing is recommended. According to this pairing however, the steady-state gain matrix for the system has determinant given by:

IKI =

(21.48)

-0.148

and the product of the diagonal elements of K is:

/J/u =(+)(+)(+)= i1

(21.49)

Thus, for this system, according to Eq. (21.45) this pairing will lead to a negative Niederlinski index and an unstable configuration. This example provides a situation in which Rule #2 disqualifies a loop pairing suggested by Rule ##1; thus an alternative loop pairing must now be considered. While the available alternatives are not especially attractive, a possible pairing would be 1-1/2-3/3-2 which has the rearranged relative gain array: A= [

~~5

;:

-4.5

~-5] 4.5

and new steady-state gain matrix:

with determinant I K I= ; 7 and Niederlinski index: N _ 4/27 _ .i. - 5/3 - 45

so that with this pairing the system is no longer structurally unstable. In higher dimensional systems, it is possible to pair on negative RGA values and still have a stable system, something of an impossibility in a 2 x 2 system. ·However, for such systems, it has been shown (see also Gagnepain and Seborg [6]) that if the loop which is paired on the negative RGA element is opened, the lower dimensional subsystem will be unstable. An example system that illustrates this (see Koppel [5] and MeA voy [7]) has steady-state gain matrix:

K = G(O) = [

o\ -2

2 -3

-0.1 ] -1

(21.50)

742

MULTIVA.RIABLE PROCESS CONTROL

with a deter mina nt,

I K I =0.53. A

=

The RGA for this syste m is:

-1.89 [ -0.13

3.59

-0.7 ]

3.02

-1.89

3.02

-5.61

3.59

(21.51)

from whic h one fmds that the only feasible pairin g has to invol ve a negat ive RGA elem ent for the first loop becau se the other pairi ng confi gurat ions are disqu alifie d by Rule #2. The sugge sted 1-1/2 -2/3- 3 pairin g gives rise to a confi gurat ion for whic h the syste m, accor ding to Niede rlinsk i's theor em, is not struc turall y unsta ble despi te pairin g on the negat ive (1,1) RGA eleme nt. Howe ver, if the first loop is open ed (i.e., the y1 - m1 varia bles are dropp ed from the proce ss mode l) the resul ting subsy stem will hav.e a stead y-sta te gain matri x relati ng the rema ining two outpu t varia bles y2 and yy to the rema ining two input variab les, ~ and my given by:

(21.52) and it is easy to verify that the y - m , y 2 2 3 - m 3 pairi ng will viola te the Niede rlinsk i theor em and thus give rise to an unsta ble syste m. Such a syste m that is stable when all loops are close d, but that goes unsta ble shou ld one of them becom e open , is said to have a low degree

integrity.

of

Thus , in the final analy sis, after we put all the foreg oing facto rs into consi derat ion, we come to the follow ing concl usion abou t using the RGA for loop pairin g:

Always pair on positive RGA elements that are closes t thereafter use Niederlinski's condition to check to 1.0 in value; the resulting configuration for structural instability. Wherever possib le, avoid pairing on negative RGA elements; for 2 x 2 system s such pairings always lead to an unstable configuration, while for :highe r dimen sional systems, they lead to a configuration which at best has a low degree of integrity. '

21.5

LOOP PAIRING FOR NON UNE AR SYSTEMS

So far, we have discussed how to obtai n the RGA essen tially from trans fer funct ion mode ls, which , as we know, repre sent only linea r syste ms. By virtu e of the fact that most chem ical proce sses of impo rtanc e are nonli near, it is pertin ent at this stage to consi der whet her Bristo l's RGA techn ique can be used for pairin g the inpu t/ outpu t varia bles of nonli near syste ms. The reade r shoul d have noted by now that intera ction analy sis by mean s of the RGA is based only on stead y-sta te infor matio n. Whil e this is often consi dered a limita tion (see Sectio n 21.8), this partic ular aspec t is in fact what make s the RGA techn ique easily appli cable to nonli near syste ms:

CHAP 21

INTERA CTION ANALY SIS AND MULTIPLE SINGLE-LOOP DESIGNS

743

Assum ing that a process model is available, any of the follow ing two approaches will enable us obtain RGA's for nonlinear systems: 1. From first princip les (i.e., taking partial deriva tives), using the steady-state version of the nonlinear model, because only steady -state models are involved, it is sometimes possible to obtain analyti cal expressions.

2. By linearizing the nonlinear model around a specific steady state and using the approximate K matrix to obtain the RGA. The second approach is usually less tedious than the first one, and is therefore

more popular.

It should not come as a surpris e that RGA's for nonlinear system s are distinguished by their dependence on specific steady-state proces s conditionS. As we may recall from Chapter 10, it is the nature of nonline ar process systems for characteristic process parame ters to be different at differe nt steady -state operat ing conditions. Thus, while RGA's for linear systems consist only of constant elements, the RGA's for nonlinear systems are functio ns of process steady-state operating conditions and will therefore change as these operat ing conditions change. We shall now presen t examples to illustrate these two proced ures for obtaining RGA's for nonlinear systems, and how such RGA's are used for loop pairing . Examp le 21.5

RGA AND LOOP PAIRING FOR NONLINEAR SYSTEMS: 1. mE BLENDING PROCESS.

The process shown in Figure 21.2 below is used to blend two process streams containing pure material A and pure material B, with respective (molal) flowrates, FA and F8 • The objective is to control both the total product flowrate f, and the produc t composition, represented as x, and calculated in terms of mole fraction of material A in the blend. The mathematical model for the process, assuming negligible mixing dynamics, is obtained from total and component mass balances as follows:

Total Milss Balance: (21.53)

Component A MRss Balance: X

Figure 21.2.

=

The blending process.

(21.54)

744

MULTWARIABLE PROCESS CONTROL Obta in the RGA for this proce ss and sugg est whic h input varia ble to pair with whic h outpu t variable to achieve good contro L Solut ion: Obse rve that for this process, the two outpu t varia bles are F and :x, and the input (or control) varia bles are FA and F,y let us refer to these inpu t variables as m and~ 1 respectively. The process modeling equations then become: (21.55)

whic h is linear, and:

(21.56) whic h is nonli near. Since this is a 2 x 2 system, we only need to obtai n the (1,1) element of the RGA given by:

il.

=

both loops open

~--'-----

(

~

(21.57)

) oecond loop closed

From Bq. (21.55), with bOth loops open:

(

~

) both loops open

=

(21.58)

Now , upon closing the secon d loop, the quest ion now is: what value will "'2 have to take, in order that, for any change in m , 1 :xis resto red to its desir ed stead y-sta te value , say x"? To answ er this quest ion calls for setting :x = X' in Eq. (21.56) and solvi ng for "'2 in terms of m1 and%'; the resul t is:

(21.59) Thus , when the secon d loop is close d, to perfo rm its task of maintaining the mole fracti on of mate rial A at :x•, m will respo 2 nd to any chang es in m1 as indic ated in Eq. (21.59). If we now intro duce this into Bq. (21.55 ) we will have:

or (21.60) an expre ssion that repre sents the chang es to expec t in F as a resul t of changes in m1, when the second loop is closed. From here, differ entia ting with respe ct to m now gives: 1

iJF ) ( iJm 1 second loop closed

1

= :x*

(21.61)

CHAP 21 INTERACTI ON ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

745

with the final result that (21.62) and therefore, the RGA for this blending system is:

x*

1- x*

1 - x*

x*

A- [

J

(21.63)

where x"' is the desired mole fraction of species A in the blend. It is now important to observe the following about this result 1.

The RGA is dependent on the particular steady-state value x"' desired for the composition of the blend; it is not constant as was the case with the linear systems we have dealt with so far.

2.

The implication here is that the recommended loop pairing will depend on our steady-state operating point

3.

Because x• is a mole fraction, it is bounded between 0 and 1, i.e., 0 ~ x"' S 1. Thus, none of the elements of the RGA will ever be negative. The implication here is that the worst we can do is to have severe interactions if the input and output variables are not properly paired; the system cannot be unstable as a result of poor pairing. A little reflection on the nature of the physical blending system will confirm this.

The loop pairing strategy can now be developed as follows: 1.

If x" is close to 1, the first implication, from Eq. (21.56), is that m1 (which represents the flowrate of pure material A) is larger than m2, the other flowrate- From Eq. (21.63) the recommended pairing in this case is F- m1 and x - m2, i.e., the larger flowrate is to be used in controlling total flow, while the smaller flowrate is used to control the composition.

2.

That this is the most reasonable pairing can be seen by examining the physical system: when the product composition is close to 1, we have almost pure A in the blend, and the total flowrate can be modified quite easily using the flowrate of A, without affecting the composition too drastically; similarly, if we wish to alter the composition, the addition of small amounts of material B can cause significant changes in the composition without altering the flowrate significantly. Thus, the flow controller will not interact strongly with the composition controller if this F- m1 and x - m2 pairing is used. By contrast, if the alternate pairing configuratio n is used, the interactions between the two control loops will be quite severe.

3.

When the steady-state product composition is closer to 0, the RGA suggests that we switch loop pairings, and use m2 to control F, and m1 to control x. Again, the physics of the problem indicates that this is the best course of action.

4.

An interesting situation arises when x• = 0.5; in this case, it matters not which input variable is used to control which output variable; the observed interactions will be significant in either case.

746

MUL TWA RIAB LE PROCESS CON TROL

Non line ar syst em models are not always as easy to deal with as the preceding example appears to indicate . In mos t cases, the nonlinear steady~ stat e mod el can be quit e tedi ous to han dle, especially in· obta inin g the den omi nato r part ial deri vati ve that alm ost alw ays requ ires analytical solutions of several nonlinear algebrai c equations. The technique of utilizing an appr oxim ate linear mod el (obtained by linearization around an operating stea dy state) is not only more convenient in such cases, it is also consistent with other aspects of nonlinear control syst em analysis and design we have presente d so far. The following example illustrates this technique. Exam ple 21.6

RCA AND LOO P PAIR ING FOR NON LINE AR SYSTEMS:

2. THE STIRRED MIX ING TANK.

The stirre d mixing tank intro duce d in Chap ter 20 (see Figure 20.1), is a multivariable syste m in whic h the cold and hot strea m flowrates are to be used in cont rolli ng the liquid level and tank temperature. The nonl inea r modeling equa tions for this syste m were obta ined in Example 20.3, and an appr oxim ate transfer function, obta ined by linearizing arou nd som e nom inal stead y-sta te liqui d level h and temp eratu re T5, has also been obta ined in 8 Example 20.5, and pres ente d in Eqs. (20.23) and (20.24). Obta in the RGA for this syste m and use it to reco mme nd whic h of the inpu t varia bles shou ld be used to cont rol the liqui d level and whic h shou ld be used to control the tank temperature.

Solu tion: From the appr oxim ate transfer func tion matr ix give n in Example 20.5, we obta in the steady-state gain matrix for the proc ess as:

(21.64)

We recall from our prev ious enco unte rs with this process that its outp ut varia bles are repre sente d as follows:

y1 = liquid level; and the inpu t variables are repre sente

m1

Y2 = tank temperature d as:

= hot stream flowrate;

m.z

= cold strea m flowrate

Either by takin g adva ntag e of the fact that this is a 2 x 2 syste m, and there fore usin g Eqs. (21.32) and (21.33), (this is reco mme nded ), or by usin g the full matr ix meth od, the RGA for this system is obta ined as:


747

(21.65)

A

where we see again that the RGA depends on the steady-state operating temperature T5' and as such, we will expect different loop pairing suggestions for different steadystate operating conditions. (It is interesting to note that the RGA is not directly dependent on the steady-state level h5 .) To illustrate these points, let us introduce numerical values for the parameters of this process as follows: Hot stream temperature, TH =650C Cold stream temperature, Tc = 150C We will now investigate loop pairing for this process under four different steady-state operating conditions: Condition 1:

T5 > 40°C (in particular T5 = ss•q

Under these conditions, observe that the operating steady-state temperature is closer to the hot stream temperature. Introducing the numerical values into the RGA gives:

A

=[

0.8 0.2

0.2 0.8

J

(21.66)

and the RGA recommends a 1-1/2-2 pairing, i.e.: Hot stream (m1) to control liquid level, (y1), and Cold stream (J!l:z) to control temperature, (y2). Upon examining the physical process, we immediately see that this is the only reasonable thing to do: the cold stream temperature (ts•q is much farther away from the steady-state operating tank temperature (55°C) than the hot stream temperature (65°C). As a result, small changes are required in the cold stream flowrate in order to produce noticeable changes in the tank temperature, making it possible for the temperature controller to do its job effectively without causing much upset in the liquid leveL Conversely, because the hot stream temperature is closer to the operating steadystate tank temperature, it can be used to control the level without causing significant changes in the tank temperature. Note therefore that the alternative pairing (hot stream to control temperature, and cold stream to control level) will cause significantly more pronounced problems due to loop interactions, and will therefore not be as effective.

Contlition 2:

.... .... .. ~ J

v llKlli8l..E PROC ESS CONTROL T1 < 40°C (In parti cular Ts = 25°Q

We now have a situa tion in which th temperature. The RG e operating steady-state temperature is clo A in this case, upon values, is: introducing the num ser erical A = [ 0. 2 0. 8] to the cold stream

0.8 0. 2 (21.67) and the RGA recom men condition 1: the 1-2/2 ds a pairing which is the reverse of th at recommended un -1 pairing, in which der : Uquid level (y ) is co 1 ntrolled by the cold stream ("'2}, and Temperature (y:z) is co ntrolled by the ho Again, from physica l considerations, thi t stream (lilt)· configuration th at s makes perfect se minimizes t."e ste ady-state interactio nse, since this is the only variables. n effects among th e process Condition 3: In this case, the op stream temperatures erating temperature is exactly equidista . The RGA in this ca nt from the cold and se is: hot

A

= [ 0.5

0. 5] 0.5 0.5

and, as we might als o expect from the ph ysics of this situatio ba d. n,

(21.68) either pairing is eq

ually

Contlifion 4: In this case, the ta Observe that the RG nk is to operate at the same tempe rature as the hot str A in this case is: eam.

(21.69) an d it suggests th at we can achieve perfect control of th wi th the temperatu e level without in re if we use the ho terac t stream for this pu obvious fact th at va rpose. It also indica ting riations in the hot tes the stream flowrate ar temperature. e not able to affect the tank

These simple ex indeed be us ed fo amples have therefore demonstrate r d sometimes recom nonlinear as well as for linear syste that the RGA can mends different pa ms. That the RG iring A confirms to us that even though the an s at different ~perating conditions linearized models, alysis has been ba this cardinal prop erty of nonlinear sy sed on approximate stems is not lost.

21.6

LOOP PAIRING INTEGRATOR MFOR SYSTEMS WITH PURE ODES

Since interaction analysis via the information, a pa RGA involves us rticularly interest in g steady-state ing situation arise processes that cont s when dealing w ain pure integrator ith show no steady sta elements, since pu re integrator elemen te. ts


749

Even though McAvoy [7] has several suggestio ns as to what might be done in such cases, we use the following example to suggest an alternativ e way of dealing with systems of this nature. Example 21.7

RGA FOR AN INDUSTRIAL DE-ETHANIZER.

The transfer function for a 2 x 2 subsystem extracted from the 3 x 3 model obtained by Tyreus [8] for an industrial de-ethanizer is given below. Obtain the RGA for this system and use it to recommend loop pairing:

G(s)

=

-4s]

1.318e-2.5s 20s + 1

-:!~___

0.038(182s + I) (27s + 1)(10s + 1)(6.5s + l)

0.36 s

[

3s

(21. 70)

Solution:

First we observe that to obtain the steady-state gain matrix K, settings= 0 in Eq. (21.70) will not work since the (1,2) and (2,2) elements contain the pure integrator

mode represented by 1/s. Let us replace this term by I, i.e.: 1

I= s

We then have:

K = lim G(s) = lim [ s-+O

/-+oo

1.3.18

-/ ]

3

(21. 71)

0.038 0.36/

Thus the relative gain parameter for this system may be obtained from Eqs. (21.18), (21.19), and (21.71) to yield: ,t = lim I-+..

(I

+ 0.03; X 0.333/) 1.138 X 0.36/

(21.72)

Since the variables I cancel, we obtain: ,t

= 0.97

and the resulting RGA is: A= [

0.97 O.QJ 0.03

0.97

(21.73)

J

(21.74)

Obviously 1-1/2-2 pairing is recommended. The approach of represent ing the integrato r element by 1, calculatin g the RGA with this as a paramete r, and finally taking the limit as this quantity tends to infinity will work as illustrate d in the example wheneve r the integratin g elements occur in such a way that the 1 terms cancel. When this is not the case, other approach es suggested by MeAvoy [7] can be used.

750

21.7

LOOP PAIRING FOR NO

MULTIVARIABLE PROC ESS CONTROL

NSQUARE SYSTEMS

Ou r discussion so far ha s centered aro un d ho w to obtain RGA's, an d ho the m for inp ut/ ou tpu t w to use pa iri ng wh en the proces s ha s an equal nu mb er an d ou tpu t variables. of inp ut Such systems are referr ed to as square systems their transfer function ma because trices are square. It is no t always true, ho wever, tha t a multivari able process must ha ve eq ua l nu mb er of inp ut an d ou tpu t variables. an Some multivariable sys nonsquare systems in the sen tems are se tha t they ha ve an un equal nu mb er of inp ut ou tpu t variables, an d an d therefore the ir transfer function matrices wi ll square. not be The mo st obvious proble m with nonsquare system pairing, there wi ll alway s is that after inp ut/ ou tpu s be a residual of un pa t ire d inp ut or ou tpu t var de pe nd ing on wh ich iables, of the se are in excess . Thus, qu ite ap art fro no ntr ivi al issue of decid m the ing wh ich inp ut variable variable, we no w ha ve to pa ir wi th wh ich ou tpu t to co nte nd wi th an oth er problem: wh ich varia sh ou ld be left redundant, bles an d which ones sho uld take active pa rt in the co scheme? These are the ntrol questions to be answere d in this section.

21.7.1 Oassifying No ns qu ar e Syste

ms

We sta rt by no tin g tha t the fewer inp ut tha n ou tpu t re are two types of nonsquare systems: systems wi th variables, an d those wi th more inp ut tha n ou variables. In ord er to tpu t be able to control the ou tpu t variables arbitra ne ed at least an eq ua l nu rily, we mb er of inp ut variables ; thu s systems tha t ha ve inp ut tha n ou tpu t variab fewer les wi ll be referred to as underdefined systems, wh the sy ste ms wi th mo re ile inp ut tha n ou tpu t varia bles wi ll be ref err ed overdefined. The former to as ha s a deficiency in inp ut variables; the lat ter excess of inp ut variable has an s. Thus, a multivariab le system wi th n ou tpu inp ut variables - whose t an d m tra nsf er function matrix di me ns io n- is underde will therefore be n x m fined if m < n, an d overd in efined if m > n.

21.7.2 Un de rd ef in ed Sy ste ms

Fo r un de rde fin ed system s, the ma in iss ue is tha t no t all the ou tpu ts ca controlled, since we do n be no t ha ve en ou gh inp ut variables. The loo p pa decision is, however, mu iri ng ch simpler in this case:

By economic considerations, n output variables are the or other such means, decide which m of the the m available input var most important; these be paired with variables will not be under iables; the less important n - m output any form of control.

will

Let us illustrate this wi th an example. Ex am ple 2L 8

RG A AN D LO OP PA IRI NG FO R AN UNDE RDEFINED NONSQUARE SYSTEM : A BINARY DISTILL ATION CO LU MN .

Co nsi der the situ atio n in which the bin ary distill ation col um n used in Ex has its sid est rea m drawample 21.3 off rat e set at a fixed am oun t tha t cannot be cha all three of its ou tpu t var nge d, and yet iables, the ove rhe ad mo le fraction of ethanol, the sid est rea m


751

mole fraction of ethanol, and the Tray #19 temperature are still being monitored, and are to be controlled. Since we have lost one control variable, the process model now

becomes:

[~;]

0.66e-2 ·6• 6.7s + I 1.11e-6.5s 3.25s + 1 -33.68e-9·2• 8.15s+l

-0.0049e-• 9.06s + I -0.012e-1. 2• 7.09s + I 0.87(11.61s + l}e-• (3.89s + 1)(18.8s + 1)

[::]

(21.75)

where the input variables m1 and m2 are, respectively, the overhead reflux rate and the reboiler steam pressure. It is impossible to control all three output variables with only two input variables; however, the sidestream mole fraction of ethanol is deemed the least important of the output variables, leaving two output variables to be controlled with the two input variables. How should these input and output variables be paired? Solution: Upon deciding to leave the control of the sidestream composition out of the control scheme, we may now rewrite the model as:

[

0.66[2•6• 6.1s + 1

-0.0049e-• 9.06s + 1

-33.68e-9 ·2' 8.15s + 1

0.87(11.6ls + 1)e-• (3.89s + 1)(18.8s + 1)

]

J

[

::

(21.76)

along with the additional relation 1.11e-6 ·5' -0.012e-1.2s + I ml + 7.09s + I m2

= 3.25s

Y2

(21.77)

The modified (square) subsystem's steady-state gain matrix is obtained from Eq. (21.76) as:

i

[

o.66 -33.68

-0.0049 ] 0.87

(21.78)

and the RGA is:

A

[ 1.4 -0.4] -0.4 1.4

(21.79)

The recommended input/ output pairing scheme is:

y1 - m1; overhead reflux to control overhead composition y3 - m2; reboiler steam pressure to control Tray #19 temperature Note that according to this control scheme, as indicated in Eq. (21.77), the sidestream composition will drift as the values of m1, and m2 change, but that is the nature of underdefined systems: we can only achieve arbitrarily good control of two out of the three output variables and accept the drift in the third one.

752

MULTIVARIABLE PROCESS CON TROL This example has sho wn that for und erd efin ed systems, the stra teg y is to choose a square subsystem by dro ppi ng off the excess num ber of out put variables on the basis of economic importa nce; the subsequent analysis is the same as for square systems.

21.7.3 Ov erd efi ned Systems The rea l cha llen ge in dec idin g on loo p pai ring s for non squ are sys tem s is pre sen ted by ove rde fine d syst ems. In this case, we hav e an excess of inp ut variables, and therefore we can actually achieve arbitrary control of the fewer out put variables in more than one way. The situ atio n her e is as follows : since ther e are m inp ut var iables to n out put variables (an d m > n), there are man y mo re inp ut var iables to choose from in pai ring the inp uts and the outputs, and therefore ther e will be several diff eren t squ are sub sys tem s from whi ch the(~~iring possibi lities are to be det erm ine d; in actua}!~ct, ther e are exactly 11 ) possible squ are subsystems, whe re we recall tha t \n) =m!/ n! (m-n)!. The stra teg y is therefore: 1. Firs t dete rmi ne, from the given pro ces s mo del, all the (:) possible square subsystems. 2. Obt ain the RGA's for each of these square subsystems. 3. Examine these RGA's and pic k the bes t sub sys tem on the basis of the ove rall character of its RGA (in term s of how close it is to the ide al RGA). 4. Hav ing thu s dete rmi ned the bes t subsystem, use its RGA to dete rmi ne whi ch inp ut variable wit hin this subsystem to pai r wit h whi ch out put var iab le. 'This is bes t illu stra ted wit h an example. Exa mpl e 21.9

RGA AN D LOO P PAI RIN G FOR AN OVERDEFINED NO NSQ UA RE SYSTEM.

A cert ain multivariable syst em has two outp uts y1 and y ,tha t can be controlled by 2 any of three available inpu ts m 1, m2, and m3• Thr oug h pulse testing, the following transfer function model was obta ined:

Which loop pair ing is expected

0.07 e-0. 3s 2.5s + 1

0.04 e-0. 03•

- 0.003e-O.ts s + 1

- O.OOle-0' 41

2.8s + 1 ] [:; ]

(21 80)

1.6s + 1

to give the best control?

Solu tion : This is a 2 x 3 system, and as such , we hav e a situ atio n in whi ch only two of the thre e cand idat e inpu t variables will be used for control, whi le the thir d inpu t variable will hav e to be set at a fixed valu e, and will therefore be redWldan t.

-LOOP DESIGNS CHAP 21 INTERACTION ANALYSIS AND MULTIPLE SINGLE

753

should be redund ant, we To determ ine which variable should be active and wwcr 3 such subsystems that = ~~1 are there tems; subsys 2 x 2 e possibl the first obtain all functio n matrices, all r transfe onding we now list, comple te with their corresp (21.80). Eq. from course, extracted, of

l

• Subsystem 1 (Utilizing m1 and m2 for control): 0.07e-0.3s o.se-0.2s 2.5s + I I + 3s [ 0.004e-O.Ss - 0.003e-()· 2• s + I l.5s + 1

[

m1 mJ

(21.81)

• Subsystem 2 (Utilizing m1 and m3 for control):

=

0.5e-O.Zs 3s + 1

[

0.04e-0.03s ] 2.8s + 1

0.004e...o.ss - 0.00le...o· 4 ' l.6s + 1 1.5s + 1

m1 [

(21.82)

mJ

l

• Subsystem 3 (Utilizing m2 and m3 for control):

[y Yz

1]

=

0.04e-0 .0 3s 0.07e...O.Js 2.8s + 1 [ 2.5s + I - 0.003e-().2s - O.OOie-0•4• 1.6s + I s + I

[mz]

(21.83)

m3

Next we obtain the RGA's for each subsystem; the results

are:

For Subsystem 1: [

0.843 0.157

J

(21.84)

J

(21.85)

0.157 0.843

For Subsystem 2: Az

= [ 0.758

0.242 0.242 0.758

and for Subsystem 3:

- [ -1.4 2.4 2.4 -1.4

11.3 -

J

(21.86)

Subsystem 1 will offer us Based on inspection of these three RGA's, it appear s that the ideal situatio n; it is to closest the best possible control, because its RGA is the 3. tem Subsys to r superio far and 2, tem somew hat better than Subsys that the input variabl es Having thus decide d on Subsystem 1, the implication is final task is to m1 and m2 are to be used for control and m3 is to be redund ant. The (21.84) we see Eq. from since pair the variabl es of this subsystem; an easy exercise, is: y1 - m1 and y2 - m2. that the best possible loop pairing within this subsys tem

754

MU LTIV ARI ABL E PROCESS CON TRO L

One mu st realize, of course, that ther e may be constraints on som man ipu late d variables or time e of the dela ys in some of the dynamics or oth er issues that cou ld be important in choosin g the bes t sub syst em or pairing. Thu s as will be discussed in more detail belo w, one mus t consider these as wel l as the RGA in the final choice of control system design.

21.8

FINAL COMMENTS ON LOOP

PAIRING AN D THE RG A 21.8.1 Loop Pairing in the Ab sen ce of Process Mo del s Wh en process models are not ava ilable, it is still possible to obta in RGA's from exp erim enta l process data . One may ado pt eith er of the foll owi ng two app roac hes in such situations: 1.

Experimentally determine the stea dy-s tate gain mat rix K, by implementing step changes in the process inpu t variables, one at a time, and observing the ultimate chan ge in each of the output variable s. Let t::..y11 be the observed change in the value of out put variable 1 in response to a change of t::..m in the jth inp 1 ut variable m1; then, by definiti on of the stea dy- stat e gain:

and , in general, the stea dy- stat e gain betw een the ith out put vari able and the jth inp ut variable will be given by:

(21.87)

The elements of the K mat rix are thus obtained. Of course, onc e this gain matrix is known, we can easi ly generate the RGA. 2. It is also possible to determin e each element of the RGA dire ctly from experiment. As we may recall, eac h RGA elem ent A-; can be dete rmi ned 1 upo n perf orm ing two experim ents; the first dete rmi nes the ope n-lo op steady-state gain by mea suri ng the response of Yi to mp ut m , whe n all the oth er loop s are ope ned ; in 1 the sec ond exp erim ent, all the oth er loop s are clo sed - usin g PI con trollers to ens ure that ther e wil l be no stea dy- stat e offs ets - and the resp ons e of y 1 to inp ut m i is rede term ined . By defi niti on (see Eq. (21.8)), the rati o of thes e gain s gives us the desired relative gain element. It sho uld be clea r that, of thes e two exp erim enta l procedUres, one is mor e time consuming, and the sec ond involves too man y ups ets to the process; for these reas ons it is not likely to be pop ular in practice. The firs t pro ced ure is to be pref erre d.


755

21.8.2 Final Comments on the RGA The following are summary comments about Bristol's relative gain array: 1. The RGA requires only steady-state process information; it is therefore easy to calculate, and easy to use. 2. The RGA provides information only about the steady-state interaction inherent within a process system; dynamic factors are not taken into consideration in Bristol's relative gain analysis. This remains the main criticism of the RGA. As a result of this apparent limitation of the RGA, several modifications, and alternatives, which take dynamic factors into consideration have been proposed (See Gagnepain and Seborg [6] and MeAvoy [7]). 3. The RGA (or any of its newer dynamic variations) only suggests input/ output pairings for which the interaction effects are minimized; it provides no guidance about other features which may influence the pairing. Thus this is the topic of the next subsection.

21.8.3 Other Factors Influencing the Choice of Loop Pairing The RGA and the Niederlinski index provide guidance on the choice of loop pairing in order to minimize steady-state interactions and avoid structural instability. However, there are other factors that influence the choice of loop pairing. Some important ones are: 1.

Constraints on the input variable: It may happen that the best RGA pairing results in a choice of input variable for Y; that is severely limited by some constraints (maximum feed concentration, maximum heater power, limited heat duty, etc.) so that it cannot carry out the control task assigned. In this case, another pairing might perform better even with increased steady-state interactions.

2. The presence of time-delay, inverse-response, or other slow dynamics in the best RGA pairing: Since the RGA is based only on stea:dy-state considerations, it can easily happen that the best RGA pairing results in very sluggish closed-loop response because of long time delays, significant inverse response, or large time constants in the diagonal elements of the transfer function for the recommended pairing. In this situation, choosing a pairing with inferior RGA properties but without these sluggish diagonal dynamics could greatly improve the control system performance. 3. Timescale Decoupling of Loop Dynamics: There are often timescale issues that can influence the choice of loop pairing. For example, in a 2 x 2 system it may be that with a certain pairing, the RGA indicates serious loop interactions. However, if at the same time, one of the loops

;:,b

MULTIVARIABLE PROCESS CONTROL res pon ds ver y mu ch faster tha n the oth er, the n the re can be timescale decoupling of the loops. This occurs wh en the fast loo p res pon ds so fast tha t the effect of the slo w loo p app ear s as a con sta nt dis turb anc e· conversely the slow loop doe s not respond at all to the hig h-frequency dist urb anc es com ing from the fast loop. This me ans tha t we pai r loo ps wit h larg e differe can safely nces in closed-loop res pon se tim es eve n wh en the RGA is not favora ble.

Let us illustrate these ide as wit h some examples. Exa mpl e 21.10 LOO P PAI RIN G FOR TH E STI RRE D MIX ING TA NK WIT H AD DIT ION AL HEATER. Sup pos e that the stir red mix ing tank of Exa mpl e 20.3 (Fig ure 20.1) has add ed an introl wit h heater pow er Q, so that the mod el becomes:

tank hea ter for tem pera ture con

The n we hav e thre e inp uts (fc' FH' Q) and only two out put s (h, 1). By line ariz ing the equ atio ns as befo re we obt ain a tran sfer fun ctio n mod el slig htly mod ifie d from Eq. (20.23):

(Tc - Ts)

Achs<s + a 22 ) whe re all' lln are give n by Eq. (20.24). Now the three possible squ are sub syst ems of this ove rdet erm ined syst em yield relative gain arra ys as follows: 1.

For FH, Fe as man ipu late vari able s, the rela tive gain arra y is giv en in Eq. (21.65).

2.

For FH, Q as man ipul ated vari

ables:

A=[~~] 3.

Fe, Q as man ipul ated vari able

s:

A=[~~] If we wan t the stea dy-s tate tank tem pera ture to be T = TH +Te -5 theR GA is: 2 - , then for Sub syst em 1,

A

[ 0.5 . 0.5 0.5 0.5

J


757

so that either Subsystem 2 or 3 using the in-tank heater would be preferred based on RGA analysis. However, suppose that the in-tank heater has limited power and can barely achieve the steady state, T" at maximum power. Thus it would not be a desirable choice for regulatory temperature control. In this case, Subsystem 1 with the poor RGA pairing would be better than using such a low-power heater. Suppose that in order to overcome the heater power limitation, a much larger heater was installed in the tank, but because of its massiveness it has a very large time delay between the control signal and the actual power delivery. The poor RGA choice of Subsystem 1 could still be the best choice because of the sluggish closed-loop response with the heater. Example 21.11 LOOP PAIRING WITH TIMESCALE DECOUPLING. Consider the in-line blending problem of Example 215. Recall that when we wish a blend which contains 20% species A, the RGA is: A

=[

0.2 0.8] 0.8 0.2

Based on RGA analysis alone, pairing the total flowrate, yp to the flowrate of A, m1, and the blend composition, y2, to the flowrate of B, m2, would not be a recommended pairing. However, if the flow measurement and valve dynamics were very rapid, then the dosed-loop response of the y1 - m1 loop would be extremely fast (on the order of seconds). By contrast, if the composition analyser had a significant time constant (on the order of minutes), then the y2 - m2 loop would be 10- 100 times slower than the y1 - ~ loop. In this case, timescale decoupling would mean the control loops could be considered virtually independent of each other, and no serious interactions would occur. Example 21.12 ANOTHER FORM OF TIMESCALE DECOUPLING. Consider the 2 x 2 system with the transfer function:

G(s) :: [

IO•> I s + 1

s_+21 ] (21.88)

- 4 I Os + 1

with RGA given by: A :: [ 0.8 0.2

0.2 ] 0.8

so that the y 1 - m 1 /y 2 - m2 pairing shown in Eq. (21.88) would definitely be recommended. The closed-loop response for a unit set-point change in y1 using this pairing and a diagonal PI controller (K, =4, -r1 =0.5; K, =- 4, -r1 =0.3) is shown in Figure 21.3. The performance is not tdo bad 'considerfug that the open-loop time constants on the diagonal are 10 minutes. However, note that in spite of an adverse RGA parameter, A. = 0.2, the reverse y1 - m2 /y2 - m1 pairing could have great advantages because with this configuration,

:'j·_:

758 MU LTW AR IAB LE PR OCESS CONTROL

0

Fig ure 21.3.

1

2

3

4

6

Set-point response for RG A recommended pai rin g

in Example 21.12 the ope n-l oop tim e con stants on the dia gon al are onl y 1 min ute . In fact, rev ers e pai rin g, the clo wit h this sed-loop res pon se for the sam e set -po int cha dia gon al PI con tro ller (Kc nge usi ng a = 10, -r 0.3; 1 Kct =20, -r Th e per for ma nce in this cis e is dra kat ica lly bet ter 1 =0.3) is sho wn in Fig ure 21.4. th'i n wit h the RGA rec om pai rin g. Th e rea son is me nde d tha t the control loo ps are able to res pon d so rap idl interactions tha t app ear y tha t the mo re slowly are easily dea lt with.

=

Th e les son he re is tha t the RGA pro vid es on ly a gu ide to ste ady -st int era cti ons an d mu st be ate use d tog eth er with all oth er eng ine eri ng considera in cho osi ng the loo p pai tions ring. 1. 5

1.0

0.5

0.0

0

Fig ure 21.4.

1

2

Set-point response for adv

3

4

erse RGA pai rin g in Exa

5

mple 21.12.

CHAP 21

21.9

INTERACTIO N ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGNS

759

CONTROLLER DESIGN PROCEDURE

21.9.1 Multiloop Controlle r Design The design of multiple single-loop controllers for multivariab le systems proceeds in two stages: 1. Judicious choice of loop pairing 2. Controller tuning for each individual loop Since our primary objective in this chapter is really the design of effective multiple single-loop controllers for multivariable systems, the amount of time invested in discussing the issue of input/ output pairing should give an indication of how important this aspect is in the overall scheme of multiloop control systems design. The crucial point is that if the input and output variables making up the control loops are poorly paired, it will be exceedingly difficult or even impossible to design a successful multiloop control system. Having covered the first half of this design task in earlier portions of this chapter, it now remains to discuss the issue of tuning the individual controllers. Before we do this, however, it is useful to indicate how the RGA analysis can be used to indicate the expected performance of a multiple single-loop control strategy. It should not be surprising that when the RGA for the process is close to the ideal, (i.e., A;; very close to 1) that multiloop controllers are likely to function very well if carefully designed. However, when the RGA indicates strong interactions for the chosen loop pairing and especially when the input and output variables are paired on values of }.ij that are very large or negative, multiloop controllers are not likely to perform well even for the best possible tuning. If, based on an analysis of the expected extent of loop interaction, the decision is made to proceed with a multiple single-loop control strategy, one must now consider how to tune the individual controllers.

21.9.2 Controlle r Tuning for Multiloop Systems The main obstacle to proper controller tuning is presented by the interaction s that exist between the control loops of a multiloop system. This is what makes it risky to adopt the obvious strategy of tuning the controllers individually, in isolation from the others, with the hope that when all the loops are eventually closed, the overall system performan ce will still be adequate. (Recall from the stability considerati on presented in Chapter 20 that the stability of individual control loops of a multiloop system when operating singly does not guarantee the stability of the complete system when all the loops are closed.) Although several tuning procedures have been proposed (cf. Niederlins ki [3] and MeAvoy [7]) the procedure that is typically followed in practice is the following:

760


1. With the othe r loop s on manu al contr ol, tune each contr ol loop indep ende ntly until satisfactory closed-loop perfo rman ce is obtained. 2. Restore all the controllers to joint opera tion unde r autom atic control and read just the tunin g para mete rs until the over all close d-loo p perfo rman ce is satisfactory in all the loops . Whe n the interactions betw een the contr ol loops are not too significant, this proc edur e can be quite useful. It is clear , howe ver, that for syste ms with significant interactions, the readj ustm ent of the tunin g in Step 2 can be difficult and tedious. It is possible to cut down on the amou nt of gues swor k that goes into such a proc edur e by notin g that in almo st all cases, the controllers will need to be made more conse rvati ve (i.e., the controller gains reduc ed, and the integ ral times increased) when all the loops are close d, in comp ariso n to when opera ting alone , with all the other loops open . Ther e are vario us ways of accomplishing this udetu ning" (e.g., Ref. [7-10]). The following proce dure is recom mend ed by McAvoy [9] for 2 x 2 problems:

1. Use any of the singl e-loo p tunin g rules (Ziegler-Nichols, Cohe n and Coon , etc.) to obtai n starti ng value s for the indiv idual controllers; let the contr oller gains be Kc;•. 2. Thes e gains shou ld be redu ced using the follo wing expre ssion s that depe nd on the relative gain param eter k

Kci =

{

(.t- ..J ,t2- .t)Kd*

b + ..J ,t2 -

.t > 1.0

A IKd* .t < 1.0

(21.89)

It may still be neces sary to "retu ne" these controllers after they have been put in oper ation ; howe ver, this will not requ ire as much effor t as if one were start ing from scrat ch. Let us illust rate mult iloop controller tunin g with the following example. Exam ple 21.13 MULTIPLE LOO P TUN ING FOR THE WOO D AND BERRY DISTILLATION COLUMN.

We wish to design multiloop controllers for the Wood and Berry distillation rolumn of Example 21.2. Recall that this 2 x 2 system has the transfer function: 12.8e-s [ 16.7.r + 1 G(s) 6.6e-78 10.9. r+ 1

-18.9 e-3• ] 2l.O.r + .1 -19.4 e-3• 14.4.r + .1

How do we tune the multiloop controllers? Solution:

From Example 21.2, the RGA was found to be: A = [ 2.0 -1.0 -1.0 2.0

J

S CHAP 21 INTERACTION ANALYSIS AND MULTIPLE SINGLE-LOOP DESIGN

0

Figure 21.5.

20

10

30

40

761

50

Closed-loop response of Loop 1 when Loop 2 is open.

so that 1-1/2- 2 loop pairing is indicated . by tuning the If we make use of the loop tuning procedu re suggeste d above, we begin

rs for the two SISO "individ ual loops with the other loops open, i.e., tune PI controlle

systems:

Y2(s)

0

Figure 21.6.

10

-19.4 e- 3• !4.4s + 1 uz(s)

20

30

40

50

Closed-loop response of Loop 2 when Loop 1 is open.

irl

r

t·

762

MUL TIVA RIA BLE PROCESS CONTROL

.,~

(a)

1. 00

0. 75 Yt

0.50

0.25

0.00

0

10

20

30

40

50 t

1. 0 (b)

0. 5

y 2 0.0

-0.6

-1.0

I

0

Figu re 21.7.

10

I

I

20

I 30

I

I

I 40

I

I

I 50

Clos ed-l oop resp onse whe n both loop s are clos ed simu ltan eous ly:

(a) Yt' (b) Y2·

Cho osin g a PI controller with para mete rs close to Ziegler-Nichols tuni ng lead s to the following parameters:

K,1 "' 0.80

-ri, = 3.26

1"~ =

9.35

With the othe r loop open , these tuning para mete rs lead to the quit e goo d indi vidu al closed-loop responses to set-point changes show n in Figures 21.5 and 21.6. If we dose both loops and mak e the sam e set-p oint chan ge in y chan ge is desi red for y ) usin g 1 (no set-p oint these tuni ng para mete rs, we see 2 that the proc ess is near ly unst able due to loop inter actions and oscillates cont inuo usly , as show n in

CHAP 21


763

1.26

1.00

y 1 0.75

0.50

0.25

0.00 0

10

20

30

40

1.0

60

(b)

0.6

Y2 0. 0

+----1

-0.5

0

Figure 21.8.

10

20

30

40

60

Closed-loop response with both loops closed and "detuned" controllers: (a) Y1' (b) Y2·

Figure 21.7. Thus we must detune both loops. Using the detuning rule in Eq. (21.89), for A.= 2 we see that the original controller gains must be multiplied by a factor of 0.59 so that the new tuning is:

K,, = 0.47

-r11 = 3.26

K,2 = -0.088 '~"J2 = 9.35

The closed-loop performance of the multiloop system with these new parameters is shown in Figure 21.8. Now the process is stable and has better closed-loop response with the new tuning parameters. The response could be further improved by continued adjustment on the tuning parameters.

764


Other techniques that may be used include optim izatio n proced ures that minim ize the weighted sum of the integral-squared error for all the loops (recall the discussion of these in Chapter 15) and model -based tuning (to be discussed in Chapter 22).

21.10 SUM MAR Y A first approach to the design of effective controllers for multivariable systems has been considered in this chapter through the use of rationa l loop pairing and multiple single-loop controllers. In adopting the multip le single-loop control strategy, we have shown that the first order of business is to determine the best way to pair the input and outpu t variables, and whether or not such a control strategy is indeed viable. It turns out that the principal loop analysis tool for answe ring both questions is the Bristol RGA, the most popul ar means of quantifying process interactions and determining the input / outpu t pairing for which the interaction effects are minimized. We have discussed the RGA in great detail, showing how it is calculated and how it is used for input / outpu t pairing for various classes of systems. The task of designing multiloop controllers for multivariable systems still remains an art, but some guidelines on to how to system atize this task have been given. These multiple single-loop controllers will work well only in situat ions in which contro l loop interactions are not very strong. For multiv ariabl e systems whose RGA's indicate very strong loop interactions, closed-loop stability will be attained only by sacrificing speed of response, for, as is well known, it will be necessary to reduce the contro ller gains in each loop significantly (i.e., "detune" the loops) under such circumstance s. When multiple single-loop controllers cannot be made to perform well, control system performance can be improved by considering other more advanced techniques; these will be discussed in the next chapter. REFERENCES AND SUGG ESTED FURTHER READ

ING Bristol, E. H., "On a New Measure of Interactions for Multiv ariable Process Control," IEEE Trans. Auto. Cont., AC - 11, 133 (1966) 2. Ogunnaike, B. A., J.P. Lemaire, M. Morari, and W. H. Ray, "Advanced Multivariable Control of a Pilot Plant Distillation Column," AIChE, 29, 632 (1983) 3. Niederlinski, A., "A Heuristic Appro ach to the Design of Linear Multivariable Interacting Control Systems," Automatica, 7, 691 (1971) ' 4. Grosdidier, P., M. Morari, and B. R. Holt, "Oose d Loop Properties from Steady-State Information," I&EC Fund., 24, 221 (1985) · 5. Koppel, L. B., "input- Output Pairing in Multivariable Control," AIChE J., (1982) 6. Gagnepain, J.-P. D. and E. Seborg, "Analysis of Process Interactions with Applications to Multiloop Control System Design," I&EC Process Des. Dev., 21, 5 (1982) 7. McAvoy, T. J., Interaction Analysis Theory and Applica tion, ISA, Research Triangle Park, NC (1983) 1.

8.

Tyreus, B., "Multivariable Control System Design for an Industrial Distillation Column," I&EC Proc. Des. Dev., 18, 177 (1979) 9. McAvoy, T. J., "Connection between Relative Gain and Control Loop Stability and Design," AIChE J.,27, 613 (1981) 10. Marino-Galarraga, M., T. J. McAvoy, and T. E. Marlin , "Short-cut Operability. 2. Estimation of Detuning Parameter for Classical Control System s," I&EC Res., 26, 511 (1987)


765

REVIEW QUESTION S 1.

In using multiple single-loop feedback controllers to control a multivariable system, what are the two iss)les to be resolved?

2.

What is the relative gain array (RGA); what is it used for?

3.

What are some of the important properties of the RGA?

4.

Why is a single element of the RGA sufficient to determine the entire array for a 2 x 2 system?

5.

Qualitatively, what is the implication for loop interaction when the ijth element of a multivariable system's RGA: • Is equal to 1? • Is equal to 0? • Lies between 0 and 1? • Is greater than 1? • Is equal to 1? • Isnegative?

6.

What is the ideal RGA, and under what condition can it be obtained?

7.

Why is the RGA for a process in which the loops experience no interactions identical to the RGA for a process in which the loops experience one-way interaction?

8.

How is the RGA used for loop pairing?

9.

What is the Niederlinski index, and how is it useful for loop pairing?

10. How is loop pairing carried out for nonlinear systems using the RGA? 11. What is peculiar about the RGA's for nonlinear systems? 12. What is a noilsquare system? 13. How is the loop pairing problem for underdefined systems fundamenta lly different from the loop pairing problem for overdefined systems? 14. What is the procedure for applying the RGA for pairing the input and output variables of an overdefined system? 15. How can the RGA be obtained and used for loop pairing in the absence of process models? 16. What is the main criticism of the RGA? 17. Even though the RGA is the most widely used loop pairing tool, what are some other important factors that should guide the final choice in this matter? 18. What is the procedure for designing multiple single-loop controllers for multivariable systems? 19. After designing each individual single-loop controller in a multiloop control system independently, why is it necessary to detune them when all the loops operate

simultaneously?

~TIV~LEPROCESSCONTROL

766

PROBLEMS 21.1

An approximate model identified for an experimental, laboratory scale continuous stirred-tank polymerization reactor is:

r'~] = [ 2.~~8! 1

L,2

0.55

4.36 + 1

0

(P21.1)

-0.013 3.56 + l

where, in terms of deviations from their respective steady states, the input and output variables are:

= Polymer production rate (gm/min) = Weight average molecular weight

y1 y2

m1 = Monomer flowrate (gm/min) m2 = Chain transfer agent flowrate (gm/min)

Not included in the model is the effect of the catalyst flowrate; for the experimental system this was permanently set at a fixed value. (a) First study the given transfer function matrix; does it show any evidence of interactions between the process variables? Which output variable is expected to be more susceptible to the effect of interactions and why? (b) Calculate the RGA for this process. What does this suggest in terms of input/output pairing and the loop interactions resulting from such a pairing scheme? Interpret your results in light of your initial assessment in part (a). 21.2

(a) Assuming a Ycm1 I y2-m 2 pairing, design two independent single-loop PI controllers for the polymer reactor of Problem 21.1 using Cohen-Coon settings. What are the recommended oontroller parameters? (b) Using your favorite simulation package, obtain the response of the overall closed-loop system to a set-point change of 100 gm/min in the polymer production rate (with no change desired in the weight average molecular weight), using the controllers you designed in part (a). Plot the responses of all the process input and output variables. Evaluate the performance of these controllers.

21.3

A pilot scale distillation column used for separating a binary mixture of ethanol and water is modeled by the following transfer function matrix~model: y(6) = G(l)m(6)

+ GJ6)d(s)

(P21.2)

with the mabices G(s) and GJ.s) given by:

G(s)

=

[

·

GJ.s) =

[

0.7e-2.6s 6.1s + 1

-O.oose-

-34.7e-9·26 8.2s + 1

0.9(11.6s + i)e-8 (3.9s + 1)(18.8s + 1)

0.14e-128 6.26 + 1 -11.5e-0.6s 1.0s + 1

8

9.ls + 1

]

(P21.3a)

-0.0011(26.3s + l)e-2·78 ] (7.9s + 1)(14.6s + 1) 0.32e-2·6s 7 .8s + 1

(P21.3b)

CHAP 21


767

The process variables are (in terms of deviations from their respective steady-state values): Overhead mole fraction ethanol y1 Temperature on Tray #19 y2 Overhead reflux flowrate m1 Reboiler steam pressure m2 Feed flowrate d1 Feed temperature d2

=

It is desired to design two single-loop feedback controllers for this process with Controller 1 to regulate y1, and Controller 2 to regulate y2 . Which input variable (m 1, m2) will be more advantageous to pair with which output variable (y1, y 2)? Support your decision adequately. Comment on how well you think the best set of single-loop controllers designed for this distillation column will perform. 21.4

The "low-boiler column" in a distillation train used to separate the products of an industrial chemical reactor has the following experimentally determined approximate transfer function model : y(s)

=

G(s) m(s) + Gj..s) d(s)

with the matrices G(s) and Gd(s) given by: -0.12 0.004s + 1 G(s)

0.011 0.055s + 1

-0.00175e-O.Is 0.00525s + 1

-0.00012e-0.2Ss 0.075 -0.013 0.00325s + 1 0.015s + 1 0.0141s + 1 0.04 0.000086 -0.0043 s s s [

Gj..s)

=

~0041 1 0.0041s+

0.185 0.37s +

-0.001 0.0002:5s + 1

0.16 0.4s + 0

]

(P21.4a)

(P21.4b)

The distillation column variables are (in terms· of deviations from their respective steady-state values):

y1 = T36 , temperature on Tray #36 (•C) y2 = Bottoms temperature COC) y3 = Reflux receiver level (ft) m1 = Reflux flowrate (lb/hr) ~ = Reboiler heat duty (NHDUt) m 3 = Condenser cooling water flowrate (lb/hr) d1 = Feed flowrate d2 = Feed temperature

t Normalized Heat Duty Units.

168 MULTIVARIABLE PROCES S CONTROL (a) Ass um ing tha t it has bee n pre det erm ine d tha t the bot controlled wit h the reboile tom s tem per atu re is to be r heat dut y, use the RGA to determine how the rem variables should be paired aining . (b) Using the Ycm I y -m 3 2 2 I y 3-m 1 pai rin g, and usi ng parameters: the following controller

Loop 1: Loop2: Loop 3:

(T36 loop); (Bottoms temperature loop); (Reflux level loop);

K,=-80.0; K,= 2.0 ; K, =-500.0;

1/- r1 =3.5 ll-r1 =6.0 ll-r1 =2.0

obt ain a simulation of the overall col um n response to a 600 lb /hr ste p increa the feed flowrate. se in No w, usi ng a "ne w con fig ura tio n" obt ain ed fro m the alte rna te pai rin g: y1 - m1 I y2 - m2 I y - m , alo 3 ng wit h the new controller 3 parameters:

Loop 1: Lo op2 : Loop 3:

(T36 Ioop); (Bottoms temperature loo p); (Reflux level loop);

K,=-500.0; K,= 4.5 ; K,= 12,000;

1/- r1 = 1.0 l/r1 =2. 0 1/r1 =0. 8

obt ain the col um n res pon se to the same dis tur ban ce (600 lb/ hr feed rate change). Compare, on cor ste p responding plots, the respon ses und er bot h the old and new configurations. the (c) Given now tha t for eco nomic and environmental reasons the tem per atu re T the mo st critical of all the is low boiler col um n variab les, and tha t ver y tight con 36 therefore required for this trol is variable, which of the two configurations investigat par t (b) is preferable? Dis ed in cuss you r ans we r in the ligh t of the RGA results obtain in par t (a). ed Given tha t Tray #36 is clo se to the top of the colum n, and tha t the reflux receiv is a 3-ft dia me ter cylind er rical dru m into wh ich em ptie s all the material fro (essentially total) conden m the ser, from a con sid era tion of the physics of the pro ble wh ich wo uld you expect m, to be the mo re effective ma nip ula ted variable to con tro llin g this tem per use in atu re: the ref lux flo wra te or the con den ser coo temperature? Sup por t you ling r ans we r adequately. Off er an explanation for the res given by the RGA. ults (d) Repeat par t (b) for a ste p inp ut disturbance of 5°C in the feed temperature. again, com me nt on which On ce configuration pro vid es tigh ter control on the critical var iab le. T36 21.5

A cer tain mu ltiv aria ble sys tem has thr ee out put s y1, y2, and y wh ich .can con tro lled by any of fou 3 be r available inp uts ml ' m 21 my and m4 • Th rou gh pul testing, the following 3 x se 4 transfer function matrix mo del wa s obtained:

[~;]

[

0.,.~~ 3s + 1

0.07e-O.Js 2.5 s + 1

0.0 4e-0· 3• 1.5 s + 1

0.00:e-O.Ss

0 -O.OOJe-0· 2• s +I

0 -0.006e-0.4s 1.6s + 1

= I. 5s + 1

0.012

(22.15) 2 again, of the form in Eq . (22.2). Fin din g the app rop ria te com bin ati on of process tot al eli mi nat ion of, or, variables tha t will lea d to at lea st a red uct ion in, the con tro l loo p intera ten ds in gen era l to be ctions, an "ar t" (see Refs. [1, 2]) ; nev later, the concept of sin gu ert hel ess , as we will see lar value decomposition can pro vid e guidelines reg ard . in this Ha vin g thu s illu str ate d the ge ne ral pri nci ple s an d con cep ts of mu ltiv ari abl e controller , we no w pro cee d to dis a cuss som e of the mu ltiv con tro lle r des ign tec hn ari abl e iqu es in gre ate r detail . In eac h case we wi att ent ion on thr ee things ll focus :

1. Principles beh ind the technique 2. Re sul ts 3. Ap pli ca tio ns, inc lud ing pra cti cal co nsi de rat ion s for eff ect ive uti liz ati on

22.2

DECOUPLING

22.2.1 Principles of De co up lin

g Us ing In ter ac tio n Co mp

en sa tor s

Fro m ou r int rod uct ion to mu ltiv ari abl e sys tem s in Ch ap ter 20, an d sub seq uen t discussions ou r in Ch apt er 21, we kno w tha t all the inp ut variab typ ica l mu ltiv ari abl e pro les of a ces s are coupled to all its ou tpu t variables; the y1 - u 1, y2 - u2, ... , Yn - un cou ma in plings are desirable (this is wh at ma kes it possib le

CHAP 22 DESIGN OF MULTWARIABLE CONTROLLERS

777

to control the process variable in the first instance); it is the y 1 - u cross1 couplings, by which y1 is influenced by u1 (for all i and all j, with i '!! J), that are undesirable: they are responsible for the control loop interactions. It is therefore clear that any technique that eliminates the effect of the undesired cross-couplings will improve control system performance. Note that the purpose is not to "eliminate" the cross-couplings; that is an impossibility, since to do this will require altering the physical nature of the system. Observe, for example, that it is not possible to stop the hot stream flowrate from affecting the tank temperature in the stirred mixing tank system of Figure 20.1, despite the fact that our main desire is to use this flowrate to control the level; neither can we prevent the cold stream from affecting the level, even though controlling the tank temperature is its main responsibility. We can, however, compensate for the noted cross-coupling effects by carefully balancing the hot and cold stream flowrates. The main objective in decoupling is to compensate for the effect of interactions brought about by cross-couplings of the process variables. As depicted in Figure 22.2, this is to be achieved by introducing an additional transfer function "block" (the interaction compensator) between the single-loop controllers, and the process. This interaction compensator, together with the single-loop controllers, now constitute the multivariable decoupling controller. In the ideal case, the decoupler causes the control loops to act as if totally independent of one another, thereby reducing the controller tuning task to that of tuning several, noninteractin g controllers. The main advantage here is that it will now be possible to use SISO controller design techniques. The decoupler design problem is that of choosing the elements of the compensator , G 1, shown in Figure 22.2, to satisfy usually one of the following objectives: 1. Dynamic Decoupling: eliminate interactions from all loops, at every instant of time.

2. Steady-State Decoupling: eliminate only steady-state interactions from all loops; because dynamic interactions are tolerated, this objective is less ambitious than dynamic decoupling but leads to much simpler decoupler designs. 3. Partial Decoupling: eliminate interactions (dynamic, or only steady state) in a subset of the Control loops; this objective focuses attention on only the critical loops in which the interactions are strongest, leaving those with weak interactions to act without the aid of decoupling. Ir-------------------~t

;;.:Yd._+'"{)(:l--E-ti'

v

Gr

• u

: ~~~P

G

teraction 1 compensator, 1 controllera DecoupHngcontroller

--------------------Figure 22.2.

A multivariable decoupling control system incorporating an interaction compensator.

22.2.2 D es ig n o f Ideal Dec

ouplers

MULTWARIABL E PROCESS CO NTROL

Even th ou gh th ere are several w ay s by which basic principles deco simplified decoup are all the same. We shall illus uplers can be designed, the ling, us in g a 2 x trate, first, the principles of 2 example an d Figure 22.1. the block di ag ra m sh ow n in

Simplified Deco upling

Let us st ar t by observing the fo llowing im po rt in Figure 22.1: an t aspe 1. There ar e tw o co

cts of the block

di ag ra m

mpensator bloc

ks g1 an d g , on 1 e for each loop. I 2 2. We ha ve ad op te d a ne w no ta ti on : the co nt designated as v ro ll er an the process rem1 d v2, while the actual control ac ou tp ut s ar e no w ains as u1 an d tion implemente u• be ca us e th e ou d on tp ut of the co 2 This distinction becomes ne cessary ntrollers an d th implemented on e co nt ro l ac ti on the process no lo to be nger have to be the same. 3. W it ho ut th e compensator , u1 =v1 an d u remains as: 2 v2, an d th e process m od el

=

Y1

Yz

= 8 u "t + 81 2" 2}

= 821 "t + 822"2

(22.16)

so th at the inte ractions persist, since u2 is still therefore affect cross-coupled w s y1 th ro ug h th it h e g 12 element, an coupling th ro ug d u1 affects y by an d h g21· 2 cross4. W it h the in teraction compe nsator, Loop 2 v 1 th ro ug h g is "i nf or m ed " 1 , so th at u of change 2 w ad ju st ed accor~ :lingly. The sa m ha t the process actually feels s in e ta sk is pe rf or - is which adjusts u m ed for Loop 1 w it h v2 inform 1 by g1 , ation. The design ques 1 tion is now pose d as follows:

What should g completely neul1 and & be ralizei! We will no w an sw

if the effects of loop interactions are to be

er this question.

First, let us cons ider

Loop 1, in Figu re 22.1 where th e proc

'

ess model is: (22.17a) (22.17b)

CHAP 22 DESIGN OF MULTWARIAB LE CONTROLLERS 779 How eve r, bec aus e of the compen sators, the equ atio ns gov erni ng the con trol acti on are:

(22.18)

Intr odu cing these equations into Eq.

(22.17) now gives: (22.19a) (22.19b)

Now in ord er to hav e only v affe ct y and to 1

eliminate the effect of v2 on y , it is now eno ugh to choose g suc 1 1 h that the coefficient of v in Eq. 1 . h . 2 (22.19a) van tS es, 1.e: 1

whi ch requires:

(22.20) A sim ilar pro ced ure carr ied out for Loop 2 lead s to the con diti on that to mak e y2 in Eq. (22.19b) entirely free of v1 requires:

(22.21)

These are the transfer functions of the decouplers nee ded to compensate exactly for the effect of loop interactions in the example 2 x 2 system of Figure 22.1. If Eqs. (22.20) and (22.21) are now intr odu ced into Eq. (22.19), we see that the overall syst em equations become:

(22.22a)

and

(22.22b)

and the syst em is com plet ely dec oup led, with v 1 affecting only Yv and v 2 affecting only y2 • The equivalent block diag ram is sho wn in Figure 22.3 whe re the loop s now app ear to act inde pen den tly and can ther efor e be tun ed ind ivid uall y.

Fig ure 22.3.

Equivalent block diagra m for system in Figure 22.2 wi th the compen implemented as in Eqs. sators (22.20) and (22.21).

It is also interesti ng to consider wh at Eqs. (22.22a,b) indica overall closed-loop system behavior at ste ady state. If the steady-ste for the for ea ch tra ns fer fun ction element g i is K tate gain 1 Eqs. (22.22a,b) become 11, observe tha t at ste ad : y state,

(22.23a)

an d

(22.23b)

If we no w recall Eqs. (21.19) an d (21.23) as the definition of the pa ram ete r 1 for the rel 2 x 2 system, we im mediately see that Eq ative gain become: s. (22.23a,b)

an d

(22.24a)

(22.24b)

implying tha t up on im plementing simplified loop steady-state gain de in each loop is the rat coupling, the effective closedio of the open-loop ga relative gain paramete in an d the r. Note tha t wh en 1 loop gains become is ve ry small,. an d contr very large, th~ effective closedol sy ste m performan jeopardized. We will ce ma y be return It is no w im po rta nt to this point below. to no te th at wh dimensions lar ge r tha en de ali ng wi th sy ste n potential of rapidly be 2 x 2, the simplified decoupling approa ms ·w ith ch ha s the coming tedious. For exam system is sh ow n in Fig ure 22.4 where there are ple, the situation for a 3 x 3 design. More general ly, an N x N system lea no w six compensator blocks to ds same principles us ed for the 2 x 2 system are to (N2- N) compensators. The cumbersome work inv also applicable, bu t the olved. re is more

CHAP 22 DESIGN OF MULTIVARIABLE CONTROU.E RS

Figure 22.4.

781

Simplified decoupling for a 3 x 3 system.

The more general approach to decoupler design employs matrix methods that are particularly convenient for higher dimensional systems.

Generalized Decoupling A more general procedure for decoupler design may be outlined as follows: 1. From Figure 22.2, we observe that: y =Go

(22.25) (22.26)

so that: (22.27)

2. In order to eliminate interaction s absolutely, y must be related to v through a diagonal matrix; let us call it GR(s). We may now choose G 1 such that: (22.28)

and the compensated input/output relation becomes:

where GR represents the equivalent diagonal process that the diagonal controllers Gc are required to control.

3. Therefore from Eq. (22.28),

MULTIVARIABLE PROCESS CONTRO L the compensator G 1 must be given by:

(22.29) 4. Clearly, the co mpensator obtain ed using Eq. (22.29) de chosen for GR. Th pends on w ha t is us the elements of GR should be sele de si re d de co up le cteq to provide th d be ha vi or w ith e th e defined in Eq. (22.2 9). A commonly em simplest possible decoupler ployed choice is: GR = Diag[G{s)] (22.30) i.e., the diagonal elements of G(s) are retained as th diagonal matrix GR e elements of the ; however, other choices may be us ed.

Relationship betw een Generalized and Simplified D ecoupling "Gener

alized" decoup th at for simplified ling m ay be related to simplified decoupling, by no decoupling ap pl ie ting d to a 2 X 2 syst transfer function matrix is given by em , th e compensat : or

(22.31)

while for a 3 x 3 sy

stem, the compens

ator matrix Gr take s the form:

(22.32)

Thus, for simplifi ed decoupling th e specific form de Eqs. (22.31) an d (2 sired for G1 is spec 2.32}, etc., an d the ified task is to find the diagonal. The fin g1. required to make by al diagonal form of G G1 G G is O n th e ot he r ha a re~ult of th~ desig nd w ith th e ge ne1 n. diagonal form fo ra l de co up lin g appr r G G 1 is specified oach, th e final as GR, an d then th this is derived fro e G 1 required to m Eq. (22.29). achieve Let us no w cons id er so m e ex am simplified an d ge ples to illustrate neralized decoup the principles of ler design. bo th Ex am pl e 22.1

SIMPLIFIED DE CO UP LE R DE SI GN FO R TH E W AN D BERRY DI OO D ST IL LA TI ON CO LU M N. Using the simplifi ed decoupling appr oach, de sig n a de di sti lla tio n co lu coupler for th eW m n wh os e trans ood an d Berry fer fu nc tio n m od Eq. (20.26). el wa s given in Example 20.6, So lu tio n:

Recalling the trans fer function matr ai d of Eqs. (22.20) ix for this proces s, we no w obtain, an d (22.21), th at th wi th th e e transfer functio are: ns for the simplifi ed decouplers

CHAP 22 DESIGN OF MU LTNARIA Bl.E CONTROLLERS

81 •

783

-18 .9e- 3• 2l.O s + 1 12.8e-• 16.7 s + 1

= -

and

6.6e-1' 10.9 s + 1 -19. 4e-3' l4.4 s + 1

811

=

8J,

= 1.48

(16. 7s + l)e- 2• (2l.O s + 1)

(22.33)

812

= 0 ·34

(14. 4s + l)e- 41 (l0. 9s + 1)

(22.34)

whic h simplify to:

and

Note that these interaction compens ator blocks each have the form of a lead /lag system with a gain term and a time-dela y term. In term s of actu al imp leme ntati on, using Eqs. (22.33) and (22.34) in Eq. (22.18) implies that the overhead reflux rate, u1 will be determined from: ul

=

VI+

[

1. 48

(16. 7s + 1) e-2•] (21. 0s + 1) V2

(22.35)

whe re, as we may recall, v and v are, respectively, the outp uts of 1 the Loop 1 (overhead composition) controller 2 and Loop 2 (bottoms composition) cont roller. The bottoms stea m flowrate will be dete rmined from:

U2

4s + 1) e- 4•] = Vz + [ 0.34 (14.(10. 9s + 1) VI

(22.36)

The next example illustrates the proc edure for carrying out generalized decoupling in practice; it also illus trates, rath er graphically, the rati onale behind referring to the first approac h as "simplified" decoupling. Exa mpl e 22.2

A GENERALIZED DECOUPLER FOR mE WO OD AND BERRY DIS TIL LAT ION COL UM N.

Usin g the gene raliz ed deco upli ng appr oach , desi gn a deco uple r for the Woo d and of Example 22.1.

Berry distillation column, the same syst em Solu tion : The transfer function matrix in ques

G(s)

=

12.8e-• [ l6.7 s + 1

6.6e -1' 10.9 s + 1

tion is:

-18. 9e-3' ] 2l.O s + 1 -19. 4e-3' J4.4s + 1

(22.37)

and if we choose GR to be diag

MU LDV ARI ABL E PROCESS CONTROL

onal elements of Eq. (22.37), then :

-19.04e-3s ] 14.4 s + 1

(22.38)

To obta in the requ ired deco uple r as indi cate d in Eq. (22.29), we mus t now eval uate the inve rse of the tran sfer function matrix in Eq. (22.37); this is give n by:

-19. 4e- 3' 1 [ 14.4 s + l G (s) - - d -6.6 e-1' 10.9 s + 1 _1

whe re 6. is the dete nnin ant of the

6. =

18.9 e-3' ] 2l.O s + 1 12.8e-• 16.7 s + 1

(22.39)

matrix:

-248 .32( 21.0 s+l) (10. 9s+ l)e-4 • + 124. 74(1 6.7s +l)( l4.4 s+ l)e- 10 • (16. 7s+ l)(l4 .4s+ l)(2 1.0s +l){ l0.9 s+l)

(22.40) By mul tiply ing the inve rse mat rix in Eq. (22.39) and GR in Eq. (22.38) as indi cate d in Eq. (22.29) we obta in the generaliz ed deco uple r for the Wood and Berry colu mn as:

(22.41) with each elem ent given as follo ws:

-248.32(21.0s+ 1)(I 0.9s+ I) 8111 = 124 .74( 16.7 s+l) (14. 4s+ l)e-6' - 248 .32( 2l.O s+l) (l0. 9s+ l)

(22.42)

-366.66(16. 7s+ l)(l0 .9s+ IV25 124 .74{ 16.7 s+l) (l4. 4s+ l)e-65 - 248 .32( 2l.O s+l) (I0. 9s+ l)

(22.43)

84.48(21.0s+ l )( 14.4s+ l)e-4 • = 124 .74( 16.7 s+l) (l4. 4s+ l)e- 6•248 .32( 2l.O s+1 )(10 .9s+ l)

(22.44)

g/12 =

g/21

and

(22.45) The actual imp lem enta tion of this deco uple r mea ns that give n v , Loop 1 (ove rhea d com posi tion 1 the outp ut of the ) controller, and v2, the outp ut of the Loo p 2 (bot tom s composition) controller, the ove rhea d reflux rate, u1 will be dete rmin ed from:

(22.46) and the bott oms stea m flowrate

will be determined from:

(22.47) with the deco uple r transfer func tion elem ents as give n in Eqs. (22.42), (22.43), (22.44), and (22.45).

CHAP 22 DESIGN OF MULTIVAR IABLE CONTROLLERS

785

These two examples clearly illustrate the advantage s and disadvant ages of simplified versus generalized decoupling: 1.

While both decouple rs provide diagonal systems for which SISO controllers can be tuned by SISO methods, the closed-loop response could be complete ly different in each case. For example, the "equivale nt" open-loop decoupled system using simplified decoupling is given by Eq. (22.22) and shown in Figure 22.3 as:

Y1

= ( 811-

Y2

= ( 822-

812821) ( I2.8e-• (18.9)(6.6)(14.4s + l)e-1 • ) ~ vi= 16.7s + 1 - (19.4}(10.9s + 1}(21.0s + 1) vi 812821)

(-l9.4e-3s

--g:- v2 = l4.4s +I

{18.9){6.6)(16.7s + l)e-9s ) + (12.8)(10.9s + 1}(21.0s + 1) v 2

which is much more complicated than the GR matrix specified in the Generalized decouplin g design, as, for example, Eq. (22.38). One would expect that the tuning and closed-loop performa nce for the Generalized decouplin g case to be much better than for Simplified decoupling. 2.

The price we must pay for the improved closed-loop performance in the Generalized decouplin g case is a more complica ted decoupler . This is easily seen if we compare the Simplified decouple r given by Eqs. (22.33) and (22.34) with the Generalized decouple r in Eqs. (22.42) - (22.45).

22.2.3 Some Limitati ons to the Applica tion of Decoup ling There are some limitations to the application of decoupling, and these must be kept in mind in order to maintain a proper perspecti ve when designing decouplers. Perfect decouplin g is only possible if the process model is perfect. As this is hardly ever the case, perfect decouplin g is unattaina ble in practice. However, even with imperfect models, decouplin g has been applied with notable success on a number of industrial processes [1]. The ideal decouple rs derived by the simplified decouplin g approach are similar to feedforw ard controller s (compare Eqs. (22.20) and (22.21) with Eq. (16.22)). Returning to take another look at Figure (22.1) will confirm why this has to be the case: observe that, as far as Loop 1 is concerned, u appears as 2 a "disturba nce" passing through the g12 transfer function element. Thus designing g1, to take the form in Eq. (22.20) is identical to the procedure for feedforward controller design. Ideal dynamic decouplers are thus subject to the same realizatio n problems we had enumerat ed in Chapter 16 (Section 16.1.2) for feedforwa rd controllers, particular ly when time delays are involved in the transfer function elements. Perfect dynamic decoupler s (whether simplified or generalized) are based on model inverses. As such, they can only be implemen ted if such inverses are both causal and stable. To illustrate, consider the 2 x 2 compensa tors in

786 MU LT NARIABLE PR OCESS CONTROL + Sin gle

Loop

I I

Decoupler 1

Process

~---------------

I I

y

1

--l

Controllers

Fig ure 22.5.

Delays

Use of additional time del

ays to allow dynamic dec oupling.

Figure 22.1, wh ere the transfer functions g an d g mu st be causal (no 1 an d stable. To satisfy e+as terms) causality for the 2'x 2 1 sygtem, we see from Eq an d (22.21) tha t an y tim s. (22.20) e delays in g11 mu st be sm all er tha n the tim e g 12, an d a sim ila r co nd delays in itio n mu st ho ld for g sec on d co nd itio n is tha 22 an d g 2r To sat isf y stability, a t g11 an d g22 mu st no t ha ve an y RHP zeros an an d g21 mu st no t ha ve d also g 12 an y RHP poles. This lea ds to the following mo co nd iti on s tha t mu st re general be satisfied in or de r to im ple me nt sim pli fie de co up lin g for N xN sys d dy na mi c tems: 1. Ca us ali ty: In ord er to en su re ca us ali ty in the co mp en sat or tra functions it is necessar ns fer y tha t the tim e-d ela y str uc tur e in G(s) be su tha t the sm all est de lay ch in ea ch ro w oc cu rs on the dia go na l. Fo sim pli fie d decoupling, r this is an ab sol ute req uir em en t; ho we ve r, it possible to ad d delays is to the inp uts utr u , ••• , u,., in ord er to 2 req uir em en t if the ori satisfy the gin al pro ce ss G do es no t co mp ly. Th is is eq uiv ale nt to defining a mo dif ied process:

wh ere D is a diagonal matrix of tim

(22.4&)

e delays:

(22.49)

Th e simplified de co up ler is the n de sig ne d by us ing the elements of rat he r tha n G, an d the Gm ma tri x D mu st be ins ert ed int o the co ntr ol as sh ow n in Figure 22. loop 5. An example below illustrates :this procedure In the case of ge ne ral . ize d de co up lin g, on e ma y us e the mo dif ied pro ce ss Gm as above, or alt ern ati ve ly ad jus t the tim e de lay s in dia go na l ma tri x, G R the in Eq. (22.29), in or de r tha t the ele me nts G1 =(G D) -1 GR are cau of sal. This is equivalent to requiring tha t G,R1 GD ha ve the smallest de lay in eac h row on the dia gonal. 2. Sta bil ity : In ord er to en su re the· stability of the co mp en sat or tra functions, it is necessar nsf er y tha t the cau sal ity co nd itio n above be satisf an d tha t the re be no RH ied P zeros of the process G(s). This is an absol req uir em en t for simpli ute fied de co up lin g an d red uces to the condition tha t

CHAP 22 DESIGN OF MUL TNARIABLE CONT ROLLERS

787

there be no RHP zeros in the diagonal elem ents of G and that the offdiagonal elements of G be stable. For generalize d decoupling this may be relaxed by adjusting the dynamics of GR in orde r that the elements of G1 =G-1 GR be stable. One may quickly verify that the Wood and Berry Column of Exam

ples 22.1 and 22.2 satisfies these dynamic decoupling conditions. However, let us provide an

example for which these conditions are not satisf ied. Exam ple 22.3

DECO UPLI NG CON TROL OF A DIST ILLA TION COLU MN WITH DIFFICULT TIME-DEL AY STRUCTURE.

Let us consi der a slight modification of the Wood and Berry proce ss wher e a process chang e has added a time delay of 3 minutes to the input u1, so that the proce ss mode l takes the form: 12.8 e-4 • -18.9 e- 3 • 16.7s + I 2I.Os + 1 [ G(s) = 6.6 e-IOs -19.4 e- 3• I0.9s + I 14.4s + 1 Note that now the small est delay in each row

decoupling compensator Eq. (22.33) becomes:

]

(22.50)

Is not on the diago nal, and the simplified

_ 148 (16.7 s +I) e+s • (2l.O s + I)

(22.51)

g/1 -

which is nonca usal and canno t be implemente d. Thus simplified decou pling canno t be used unless we make use of a delay matrix, D(s). Let us desig n D(s) to add a time delay of 1 minu te to the input~' i.e.:

(22.52)

so that the modified process is:

G

m

= GD =

[

12.8 e- 4 s 16.7s + I

-18.9 e-4 • 2l.Os + 1

6.6 e-IOs -19.4 e-4s 10.9s + 1 14.4s + 1

]

(22.53)

Now we have satisfied the requi remen t that the small est delay be on the diago nal and the simplified decoupling compensators becom e: [ 1.48 (16.7 s + 21.0s + I

t)J

_ [0.34 (14.4 s + 1) e- 6 •] (10.9 s + 1)

g/2 -

788

MULTIVARIABLE PROCESS CONTRO L

and these are implemented together with the delay matrix, D(s). Note that this decoupling comes at a cost of an additional delay of one minute in Loop 2. If we use generaliz ed decoupli ng for the process in Eq. (22.50), all the compensators are causal except for: g 112

=

-366.66(1 6. 7s+ 1)( 10.9s+ 1)e+s 124.74(1 6.7s+l)(1 4.4s+l)e- 6•- 248.32(2 1.0s+l)(l 0.9s+l)

which cannot be implemented. To remedy this we could use the D(s) given by Eq. (22.52) and let: Alternatively, we could let D(s) =I and increase the time delay in the (2,2) element of GR to 4 minutes and have the same generalized compensator as in Eqs. (22.42)-

(22.45), but now g/21

84.48(21.0.s+ 1)(14.4s+ I )e-7• 124. 74(16. 7s+ I)( 14.4.s+ I )e-6• - 248.32(21.0s+ 1)(10.9s+ l) -248.32(2 1.0s+ I )(10.9s+ l)e-•

Kin

= 124.74(1 6.7s+l)(l 4.4s+l)e 6•- 248.32(2 1.0s+l)(l 0.9s+l)

The problem s at the end of the chapter provide addition al example s of dynamic decoupling. Because of complexities in impleme ntation or the lack of a high-qua lity dynamic model, it is often necessary to make use of simplifications of the ideal decouple r. For example, steady-state decoupling requires only the matrix of steady-s tate gains as a model. Alternatively partial decoupli ng reduces the dimensi onality of the problem by requirin g decoupl ing in only a few of the control loops. Often one of these choices is employe d as the best balance between complexity and closed-loop controller performance. Let us now discuss some of these simpler types of decoupling.

22.2.4 Simple r Types of Decoup ling When ideal dynamic decoupling is not desirable or possible, useful results may be obtained by using the less ambitious approaches of Partial Decoupl ing and Steady-State Decoupling.

Partial Decoupling When some of the loop interactions are weak or if some of the loops need not have high performance, then one may consider partial decoupling. In this case, attention is focused only on a subset of the control loops where the interactions are importan t and high performance control is required. It is typical to consider partial decoupling for 3 x 3 and higher dimensio nal systems , since the main advanta ge lies in the reductio n of dimensi onality; however , partial decoupling is also applicable to 2 x 2 systems, in which case, one of the compensator blocks in Figure 22.1 is set to zero for the loop that is to be excluded from decoupling. The following example illustrates the application of partial decoupl ing for a 3 x 3 system.

789

CHAP 22 DESIGN OF MULTWARIAB LE CONTROLLERS

Example 22.4

PARTIAL DYNAMIC DECOUPLING FOR THE HULBERT AND WOODBURN WET-GRINDI NG CIRCUIT.

Hulbert and Woodburn [3] have reported the control of a wet-grinding circuit whose transfer function model, obtained by experimental means, is given by:

[m

153 337s + 1 0.000767 33s + 1

-21 lOs + 1 -0.00005 lOs + 1

,-667e-320s 930 500s + 1 166s + 1

_-1033 41s + 1

119 217s + 1 0.00037 500s + 1

[~

(22.54)

The time delay and time constants are in seconds; and the indicated process variables are given in terms of deviations from their respective steady-state values as: y 1 = Torque required lu turn the mill (Nm) y 2 = Flowrate from the mill (m3/ s) y3 = Density of the cyclone feed (kg/m3) u1 = Feedrate of solids to the mill (kg/s) u2 = Feedrateofwa terluthemill(k g/s) u3 = Feedrateofwa terluthesump( kg/s)

A relative gain analysis of this system (see Problem (22.2)) recommends the 1-1/2-

2/3-3 input/ output pairing configuration; it also indicates that if three independent

single-loop feedback controllers are used, the system will experience significant loop interactions. A schematic diagram for this process is shown in Figure 22.6. Since the least sensitive of the output variables for this system is y 2 (the total flow from the mill), and since from experience, it is known that the greatest amount of interaction is between control Loops 1 and 3, it is desired to design decouplers for only these loops, leaving Loop 2 to operate without decoupling. Obtain the transfer functions required for these decouplers. Solution: Under the indicated conditions, the transfer function matrix for the subsystem of interest is now given by:

-21 ] 119 1. ["t] [Yt] = [ 217:s930+ 1 lOs+ u3 -1033 Y3

(22.55)

500:s+1 47s+l and if the decoupler transfer functions are represented as g1 for Loop 1, and g1 for 3 Loop 3, then using the simplified decoupling approach, we hav~, from Eq. (22.55):

81,

_2_1_ lOs + 1 119 217s + 1

790

MU LTI VAR IAB LE PROCES S CON TRO L

Torque,y 1 Y2

Mill

Flow from mill

Sump wat er feed rate

Wat er to the mill Density of cyclone feed

Fig ure 22.6.

Schematic diag ram of the wet -gri ndin g circ uit und er inde pen den t mul tipl e single-loop feedback controL

and g/3

=

930 500 s + 1

_j_ Qll _ 41s + 1

whi ch sim plif y to: g/1

and

gI,

0.17 6{2 17s + 1} (lO s + l) 47s + 0 = 0.9( (50 0s+ l)

each imp lem enta ble by lead /lag units. Not e that this con trol sche me is equ ival ent to letti ng g = • 113 g1 and g1 22.4 wtt h all the othe r g ~ =0. I 31 1

(22. 56)

(22. 57)

=g13 in Figure

Steady-State Decoupling The only difference between dyn amic decoupling and steady-stat e decoupling is tha t while the former uses the complete, dynamic version of eac h transfer function element in obtaining the decoupler, the latter uses only the steadystate gain portion of these tran sfer function elements. · Thus, if each transfer functio element g;is), has a steady-stat Kw and if the matrix of steady-nstat e gai n term e gains is rep res ent ed by K, steady-state decoupling results the n the equivalent to those obtained in Section 22.2.2 are now summarized: Sim plif ied steady-state decoup

ling for a 2 x 2 system

Here: (22. 58)

CHAP 22 DESIGN OF MUL TIVARIABLE CONTROLLERS

791

and (22.59)

will take simple, constant, num erica l values, and will therefore always be realizable, as well as implementable. Generalized steady-state decoupling

In this case, the decoupler matrix will be

given by: (22.60)

whe re KR is the steady-state version of GR(s). Since the inversion indicated in Eq. (22.60) now concerns only a matrix of num bers , Eq. (22.60) will always be realizable and easily implemented. The main adva ntag es of steady-state deco upli ng are, therefore, that its desi gn invo lves simp le num erica l com puta tion s, and that the resu lting decouplers are always realizable. Example 22.5

SIMPLIFIED AND GENERALIZED STEADY-STATE DECOUPLERS FOR THE WOO D AND BERRY DIST ILLA TION COL UMN .

Desi gn stead y-sta te deco upler s for the Woo d and Berry column using both the simplified and the generalized approa ~

Solution: The simp lified stead y-sta te deco upler s are easil y obtai ned by retai ning only the steady-state gains of the transfer function elements; the results are: -18.9

g/1

= - 12.8 =l.48

(22.61)

8J,

=

6.6 --19 .4 = 0.34

(22.62)

and

and these will be impl emen ted on the real

process as: (22.63)

and (22.64) of:

For the generalized deco upler on the other

hand , we have, upon takin g the inverse

K = [ 12.8 -18.9 6.6 -19. 4

J

(22.65)

792

MULTIVARIABLE PROCESS CONTROL and multiplying by: K

= [ 12.8 0

R

0 ] -19.4

we would obtain: Gl

=[

2.01 2.97 0.68 2.01

J

(22.66)

(22.67)

and these will be implemented as: 2.01 v 1 + 2.97 Vz

(22.68)

and (22.69)

Note that these same results could have been obtained by settings = 0 in the dynamic decouple r results obtained in Example 22.2. In principle we would expect the simplified approac h and generali zed approac h to give the same result for steady-s tate decoupling. That this is true may be seen by dividing Eqs. (22.68) and (22.69) by the common factor 2.01 to obtain Eqs. (22.63) and (22.64). Howeve r, the single-loop controller gains in each case will also differ by the factor 2.01.

As the exampl e illustrat es, steady-state decoupl ers are very easy to design and straigh tforwar d to implem ent. This advanta ge in simplic ity can be so compel ling that, quite often, in dealing with control loop interact ion problem s in practice, steady- state decoup ling is the first techniq ue to try; only when the dynami c interact ions prove to be persiste nt will dynami c conside rations be enterta ined. Observ e that whenev er the dynami c aspects of the transfe r functio n elemen ts in each row of the transfer function matrix are similar, the dynami c decoup ler will be very close to its steady- state version. It can be shown (and this is left as an exercise to the reader) that for the stirred mixing tank system whose approxi mate transfer function matrix was given in Eq. (20.23), whethe r we use the simplif ied or the general ized approac h, the dynami c decoup ler obtaine d in each case is actually identica l to the corresp onding steady- state decoupler. · Because steady- state decoup ling often leads to great improv ements in control system perform ance with very little work or cost, it is the decoup ling techniq ue most often applied in practice. Thus in the next section we will discuss the design of steady- state decoup lers and their perform ance in some detail.

22.3

FEASIBILITY OF STEADY-STATE DECOUPLER DESIG N

One would think from the discussion in the last section that design of a steadystate decoup ler is trivial. One simply finds the matrix of steadystate process gains, calcula tes the matrix inverse , and program s these number s into the control comput er. Operati onally, this is true- no further compen sations are


793

required on-line. However, the successful implement ation of a steady-stat e decoupler requires answers to further questions: 1.

Are there constraints on some of the inputs (e.g., valve range, heating or cooling power, etc.) which are reached? If so the decoupler cannot be used beyond the point of these constraints because the constrained controllers are no longer responding to error signals and the decoupler could even make the feedback control scheme drive the process unstable once an input constraint is reached.

2. Is the steady-state model of the process sufficiently accurate to permit steady-state decoupling? Depending on the degree of "ill-conditi oning" of the true steady-stat e gain matrix, an extremely accurate process model or only a rough approximat ion might be required for successful implementa tion of decoupling. 3.

Is the steady-state gain structure of the process so "ill-conditionedH that even with a perfect model, successful decoupling is impossible. As we will demonstrat e below, there are "impossibl e situations" in which the true process model is so ill-conditio ned that decoupling control is useless. Unfortunat ely, such situations are not rare in industrial practice.

Let us now discuss these questions of feasibility of the design of steady-stat e decoupling controllers in more detail.

22.3.1 Effect of Controlle r Constrain ts In process control practice there are always constraints on some of the process input variables because valves cannot go beyond full open or full shut, heaters cannot go beyond full power or zero power, etc. Thus, the important question in evaluating this factor is whether or not one or more of the inputs reaches a constraint during normal operation. If the controller gains are such that none of

the constraints come into play, then decoupling control is not affected by these constraints . However, if even one of the inputs reaches a constraint then the control system can no longer send the combinatio n of control actions required to achieve decoupling , and extremely poor (or even unstable) responses could result. Thus more sophisticat ed controller strategies, which explicitly recognize and take constraints into considerati on, are required when input constraints come into play. Let us illustrate with an example. PERFORMANCE OF STEADY-STATE DECOUPLER WITH INPUT CONSTRAINTS.

Example 22.6

Let us consider the implementation of the simplified steady-state decoupler for the Wood and Berry Distillation Column of Example 22.2. Recall that the decoupler is given by Eqs. (22.61) and (22.62). When this is implemented together with diagonal PI controllers with tuning parameters K, = 0.30, 1/-r1 = 0.307, K, =- 0.05, 1/-r1 = 0.107, one sees from Figures 22.7-22.8 t'hat the I

2

2

794

MU LTIVARI ABL E PROCES S CONTROL

1.0

0.5

0.0

0

Figu re 22.7.

10

20

30

40

50

Woo d and Berry Distillation Colu mn: closed-loop resp onse of y 1 and y2 und er steady-state decoupling cont rol with unconstrained inputs.

dos ed-l oop resp onse is bett er than wha t was obta ined with mul tiple single-loop designs (cf. Example 21.13). Thu s a steady-state decoupler seem s very attractive. Now supp ose that an ener gy reva mp of the colu mn has redu ced reflux drum and reflux control valv the size of the e so that u1 has a more limited max imu m flowrate, 0 ~ u1 ~ 0.15. Whe n the sam e decoupling control scheme is appl ied to the colu mn with the new reflu x flow rate cons train t, Figures 22.9, and 22.10 sho w that the clos ed-l oop resp onse is very poo r once the reflu x valv e is full ope n and the syst em beco mes unstable. This is beca use u is pegg ed at its max imu m valu e tryin g 1 to increase y1 and Loo p 2 is tryin g to com pens ate ima gine d control actions u , whi ch are com pute d, but 1 not implemented. 0.5

0.4

0.3

0.2

/ -~~

0.1

..,

.,,

....._______~~----------------------------------·-"'··---------------------------o.o -r. -.- .-. .~-..-.-.-~~0

Figu re 22.8.

10

..-.-.,~-.-..

20

30

1 40

-.-. .

50

Unc onst rain ed man ipul ated vari ables, u 1, u2, for closed-loop resp onse in Figure 22.7.

CHAP 22 DESIGN OF MULTIVARIABLE CONTROLLERS

0

Figure 22.9.

20

10

795

30

40

50

Wood and Berry Distillation Column: closed-loop response of y1 and y2 under steady-state decoupling control with constrained u1, 0 S u1 S 0.15.

Thus steady-state decoupling cannot be applied in this case; however, more sophisticated, constraint following controllers could be implemented (cf. Chapter 27 for more discussion of these).

22.3.2 Decoupling Control of lll-Conditioned Processes There are processes whose structure make them poor candidates for decoupling control. Such processes fall under the general category of "ill-conditioned" processes. Although there is a unifying structure to "ill-conditioned" processes, let us begin the discussion by considering the various manifestations of this illconditioning. 0.5

0.4

0.3

/"2 0.2

,_,~:;~c··':~: .......... ~'-·························

0. 0

I

0

I

I

I

I

10

I

I

I

I 20

I

I

I

I

I 30

I

I

I

I

I 40

I

I

I

I

I 50.

Figure 22.10. Manipulated variables u1, u2 for closed-loop response in Figure 22.9 when u1 is constrained 0 S u1 S 0.15.

796


High Sensi tivity to Model Error Consi der the system whose steady-state gain matrix is K, so that its steadystate model is given by: y = Ku (22.70) If steady-state decoupling is applied on such a system , we now know that the outpu ts of the independent, multiple single-loop controllers, represented by the vector v, will be related to the actual vector of process input variables u by: (22.71)

where G 1 is the decoupler matrix. For the system in Eq. (22.70), we have established in the previous section that the generalized decoupler is given by: (22.72)

where KR is the diagonal gain m!ltrix of GR(s). Therefore at steady

state, if the model is perfect, the relationship between v and the process outpu t will be: (22.73)

or

(22.74)

and since KR is diagonal, G 1 will have achieved decou pling; there will no longer be cross-coupling between the control loops. We know, howev er, that no process model (not even the steady -state portio n alone) is 100% accurate; let us then consider the situation in which there is an error .6.K in the estimate of the steady-state gain matrix. In reality, therefore, the true steady-state model should be: y = (K +.6.K)u

(22.75)

The decou pler in Eq. (22.72), designed on the basis of K, will now provid e the following relationship between v andy: y

= (K+.6.K) K- 1 KRv

or (22.76)

Observe that there will now be a deviation from the ideal situation indicated in Eq. (22.74): (22.77)

which represents the amoun t of error introduced into y as a result of carrying out decoupling on a system model ed as in Eq. (22.70), but with the gain matrix subject to modeling error in the amount represented by .6.K.

797

RS CHAP 22 DESIGN OF MULTWARIABLE CONTROllE

From -the definition of the inverse of a matrix, Eq.

(22.77) becomes: (22.78)

matrix, I K I, is very small, We now observe that if the determinant of the gain its reciprocal will be very large, and: very 1. Small modeling errors will be magnified into

large errors in y.

result in large errors 2. Small changes in controller outpu t v will also iny. ling errors, and it is clear Such systems are said to be very sensitive to mode ermore, observe from Furth . cases that decoupling will be a risky ventu re in such farth er away from the fly error the r large Eqs. (22.76) and (22.77) that the a more difficult has control being decoupled the syste m is, and thus feedback time correcting for the error. h decoupling will prove We therefore have the first condition unde r whic diffic ult:

is very sTIUlll, the Whenever the determimmt of the process gain matrix and decoupling errors ling mode to ive sensit ely system will be extrem inant completely will be difficult to achieve; in the limit as the determ · sible. impos ly vanishes, decoupling will be entire dedu ced from the elements Process sensitivity to modeling error can also be the RGA are intimately of nts of the RGA of the syste m because the eleme x. We recall from the matri gain ss proce the of relate d to the deter mina nt lating RGA elements, that if matri x meth od prese nted in Chap ter 21 for calcu of the gain matrix K), then nt Cq represents the cofactor of Kq (the i,j th eleme by: given nts the RGA of the system has eleme KqC11

.il.q =

IKI

(22.79)

alde d by the nearness of I K I Therefore large sensitivity to modeling errors -her RGA values. to zero - will be manifested as inordinately large in addit ion to provi ding that stated is it [4] r pape RGA al origin In Bristol's des a measure of a provi also RGA a meas ure of control loop interactions, the also prese nted has 6] [5, key Shins s. error syste m's sensitivity to mode ling sensitive to very are values results show ing that systems with very large RGA s. value RGA asing incre decoupler error, with sensitivity increasing with Thus we may conclude that:

whose input/output Decoupling will be difficult to achieve for a system a system will also such ; values RGA variables are paired on very large be very sensitive to modeling errors. Let us illustrate this with an example.

798

Example 22.7

MULTIVARIABL£ PROCESS CONTRO L PAR TIAL STE AI)Y -STA TE DEC OUP LING OF THE PRE 'IT AND GAR CIA HEA VY OIL FRA CTIO NAT OR.

Pret t and Garc ia pres ente d in Ref. [7] a 7 x 5 trans fer func tion mod el for a heav y oil fract iona tor colu mn. Here we will cons ider the parti al deco uplin g of this syste m by focu sing on a 2 x 2 subs ystem invo lving the top end poin t, and the inter med iate reflux temp eratu re as the two outp uts, with top draw rate and the inter med iate reflu x duty as the two inpu ts. This subsystem has the following transfer func tion mod el:

4.05e-27s 1.20e-27s ] [ 50s+ 1 45s + 1 G(s) = 4.06e-& __!.,_!L 13s +1 l9s +l

(22.80)

The stead y-sta te gain s are dimension less, and the time delay s and time cons tants are in minu tes. Ana lyze the sens itivi ty of this syste m to mod el erro rs, and asses s the feasibility of deco uplin g. Solu tion: The stead y-sta te gain matr ix for this

subs ystem is:

K = [ 4.05 1.20 4.06 1.19 with dete rmin ant

IKI

= -o.os2s

A

= [ -91.8

J

(22.81)

(22.82) whic h is very close to zero , indic ating that deco uplin g will be very diffi cult. The RGA for this syste m calcu lated in the usua l man ner, is:

92.8

92.8 -91. 8

J

(22.83)

Thus , beca use of the smal l valu e of the dete rmin ant of the gain matr ix, and the inord inate ly large valu es of the RGA elem ents, we conc lude that deco uplin g will be extre mely difficult for this system. Belo w, we will pres ent othe r reaso ns why deco uplin g: will in fact be next to impo ssibl e for this system.

'

Degeneracy When the determinant of a matrix is clos e to zero , the· mat rix is said to be nearly singular; and one of the man ifestations of singularity in a matrix is linear dependence between the colu mns and/ or the rows, or degeneracy of the matrix. To further illus trate this poin t, con sider the prob lem of solv ing the following system of two algebraic equa tions: xl + 2x2 2x 1 +4x2

=1 =3

(22.84) (22.85)


799

It is clear that there is no solution, and the reason is that the elements of the LHS of both equations are linearly dependent; one is exactly twice the other. If these two equations are written in matrix form, we would have:

[ ~ ~ J[::J = DJ

(22.86)

or Ax= b

(22.87)

and ordinarily, the solution to Eq. (22.87), will be: (22.88)

Observe, however, that in this case A is: (22.89)

which is singular; hence its inverse does not exist, and there is therefore no solution. Eqs. (22.84) and (22.85) are called a degenerate set of equations; the determinant of the system matrix given in Eq. (22.89) is zero because the two columns (and the two rows) are linearly dependent. If now the equations are modified slightly to read: x 1 + 2.01~

=1

(22.90)

2.01x 1 + 4x2

=3

(22.91)

so that: A- [

1 2:1 2.01

J

(22.92)

in principle, these equations now have a solution; however, small errors in the b vector will result in very large errors in the solution. These system of equations are now very nearly degenerate, and the solutions are not very reliable. The same issue of degeneracy illustrated above applies to decoupling; for observe that what we are trying to do with decoupling is exactly the same as solving the system of linear equations: (22.93) to determine the decoupled process input vector u; therefore, if K is degenerate, it will be extremely difficult to achieve decoupling. To illustrate this issue of degeneracy in decoupling, and what it means in physical terms, let us return to the Prett and Garcia hot oil fractionator subsystem of Example 22.7; the steady-state model for this system is:

fy·] =[ l.Yz

4.05 1.20] 4.06 1.19

["·1 uJ

(22.94)

MULTIVARIABLE PROCESS CONTRO L

Note that the rows of the stead y gain matrix are almo st identical, and if we writ e out these equations in full to obta in: y1

and

= 4.05u 1 + 1.20u2

y 2 = 4.06u 1 + l.l9u 2

we obse rve that the equation for y is almost identical to y2 • Thus the syste m 1 is dege nera te. The objective of decoupling is to mak e it possible for the control syste m to regu late y1 and y2 inde pend ently ; how ever , the natu re of this syste m is that whatever y1 does, y2 mus t do, also. It is struc tural ly impossible to control these varia bles inde pend ently ; thus for all inten ts and purp oses , deco upli ng is impo ssibl e. The dege nera cy in this syst em has been brou ght abou t by the linea r depe nden ce of the output variables; the next exam ple illustrates dege nera cy brou ght abou t by near-linear dependen ce in the inpu t variables. Exam ple 22.8

DEGENERACY AND DEC OUP LING CON TRO L OF THE STIRRED MIX ING TAN K SYSTEM.

Let us cons ider the stirre d mixi ng tank expa nded with an addit ional heate r as in Exam ple 21.10. H we fiX the heate r at a cons tant powe r and only cons ider the hot and cold strea m flowr ates as input s, the linea rized mode l is given by Eq. (22.95). When some speci fic value s are intro duce d for the phys ical dime nsion s of the stirre d mixin g tank system:

G(s)

[

0.261 43.5 s + I 0.012 35 (T H- T,) 21.7 s + 1

0.261 ] 43.5 s + l 0.01 235( T c- T,) 21.7 s + l

(22.95)

(These phys ical dime nsion s corre spon d to those for the stirred mixin g tank syste m used for instr uctio n at the Univ ersity of Wisc onsin -Mad ison. ) Inves tigate the pOSSlble dege nerac y for this syste m unde r two conditions:

• Condition 1 Hot Strea m Temperature: TH 53°C . Cold Strea m Temperature: Tc 13°C Oper ating stead y-sta te tank temp eratu re:

= =

• Condition 2

Hot Strea m Temperature: TH = 28.SOC Cold Strea m Temperature: Tc = 28°C Oper ating steady-state tank temp eratu re: Obse rve that for Cond ition 2 the hot strea m has coole d off subst antia lly, and the cold strea m has heate d up significantly so that the desir ed stead y-sta te is abov e both feed temp eratu res. This is only possi ble beca use of the auxil iary heate r.


801

Solution: Under Condition 1, the steady-state gain matrix is given by: K

=[

0.261 0.261 0.247 -0.247

J

(22.96)

whose determinant is- 0.129; the RGA for this system is obtained as: A = [ O.S O.S

o.s

J

(22.97)

0.5

and even though the indication is that there will be significant interactions, we observe that decoupling is very feasible in this case. (We will show later what is required to achieve decoupling under these conditions.) Note that because of the -D.247 element, the rows and the columns of Eq. (22.96) are; in fact, not linearly dependent even though they may appear to be so. The gain matrix under Condition 2 is:

K _ [

0.261 0.261 -O.OSS6 -0.0618

J

(22.98)

and we observe immediately that if we rounded all the elements off to only the second decimal place, Column 1 will be identical to Column 2.

The determinant of this matrix is- 0.00162, and the RGA is: A = [ 10 -9 -9 10

J

(22.99)

and it is now clear that the system will be easier to decouple under Condition 1 than under Condition 2. There is a physical reason for the increased. degeneracy under Condition 2: observe that the difference between the hot and cold stream temperatures is only half of a degree. The implication is therefore clear: we no longer have two input variables, we essentially have only one (since the hot stream is almost identical to the cold stream), and as a result we have lost one degree of freedom in controlling the process. Even though it is not crystal clear from the discussion given above, note that the degeneracy will affect only our control of the temperature; in no way does it affect our control of the level.

Less Obvious Ill-Conditioning So far, we have seen that one possible measure of the feasibility of decoupling is how small the determinant of the process gain matrix K is, or, equivalently, how large the RGA elements for the process are. We have also seen that under circumstances in which I K I "'0, or A;/S are very large, there is usually something wrong with the physical system itself; either the output variables are not independent (as was the case with the Prett and Garcia fractionator) or

802


the input variables are not independen t (as with the water tank system of Example 22.8 under Condition 2). There are however some situations in which the determinan t is not too small, neither are the RGA elements too large, and yet decoupling is not feasible. Consider for example the process whose gain matrix is:

K

= [ -60

0.05 -40 -0.05

J

(22.100)

The determinant for this matrix is 5; and the RGA is: A= [ 0.6 0.4] 0.4 0.6

(22.101)

neither of which is unusual. Nevertheless, a trained eye will spot the problem with this system upon inspecting Eq. (22.100) closely: observe that if u1 and u 2 are of the same order of magnitude (i.e., same scaling), this matrix indicates that u 1 is the only ·effective control variable; the effect of u2 on the process output variables is negligible in comparison. Thus, we have a situation somewhat similar to that illustrated with the stirred mixing tank in Example 22.8, under Condition 2; the system in Eq. (22.100) essentially has only one input variable; decoupling cannot be achieved in two output variables using only one input variable. Compare this now with the process whose gain matrix is: K =

[ -3 1

-2 -1

J

(22.102)

The two processes have identical determinants and identical RGA's, but while decoupling is not feasible in Eq. (22.100) it is entirely feasible in Eq. (22.102). It is obvious from this example that neither the determinant nor the RGA is a reliable indicator of ill-conditioning; in this case, what indicates the difference between the two processes in Eqs. (22.100) and (22.102) are the eigenvalues of the two matrices. : The eigenvalues of the matrix in Eq. (22.100) are:;

A-1

= 59.965

~

= 0.0835

(22.103a) (22.103b)

while those for the other process in Eq. (22.102) are:

.t. ~

= -2 +j = -2-j

(22.104a) (22.104b)

We may now note that while the product of each set of eigenvalues is the same (the product of the eigenvalues of a matrix is equal to its determinant) for the difficult-to-decouple system, one eigenvalue is about 720 times larger that the

803


other; for the easy-to- decouple system, the two eigenval ues have identical magnitudes. Unfortu nately, eigenva lues are not always reliable indicato rs of illconditioning, as illustrate d by the following process having a gain matrix:

K

=[

1 0.001 1 100

J

(22.105)

The determin ant for this matrix is 0.9; and the RGA is: A = [ 1.11 -0.11 -0.11 1.11

J

(22.106)

both of which give absolute ly no indicatio n of the serious ill-condi tioning problem s afflicting the process. The eigenval ues for this matrix are:

A.l ./1,2

= 1.316 = 0.684

(22.107a) (22.107b)

no and the larger is less that twice the value of the smaller, again giving indicatio n of any problem s with decoupling. Howeve r, upon critical examina tion, we see that the problem with this 100 process is that u 1 exerts 1000 times more influence on y1 than does u2 ; and one times more influence on y2• Thus once more, we have a situation in which of input variable is significantly more influential tltan tlte other: anotlter case ill-condi tioning. As mention ed earlier in our introduc tory discussio n in Chapter 20, the most reliable indicato rs of ill-condi tioning in a matrix are its singular values are defined as the square root of the eigenvalues of the matrix KTK. (Since we general more the use to need not now dealing witlt constant matrices, we do definition involvin g the transpos e of the complex conjugate.) For the process in Eq. (22.105) the singular values are: (22.108a)

02 = 0.009

(22.108b)

the The ratio of the largest to the smallest singular value of a matrix is called ning conditio tlte of r indicato reliable most single the is it and condition number, the of a matrix: the larger the conditio n number, the poorer the conditio ning of matrix. The condition number for the process currently under investigation is: 1C

= 1.113 X 104

which is really quite large, clearly indicatin g its poor conditioning.

MU LT IVA RIA BL E PR OCESS

CO NT RO L Let us fur the r illu str ate the use of sin gu lar val of a pro ces s by ref err ing ues to assess the con dit ion ing bac k to the processes of the ear lie r examples. us co nsi de r the Pre tt First, let an d Ga rci a fra cti on ato r [7]. Fo r thi s pro ces s, the gai n ma trix is: rec all tha t

K = [ 4.05 1.20 4.06 1.19 wh ich has sin gu lar val

J

(22.109)

ues Oi = 5.978, a = 0.0087 2 8 an d a con dit ion 1C

5.978

= 0.00878

nu mb er:

= 680.778

Th is de arl y ind ica tes ser iou s ill-conditioning .

Sim ila rly1 the sti rre d mi xin g tan k un de r Co nd itio n 1 sti pu lat ed in Ex am ple

22.8 ha s a gai n ma trix :

K = [ 0.261 0.261 0.247 -0.247

wh ich has sin gu lar val

J

ues a1 = 0.3695, n

CHAPTER

23 INTRODUCTION TO SAMPLED-DATA SYSTEMS The universal drive for improved efficiency in the operation of industrial processes has led to more stringent demands on the performance of control systems demands that can no longer be met by relying solely on the traditional, analog control hardware. Fortunately, because of the dramatic changes in computer technology, the digital computer allows almost routine realizations of the most complex control algorithms at very low cost. As a result, the use of the digital computer to implement control schemes has now become so widespread that virtually all new plants are designed to operate under computer control in one form or another. In our discussions up to this point, we have assumed that the process· measurements available to the control system were continuous in time and the controller made continuous changes in control action. By contrast, when the process is being monitored and controlled by a digital computer, data are sampled only at discrete points in time, and changes in control action are taken only at these discrete times. Thus the sampled process and controller are Jr.nown collectively as a "sampled-data system." Processes operating in this way acquire a new character, completely different from anything we have encountered so far. Our study of computer process control therefore begins in this chapter with an introduction to the characteristics of a process induced by its association with the digital computer.

23.1

INTRODUCTION AND MOTIVATION

23.1.1 The Digital Computer as a Controller Reduced to its most fundamental essence, a controller is no more than a device that, given information about the error signal e from a process, computes the control action required to reduce the observed error, using a predetermined control "law."

821

822

COMPUTER PROCESS CONTROL

Viewed in this light, we see that the classical analog feedback controllers - which operate according to the PID control laws discussed earlier - are just one possible class of controllers; one can see that there are other devices, operating according to other control laws, that can also function as controllers. The digital computer is perfectly suited to do the job of a controller with greater flexibility and versatility than the classical controllers because of the following advantages: 1. Data acquisition from the physical process is very easily and rapidly done with the digital computer.

2. A tremendous capacity for mass storage (and rapid retrieval) of the collected process information is readily available. 3. The control "law" for calculating corrective control action no longer has to be restricted to a hardwired analog circuit (e.g., for PID control) because any control law, no matter how unconventional or complicated, can usually be programmed on a computer. 4. The required computations, no matter how complex, can be carried out at very high speeds. 5. The cost of digital computers and ancillary equipment has reduced drastically in the last few years. All these factors combine to make computer applications to the on-line implementation of process control schemes very attractive.

23.1.2 Idiosyncrasies of Computer Control If we recall our discussion in Chapter 1, the three main tasks of a control system

are: • Monitoring process variables by measurement • Calculation of required corrective control action • Implementation of the calculated control action For a computer control system, all three tasks are carried out exclusively by the digital computer. Observe that the use of the computer to carry out the second task is no different from its traditional use as a fast calculator; it is in carrying out the first, and the third, tasks that the unique character of the computer is brought to bear on the physical process in ways that require some attention.

Computers and Data Acquisition Computers, by nature, deal only in "digitized" entities: integer numbers. The output signals from a process, however, are usually continuous (e.g., voltage signals from a thermocouple). Since the computer can only access information in digital form, computer control applications therefore require that the continuous process output signal be converted to digital form.

CHAP 23 INTROD UCTION TO SAMPLE D-DATA SYSTEM S

823

y(t)

Time

r----- -----,·

I

_Y_______

I

-T:_.~ I

------- -----I

SAMPLE R

Figure 23.1.

A continuous signal discretized.

As we discuss ed in Chapter 2, this "digitiz ation" of a continu ous signal is carried out by an analog-to-digital {A/D) converter. The continu ous signal, y(t), is sample d at discrete points in time, t0, tlt t2, ••• , to obtain the "sampl ed" data, y(t1), k = 0, 1, 2, 3, ..., which is then digitized. As indicate d in Figure 23.1 below, the sampler of the AID convert er consists of a switch that closes every At time units, theoretically for an infinite simally small period of time, to produce "samples" of y(t) as y(t ) "spikes ." 1 Let us therefore note the following about computer control systems:

The use of the computer for data acquisition introduces a new perceptio n of the process: a quantized view, in which the system is no longer seen as giving out continuous information, continuously in time; process information now appears to be available only in discrete form, at discrete points in time. It is customa ry to refer to At, the length of the time interval betwee n each success ive sample , as the sample time or sampling period; natural ly, its recipro cal, 1/ At, is called the sampling rate; and multipl ying the samplin g rate by 27t gives the sampling frequency.

Computers and Control Action On the other side of the comput er process control coin is the fact that the comput er also gives out information in digital form, at discrete points in time. This has significant implications on control action implem entation , as we now illustra te. Conside r the situation in which the comput er gives out control comma nd signals as indicate d in Figure 23.2(a); note that the comma nd signal takes on the value u{t1) at time instants t0, t1, t2, ••• , and is zero in betwee n these sample points. In implementing such a control action sequence on a process whose final control elemen t is, for example, an air-to-open control valve, we will find that the valve opens up by an amount dictated by the value that u(tk) takes at each

824


u(t)

/

...

/ ·--'-\ .., ______/ I t---

0

(a

Figure 23.2.

(b

A discrete signal from the computer and the intended continuous version.

sampling point, and shuts off completely in between samples. Quite apart from the unusual wear and tear that such control action will have on the valve, it is obvious that the intention is not to have the control valve open up only at sampling points; the intended control action is shown in the dashed line in Figure 23.2(b ). Since the computer is incapable of giving out continuous signals, what is required is a means of reconstructing some form of "continuous" signal from these impulses. The device for doing precisely this is known as a digital-to-analog {D/ A) converter. The key elements in D/ A converters are called holds; they are designed to simply "hold" the previous value of the signal (or its slope, or some other related function) until another sample is made available. We will discuss these further later. The data acquisition and control action implementation aspects of the computer control systems may be combined to produce Figure 23.3, a typical block diagram for a computer control system - an equivalent of that shown in Figure 14.2 for the analog feedback control system.

Mathematical Descriptions As a result of the effects of using the computer for data acquisition and control

action implementation, computer control systems are often referred to by such terms as sampled-data systems, discrete-time systems, or digital control systems, to indicate explicitly this quantized characteristic bestowed upon it by the digital computer. Such systems must now be described, mathematically, in a manner that reflects their "new" character of giving out, and receiving, · Process ~-------,

d

Process control computer

-----------------, 1 I

I

I

D/A Converter

I

Setpoint

----!-/.,+.()(}-.-t Yd

I

L----

,.-----, I

'----,li--"' 1

I y(t) ---c~:-------1

L _____

Measuring :Analog '---D_e_vt_·c_e_ _, Measurement

.J

AID Converter

Figure 23.3.

Typical block diagram for a computer control system.

CHAP 23 INTROD UCTION TO SAMPLE D-DATA SYSTEM S

825

systems are informa tion only at discrete points in time. Compu ter control of the type therefore more conveni ently represe nted by discrete-time models 4. Chapter of briefly introdu ced in various sections control The three major issues raised by the use of digital comput ers for follows: as ized system implementation may therefore be summar 1. Sampling (and conditioning) of continuous signals 2. Continuous signal reconstruction 3. Approp riate mathem atical description of sample d-data systems

We shall now consider each of these importa nt issues in tum.

23.2

SAMPLING AND COND ITION ING OF CONT INUO US SIGNALS

23.2.1 Sampl ing aid of a Sampli ng a continu ous signal is achieve d, in practice , with the like a sampler: a device that, as indicate d in Figure 23.1, operate s physically ; this seconds t" period, finite, but short, switch that stays closed for a very . seconds At every d action is repeate

Quantitative Description of Sampled Signals seconds , at Conside r the arbitrar y, continu ous function, f(t), sample d every At to consider ry customa is It . . . 2, 1, 0, = k tk; by ted represen specific points in time = t At; = t 2 2At; and in the samplin g as commencing at time t = 0 such that t0 = 0; 1 general : (23.1)

Let us now It is also custom ary to represe nt the sample d version off(t) asf"(t). and its f(t), ous continu the s, function two derive an expression for relating these r 24 Chapte in d require be will ship relation this because f"(t), , sample d version . when we take up the analysis of sampled -data systems g Note that the action of the sampler is such that during the kth samplin t" for f(tt)), (i.e., f~:) = f(t value the takes f*(t) output r sample the interva l, Figure 23.1). seconds; immedi ately afterwa rds, the sampler output is zero (cf. that of the match values whose s impulse of train a is Thus the comple te f"(t) at zero continu ous functio n f(t) only at the samplin g points, and remain is ement measur which in n operatio r elsewhere. If we assume ideal sample tk is time at output r sample single a then 0) e-+ = (t" neously provide d instanta as follows: most conveniently modele d using the ideal delta (impulse) function (23.2)

possible is The key propert y of the delta function that makes this descrip tion x(t): shown below for any arbitrary function

826


f

t+£

x(t) o(t- 6) dt

{x((J)·

t

=6

0,

t

*

= ..

,_ £

(23.3)

6

That is, the operation of l>(t- 9) on any function combined with the indicated integration has the net effect of "wiping out" the original function, leaving a single, surviving "spike" at the point t = 6 whose magnitude coincides with the value of the original function at this single point. Since what we observe as the sampler output is a sequence of such impulses as that represented in Eq. (23.2), then.f(t) can be represented as: J*(t)

= [it(t)8(t-tk )dt o

(23.4a)

k=O

or (23.4b)

It is important to recognize that each of the entities indicated in this sum

remains permanently at zero until t equals the argument in the parentheses, when it instantaneously takes the indicated value; thereafter, it promptly returns to zero once more. Thus, for example, when t t1, everythlng else but .f(t1) remains at zero. With this in mind, we may now write Eq. (23.4) more compactly as follows without the risk of confusion:

=

f*(t)

=

"'LfCtk) O(t- tk)

(23.5)

k=O

This is the quantitative representation of the output signal of an ideal impulse sampler whose input is the continuous function f(t).

Choosing Sampling Time, L1t The process of sampling a continuous signal is, in practice, not as straightforward as it appears at first. The main question to be answered is: how frequently should we sample? That is, how should At, the sampling interval, be chosen? Surprisingly, this deceptively simple question can be very difficult to answer conclusively. · Let us illustrate the issues involved with choosing At by considering the following two extreme situations: 1.

Sampling too rapidly: The sampled signal, y(tk), very closely resembles the continuous signal, y(t); but this will obviously require a large number of samples and a large amount of data storage. In addition, this increased work load can limit the number of other loops the computer can service.

2.

Sampling too slowly. In this case, fewer samples will be taken, and as a result, there will be no unusually heavy burden on the computer. However, the main problem is that y(tk) may no longer resemble the original, continuous signal, y(t). This problem is known as aliasing, and

CHAP 23 INTRODUCTION TO SAMPLED-DATA SYSTEM S

827

a classic examp le may be used to illustr ate its effect in Figure 23.4. If the origin al sinuso idal function, shown in Figure 23.4(a) , is sampl ed four times every three cycles, the result is the sampl ed signal shown by X's in Figure 23.4(b). If we attem pt to recons truct the origin al sinuso idal signal from this sampl ed data, we obtain the appare nt sinuso idal signal shown in Figure 23.4(b), whose freque ncy is clearly seen to be lower. Of course, typical process signal s are not sinuso idal, but the examp le serves to illustr ate the proble m of captur ing highfrequency inform ation when sampl ing too slowly. The other proble m associ ated with sampl ing too slowly is that the effecti veness of the contro l system is often substantially reduce d. This stems from the fact that in betwe en sampl es, the contro l system actual ly operat es as an open-l oop system since no control action is taking place during the entire period in betwe en samples. Thus if the sampl ing interv al is very long the perfor mance will clearly deteriorate. From such considerations, it is thus clear that the "optim um" sampl ing time must lie somew here in betwe en these two extremes. Unfor tunate ly, there are no hard and fast rules for choosing At. A pruden t choice of At is clearly one which is small enoug h to guaran tee that no significant dynam ic inform ation is lost in · sampling, but which is not so fast that the compu ter will becom e overloaded. If the fastest dynam ics one wishes to captur e by sampl ing is known , say, this is freque ncy a>max, then one must have a sampling frequency: 0)

s

=21t At

(23.6)

(a)

(b)

Figure 23.4.

Effect of aliasing resulting from sampling a sinusoidal functio n too slowly: x-tJ.t =3T/4; o-tJ.t =T/2; tJ.-tJ.t =T/2.

828


which satisfies the sampling rule: (23.7)

Alternatively, for any given sampling frequency ro5, the largest frequency that can be extracted from sampled data, called the Nyquist frequency OJN, can be determined from the equality in Eq. (23.7) as: (23.8)

The validity of the sampling rule in Eq. (23.7) can be seen from Figure 23.4 where we observe that if we want to capture a sine wave, which has period T and frequency, OJ= 2.1t!T, then we must sample with frequency: OJS

>

41t

T

(23.9)

By comparing Eqs. (23.8) and (23.9), we see that this means choosing M < T /2 in Figure 23.4. Note that if we were to choose At =T /2 and had our first sample at T I 4, then we obtain the sequence of samples shown with circles in Figure 23.4, and we have captured the sitte wave in sampled form. However, choosing M to barely satisfy the sampling rule, At = T /2, does not capture all the details of the sine wave. It only guarantees that one obtains the correct frequency, but the amplitude could be totally incorrect. For example, suppose one sampled with At = T/2, but took the first sample at T/3 as shown by triangles in Figure 23.4. In this case the sampled sine wave has reduced amplitude; in fact if one took the first sample at T/l, the resulting signal would appear constant because the sampled amplitude is zero. The lesson here is that one must consider carefully the specification of OJmax' the highest frequency one wishes to capture, and allow sufficiently rapid sampling to uncover the details of the signal. Obviously this means that the choice of At will very much depend on the particular application at hand. For digital computers that are simply pieces of hardware emulating continuous analog controllers, rapid sampling is important. However, digital control algorithms based on less frequent sampling are usually used. In this case the following rule-of-thumb has been found to provide a good starting choice for sampling time: ·

For a process with a dominant time constant that the sample time, .dt, be chosen as:

'~'dom'

it is recommended (23.10)

This rule will ensure, for example, that for a first-order system with time constant -r, no more than 20-25 samples will be required to represent a step response.

CHAP 23 INTRODUCTIO N TO SAMPLED-DA TA SYSTEMS

829

23.2.2 Signal Conditioni ng The sampled signal is seldom in a condition to be used directly by the computer. For example, temperature measurement signals from a thermocouple are usually in millivolts. Such low-voltage signals are often too weak to be transmitted over some distance to the A/D converters; they will be susceptible to corruption by noise, and will have very low resolution by the time they reach the A/D converters. Therefore, a certain degree of "conditioning " is needed before the sample signals can be useful. As discussed in Chapter 2, signal conditioning usually involves the following: • Amplification of weak signals • Noise suppression by filtering • Possible multiplexing Even though the usual practice is for instrumentati on engineers to take care of the issue of signal conditioning at the hardware design and selection stage, this only ensures that whatever signal was produced by the measuremen t device reaches the computer intact. Once the data are in the computer, additional digital filtering may be required to enhance the precision of the measurements. The simplest, and perhaps the most popular, digital iilter is the exponential filter which works according to the following equation: y(k) = /Jy(k-1)+(1 -/J)y(k)

(23.11)

where: y(k) y(k)

f3

is the filtered value of the signal at the sampling instant k (when t = tk) is the measured signal value at the sampling instant k is the filter constant 0 < f3 < 1

This filter is classified as a low-pass filter because it allows only the low-

frequency components of a signal to pass through, the higher frequency components having been substantially reduced. Other types of digital filters are also used in practice. It is important to note that while filters help suppress the effect of highfrequency fluctuations, they may also introduce additional dynamics into the control system, and hence can affect the dynamic behavior of the overall control system. For example, the exponential filter of Eq. (23.11) can be shown to be the discrete-time version of a first-order system with a steady-state gain of 1, a time constant related to fJ and flt, and having as input the "noisy" signal y(t). If such a filter is used in a feedback control system, it will be important to take the additional filter dynamics into consideratio n if the filter time constant is comparable to the process time constants.

830


23.3

CONTINUOUS SIGNAL RECONSTRUCTION

The other main consequence of computer control that must be carefully - addressed is that, as illustrated in Figure 23.2(a), control commands generated by the computer are in the form of individual impulses, issued at discrete-time points t(Y t 1, t2, .•• , tk, . . . Keep in mind that such impulses take on nonzero values only at the indicated discrete points in time; in between samples, they are zero. As explained earlier, the objective is clearly not to implement control action in terms of such impulses, but as the continuous function indicated in the dashed line in Figure 23.2{b). Unfortunately, only the sampled versions of the desired continuous function - the indicated "spikes" - are all that are available. The problem is now clearly seen as that of reconstructing a continuous signal, given only the values it takes at certain discrete points in time. Note that this is the reverse of sampling. With sampling, the continuous signal y(t) is available; by choosing an appropriate value forM- the time interval between samples - the sampled version y(tk) is obtained using a sampler. In this case, however, what we have available is the discrete control command signal, u(tk), from which we must now reconstruct a continuous signal, u(t). The first point to note about continuous signal reconstruction is that there is no unique solution to the problem: there are many different ways by which one can make a discrete signal take on a continuous form. Thus depending on what strategy is adopted, several different kinds of "continuous" versions are possible for the same discrete signal.

Holds The basic essence of the signal reconstruction problem is really that of connecting discrete points with some curve to give the otherwise discrete signal a semblance of being continuous. From a purely mathematical point of view, it would appear as if the best way to connect a smooth curve through discrete points is by employing splines or other interpolation device. This approach is, however, not very practical, for two reasons: 1. In practice, the discrete points to be connected by the smooth curve are control commands issued sequentially by the computer. This goes

directly against the principles underlying the application of splines, since they are typically used to connect a "static" collection of points, not a set of points made available sequentially. · 2. Even if the process of connecting points with splines could be made sequential, the effort required to make this possible appears disproportionate to any potential benefits to be realized. A very simple, and intuitively appealing, approach to this problem is to simply hold the value of the discrete signal constant at its previous value over the sampling interval until the next sampled value is available; the control variable then takes on this new value that is also held over the next sampling interval, etc. Mathematically, this means that u(t) and u(tk) are related according to:

CHAP 23 INTRODUCTION TO SAMPLED-DATA SYSTEMS

Figure 23.5.

831

Continuous signal reconstruction using a zero-order hold.

u(t)

{

= tk

u(tk);

t

u(t~c) ;

t" < t < t" +

u(tk +I)

;

t = tk

1

(23.12)

+ I

As shown in Figure 23.5, this leads to the "stair-case" function, which, even

though not an exact representation of the continuous function in dashed lines, is good enough, and very practical Physical devices which make it possible to generate a signal as illustrated above by holding the value of the discrete signal in a certain fashion over the sampling interval until new information is available are referred to as holds. In particular, what is indicated in Eq. (23.12), and illustrated in Figure 23.5, is achieved by a zero-order hold (ZOH), so called because of the "holding" policy of simply maintaining the previous discrete value over the sampling interval. The first-order hold, on the other hand, "holds" at tk by linearly extrapolating between u(tk_ 1) and u(tk) until u(tk + 1) becomes available, i.e.:

(23.13)

A reconstruction of the signal in Figure 23.2 using a first-order hold element is shown in Figure 23.6. Observe that we can extend the foregoing id~as and develop equations for the operation of second, third, and higher order hold elements. However, this

Figure 23.6.

Continuous signal reconstruction using a first-order hold.


832

u(t)

Tme

Figure 23.7.

Apparent signal delay induced by the ZOH element.

has the disadvantage that the implementation of a first-order hold requires two initial starting values from which to begin the linear extrapolation, and in general, the nth-order hold element will require n + 1 initial starting values. Furthermore, it is not always true that the reconstructed signals obtained using higher order holds are better approximations of the actual continuous signal. In fact, there are situations when, for example, the first-order hold will give a poorer approximation than the simple ZOH. Thus, because higher order holds require more starting values, put more computational burden on the computer, and do not offer much improvement over the simple ZOH, most process control applications use the ZOH element. Unless otherwise stated, when we refer to a hold element in our subsequent discussion, we are referring to the ZOH. One final point about the ZOH: because of its strategy of holding the value of the discrete signal constant over the sampling interval, the ZOH generates an apparent signal that lags behind the true signal by approximately 1/2 of the sampling time (see Figure 23.7). This fact is sometimes important in the design of digital feedback controllers, especially when At is large.

23.4

MATHEMATICAL DESCRIPTION OF DISCRETE-TIME SYSTEMS

From our discussions, it should be clear that it is no longer appropriate, under a sampled-data computer control strategy, to represent a process by the continuous differential equation, or transfer function models used·. up to now. An appropriate mathematical representation must take cognizance of the fact that the process gives out, and receives, information at discrete' points in time, no longer continuously. Thus, as introduced in Chapter 4, discrete-time models for sampled-data systems take the form of difference equations (instead of differential equations), pulse transfer functions (instead of Laplace domain transfer functions), and discrete impulse-response models involving summations, instead of integrals. The development of such discrete-time representations is the main concern of these final sections of the chapter.

23.4.1 Discrete-Time Difference Equation Models Discrete-time difference equation models were briefly introduced in Chapter 4; however, here we will generalize the model further to the form: p

.

q

x(k + 1) = fl>x(k) + ~u(k) +):: P;u(k -D)+ rd(k) +):: rjd(k -M) "'d

J=l

(23.14)

CHAP 23 INTRODUCTION TO SAMPLED-DATA SYSTEMS

y(k)

833

= Cx(k)

(23.15)

Here we see that Eqs. (23.14) and (23.15} have terms u(k-D;) that allow delays in the control inputs and terms d ( k - Mi) that allow delays in the disturbances. For example, if there were a time delay a in the inputs, this corresponds to D = a/ t:..t being subtracted from k in the input argument. It would also be possible to include terms for delayed states or delayed measurements, but we will not deal with that case here. These models are either derived from the corresponding continuous model or constructed directly from experimental data (see the next two sections). The premise behind Eqs. (23.14) and (23.15) is that they represent the state and output dynamics at sample times tk = kM when the controls u and disturbances d are applied with a zero-order hold (ZOH). From our discussions in the last section, it is clear that having u enter the process with a ZOH is straightforward. However, the process disturbances, d, are really continuous variables, and approximating them as constant over a sampling interval, t:..t, may not always be appropriate. This point will be discussed in more detail in subsequent chapters. In order to develop better insight into difference equation models, a few common ones are listed in Table 23.1. These forms are explicitly derived below and in the problems at the end of the chapter.

23.4.2 Discrete-Time Impulse-Response Models The discrete-time impulse-response model can be found from the difference equation form by letting u{k) = 3~0 (a unit pulse at k = 0) in Eqs. (23.14) and (23.15). This leads to g(k ), and g4 ~k) which can be convoluted with the inputs u and d respectively, to yield the process outputs from: k

k

I, g(k

y(k)

r= I

- i)u(i) +

I, gik

r= I

- i)d(i)

(23.16)

for the SISO case or: y(k)

=

k

k

_L G(k- i)u(i) + _L Gik- i)d(i)

(23.17)

r= I

r= I

in the multivariable case. The convolution identity: k

k

.L g(k r= I

i)u(i)

.LI g(i)u(k = r=

i)

(23.18)

which is valid for any convolution operation may be used to make the computations more convenient. Remember that these impulse-response models include a zero-order hold. Impulse-response models for some common cases are also shown in Table 23.1.

23.4.3 Discrete-Time Transform-Domain Models The discrete-time analog to the Laplace transform model is known as the pulse transfer function model or the z-transform model. These models are discussed in detail in Chapter 24 so that here we only indicate their form for some common cases in Table 23.1.


834

Table23.1 Some Common Forms of Discrete-Time Models..

Continuous Model

Difference Equation (t = kt.t)

Impulse Response g(k)

z-Transform

K(- 1 (1-,)

K {1- ,)z-1

g(z)

First Order:

'l"dJft) + y(t)= Ku(t)

y(k+ 1) = ~(k) + K {1- ~) u(k)

K

1- ,z-1

t/J=e-MI~

g(s)=-'!"S+l

First Order with Time Delay:

-r ~+y=Ku(t- S) Ke-lis

!l{k+1)= Mk> +K (1- t/l) u(k-m)

K,k-m-1 (1 - ¢)

K (1 - ¢) z-1 - m 1- tflz-l

¢= e-li.l/~

g{s)=-'!"S+l

o=mM

Second Order: + 21;, 4Yill. i'- !&ill 'I" dt dt2 +!J = Ku(t) g(s) =

K

y{k+l)= 2afJyk)- a2y (k -1) + b1 u(k) + b2 u (k-1)

(cf. Appendix C)

(blz-1 + bzz-2) 1 - 2afjz-1 + dlz-2

where

where

i'-s2 + 2('!5 + 1 b1 =11-a (P+

bz=1 £1+a

~r)J

b1 =K[1-a

(~r-P)J

x(P+

~r)J

b2 =K[£1+ a

x(~r-P)] (J}t=~ a=e-

~

z li'ttJ)

I I

Figure 24.10. Interrel ationsh ips betwee n continu ous-tim e and discrete -time process models.

CHAP 24 TOOLS OF DISCRE TE-TIME SYSTEMS ANALYSIS

24.5

SUMMARY

877

This cha pte r has bee n con cerned primarily wit h pre sen ting the z-tr ans fon n and the pulse transfer function as the basic tools of discrete sys tem s analysis. These ma y be der ive d from fam iliar con tinu ous ana log s, bu t the analysis req uir es gre at car e bec aus e the tran siti on from con tin uou s-ti me to dis cre te time is fra ugh t wit h pot ent ial pitf alls wh ich the rea der can avo id by kee pin g the essentials in min d:

1. The z-transform does not exist for a continuous functio n; it exists only for its discretized version.

2. The inv ers e z-t ran sfo rm pro vid es onl y the dis cre te fun ctio n; the original, con tin uou s functio n can not be rec ove red fro m the inv ers e z-tr ans fon n. 3. The app rop ria te pul se tra nsf er fun ctio n, g(z ), for a sam ple d-d ata sys tem must inc lud e a hol d ele me nt alo ng wit h the pro ces s tran sfe r function. 4. Par ticu lar care mu st be taken in obt ain ing g(z); it cannot be obt ain ed directly from gNH(z), the isol ated process pul se transfer fun ction; it mu st be obt ain ed as ind ica ted in Fig ure 24.10, eith er fro m a dif fer enc e equ atio n mo del , or thr oug h g(t), the inv ers e Lap lac e tra nsf orm of g{s)/s. The next logical ste p in our discussions of com put er con trol systems is tha t of em plo yin g the tools of thi s cha pte r to ana lyz e the dyn am ic beh avi or of discrete-time systems; this is the subject of the nex t cha pter. REFERENCES AN D SUGG

ESTED FURTHER RE AD

ING

Detailed treatment of additio nal issues involved wit h dig ital control systems in general available, for example, in the are following books devoted enti rely to this subject matter: 1. Franklin, G. F. and J.D . Powell, Digital Control of Dyn amic Systems, Addison-Wesle Reading, MA (1980) y, 2. Ogata, K, Discrete-Tim e Control Systems, Prentice-Halt Englewood Cliffs, NJ (1981) 3. Astrom, K. J. and B. Wit tenmark, Computer Controlled Systems, (2nd ed.) Prentice-Hall, Englewood Cliffs, NJ (1992)

REVIEW QU ES TIO NS 1.

Wh at is the z-transform?

2.

Wh y is the z-transform par ticularly useful in analyzing the behavior of discrete-tim systems? e

3.

In wh at way is the z-transform

a special case of the Laplace

transform?

L COMPUTER PROCESS CON TRO

878 4.

5.

a cont inuo us one can obta in the z-tra nsfo rm of Why is it incorrect to assu me that function? rms to discretelved in the application of z-transfo Wha t are the broa d concepts invo time control systems analysis?

6.

Wha t are the various means by whic

7.

Wha t is a puls e transfer function?

h z-transform inverses can be obtained

?

nding sam pled continuous process to the correspo In relating the transfer function of a hold element? include the dynamics of a zero-order data version, why is it necessary to tion of a sam pled for obta ining the puls e transfer func 9. Wha t is the actual proc edur e function? fer trans ain dom given its Laplace data process with a ZOH element, to a puls e transfer ertin g a difference equation mod el 10. Wha t is the proc edur e for conv function model? tion process model tion derived from a difference equa 11. Why does the pulse transfer func ? ated rpor inco ent alrea dy occur natu rally with the ZOH elem a puls e transfer ining a difference equation model from 12. Wha t is the proc edur e for obta function model?

8.

PROBLEMS 24.1

sam plin discrete function f(t) gene rated by Establish that the z-tra nsfo nn of the by: n give Mis sample time cosine function f{t) =co s (rot) with a

g the

(P24.1)

24.2

ace dom ain transfer function is give For a system who se continuous Lapl

n as: (P24.2)

g(s)

sam ple time g(z), for the sampled proCess (with show that the puls e transfer function, by: n give is ent, elem ) (ZOH Ill) incorporating a zero-order hold g(z) =

1 + a 1z- 1 + a 2z- 2

where, if we define:

then the indicated coefficients are give

(P24.3)

(P24.4a)

n by: (P24.4b)

MS ANAL YSIS CHAP 24 TOOL S OF DISCR ETE-T IME SYSTE

879 (P24.4c)

and (P24.4d)

(P24.4e)

24.3

contin uous Lapla ce doma in transfer Given the invers e-resp onse proce ss with the function: g(s)

2.0(-3 s + 1) (2s+ I )(5s+ I)

(P24.5)

with time constants in minutes: pulse transf er functi on, g(z), for the (a) Use the result of Proble m 24.2 to obtain a (ZOH) eleme nt, when the sampl e sampl ed proce ss incorp oratin g a zero-o rder hold time is specified as M =1 min. (b) Repea t part (a) forM = 7 min. ed in parts (a) and (b), derive the (c) From the two pulse transf er functi ons obtain the sampl ed proces s opera ting under corres pondi ng difference equati on model s for response obtain ed from each model. step unit the plot the two different sampl e times; systems, comm ent on the nature of In light of what you know about inverse-response perceived behav ior of contin uous the on ing sampl of effect the two respon ses and the processes. 24.4

a CSTR in which a first-o rder By linear izing the first-p rincip les mode l for ximat e mode l for the reactor appro ing result the place, taking exoth ermic reacti on is flowra te is given by the water g tempe rature respo nse to chang es in jacket coolin on: functi er transf in doma e following contin uous Laplac

-0.55 ("F/gpm)

g(s) = (5s- 1)(2s + 1)

(P24.6)

on

corres pondi ng pulse transf er functi The time consta nts are in minut es. Obtai n the sampl e time eleme nt given that the for the sampl ed proce ss incorp oratin g a ZOH Smin. M=O.

24.5

, the transf er functi on mode l for the From an open- loop identi ficatio n exper iment steam rate changes has been obtain ed to se respon in er level in a therm osipho n reboil as:

() ys

=

0.9 (0.3s - I)() s (2. 5s + I) us

(P24.7 )

input and outpu t variab les are, The time const ants are in minut es; and the lized level, define d in terms of norma and rate, steam lized norma respectively, the F5 * (lb /hr), as: and (in), h* nomin al steady -state opera ting conditions,

880


= [u(l) + u(2 )] so that in general:

y(k) ,;.

b[i/-

1u(k -i)]

c

(25.13)

1

RETE-TIME SYST EMS CHAP 25 DYNA MIC ANAL YSIS OF DISC

887

ete first- orcie r proc es;s to any Thus, Eq. (25.13) gives the response of the discr arbit rary inpu t sequence u(k), k =0, 1, 2, ....

Response to an Arbitrary Inpu t Sequence: z-Domain Ana lysis is parti cular ly usef ul for t:his type The convolution prop erty of the z-transform t of Eq. (24.16) that: of analysis. Recall from the convolution resul (24.16)

This imm ediat ely implies that: (25.14)

versi on of the conv oluti on resul t Obse rve now that this inverse z-tra nsfor m tion of the inve rse trans form ation senta prov ides us an immediate, compact repre Eq. (4.22), the conv oluti on result of a prod uct of two z-transforms (compare with of Laplace transforms). pow erful resu lt for open -loop The most impo rtant impl icatio n of this s: follow as d z-dom ain analysis may be state nsfer funct ion mod el: For any discrete process represented by the z-tra (25.15)

y(z) = g(z) u(z)

Let Z'1 {g(z)} = g(k), k = 0, 1, 2, ... , and is imm ediat ely obtai ned as: y(k)

z-1{u(z)} =u(k), k = 0, 1, 2, ... , then, y(k)

.L g(i)u (k- i) = 1=0

There are a few impo rtant point s we must note

(25.16)

here:

funct ion for the proc ess whos e 1. g(k) is the discrete impulse-response when u(k) is the Kronecker z-tra nsfer funct ion is g(z). Obse rve that by: ed delta function O(kllt) defin

fi..kllt) =

Jl; k=O lo; k;e o

(25.17)

nuou s-tim e unit impu lse (i.e., the discrete-time equiv alent of the conti nsform, u(z) = Z[l5(kAt)] is function, 5(t)), it is easy to see that the Z"tra y(k), the respo nse, will fore there 1 in this case; hence, y(z) = g(z), and be exactly g(k). that the process response to any 2. The expre ssion in Eq. (25.16) indicates obtained by the convolution of be can other arbit rary sequence of input s inpu t sequence, u(k). the with g(k) ion funct this impulse-response


88

; it was enco unte red earli er in 3. This expr essio n is, in fact, not new refer red to as the discrete Chap ter 4 and in Chap ter 23, wher e it was fore, that Eq. (25.16) is a there rve, Obse . impulse-response mod el form ely equi valen t to Eq. (25.2) as valid, alternative mod el form, being entir we have now established. lem of finding the response of the Let us now appl y Eq. (25.16) to the current prob . From Eq. (25.2), we observe ence first-order process to an arbitrary inpu t sequ by: n give is that g(z)

(25.18)

1 function, first, let us factor out the zTo obta in the inverse z-transform of this term in the num erato r to obtain:

g(z)

=

C

_b41

z_,)z-I

(25.19)

We now recall that: wing 1. The inverse transform of the terms in

ed brackets is bqf.

nsforms give n in Eq. (24.18), the 2. From the time-shift prop erty .of z-tra 1 ly to intro duce a time delay of mere is effect of the additional z- term one samp le time. Thus, we have:

k < 1

;

(k) =

g

~ 41-

1;

may now use Eq. and for the general inpu t sequence u(k), we desired response as: y(k) == I b4';- 1 u(k- i) I=

(25.20)

k~l

(25.16) to obtain the (25.21)

I

because, from Eq. (25.20); g(k) is zero wher e the inde x mus t now start from 1 ling that u(k) is zero fork < 0, so that until then. By factoring out b, and recal 1) becomes: u(k- t) will vani sh for any i > k, Eq. (25.2 (25.22)

whic h is, of course, identical to Eq. (25.13).

ess 25.1.2 The Gen eral , Linear Discrete Proc process were give ni.e models for the general, linear discrete

n in Chap ter 24 as:

k- n) y(k) + a 1y(k- 1) + a7J( k- 2) + ... + a.;y( 2) +... + bmu (k- m) u(kb + 1) u(k2 b + u(k) b 1 = 0

(24.106)

CHAP 25 DYNAMIC ANALYSIS Of DISCRETE-TIME SYSTEMS

889

in the time domain, and,: (24.110)

in the z-domain. r of these If all we are interest ed in is numeri cal analys is of the behavio med on program easily being e, purpos this for ideal is systems, then Eq. (24.106) u(k), and e, sequenc e discret the of values accept to ter compu a digital the general sequen tially comput e the resulta nt y(k). To gain insight into hat more somew is s analysi the r, howeve , characteristiCs of these systems in. z-doma the in out carried ently conveni

Unit Step Respo nse introdu cing The unit step response of this general, linear system is obtained by 1/(1-z -1) for u(z), giving: (25.23)

domain The z-transform inversion required for conver ting Eq. (25.23) to the time 24.3 of is best carried out by partial fraction expans ion as discuss ed in Section Chapte r 24. , real Assumi ng once again that the denomi nator polyno mial has n distinct on, expansi fraction roots, then recalling Eq. (24.67), we see that by partial Eq. (25.23) becomes: (25.24)

1/(1 _ z-1) where the first term on the RHS arises because of the additio nal ial, and polynom nator denomi the of roots the are n p , ••• , term. As before, p1, p 2 fashion. usual the in ned determi are Rn , ... , R ~ ts the constan 1 Taking inverse z-transforms in Eq. (25.24) now yields: (25.25)

e: It is importa nt to note a few things about this general unit step respons 1. If, for every P;, i == 1, 2, ... , n in Eq. (25.25): (25.26)

then observe that as k ~ oo, all the terms except R0 vanish and y(k) attains a finite, steady-state value, y(oo) == R0 •


890

fy Eq. (25.26), then this port ion of 2. If just one of the roots pi fails to satis indefinitely as k ~"", and the the expression in Eq. (25.25) will grow single term. this of use beca d response will be unbounde of the roots of the denominator 3. It is therefore clear that the natu re tion determines ,the character func fer polynomial of the z-domain trans isely the case with s-do main of the response. Recall that this was prec transfer functions. nomial of the z-tra nsfe r function Being the roots of the deno min ator poly whe n z =pi' i = 1, 2, ... , n; therefore, implies that this polynomial will be zero analysis, these roots pi are know n as as was the case with s-domain dynamic r in Section 25.2 we shall have more to the poles of the z-transfer function. Late of the z-transfer function. say about the poles (and also the zeros)

ce Response to an Arbitrary Inp ut Sequen rete syst em to an arbi trary inpu t The response of the general, linear disc olution result. Since the z-tra nsfe r sequence is best obtained using the conv function for this process is: (25.27)

z-transform inversion (precisely as in by partial fraction expansion followed by Section 24.3.3) we obtain: (25.28)

poles of the z-transfer function. If we where pi' i = 1, 2, ... , n are, once again, the now be obtained as: now use Eq. (25.28) in Eq. (25.16), y(k) may

..

y(k) =

"J: C

0 1

+

-

..

(p 1iu(k -i) +

"J: C

0 2

(p 2)iu( k-i) + ...

.L en (pni u(k- i)

(25.29)

r=O

which may be further simplified to: n

y(k) =

L,

..

u(k- z') .L ci (p/ -

(25.30)

j= I r=O

even in this general case, the ultimate The important point to note here is that again, solely by the natu re of the once behavior of the response is determined, that is consistent with continuous-timepoles of the z-transfer function, a result domain results.

tem s 25.1.3 Extension to Mu ltiv aria ble Sys

to mul tiva riab le syst ems is fairly Exte ndin g the foregoing disc ussi on scalar u(k) and y(k) by appropriately straightforward; we simply replace the

891 S IS OF DISCRETE-TIME SYSTEM CHAP 25 DYNAMIC ANALYS rop riat e the mo del par am ete rs by app ed vectors u(k) and y(k), and

dimension matrices. tim e del ays ) is pro ces s (in the abs enc e of The typ ica l mu ltiv aria ble the equation: modeled in discrete-time by (25.31) 1) y(k) = clJy (k- 1) + Bu (knts Y;(k); i = 1, 2, ..., ing of v1 out put variable eleme ui(k); i = 1, 2, ... , where y(k) is a vector consist nts me sisting of v2 inp ut variable ele v 1, and u(k) is a vector con v2, for a v1 X v2 system. mo del is easily obtained to be: The corresponding z-domain (25.32a) ged to give: This may be further rearran (25.32b)

y(z) = [zl - clJ r'B u(z }

the continuous-time, s-transfer

an expression which mimics ver y well. Example 25.1

function matrix form

A 2 x 2 SYSTEM. DISCRETE-TIME MODEL FOR

Given a two-input, two -ou tput

syst em modeled by the two equ

ations:

(25.33) (25.34)

observe that by defining:

y(k) = [

Yt( k)]

;

Y2(k)

this model becomes:

y(k )

= ~y(k- 1)

+ Bu (k- 1)

as give n in Eq. (25.31).

it is easy to sho w to the laws of matrix algebra, By pay ing careful attention ble sys tem in aria ltiv response of the discrete mu the i and for (see Problem 25.4) tha t the all for 1, = all the v 2 inp uts (i.e., ui(k) Eq. (25.31) to a uni t step in 1, 2, ...)is given by: all discrete-time step s k = 0, (25.35) rely a matrix versio Observe tha t Eq. (25.35) is me

n of Eq. (25.8).


892

It can be shown that provided all the eigenvalues 1 in absolute value, i.e.: IA(~)I

of~

are strictly less than

< 1

(25.36)

then (25.37)

and the output vector attains to the final, steady-state value: (25.38)

The response to the arbitrary vector sequence of inputs, u(k), is easily shown to be: y(k)

= [i~i-l Bu(k- i)J

(25.39)

'"''

which is recognizable as the matrix version of Eq. (25.13).

25.1.4 Effect of Time Delays So far, we are yet to consider formally the dynamic behavior of discrete systems having time delays. Observe, however, that the presence of a time delay merely causes a time shift in the response of the "undelayed" version of the process; thus, deriving the time response of time-delay systems creates no additional problem.

First-Order System with Time Delay When a first-order system has an associated delay of D sampling periods, the model in Eq. (25.1) is modified to read: y(k)

+ ay(k-1)

=

bu(k-D-1)

(25.40)

and the corresponding pulse transfer function model with (rp =-a) is: bz-'(z-0~

y(z) = 1 _ rpz- u(z)

(25.41)

It is important to note that the presence of a time delay will always be announced by the appearance of a z-D term common to all elements in the

numerator of the z-transfer function. This can therefore be factored out, as indicated in Eq. (25.41). , Deriving the dynamic response of this system should proceed as follows: 1. Recast the z-transfer function as g 1(z) z-D, where: g,(z)

bz-• rpz-'

= 1-

the "undelayed" portion of the transfer function.

CHAP 25 DYNAMIC .t'.NALYSIS OF DISCRETE-TIME SYSTEMS

893

2. Let y1(z) = g 1(z) u(z), so that: y(z)

= Y1(z) Z-D

(25.42)

Because g 1(z) has no delays, y1(k) is obtained as before in Eq. (25.8); 3. Now recall Eq. (24.18) for the effect of time shifts on z-transforms; in particular, its reverse, in Eq. (24.119), is used to derive the desired y(k) from y 1(k), using Eq. (25.42): in this specific case, the result is: k m; and we will represent the polynomial-order difference (n- m) by the variable l. This form is sometimes known as the "backward shift" form for the following reason: only powers of z-1 are involved, and since z-1f(z) in the time domain is f(k -1) (which is f(k) shifted backwards in time by one unit) we may then observe that z-1 actually behaves like a unit "backward shift" operator, hence the name.

The "Forward Shift" Form The pulse transfer function in Eq. (25.46) may be converted to a form involving only positive powers of the variable z as follows: By factoring out z-t" in the numerator and r" in the denominator, we obtain: z-m(bozm + blzm-1 + b2zm-2 + ... + bm) z-D g(z) =

z-"(z• + a lzn-1 + a2zn-2 + ... + a,)

(25.47)

and introducing 1= n - m , Eq. (24.47) becomes: z'(bo:z:• + blzm-1 + b2zm-2 + ... + bm) z-D z" + alzn-1 + a2zn-2 + ... +a, g(z) =

(25.48)

If we choose not to confound the z-D term, indicative of the presence of a time delay of D sample times, with the z1 term, indicative of the denominatornumerator polynomial-order excess, then it is possible to write Eq. (25.48) as the following ratio of polynomials involving only z: zl(bozm + blzm-1 + b2zm-2 + ... + bm) g(z)

or

zD(z" + a1z"- 1 + a2z•-2 + ... _ tB'(z) g(z) - zDA'(z)

+a,)

(25.49)

(25.50)

where B'(z) and A'(z) are the numerator and denominator polynomials in z indicated in Eq. (25.49). Since this form is in terms of the variable z (which is the inverse of the unit "backward shift" operator .z-1), it is sometimes referred to as the "forward shift'' form in direct contrast to the "backward shift" form. If we choose to combine the z' and the zD terms, g(z) will have the form: (25.51)

CHAP 25 DYNAMIC ANALYSIS OF DISCRETE-TIME SYSTEMS

895

with N = D-1

(25.52)

We must now note the following very crucial points about the relationship between the two forms available for representing the pulse transfer function: 1. The polynomial equations: (25.53)

found in Eqs. (25.49) and (25.50), and the corresponding (25.54)

found in Eq. (25.46) have identical sets of roots for z, provided z may therefore be considered equivalent 2.

* 0, and

The same is true for the polynomials B(z-1) and B'(z): (25.55)

and 0

(25.56)

have identical sets of roots for z. 3.

In computer process control, the pulse transfer function almost always

occurs either by taking z-transforms of a discrete-time model, or by converting (also via z-transforms) a Laplace transfer function. In either case, since the z-transform naturally occurs in terms of z- 1, such pulse transfer functions will naturally occur in the "backward shift" form. Thus, the "forward shift" form usually arises from a rearrangement of the "backward shift" form. 4. In the event that a pulse transfer function is given in the form shown in Eq. (25.51) where the time delay D and the numerator-denominator polynomial-order excess l are combined, the value of l can be deduced from the respective orders of B '{z) and A '{z), and hence the actual value of D can be determined from Eq. (25.52).

Before going on to study the characteristics of pulse transfer functions, the following example is used to further clarify the preceding discussion. Example 25.2

PULSE TRANSFER FUNCTION FORMS FOR A SECONDORDER SYSTEM WITH A ZOH.

The pulse transfer function for a second-order system with a ZOH element can be shown to be given by: blz-1 + b2z-2 (25.57) g(z) = I + a I z-1 + a2z- 2

''


896

Express g(z) in the form given in Eq. (25.46); identify D and I. Convert Eq. (25.57) to a function of z only by multiplying its numerator and denominator by z'l; interpret the resulting function.

1. 2.

Solution:

1. By observing that there is a common .z-1 factor in the numerator, Eq. (25.57) may be rearranged as:

(25.58)

and now, since the order of the remaining numerator polynomial (in parentheses) is 1, and the order of the denominator polynomial is 2, we conclude that Eq. (25.58) is in the form given in Eq. (25.46) with I = 2- 1 = 1, and D = 1. 2.

Multiplying by z2 as required, we obtain:

g(~)

= z2

b 1z + b 2 + a 1z + a 2

(25.59)

Let us now note that if we had been given Eq. (25.59) directly as the pulse transfer function for the second-order system with a ZOH, a very common mistake is to inspect Eq. (25.59) -which is entirely identical to Eq. (25.58) -and conclude that there is no delay involved in this system. However, since this is a ratio of a first-order and a second-order polynomial, obviously I '" 1 and D = 1.

To summarize, we note that the pulse transfer function may be represented in any of the following entirely equivalent forms: 1. The "Backward Shift" Form

(b 0 + b,z-t + b2z- 2 + ... + bmz-m) z-D 1

2.

+ a 1z- 1 + a 2 z- 2 + ... + a,.z-n

(25.60)

The "Forward Shift" Form

- ....!lliL zNA '(z)

g(z) -

(bozm+b,zm-1 +bzzm-2+ ... +bm)

zN( z" + a1z"- 1 + a2z"- 2 + ... + an)

(25.61)

or, when derived from the "backward shift " form, and the delay term D has not been combined with the numerator-denominator polynomial-order excess I:


897

_ iB'(z) z0 A '(z)

g(z) -

i(bozm + blzm- I + b2zm- 2 + ... + bm) =

zD(z" + a1z"- 1 + a2z"- 2 + ... +a,.)

(25.62)

We are now in a position to study the characteristics of the pulse transfer function.

25.2.2 Fundamental Characteristics By analogy with the Laplace transfer function, the pulse transfer function (expressed either in the "backward shift'' or the "forward shift'' form) is also characterized by its order, time delay (dead time), steady-state gain, poles, and zeros. The order of a pulse transfer function is the same as the order of the denominator polynomial A(z-1) in the "backward shift" form, or A'(z) in the "forward shift" form, i.e., n. As previously noted, the time delay is D. Note that the order and associated time delay are more easily identified in the ''backwm:d shift" form in Eq. (25.60). A physical system is said to be of the order of the transfer function that represents its dynamic behavior. Thus the process in Example 25.2 is a secondorder system.

Steady-State Gain Let S(k) represent the unit step response of the system whose pulse transfer function is g{z). Observe that S(k) will be given by: (25.63)

where .5(z).

l )-1 = g (1z-z

(25.64)

Provided there is a finite, steady-state value, it may be found by the final value theorem, i.e.:

lim S(k) = lim [ ( 1 - z- 1).5(z)] k-+oo

(25.65)

z-+l

and by introducing Eq. (25.64) into Eq. (25.65) we obtain:

lim S(k) k-+oo

= lim

[g(z)]

(25.66)

z-+l

H we now recall that the final steady-state value of a system's unit step

response is identical to its steady-state gain, we have the final result that K,

COMP UTER PROC ESS CONT ROL

898

its pulse transfer function the system's steady-state gain, is obtained from according to: (25.67) K = lim [g(z)] z ->I

s Recall that the corresponding result for the continuou

system is: (25.68)

K = lim [g(s)] s ->0

the pulse transfer function, we Applying Eq. (25.67) to either representation of e, K will be given by: choos we on find that regardless of which representati K=

bo + bl + bz + ... + bm 1 + a 1 + a 2 + ... + an

(24.69)

Poles and Zeros poles and zeros of a pulse We will first give the formal definition of the the "forw ard shift" form in trans fer function; this requi res the use of Ref. [2], p. 44). Thereafter we Eq. (25.61} (cf. for example, Ref. [1], p. 52, and ward shift" form so that "back the of xt will interp ret the results in the conte more natural form. this from d mine deter ately poles and zeros may be accur shown in Eq. (25.61), by Given a pulse transfer function expressed in the form of the denominator roots the ion, analo gy with the Laplace trans fer funct of the nume rator roots the while g(z) of polyn omia l A '(z) are the poles addit ional poles at z = 0 are polyn omia l B'(z) are the zeros of g(z). TheN nmark [1], p. 51). (In actual Witte contributed by the time delay (cf. Astrom and ibutes D poles at z = 0, but contr delay time fact, as indicated in Eq. (25.62), the excess resul ts in I zeros at the deno mina tor-n umer ator polyn omia l order llation, only N = D -1 of cance precisely the same location z = 0. By pole-zero the poles at z = 0 actually survive.) Thus, g{z), an nth-order pulse transfer function, has: deno mina tor polyn omia l • n poles located at the roots of the nth-o rder ' A '(z), rator polyn omia l nume rder • m zeros located at the roots of the mth-o B'(z), and , and a pole-zero excess of • When there is a time delay of D sample times located at z = 0, the n - m = l , there will be N = D -1 additional poles origin of the complex z-plane. 1

forward shift form and A(z-1) However, we had noted earlier that A'(z) in the sets of roots; simil arly, B'(z) and in the back ward shift form have identical implication for poles and zeros of The B(z-1) also have identical sets of roots. tion is now given as follows: g(z) in terms of the backward shift representa of g(z); (a time delay 1 1. Poles : Then roots of A(z- ) = 0 are then poles of the complex origin the 0, = z at poles contr ibute s D addit ional z-pla ne).

.-/.'k,.

899 DISCRETE-TIME SYSTEMS CHAP 25 DYNAMIC ANALYSIS OF er n > m, 1 0 are the m zeros of g(z • i whenev 2. Zeros: The m roots of B(z- ) (If D > 1 0. = z at os m) additional zer there will be an excess of l (n time the by d ute trib con s s are canceled with l of the D pole

= =

these zero dela y).

the following exampleso. Let us illustrate these ideas with ERA L P'LI LSE TRA NSF ER CHA RAC TER IST ICS OF SEV FUN CTI ONS . s and zeros) of order, delay, steady-state gam pole Obtain the main characteristics (i.e., : the following puls e transfer functions

Exa mpl e 25.3

b

(a)

g(z)

= l - az- 1

(b)

g(z;)

=1-

(c)

(d)

(25.70)

bz- 1 az- 1

(25.71)

bz-1-M I - az- 1

(25.72)

g(z)

g(z)

1+

blz- 1 + b'lz-2 a 1z- 1 + a 2z- 2

(25.73)

Solu tion : (25.70), that n =I, observe, first directly from Eq. (a) For this transfer function, we characteristics as ired requ the in obta we now m =0, and D =0. Thu s l = 1 and follows: , ther e is no ZOH function, because n = 1; how ever • This is a first-order transfer • The delay is zero, because D = 0 -a) • By setting z =1 we obtain K = b/(1 = 0 since z =a; the only zero is the one at z at pole one has tion • The transfer func l=la ndm =O 0) by z /z we obtain:

Of course, multiplying in Eq. (25.7

bz g(z) = z- a

(25.74)

to see the poles and at z =a. Thu s it is muc h easi er with a zero at z = 0, and a pole zeros in the forward shift form. erator, the transfer presence of the .z-1 term in the num (b) In this case, recognizing the function is first "rea rran ged" as: g(z;)

(25.75)

a unit delay. er transfer function in part (a) with and we see that this is the first-ord dy-state gain stea 1, = D y dela with function, Thus we have: a first-order transfer use l = 1. beca 0 = z and in principle an excess zero at the unit K =b/(1 -a), a pole at z =a, n: agai non ome phen ion ellat pole-zero canc How ever , we now obse rve the


900

delay contributes an extra pole at z = 0 that cancels out the excess zero located at the same point. In fact, the delay associated with this process is only noticed if the backward shift form Eq. (25.75) is used: multiplying both the numerator and denominator by z irrunediately gives: g(z) = z (~)a

(25.76)

which still shows the pole at z = a but fails to show either the excess zero or the single pole contributed by the delay because they have been canceled out, being opposing terms located at the same point. (c) In this case, the transfer function clearly has the same form as that in part (b) but with an additional delay term; thus it has all the characteristi cs identified with part (b), but now D=M+1. We must again note that one of the M + 1 additional poles contributed at z = 0 by the delay term will be canceled out by the excess zero at the same location, so that if Eq. (25.72) is put in the forward shift form, it will appear as if it has only a delay of M,i.e.: (25.77)

(d) Rearranging the transfer function as follows:

(b 1 + b2z- 1)z- 1 g(z)

= 1 + a 1z

1

+ a 2 z- 2

(25.78) -:.

which we recall from Example 25.2 is a second-orde r process with a ZOH. To verify this, observe that n = 2, m = 1, D = 1; so that I= 1, and the characteristics of Eq. (25.78) in this case are: • A second-order transfer function (n = 2) • A unit delay (D = 1) • A steady-state gain K = (b1 + b2 )I (1 + a1 + ~) • Two poles at the roots of the denominator polynomial, i.e., z = p1, p2, where:

(25.79)

• One zero at z = - (b2 /b 1 ) with an excess zero at z = 0 because I = 1 Observe again that the excess zero gets canceled out by the additional pole at z = 0 contributed by the unit delay. In the forward shift form, Eq. (:2~.73) reads: b 1z+b 2 g(z) = z2 + a,z + "2

(25.80)

in which the proper zero at z =- (b2 /b 1 ), and the two poles are preserved, but the excess zero and the additional pole due to the delay have been canceled out.

Recall from Chapter 23 that we indicated that the presence of a ZOH induces a time delay in the process. This is seen in parts (b), (c), and (d) of Example 25.3 as a factor z-1 •


901

25.2.3 Relation ship with g(s) Let us keep in mind that in computer process control, the process model may often be available first as a Laplace transfer function; the convenience of the z-transform (over the Laplace transform) in analyzin g discrete-time systems is what prompts its replaceme nt with a z-domain pulse transfer function model. Since we are therefore often engaged in converting a Laplace transfer function to its z-domain counterpa rt, it will, of course, be useful to be able to relate g(z) to the g(s) from which it was derived. It turns out, however, that in going through this exercise our ultimate reward is something more fundamen tal: we find that we are, in fact, able to relate the entire complex s-plane - the familiar arena in which continuou s analysis takes place - to the entire complex z-plane, the (yet unfamiliar) correspon ding arena for discrete analysis.

A Fundam entals-p lane

H

z-plane Relation ship

When we sample a continuou s process whose Laplace transfer function is g(s), the Laplace transform of the sampled version, i.e., g•(s), is given by: g*(s)

-

I, g(kM)e...u41 = L{g*(t)} = f=O

'

(25.81)

where g*(t) is the sampled version of the continuou s impulse response g(t). The correspon ding g(z) is obtained, in principle, by taking z-transforms of g*(t); by definition of the z-transform, therefore, we have: g(z) = Z{g*(t)}

-

= I. g(kt:.t)z-1

(25.82)

I"' 0

and by comparin g Eq. (25.82) with Eq. (25.81) we see that the sampled s-transfer function is related to the z-transfer function by the relation:

z or, conversely:

=

esAt

I s = -lnz M

(25.83)

(25.84)

Thus if one wishes to relate a z-transfer function to the Laplace transfer function from which it arose by "sampling at time intervals of At," these results are essential. Eq. (25.83) actually allows us to relate the Laplace transform of any arbitrary discrete function to its z-transform. Thus Eq. (25.83) is, in fact, a fundamen tal relationsh ip between the complex variables sand z that holds regardless of the specific function whose z-transform is being compared to the corresponding Laplace transform. In particular, Eq. (25.83) can be used to relate all possible values of s in the familiar s-plane to the correspon ding values of z in the less familiar territory of the z-plane. This will enable us to develop an analogous set of results by which discrete counterpa rts of continuous processes can be systematically analyzed.

COMPUTER PROCESS CON TROL

902

z-plane

•-plane

(a)

.Imagi nary

Imaginary

-1

Real

Imaginary

1 Real

(b)

Real

Figure 25.2.

z-plane. Corresponding regions in the s-plane and the

jiD

Sro

2 -tary j-" lemen-~ Comp~strip

~tl\\'l~tl.\\'1~ Complementary strip

8

plane

3ID j -2·

Im

z plane

ro

· ...., ~~~·J-2-

Prima ry --l- ---1 .,. (J strip 0 (0 - I \Wl\\ll\ll!l\\1~~ -JT Complementary strip

3ID

~~M~-iT Complementary

. Sro "',_,...s..,tn...;'p;.,.,.,"'""'~ -}2 1

Figure 25.3.

into the unit circle in the Periodic strips in the s-plane that all map z-plane (adapted from Ref. [2]).

E~ IS OF DISCRETE-TIME SYST CHAP 25 DYN AM IC ANA LYS

903

pl an e Relating the s-plane and the z-

In general, any poi nt in the com

plex s-plane can be rep res ent

= a+ jm

s

ed by: (25.85)

Eq. (25.83) we see is the imaginary par t. Fro m whe re CJ is the real par t and m by: x1 give the z-plane will be tha t the corresponding poi nt in (25.86) z e 0 for all co) From Eqs. (25.89) and (25.90), since CJ is positive in this case, we easily obtain that the corresponding set of points in the z-plane are now represented by: lzl

= ecrAI

> 1, for all L.z;

(25.97)

and this is the set of all complex numbers with magnitude greater than 1; this is clearly the set of points lying in the exterior of the unit circle, i.e., the complement of the set represented in Eq. (25.%). Thus:

The entire right-half plane in the s-plane corresponds to the exterior of the unit circle in the z-plane. Primary and complementary strips in the left-half plane Even though we have now completely matched the s-plane to the z-plane, there is one last point to be made: we start by considering a specific situation of special significance involving two points in the s-plane .represented by: (25.98)

and (25.99) Observe that these two points have the same real parts, and that their imaginary parts differ by an integral multiple of 27t/ t:.t, which is recognizable as the sampling frequency (recall that a sampling interval of t:.t corresponds to a sampling frequency co5 = 27t/ t:..t).


905

Let us obtain the points in the z-plane correspon ding to these two s-plane points. Using Eq. (25.92) we have: z1 =

r[cos( m1At) + j sin( m1At)]

(25.100)

and (25.101)

which simplifies to give: (25.102)

From the periodicit y property of the sin and cosine functions , Eq. (25.102) becomes: (25.103)

so that points z 1 and z 2 are identical in the z-plane. We therefore have the following result:

If two points in the s-plane have the same real part, but their

imaginary parts differ by integral multiples of W 6 , the sampling frequency, they map into the same point in the z-plane. The following are some of the important consequences of this result: 1. If the s-plane is divided up along the imaginar y axis into strips of width W5 , starting with the "primary " strip represent ed by a- j (W/2)::; s ::; a+ j (W/2), and then "complem entary" strips of width ±j (2n + l)w5 /2, n = 1, 2, ... ,then the points in each of these strips map into the interior of the unit circle in the z-plane (see Figure 25.3). 2. In mapping the entire left half of the s-plane into the z-plane, only the points within the primary strip are mapped into unique points within the unit circle in the z-plane; any other points outside the primary strip in the s-plane will map to points in the z-plane that are repetition s of these primary points. 3. The relationsh ip between the z-plane and the s-plane is not unique; a given point in the s-plane correspon ds to a single point in the z-plane, but a given point in the z-plane correspon ds to an infinite number of points in the s-plane. There is a very importan t practical reason for the nonuniqu eness in the secondar y strips where the frequency in the s-plane is larger than half the sampling frequency, i.e., wsf2. Recall from our discussion in Chapter 23 that we defined the Nyquist frequency, wN = w.f2, as the largest frequency for which informati on can be extracted from sampled data having sampling frequency Figure 25.3 is the locus of all (1} = 2n/ Lit. Thus the primary strip shown in 5


906

thus less than the Nyq uist frequency and points in the s-plane with frequency All ane. z-pl rete disc the into m properly that defines those points that transfor to be high too y uenc freq a e hav s the poin ts in the complementary strip a in fact may corr upt the sam pled -dat con tain ed in the sampled data and . 23.4 as seen in Figure results by producing aliased data such

and Zeros Ma ppi ng of Transfer Function Poles in the s-plane is related to the z-plane Hav ing examined the issue of how s zero and s pole the n give : issue ific e spec general, we now wish to address a mor g(z)? g ndin s and zeros of the correspo of g(s), wha t can we say about the pole Transfer functions without time delays rs as a that the Laplace transfer function occu In the absence of delays, we recall ratio of two polynomials in s:

(25.104)

N(s)

g(s) = D(s)

s of denominator polynomial are the pole and that the n roots of the nth-order s. zero the are er numerator polynomial g(s), while them roots of the mth-ord ler to decompose Eq. (25.104) into simp ible poss is We also recall that it terms by partial fraction expansion, viz.: (25.105)

By Laplace inversion, we obtain, from where s1, s2, ... , sn are the n poles of g(s). Eq. (25.105) that:

(25.106a)

points ktlt in time, we obtain: By evaluating this function at discrete + Anes.Mt g(kll t) = Ale••k.dl + A21f2k.dl + ...

(25.106b)

(25.106a) becomes: from which the discrete version of Eq.

c*

=

:L g(k&t) C K•,, the root falls outside the unit circle, and the system becomes unstable.

This last observation is quite significant recall that a continuous first-order system under continuous proportional control am never be unstable; we have just shown that ~e same system under digital proportional control can only be stable for controller gain values less than the critical value shown in Eq. (25.207).

Let us therefore note very carefully from this example that It is possible for a conHnuous system that is closed-loop stable under

continuous control to become closed-loop unstable under digital control, even with the same controller parameters. It must also be observed that in the simple example just:considered, not only is it possible for this system to go unstable, the critical controller gain value required for this system's closed-loop stability depends on the sampling time (through ~}. The influence of sampling time on the stability of this system is illustrated in the next example. Example 25.8

EFFECT OF SAMPLING TIME ON THE CLOSED-LOOP STABILITY OF A FIRST-ORDER SYSTEM UNDER PROPORTIONAL DIGITAL CONTROL

Examine the influence of sampling time on the stability properties of the closed-loop digital control system of Example '15.7.

CHAP 25 DYNAMIC ANAI.YSIS OF DISCRETE-TIME SYSTEMS

937

Solution : root of the From the expressi on in Eq. (25.206) for the location of the single K,, the gain r controlle of value fixed a for that observe we , equation characte ristic by: given is -1 critical value of 'for which the root becomes 'critical

=

'
0.2688;

and

0.9255 > 0.5575

indicating that no roots of the characteristic equation lie outside the unit circle, from which we conclude that the system is stable. It is, perhaps instructive to note that P(z) may be factored as follows: P(z)

= (z -

0.8)(z + O.S)(z - 0.1 )(z + 0.8)

(25.216)

from where we see that indeed no root lies outside the unit circle.

As with the Routh test, it is possible to conduct the Jury test with unspecified values for controller parameters. The results of such tests will be the range of values of the controller paramete rs over which the closed-loop system will be stable (see Problem 25.10).

Other Methods It should be noted that other methods exist for studying the stability properties of discrete-time systems. Some of these other methods include:

1. The Root Locus Method (see Ref. [3]) 2. The Nyquist method (see Refs. [1, 3]) 3. Lyapunov's direct method (Ref. [1]). Another obvious choice is to use polynomial root extraction computer programs. When all the paramete rs in the characteristic equation are specified, and we are only interested in the roots, this is perhaps one of the most efficient ways to investigat e closed-loop stability.

RETE-TIM£ SYSTEMS CHAP 25 DYNAMIC ANALYSIS OF DISC

25.5

943

SUMMARY

vior of in som e deta il, the dyn ami c beha In this chap ter, we have expl ored g the usin s, ition cond oop ed-l clos as as well discrete processes und er open-loop pter Cha in d e tran sfer func tion intro duce tools of the z-tra nsfo nn and the puls 24. Lap lace tran sfer func tion , the By ana logy with the con tinu ous the tion wer e disc usse d extensively and characteristics of the pulse transfer func e Thes ed. blish esta tion ain tran sfer func rela tion ship to the cont inuo us dom open the ying stud in ks bloc ding buil as z-do main tran sfer functions wer e used ram diag k bloc pled -dat a proc esse s via and clos ed-l oop beha vior of sam rete in dete rmin ing the stab ility of disc lved invo es issu key analysis; then the systems were discussed. be summa..-ized as follows: The mai n lessons of this chap ter may bly simi lar to thei r cont inuo us-t ime 1. Disc rete syst ems are rem arka been have s tion ifica necessary mod dom ain counterparts so that once the char acte risti cs, nearly all of the ial spec r thei for unt acco mad e to ch are appl icab le in cont inuo uswhi lts concepts, techniques, and resu disc rete -tim e analysis. time analysis carry over dire ctly into arise by the proc ess of sam plin g a 2. How ever , whe n disc rete syst ems have a discrete syst em existing in a cont inuo us process, the fact that we analysis be don e with grea t care , cont inuo us envi ronm ent requires that . as illus trate d by seve ral examples ain resu lts that do not carr y over into 3. The re are certain cont inuo us dom the area of stab ility of clos ed-l oop in the disc rete dom ain, especially e is that whi le som e cont inuo ussyst ems . The mos t imp orta nt of thes ally , can nev er be clos ed-l oop time syst ems exist whi ch, theo retic -data· syst ems beca use of the pled sam unst able , this is not the case for the hold element. unit sam ple time delay intro duce d by red in this chap ter will become even The imp orta nce of the mat eria l cove desi gn we take on the issu es invo lved in the mor e evid ent in the next chap ter as of digi tal controllers. REFERENCES AND SUGGESTED

FUR THE R REA DIN G

, puter Controlled Systems, Pren tice- Hall Astr6m, K. J. and B. Wittenmark. Com Englewood Cliffs, NJ (1984) ) ms, Prentice-Hall, Englewood Cliffs, NJ (1981 2. Ogata, K., Discrete-Time Control Syste Control of Dynamic Systems, Addison-Wesley, al 3. Franklin, G. F. and J.D. Powell, Digit Reading MA (1980) ms, Discrete-Time and Computer Control Syste 4. Cadzow, J. A. and H. R. Martens, ) Prentice-Hall, Englewood Cliffs, NJ (1970

1.

944


REVIEW QUESTIONS 1.

What determines the order of a pulse transfer function?

2.

How are the poles, the zeros, and the steady-state gain of a pulse transfer function determined?

3.

What does the time-delay term contribute to the pole-zero distribution of a pulse transfer function?

4.

What aspect of a pulse transfer function is responsible for contributing additional zeros at the origin?

5.

What is the fundamental relationship between the complex Laplace transform variable s and the z-transform variable?

6.

1he imaginary axis in the complex s-plane maps into what region in the z-plane?

7.

The entire left half of the complex s-plane maps into what region in the z-plane? What does this imply about the region in the z-plane into which the entire right-half plane is mapped?

8.

Why is the relationship between the z-plane and the s-plane not unique?

9.

How are the poles and zeros of a Laplace domain transfer function related to the poles and zeros of the corresponding z-domain transfer function?

10. Why is the number of zeros of a continuous process usually different from the number of zeros of the transfer function of the equivalent discrete process incorporating a ZOH?

11. A discrete process whose pulse transfer function has a zero outside of the unit circle will exhibit what type of dynamic behavior? 12. Why does the process of sampling a continuous inverse-response system not always produce a discrete system that exhibits inverse response? 13. How do the following characteristics of a discrete system's pulse transfer function affect its dynamic behavior? (a) Positive real poles within the unit circle (b) Negative real poles within the unit circle

(c) (d) (e) (f)

(g) (h)

(i)

G)

Real poles on the unit circle Complex conjugate poles within the unit circle Complex conjugate poles outside the unit circle Complex conjugate poles whose real parts lie within the unit circle but whose imaginary parts are multiples of± 1r Complex conjugate poles on the unit circle A pole outside the unit circle A zero outside the unit circle A time delay

14. What distinguishes the block diagram of a sampled-data feedback control system from that of a continuous feedback control system?


945

15. What is the "starred " Laplace transform, and what is it used for? the same as those 16. Which components of the sampled-data feedback control system are common to both are which system; control feedback us ccntinuo nding of the correspo unique to the ely complet are which and but modified for the sampled-data system; sampled-data system? s for sampled-data 17. What is the procedu re for obtaining closed-loop transfer function systems? 18. When is a discrete system stable? p unstable 19. Can a continuous, open-loop stable system give rise to an open-loo system? data

sampled-

us closed-loop controlle r identical with even control, digital under control to become unstable parameters?

continuo 20. Why is it possible for a continuous system that is stable under

techniqu e for 21. What is the basic principl e behind the bilinear transfor mation determining the stability of a discrete system? 22. What is the procedure for applying the Jury stability test?

PROBLEMS 25.1

er system Given the following pulse transfer function model for a first-ord incorporating a ZOH element

y(z)

=

2.0(1- 1/!)z- 1 1 - ,z-1 u(z)

(P25.1)

Obtain the system's unit step response for the following values of 1/J : (a) (b)

(c) {d)

¢= 0.5 ¢=0.8 ¢=-0.5 ¢=1.5

of a firstPlot these responses and comment on the effect of ¢on the step response order system.

u and output y of a 25.2 Each of the following pulse transfer functions relates the input case: each For element. ZOH a ating incorpor process sampled (a) Evaluate the process steady-state gain (b) State the location of the poles and zeros (c) Write the corresponding difference equation process model {d) Obtain the process unit step response; and (e) Identify the process type (e.g., first-order, inverse-response, etc.)


946

(3)

0.5z- 1 g(z) = 1- o.sz- 1 o.sz- 1 g(z) o.sz- 1 g(z) 1 - z- 1

(4)

g(z)

(5)

. g(z)

(6)

g(z)

(7)

g(z)

(8)

g(z)

(1) (2)

25.3

4.5z- 1 - 4z- 2 I- O.Sz 1 6.0z- 1 - 0.375z- 2 I + z- 1 + 1.25z- 2 - 0.43 ~z-t + 0.6093z- 2 I - 1.4252z- 1 + 0.4965z 2 (0.0995z- 1 + 0.0789z- 2)z- 6 1 - 1.4252z- 1 + 0.4965z- 2 0.5z- 1 I - 1.2z- 1

In Chapter 7, it was established that a system composed of two first-Qrder systems in opposition , i.e., whose transfer function is given by: (P25.2)

will exhibit inverse response, for K1 >

~~

when:

(P25.3) (a) Obtain the equivalent discrete-time result by deriving the conditions under which the system whose pulse transfer function (with ZOH element) is given below will exhibit inverse response: btz-1 g(z)

=

b2z-t

1-ptZ-1 -1- P2Z-I

(P25.4)

(b) By relating Eq. (P25.2) to Eq. (P25.4), show that in the limit as the sampling time ll.t tends to zero, the conditions derived in part (a) reduc~ to the conditions in Eq. (P25.3).

25.4

Establish the result given in Eq. (25.35).

25.5

For each of the block diagrams shown in Figure P25.1, determine the open-loop transfer function between y(z) and u(z). In each case:

1 s 5 3s + 1

the unit of time is minutes, and the sample time M =0.5 min.

947

SYSTEMS CHAP 25 DYNAMIC ANALYSIS OF DISCRETE-TIME y"(t)

u*(t)

t(t)

~--~ t( s)

lit

y"(s)

u*(s)

y"(t)

u*(t) t( s)

lit

....

y"(s)

u•(s)

Figure P25.1. 25.6

valve and measu ring device A process whose transfer function model (including as: ed dynamics) has been obtain 1.5 5.0 (P2S.S) y(s) 4.5s + 1 u(s)+ 2.ls +Td(s ) ck controller, including a ZOH 25.11 for this process and Figure in type the of m diagra block a Draw element. ns. relatio n obtain the closed-loop transfer functio

is to be controlled using a digital proportional feedba

l system. Obtain the closed-loop 25.7 Figure P25.2 shows a digital cascade contro en y(z) and d2(z). betwe and ;.z), y and y(z) n betwee transfer functions

Figure P25.2.

A digital cascade control system.

a thermo 25.8 In Problem 24.5, the transfer function model for as: () ~ 0.9(0 .3s-l ) () Y s - s (2.5s + 1) us

siphon reboiler was given

(P24.7 )

with the input and output variables defined as:

"

:::

y

=

(F s- F s *) (lb/hr) F s * (lb/hr) (h - h*) (ins) h* (ins)

-state operating conditions. Here, h• (ins) and F5* (lb/hr) represent nominal steady The time constants are in minutes.

948

COMPUTER PROCESS CONTROL Under digital proportional feedback control incorporating a ZOH element, with a sample rate of Ill = 0.1 min, obtain the dosed-loop pulse transfer function given that Kc = - 0.5. From the closed-loop transfer function, write the corresponding difference equation and obtain the process response to a unit step change in the normalized level set-point. Plot this response.

25.9

A proportional-only controller was designed for a first-order process by direct synthesis, in the continuous-time domain; the controller gain has been specified as: '!" (P25.6)

K'!",

where the K is the process steady-state gain, '!"is the process time constant, and '~"r is the time constant of the desired dosed-loop reference trajectory. (a) In attempting to implement the control system on a digital computer, a young engineer selected '!"/ '~"r = 3.0. A more experienced control engineer recommended that with such a selection, the sampling time, Ill, should not exceed 0.7'!", otherwise the closed-loop system might become unstable. Confirm or refute this recommendation. (b) Because of hardware limitations, it is not possible to sample this process any faster than l!t = '!". What is the maximum value of '!"/ '!", for which the closed-loop system will remain stable if At = '!"? 25.10 A process consisting. of two CSTR's in series is reasonably well modeled by the followi."1g transfer function model: y(s)

= (12.0s+

1.5 1)(5.0s+ l)u(s)

(P25.7)

The time constants are in minutes, and the valve and sensor dynamics may be considered negligible. (a) Show that under proportional feedback control in continuous time, the closedloop system is always stable, for all values of proportional controller gain KC" (b) Under digital proportional feedback control, incorporating a ZOH, find the range of Kc values for which the closed-loop system will remain stable if the sample lime Ill =1 min.

(c) Repeat part (b) for At= 5 min. 2-5.11 An approximate model for the reactor temperature response to changes in jacket cooling water flowrate was given for a certain reactor in Problem 24.4 as: -0.55 ("F/gpm) + 1)

g(s) = (5s- 1)(2s

(P24.6)

The time constants are in minutes. First show that the closed-loop system is stabilized under proportional feedback control in continuous time, with Kc = -4.0. Next, investigate the closed-loop stability of the digital control system employing the same controller (incorporating a ZOH element) with Ill = 0.5 min. 2.5.12 The dosed-loop transfer function for a digital feedback control system has been obtained as:

+ gjz)Kc

(P25.8)

CHAP 25 DYNAMIC ANALYSIS OF DISCRETE-TIME SYSTEMS with Kc = 2.0, and:

949

0.15z- 1 - 0.29z- 2 + 0.32z- 3 - 1 - 1.6z- 1 + 0.5z- 2 - 0.4z 3

,1 ) _

8J'z

(P25.9)

Investigate the stability of the closed-loop system: (a) Using the method of bilinear transformations and the Routh (b) Using the Jury test

array

25.13 For the first-order system with the transfer function:

0.66

g(s) = 6.1s +

(P25.10)

with the following a continuous-time-domain design has yielded a PI controller parameters:

6.7 0.66.il 6.7 rating a ZOH element, If this controller is now implemented in digital form, incorpo tuning parame ter .il the of values ble admissi of range the and with .O.t = 1, find digital PI controller the for function required for closed-loop stability. The transfer is: (P25.11)

a ZOH element, and first25.14 A closed-loop system consists of a proportional controller, function betwee n y(z) transfer pulse op open-lo The system. -delay order-plus-time by: given and u(z) (incorporating the ZOH) is (P25.12)

(a) What is the characteristic equation for this closed-loop system? the maximu m value of (b) For the specific case in which b = 0.09, and~= 0.86, find 0. = m if Kc for which the closed-loop system remains stable of time delays on (c) Repeat part (b) form= 1, and form =2. Comme nt on the effect . stability oop closed-l the range of Kc values required for ced in Problem 8.5, is the 25.15 The process shown below in Figure P25.3, first introdu facility. production end of a certain polymer manufacturing the polymer product The main objective is to control the normalized melt viscosity of s are defined as variable process t pertinen The . F to fy of ratio the 5 g adjustin by values: tate steady-s ate appropri from s follows, in terms of deviation ) u = ratio of the transfer agent flow to inert solvent flow (Fy/F5 ing pipe v = concentration of transfer agent measured at the inlet of the connect inlet reactor the at d measure agent transfer of ration w = concent y = normalized melt viscosity of polymer product at reactor exit


950

Catalyst

Monomer

PIPE PREMIXER

REACTOR

Figure P25.3. The following transfer functions have been given for each portion of the process:

=

k:t..1_ 1 Os + 1

Premixer.

gl(s)

Connecting pipe;

g2(s)

e-6s

Reactor:

g3(s)

_2_5_ 12s + 1

the time unit is in minutes, and all the other units are normalize d units conforming with industry standards for this product. (a) We want to control this process using a digital controller, gc(z), incorporating a ZOH. Draw a block diagram of the type in Figure 25.12 for this process and obtain the closed-loop transfer function. (b) Under proportion al control with L!.t = 3 min, find the maximum value of Kc for which the closed-loop system remains stable. (c) By increasing the sample time to M =6 min. the delay is reduced to one sample time. Repeat part (b) for this choice of the sample time. In terms of the range of Kc values required for closed-loop stability, is there an advantage to increasing the sample time in this manner in order to reduce the delay time?

CHAPTER

26 DE SI GN OF DI GI TA L CO NT RO LL ER S r control ventional analog and digital com pute The major distinction between con by the ed rict conventional con trol ler is rest syst ems is that the con tinu ous , as the h suc s, form on simple, prespecified available hard war e elements to take ed mit unli y uall virt has digi tal con trol ler P, PI, or PID; by con tras t, the . take can rithm trol algo flexibility in the possible forms a con We digital controllers are designed. how with ed cern con is pter cha This a with g alon lf, digi tal con trol ler itse the star t with an intr odu ctio n to the hes: roac app ign des ler digital con trol discussion of the two fund ame ntal rete app roac h invo lves con stru ctin g disc rect indi The ct. dire gn indirect and the desi ous trollers obta ined by con tinu app roxi mat ions to con tinu ous con this er und fall that trollers and the othe rs methodologies; discrete Pill con discrete the direct app roac h, a com plet ely h Wit . first category are disc usse d ctly to dire s lead gn desi ler trol con ess, and repr esen tatio n is used for the proc full take ally techniques in this category usu e discrete controllers. The desi gn quit e efor ther and r by the digital com pute adv anta ge of the latit ude offered the than e ctur stru in ed icat mor e sop hist often resu lt in controllers that are on of cha pter concludes wit h a discussi The ers. troll con l sica discretized clas s. esse proc le tal controllers for multivariab issues involved whe n designing digi

26.1

NS PRELIMINARY CONSIDERATIO

shall book, unle ss othe rwis e stat ed, we As we hav e don e elsewhere in this ng suri final con trol elem ent and the mea con side r that the dyn ami cs of the one into ess with the dyn ami cs of the proc device hav e all bee n com bine d outp ut be designated as g(s). The inpu t and to overall process transfer function vely , ecti resp , y(t) and , desi gna ted as u(t) of this "ov eral l proc ess" shal l be in case the lly usua is This y(s). and sforms u(s) with corr espo ndin g Laplace tran data put a proc ess mod el by inp ut/ out prac tice bec aus e in iden tify ing valve separate out the contributions of the to correlation, it is usu ally difficult Ot::1

·~ 952


~~Jl

dynamics, and those of the measuring device, from the collected input/outp ut data; thus it is precisely this overall transfer function, g(s), that is identified. Under digital feedback control, recall that the continuous input, u(t), received by the overall process is actually a discrete signal from the discrete controller that has been "made continuous " by the hold element; in particular, with a ZOH element, u(t) will be the piecewise constant input signal generated from the discrete sequence u(k), the control command signal from the controller. The controller itself is considered as receiving discrete error information, E(k), the difference between the sampled, desired set-point value, Yd(k), and the sampled process output, y(k). This digital feedback control system is shown in Figure 26.l(a). (The sampler at the input of the ZOH element is not explicitly shown because the block represents a combinatio n sample-and -hold device.) Observe that with the exception of u(t) connecting the hold element and the overall process, every other signal indicated in this diagram is sampled. If we now combine the ZOH element and the process, and refer to the composite transfer function as g~s), i.e.:

I

gn(s)

H(s) g(s)

l _ = (

e-'At) g(s)

s

(26.1)

then y(s)

= gn(s) u*(s)

(26.2)

and by taking "star" transforms, we obtain: y*(s) = gif.s) * u*(s)

and the equivalent

~-transform

(26.3)

representation is easily obtained:

y(z)

= g(z) u(z)

(26.4)

With this maneuver, all the explicitly indicated signals will now be discrete, and Figure 26.1(a) can be represented in the completely equivalent form shown in Figure 26.1(b). Once again note that we use g(z) with no subscripts to denote the discrete-time process having a zero-order hold, because this is the true equivalent of g(s) in control system design. Henceforth in our discussion, a reference to the "process" is to be understood to mean that entity represented by g, the overall process (plus valve and measuring device dynamics) incorporati ng the ZOH element. In a digital feedback control system, the only other component in the loop (as indicated in Figure 26.l{b)} will be the digital controller g, with discrete input e(k) and discrete output u(k). With these ideas in mind, let us now proceed to the task at hand: the design of digital controllers.

953

CHAP 26 DESIGN OF DIGITAL CONTROllERS (<sl

u(k)

u(t)

ZOH

g(l)

u(o)

u*(a)

(a)

y d(k)+ y . 1

Kc(~}

along with:

61 = - K,

c:~ + 1) ;

6;=0 ;i>2

l PID controller: The following points may now be noted about the digita or the velocity form. 1. It is physically realizable in either the positi on and the deriv ative in 2. If the appro ximat ions used for the integr al ple, we had used the exam for Eq. (26.24) had been differ ent (say, ard difference for the backw the and al, integr the trapez oidal rule for have been obtained derivative) completely different expressions would for the digita l PID controller. the positi on form; the 3. The velocity form has certai n advan tages over windu p. reset of most obvious is that it eliminates problems l PID controller may be A discussion of several opera tional aspects of the digita found in Ref. [1], p. 183, and Ref. [2], pp. 616-619.

964

COMPUTER PROCESS CONT ROL

26.3.2 Digi tal PID Controller Tun ing The most comm on appro ach to twtin g digit al PID controllers is to empl oy the same heuri stic rules used for tunin g conti nuou s PID controllers, i.e., the Cohe n and Coon , or the Zieg ler-N ichol s tunin g form ulas. The fund amen tal justi ficat ion is that for smal l samp ling times , the digit al PID contr oller beha ves much like the conti nuou s controller; thus conti nuou s controller settin gs may be conv erted for use with the digit al versi on. How ever, it is impo rtant to reme mber that the inclu sion of the ZOH adds an effective time delay of t:.f/2 to the proce ss time delay. For the situa tions when 6.t is not small, it is neces sary to adjus t the process time delay from a to (a+ t:.f/2) before carry ing out the conti nuou s design; this will ensu re that when the conti nuou s contr oller is discr etize d and impl emen ted in digit al form , the desta biliz ing effect of samp ling will be taken into account. It shou ld be noted that if the proce ss react ion curve is gene rated from the samp led-d ata proce ss befo re using the Cohe n and Coon tunin g meth od, the effect of the samp le-an dhold opera tion will alrea dy be inclu ded in the expe rimen tally gene rated data. Thus , we have that:

In general, digital PID controllers are tuned using the same classical techniques used for tuning continuous PID controllers, but with additional compensation for the effect of sampling time delay. Rega rdles s of the meth od used for tunin g, it is often good practice to have a first evalu ation of the controller perfo rman ce via simu lation when ever possi ble.

26.4

OTHER DIGITAL CONTROLLERS BASED ON CONTINUOUS DOM AIN STRATEGIES

The strat egy of desig ning digit al contr oller s for samp led-d ata syste ms by conv ertin g conti nuou s controllers originally desig ned for conti nuou s syste ms is not limit ed to PID contr oller s alone. In this section, we take a more gene ral appr oach to the subject: we pose the prob lem as that of obtai ning digit al appr oxim ation s to any arbit rary s-tra nsfer funct ion; the appr oxim ation form ulas gene rated there by may then be used for the specific appli catio n of conv ertin g conti nuou s contr oller trans fer functions (obta ined via conti nuou s dom ain designs) to digit al forms for use with samp led-d ata systems. I

26.4.1. Tech niqu es for Digi tal Approximat ion of Con tinu ous Transfer Functions · In Chap ter 25, we estab lishe d that the Lapla ce trans form varia bles (on whic h conti nuou s transfer functions are based ) and the z-transform variable (on whic h discr ete pulse transfer functions are based ) are relate d by: (26.42)

In usin g this to obtai n the discr ete equiv alent of an s-tran sfer function, g(s), it shou ld be recalled that the proc edur e calls for a Laplace inver sion to obtai n g(t), follo wed by discretization, to obtai n g*(t); only after this step can we take

CHAP 26 DESIGN OF DIGITAL CONTROLLERS

965

z-transforms, a process which we have shown to be exactly equivale nt to taking Laplace transfor ms to obtain g*(s), and then converti ng to the z-domai n via Eq. (26.42). Such a procedu re provides exact discrete-time equivalents, but it is clearly very tedious. It turns out, howeve r, that by replacin g Eq. (26.42) with approximations for e-sA1, we obtain relations between s and z that may be used to convert from the s-domai n to the z-domai n by direct substitution. Three of the most common approximants are now presented below: 1. Expansion of e-5~ 1 : A series expansio n for e-sM truncate d after the first two terms gives:

so that, Eq. (26.42) may be approxim ated as:

z- 1 =1-sM which rearranges to give: (26.43)

2. Expansion of 1/e~t: An alternative approxim ation may be obtained from a series expansion for the reciprocal function tfAt instead, viz.: ~ .. I +sM

(26.44)

so that e-slll is approxi mated by the reciproc al of the right-ha nd side of Eq. (26.44), with the result that z-1 =

1 +sM

which rearrang es to give: (26.45)

3. Pade Approximation: A final alternative may be obtained by using a firstorder Parle approxim ation for e-sll.t, i.e.: e-sAI ,

1 - s11tl2 1 + s!J.tfl

e-sAI ,

2-sM 2 + s/1t

(26.46)

2-sM 2 +sM

(26.47)

or

so that: z-1

COMPUTER PROCESS CONTR OL

966

and solving for s in terms of z, we obtain the Tustin approx

imation [3]: (26.48)

las may be obtained A more rigorous justification for these approximating formu differential equati on by considering the problem of numerically integrating the ce appro ach is differen rd backwa repres ented by the s-transfer function; if the d difference forwar the if result; the is (26.43) used, the appro ximat ion in Eq. and if the result; the is (26.45) Eq. in ion ximat appro the appro ach is used, ion in ximat appro the trapezoidal numerical integration appro ach is used, Eq. (26.48) is the result. ts have been The variou s prope rties of each of these appro ximan note that merely We 7). thorou ghly investigated in Ogata (Ref. [3], pp. 308-31 the most also is it te; accura most the Tustin 's approximation (Eq. (26.48)) is (26.43) is Eq. in n imatio approx nce differe ard backw complicated to use. The transfer pulse stable a easy to use; and, for a stable g(s), it will always produ ce function. known as Euler's The forward difference approximation in Eq. (26.45) (also er, it can be shown approximation) is almost as easy to use as Eq. (26.43); howev s-plan e into the the of half that because it maps certain regions of the left replaced by an be might g(s) stable a e, exterior of the unit circle in the z-plan arity with famili some with reader (The on. functi er unstab le pulse transf nown well-k the with ction numer ical metho ds will immediately see the conne ry ordina g solvin of d metho s Euler' stabil ity proble ms associ ated with differential equations.) ard difference Thus it is advisable to use either the Tustin or the backw differe nce ard backw the only appro ximat ion; for simplicity we will use s. follow what in approximation Eq. (26.43)

26.4.2 Trans lating Analog Cont roller Desig ns to Digital Equiv alent s la to translate some We will now use the backward difference conversion formu lents. equiva digital typical continuous controllers into

The PID Controller Given the PID controller, with the transfer function:

), we obtain the by replac ing s accord ing to the expres sion Eq. (26.43 appro ximat e digita l equivalent as: g ( z) = Kc [ 1 c

_ 1- + -,;D -I:J.t ( 1 - z 1) -rl + t:..t 1 - z-1

J

(26.49)

967

LERS CHAP 26 DESIGN OF DIGITAL CONTROL

tran sfer func tion obta ined earli er in whic h is, in fact, iden tical to the a verifiably appr opri ate approximation Eq. (26.34). The fact that Eq. (26.49), the Laplace transfer function shou ld from y to g,(s), coul d be obtained so easil the reasons why this appr oach is very not be lost on the reader; it is one of attra ctiv e.

The Direct Synthesis Controller t ral form ula for gene ratin g the direc Recall that in Cha pter 19, the gene be: to d foun ence trajectory was synthesis controller with a first-order refer

1 1 g (s) = 'f,S g

(19.6)

c

e the process with open -loo p transfer This is the controller whic h will caus ctory represented by a first-order lag function g(s) to follow the closed-loop traje with a time constant 'f,. me in Eq. (26.43), the digi tal Acc ordi ng to the appr oxim ation sche equivalent of this controller is: g (z) c

~ (-1-8 )1 • At t, 1 - z

=

=

1-z-•

(26.50)

/il

particularly simple: we take the process To apply this in a specific situation is lt = (1 - z- 1 ) I At, and inse rt the resu tran sfer func tion , g(s), subs titut e s trates this procedure. in Eq. (26.50). The following example illus Example 26.2

DIRECT SYNTHESIS DIGITAL APPROXIMATION TO THE CESS. PRO DER T-OR CONTROLLER FOR A FIRS

oller the cont inuo us direct synthesis contr Obtain the digital approximation for is given by : ion funct fer trans e whos m syste designed for a first-order g(s)

=

K TS

+ I

Solution: rearrangement Substituting s =(1- z-1) I M gives, upon some

..M L + 6t

T

g(s>l. = (' ·~r) =

{26.5 1)

(_T)z-1

I

1'

+ l'lt

desired controller as: and, by applying Eq. (26.50), we obtain the _

g,(z) -

1'

+

K

M(I- bz-

T,

l

- Z

-I

1)

(26.5 2)


968 where

b

=

The time-domain realization of this controller is easily obtained as:

+lit)

("' )

1' - e(k)- u(k) = u(k-1}+ ( -

K1',

KT,

e(k-1}

(26.53}

which is recognized as a PI controller in the velocity form; it is, of course, also of the general digital controller form in Eq. (26.13).

The Digital Feedforward Controller In Chapter 16, we obtained the result that the controller required for feedforward compensation is given by: gj..s)

gjs)

= - g(s)

where g4(s) is the disturbance transfer function, and g(s) is the process transfer function. By simply substituting Eq. (26.43) for s in this expression, we would obtain gu(z), the digital feedforward controller. In particular, for the purpose of illustration, when g 4(s) and g(s) are assumed to take the first-order-plus-time-delay form, we recall that gff reduces to a lead/lag element, possibly with a time delay, i.e.: (16.20)

Assuming that the indicated time delay is an integral multiple of At, say, r =MAt, then, by observing that the substitution of Eq. (26.43) for ('B + 1) results in something of the form a (1 - bz-1), it is a trivial exercise to establish that the digital lead/lag form for the feedforward controller will be: _ K8 z-M(l - bz- 1) gjz) -

(1 - cz-t)

(26.54)

It is left as an exercise for the reader to ded_uce what the c~tants Kff, a, b, and

c are. Note that even this digital feedforward controller transfer function is a special case of the general form given in Eq. (26.6).

The Discrete Smith Predictor Algorithm The Smith predictor, first introduced in Chapter 17, is one of the earliest model-based control schemes. As we recall from the earlier discussion, with tb.e continuous domain control law given by: (26.55)


969

the Smith predictor improves the closed-loop behavior of processes with transfer function g e-as by eliminating the delay from the closed loop. Here, g, is the classical controller designed for g, the process without the delay. It should be noted that this control law is not easy to implement in continuous time (cf. Ref. [1J, p. 238, and Ref. [4], p. 228). By substituting Eq. (26.43) for s, however, Eq. (26.55) can be converted to a digital control law that is quite easy to implement on the digital computer. To accomplish this, first rearrange Eq. (26.55) to give: u(s)

[1 +g._.g(!- e~]

= gAs)

(26.56)

Let us now consider, for the purpose of illustration, the situation in which the process transfer function is of the first-order-plus-time-delay type, so that g, the transfer function without the delay, is: g(s)

= - K- -rs + I

Further, suppose g, has been designed for g such that: 1 gg, = f:,S

(from the discussion in Chapter 19, we will recall that such a g, will be a PI controller with parameters as indicated in Eq. (19.11)). Under these conditions, Eq. (26.56) reduces to: u(s\ [ 1

,

+

(I -

1 e-as)] = -E(s)

r,s

gT:,s

which rearranges to:

•

(26.57)

If we now substitutes= (1-z-1)/ I::J from Eq. (26.43) and assume the time delay a = Ddt, where D is an integer, the digital approximation to Eq. (26.57) is obtained straightforwardly as: (26.58)

with. time-domain realization:

(26.59) Note once again that this control law is of the general form given in Eq. (26.13) and is very easy to implement on the digital computer.


970

26.5

DISCRETE DIGITAL CONTROLLERS BASED ON DOM AIN STRATEGIES

lating ·analog designs to digital As note d earlier, the problems with trans control of sampled-data systems are equivalents and employing these for the to take the effect of the sample-andfail twofold: (1) pure ly continuous designs the trans latio n proc ess invo lves hold oper ation into cons idera tion; (2) digital controllers may work well if approximations. Thus, these "translated" ling is faster and the approximations the sampling time is short because samp ons will be more accurate; however, tituti involved in the s and z variable subs continuous originals. they are not likely to perform as well as the takin g the direct appr oach , i.e., by lts resu r bette in obta It is possible to discrete framework, either in the in carrying out digital controller design with Some· of the approaches for ain. dom ime the z-domain, or in the discrete-t in this section. carrying out such designs will be discussed

in 26.5.1 Direct Syn thes is in the z-D oma domain results derived in Chapter Rather than approximating the continuous borrow the concepts and use them to 19, as was done in the last section, let us r from first principles. deriv e a digital direct synthesis controlle m show n in Figure 26.1(b) whic h syste rol cont ata Consider the sampled-d een y(z) and yiz) as: has the closed-loop transfer function betw y(z) =

g(z) Sc(Z) y tz) I + g(z) g c(Z) t1'

(26.60)

p behavior (the reference trajectory) If we now represent the desired closed-loo q(z), then by following precisely the by the puls e transfer function y(z) /yiz) = 19.1, we arrive at the conclusion that same arguments as presented in Section to prod uce the desir ed closed-loop the digit al controller which is requ ired response will be given by: q(z) I Kc(z) = g(z) 1 _ q(z)

(26.61)

transfer function q(z) is, of course, The choice of the desir ed closed-loop cont rolle r mus t be phys icall y restr icted by the fact that the resu lting reali zabl e. rical note at this point. One migh t It may be inter estin g to inject a histo el-based controllers that these direct conclude from the current literature on mod the conc ept of usin g refer ence synt hesi s idea s are relat ively new and rse, 1/g( z), mdesigning high · inve el mod trajectories, q(z), and the process lopm ent. In fact, this is a design perf orma nce controllers is a rece nt deve for one can find the curr ent procedure that is m·are than 35 years old, in the 1958 textbook by Ragazzi 1) philosophical discussion and even Eq. (26.6 al computers in the mid-1950s, it is a and Franklin [5]! Given the state of digit nced. won der the controller designs were so adva roller design techniques that fit into cont al digit rent diffe ral There are seve differ only in their choice of they this gene ral direc t synthesis scheme; reference trajectory, q(z).

971

TROLLERS CHAP 26 DESIGN OF DIGITAL CON

Deadbeat Controller for any is base d on the requ irem ent that, The dea dbe at controller (d. Ref. [6]) onse sho uld exhJ."bit: set-point change, the closed-loop resp • Min imu m rise time

• Zer o steady-state offset • Finite settling time

set-point syst em is requ ired to follow the in othe r wor ds, the closed-loop is past, y dela e any peri od of una void able exactly at each sampling poin t onc i.e.: (26.62) r 1y,f..z) y(z)

=

that this bee n incorporated. Thus, observe where the unit sampling delay has the form: translates to requiring that q(z) take q(z)

(26.63)

= z-1

and the resulting deadbeat controll

er will be given by:

z-1

I

gc(z.)

= g(z)

1-

(26.64)

z- I

for q(z) of D sample periods, the expression When the process has a time delay : read to y itional time dela is modified to account for the add (26.65) z-D- 1 rt_z.)

=

and the dea dbe at controller will be: I

Bc(Z)

= g(z)

z-D- 1

1 _ z-D -I

(26.66)

a troller for a first-order system with Thus, for example, the dea dbe at con ZOH will be obtained by using: g(z)

=

in Eq. (26.64), whe re we recall that controller:

K( l- f)C 1 1- ;z.-1

'=

e-At/-r.

This prov ides the dea dbe at

1- fz- 1 ;)( 1 - z- 1) K(l g.(z.)

(26.67)

ion as: (26.67) is obtained in the usu al fash The time-domain realization of Eq. (26.68)

972


whic h, again has the form given in Eq. (26.13 ). A very impo rtant point to note abou t the deadb eat contr oller (and indee d all other contr ollers based on the direc t synth esis schem e) is that it depen ds on the inver se of g(z); thus the zeros of g(z) becom e the contr oller' s poles . This imme diate ly creat es a probl em for proce sses with inver se respo nse (zero outsid e the unit circle) becau se the proce ss zero becom es an unsta ble pole in the contr oller. Thus dead beat contr ollers canno t be used for proce sses with inver se respo nse. An intere sting situat ion arises when the proce ss has a zero locat ed at z = - [3, wher e 0 < f3 < 1, so that the zero, even thoug hg(z) negat ive, lies withi n the unit circle . Unde r these condi tions , this negat ive proce ss zero becom es a nega tive contr oller pole. The contr ibutio n of such a pole to the dyna mic respo nse of the contr oller will be of the type 4 discu ssed in Chap ter 25, and illust rated in Figur e 25.5; conse cutiv e value s of the contr oller outpu t will altern ate in sign, a situat ion referr ed to as "ring ing." The effect of contr oller ringing on the proce ss outpu t y(k) is to produ ce intersample rippling; i.e., oscill ations in the outpu t betw een samp les. This can be illust rated with the follow ing exam ple. Example 26.3

DIGIT AL DEADBEAT CONTROLLER RING ING AND

PROCESS OUTP UT RIPPLING.

Consider the continuous underdamped second-order process: g(.s)

=

1

82

+ .s +

(26.69)

If we convert this to a discrete-time process havin g a zero-order hold and sample time, Ill =0.3, the pulse transfer function is:

g

( ) _ 0.040 52 z- 1 + 0.036 65 z- 2 z - 1- 1.664 z- 1 + 0.740 8 .:- 2

(26.70)

Note that this has poles Pi= 0.8318 ±0.221 i and a zero at Z; =- 0.9045. Design a deadbeat controller and determine the closed -loop response to a unit step set-point change. Solution: Since there are no time delays (other than the unit delay due to the hold), the deadbeat controller requires a reference trajectory given by Eq. (26.63). Employing the design Eq. (26.64), one obtains the controller design : gc(z) =

)(1-

1 ( z- . 1 1 - z-

1.664 z- 1 + 0.740 8 z- 2) 0.040 52 z- 1 + 0.036 65 z- 2

(26.71)

which can be realized by inverting the pulse transfe r function to: u(k)

=

24.68 [0.0039 u(k- 1) + 0.03665 u(k- 2)

+ ff..k)- 1.664 ff..k- l) + 0.7408 ff..k- 2)] (26.72) When this controller is applied to the process, the closed -loop respon se, in terms of the continuous and discrete outputs y, well as the discrete mput u, is shown in Figure 26.3. Note that the controlfer pole as at Z; =- 0.9045 (coming from the process


973 a)

y

0.5

0.0 0

5

6

40 b)

;·

' 20

' '

u(ld 0

'

-20

·, ' '.

. ·-

-·

'

-40

0

Figure 26.3.

.

,·.'

''

3

4

An illustration of controller "ringing" and intersample "rippling."

zero) causes ringing of u(k), and this produces the rippling of the continuous output between samples. Obviously if we did not measure the continuous output and had only the sampled output we would think the controller was outstanding because it caused the sampled process to reach the new set-point in one sample time. This example illustrates one of the major problems with deadbeat controller s and indeed any other controller with very aggressiv e controller action and poles close to -1: the sampled process may follow the discrete-time reference trajectory perfectly and yet the actual process is wildly oscillating. See Refs. (3, 6, 7-9] for more discussion of deadbeat controllers.

Dahlin Algorith m It is possible to improve the performan ce of the deadbeat controller by choosing a less aggressiv e reference trajectory ; this is the purpose of the Dahlin algorithm . With Dahlin' s algorithm , the closed-loo p behavior is desired to be of the first-orde r-plus-tim e-delay type, with the familiar continuou s-time represent ation:

If the delay is an integral multiple of the sampling time (i.e., then this translates to a q(z) given by: q(z) =

(I _

e-IJ.tt~,)

I - e

r= MM, say),

z-M-l

All~, z-l

(26.73)

COM PUT ER PRO CESS CON TRO L

974

with the first -ord er-p lus- time -del ay whic h incl udes the ZOH elem ent be obtained from Eq. (26.61) as: dynamics. The Dahlin controller may now 1 [

Kc(Z)

= g(z)

(1- e·d/IT,) z-M-1

z-M-1 1_ e-611r, z-1 _ ( 1 _ e Atlr,)

J

(26.74)

abou t the Dahlin algorithm: The following are some points to note time delay M, and the time constant 1. It has two tunin g parameters: the is chosen so that the controller is term -r, in the reference trajectory. M delays; -r, determines the spee d time ess realizable whe n there are proc of the closed-loop response. , we have the oppo rtun ity to avoi d 2. It is clear that with this algorithm that is possible with the dead beat n actio the type of excessive control as conservative a controller as controller since we can choose -r, to give the inve rse of g(z) mea ns the we wish; however, the depe nden ce on to cont rolle r ring ing. It is le Dah lin algo rithm is still susc eptib effects of ring ing in the Dah lin som etim es possible to mini mize the othe r tech niqu es for avoi ding algo rithm by choo sing -c, carefully; [8]. ringing may be found, for example, in Ref. ess inverse, any process zeros outside 3. Because the controller uses the proc pole s in the Dah lin controller. the unit circle will become unst able used with proc esse s havi ng Thu s the Dah lin cont rolle r cann ot be inverse response. ible with the Dah lin algo rithm by Let us illus trate the detu ning poss considering the following example. Exam ple 26.4

DAHLIN IMPROVED RESPONSE WITH THE CON TRO LLE R.

secon d-ord er syste m of Exam ple 26.3, Let us again cons ider controlling the time using the Dahlin algorithm.

but this

Solu tion: is only slightly less agres sive than that If we choose a reference traje ctory that es: 0, -r, = 0.3 =At, so that Eq. (26.73) becom used by the dead beat controller, i.e.,

r=

0.6321 z- 1 0.36 79 z- 1 1 q(z) =

(26.75)

then the Dahl in controller is:

21 z- 1 ( 0.63 z- 1 I = Ec(Z)

2 1.66 4 z- 1 + 0.74 08 z- ) 2 z1 665 0.03 + 0.04 052 z-

)(•-

(26.76)

which can be realized as u(k)

=

2) + 0.0039 u(k- I) 15.60 [e(k )- 1.664 E(k- I)+ 0.7408 e(k-

+ 0.03665 u(k- 2)]

(26.77)

975

ITAL CONTROLLERS CHAP 26 DESIGN OF DIG a)

1.0

y 0.5

b)

40

20

··-,

.- ...

' ''

:'

: ·__. :.FLru·-~--

'

u(k) 0

··' -20

-·

--·

second-order process of Dahlin controller for the nge , (b) controller action Closed-loop response of resp cha nt onse to set-poi Example 26.3; (a) out put requiied. lier dea dbe at con trol ler Eq. (26.77) wit h the ear No te tha t by com par ing only a detuned deadbeat is ler > 0, the Dahlin control -r, h wit t tha s see one 72), Eq. (26. (26.77) is shown in controller. the Dahlin controller in Eq. The closed-loop response of rippling, the effects are less le there is still ringing and Figure 26.4. Note that whi However, the response rov output response is imp ed. ous tinu con the and d, nce pronou is still not very good.

Figure 26.4.

trollers to permit them

deadbeat and Dahlin con g. This It is possible to modify the se and · to eliminate ringin

inverse respon to han dle processes with ller does not strive rs so tha t the resulting contro filte ng uni det u and Morari [9] involves add ing irio Zaf r Edgar [8} and late and gel Vo os. zer s ces pro to cancel se dehming filters. We roaches for the design of the app c cifi spe ed pos pro e hav and Edgar controller here. will discuss only the Vogel

Vogel-Edgar Controller Dahlin's algorithm by pro pos ed modifications to would me an Vogel and Ed gar [8] have ics from the controller. This am dyn tor era num del mo the Vogel-Edgar removing the cel the process zeros; thu s can to try not s doe ller tro the con cesses with inv ers e ringing and can han dle pro to t jec sub not is ller tro trajectory so tha t con is add ed to the reference response. Th e modification Eq. (26.73) becomes:


976 "') _ -

tt\Z

(1-

(

1_

e-IJ.ttT,) e-IJ.tiT,

z-•)

B'·-•'] B(l)

[~

z

-M-1

(26.78)

1

I

where

is the numerator polynomial of the process model given in

B(z-1)

Eq. (25.46).

With this modification, the Vogel-Edgar controller will be: (26.79)

Let us illustrate with an example. 1.5

a)

1.0

y

0.5

0.0 -f--,-,-,-,-,--,-,-,-.,....,. --,-T"T".,--,-,-T"T"'"T" ""11-,-I.,-,.-,-IT"T"I.....-.--,or-,1 6 5 4 3 2 1 0 b)

----------------------------..·--------------.

0

Figure 26.5.

1

2

3

4

5

6

Closed-loo(> response of the Vogel-Edgar controller for the second-order process of Example 26.3; (a) output response to set-point change, (b) controller action required.

'

977

CHAP 26 DESIGN OF DIGITAL CONTROLLERS Example 26.5

CLOSED-LOOP RESPONSE WITH THE VOGEL-EDGAR CONTR OLLER .

Let us again control the s..ocond-order system of Example 26.3, but now EdS:U controller.

use the Vogel-

Solution : 1 d in If we modify the reference trajectory by the factor B(z- )/B(l) as indicate Eq. (26.78), then forM = 0, 'fr = 0.3:

0.6321 z- 1 () _ q Z - 1-0.36 79

(0.0405 2 z- 1 + 0.03665 z- 2 ) 0.07717

ZI\

(26.80)

and the Vogel-Edgar controller becomes: 8c(z)

=

8.19- 13.6 z- 1 + 6.07 .z- 2 1 - 0. 700 z- 1 - 0.300 z 2

(26.81)

which can be realized as: u(k)

= 0.700 u(k- 1) + 0.300 u(k- 2)

+ 8.19 £(k) - 13.6 E(k- 1) + 6.07 £(k- 2)

(26.82)

much The closed-loop response to a unit step in set-poin t (Figure 26.5) shows for Thus time. rise delayed slightly only and ringing no with improve d control action real parts, the discrete processes with zeros outside the unit circle, or with negative algorithms. Vogel-Edgar controller is far superior to the original deadbea t or Dahlin

Zafiriou and Morari [9] provide a review of these and other SISO digital controllers.

26.5.2 Time- Doma in Design s discrete-time It is possible to design the discrete controller directly in the

general domain , using the discrete-time difference equatio n model. The model: process procedure involves starting from the y(k) + a 1y(k- 1) + Q.j)(k- 2) + ... + a~(k- n) - = b0 u(k) + b 1u(k- 1) + b2u(k- 2) + ...

+ b,.u(k- m)

given in with known parameters, and entertaining a control law of the type law is control This ters. parame er Eq. (26.13), with yet undetermined controll obtain to d require ters parame er controll the and model the into then introduced of the a prespecified process response derived. For example, if the control law type: u(k)

= u(k-1) +K0 E(k)+K1 E(k-1)

(26.83)

is to be used for a first-order system whose model is given by: y(k)

= ay(k- 1) + bu(k- 1)

(26.84)

L COMPUTER PROCESS CON TRO

978

observe, from Eq. (26.84) tha t whe re a and b are given, first we 2) + bu( k-1 )- bu( k- 2) y(k )- y(k - 1) = ay( k- I) -ay (k-

(26.85)

kwa rds by wit h its time arg ume nt shifted bac and upo n sub stitu ting Eq. {26.83), one, and rearranging, we obtain: (26.86) + bK0 e(k - 1) + bK1 e(k - 2) y(k) = (1 +a) y(k - 1)- ay( k- 2) finally y(k)) for e(k) and rear ran ge to obt ain We may now intr odu ce (yi k)as: ems closed-loop syst the discrete-time equ atio n for the (26.87)

) is constant. whe re we hav e assu med that yik e will be type s of designs here; two of thes eral sev loy Now , we cou ld emp disc usse d in mor e detail.

Direct Syn the sis Design dom ain; for traj ecto ry in the disc rete -tim e Sup pos e we cho se a reference wit h tim eer ord first the transfer function for exa mpl e, by inve rtin g the pul se obtains: dela y response in Eq. (26.73) one (26.88)

term by term pari ng Eq. (26.87) wit h Eq. (26.88) whe re 4Jr = e-411-r,_ Thu s by com des ired firstthe ieve ach to select K0 and K1 for the case whe n M =0 one can this mea ns selecting: ord er response. In the present case

e exactly. so that Eqs. (26.87) and (26.88) agre

Pole Placement Design If we take the z-transform of Eq.

(26.87), one obtains: b(K0 +K 1 )

y(z)

= 1- (1 +a -b K 0 )z- 1 +( a+ hK 1 )z- 2 Yd

(26.89)

roo ts of the tran sfer function who se pole s are indi cati ng a closed-loop pul se cha ract eris tic equ atio n: (26.90) ctly from the atio n can also be obt aine d dire Not e that this characteristic equ y(k - 2) ng the coefficients of y(k), y(k - 1), discrete-time Eq. (26.87) by inspecti

979

S DIGITAL CONTROLLER CHAP 26 DESIGN OF

. pu lse transfer function e ne ed not calculate the to on us ble Th ssi ). po .90 is (26 it . n, Eq to yield tic eq ua tio poles closed-loop characteris to place the closed-loop Ha vin g ob tai ne d the as so K I KCJ 1 rs ete ram the pa er by oll d ntr ne is de ter mi choose the two co closed-loop res po ns e the c mi ce na Sin dy . ed sh sir wi e the de wh ere ve r we place the poles to achiev can is e we , iqu les hn po tec se lar the cu of This pa rti location un de r feedback control. ned pa ram ete rs are determi behavior of the system er oll ntr co the se cau be ral ent ne cem ge A mo re referred to as pole pla in desired locations. les po op er -lo oth sed ere clo ter 10, wh by "p lac ing " the ailable in Ref. [1), Ch ap av is nt me ce pla le po discussion of o discussed. design methods are als

es to Di 26.5.3 Ot he r Approach

gi ta l Co nt ro lle r De sig

n

ous systems can be techniques for continu n sig de er oll ntr co er ital controller design. Several of the oth ectly applicable for dig dir be n tems. ca y the t tha plicable to discrete sys modified so example, is directly ap ital for dig d, for tho me res us du ce loc t The roo main design pro -do cy en qu fre l er tro oth d con optimal Discussion of this an e-domain techniques of tim e Th . [3] f. trix Re in Ma c nd systems may be fou in Refs. [1, 3]. Dynami lei these are discussed oth er model-based are also directly applicab c Control (MAC), an d mi rith go Al l de Mo ), Control (DMC in Ch ap ter 27. thms will be described digital controller algori

26.6

S IABLE CONTROLLER DIGITAL MULTIVAR

n t that with the exceptio le systems, it turns ou ab re ari du ltiv ce mu pro th n wi sig de ng When deali ional controller these systems, any rat classical of the mo st trivial of ble to implement wi th ssi po im lly ua us are t st be tha mu ollers leads to controllers all mu lti va ria ble contr lly tua vir , ore us ref Th the . is ers able control an alo g controll The issue of multivari m. fo r for l ers ita oll dig ntr in co d igi tal " implemente tha t of de sig nin g "d th wi us mo ny no sy alm os t y be obtained multivariable systems. s, digital controllers ma tem sys ble ria -va gle signs carried ou t As wi th the sin ctly by discretizing de ire ind r he eit ms ste sy signs in the discrete for multivariable ly by carrying ou t the de ect dir or in, ma do s ou ce du re of s, z variable in the continu approach, the same pro ct ire ind the dir ec t the ith W domain. ap ter ap pli es; wi th ch s thi in er rli ea sign in the su bs tit uti on pr es en ted on to carry ou t the de mm co re mo en oft is it n mo de l as basis. approach, however, trix difference eq ua tio ma the ng usi in, ma ssed in Refs. [1, 3]), discrete-time do omain techniques (discu e-d tim le, ab ari ltiv mu Some of these practitioners. lar with process control oth er have not been very po pu MAC, an d IMC, on the C, DM as ch su es iqu hn tec ed no tab le Model-based control cesses an d ha ve enjoy pro le ab ari ltiv mu to in designs ba sed ha nd , are applicable the principles involved s; ion cat pli ap al tri us acceptance in ind xt chapter. ll be discussed in the ne on these techniques wi

26.7

SUMMARY

presented tw o digital controller an d the ed uc rod int ve ha pro ac h, in wh ich In this ch ap ter we sign. The indirect ap de its for ies ph so lm by simple va ria ble ap pr oa ch ph ilo ted" into the digital rea sla an "tr are ns sig de continuous

980


substitution, makes it possible to convert to digital form virtually any controller designed by the controller design techniques discussed in Part IVA. By this method, continuous PID controllers, feedforward and cascade controllers, time-delay compensation controllers, direct synthesis controllers, ratio controllers, etc., are all easily digitized, and may be used on the sampleddata system. Despite the convenience, this approach has the disadvantage of requiring special compensation for the effect of sampling if the sample time is large; also, the approximations involved in the digitizing process usually result in some loss of performance. With the direct approach, the design is carried out directly in the discrete domain, either using transfer functions (as with the deadbeat and the Dahlin algorithms) or the difference equation model in discrete time. While these tecbniques implicitly take the effect of sampling into consideration, they are unable to deal directly with the fact that the process is inherently continuous; the result, as was demonstrated with the discrete domain direct synthesis tecbniques, is the possibility of a closed-loop system that achieves the desired behavior at sampling points but that exhibits unacceptable intersample response. It is important to reiterate, in closing, that the real advantage of digital over analog control is that control algorithms that are impossible to implement because of analog hardware limitations are easily implemented on the digital computer. It is this versatility of the digital computer, more than anything else, which makes the application of advanced control schemes possible. REFERENCES AND SUGGESTED FURTifER READING 1. 2.

3. 4. 5. 6. 7.

8. 9.

Astrom, K. J. and B. Wittenmark, Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, NJ (1984) Seborg, D. E., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, J. Wiley, New York (1989) Ogata, K., Discrete-Time Control Systems, Prentice-Hall, Englewood Cliffs, NJ (1981) Deshpande, P. B. and R. H. Ash, Elements of Computer Process Control with Advanced Control Applications, lSA, Research Triangle Park, NC (1981) Ragazzi, J. R. and G. F. Franklin, Sampled-data Control Systems, McGraw-Hill, New York (1958) Isermann, R., Digital Control Systems, Springer-Verlag, Berlin (1981) Kuo, B. C., Analysis and Synthesis of Sampled-data Control Systems, Prentice-Hall, Englewood Cliffs, NJ (1963) Vogel, E. F. and T. F. Edgar, "A New Dead Time Compensator for Digital Control," ISA/80 Proceedings (1980) Zafiriou, E. and M. Morarl. "Digital Controllers for SISO Systems: A Review and a New Algorithm," Int. J. Control, 42, 855 (1985)

REVIEW QUESTIONS 1.

What is a digital controller, and what are some of the characteristics that distinguish it from a conventional analog controller?

2.

What is the general z-domain form of the digital controller, and how is the corresponding time-domain realization obtained?

3.

Why is the concept of "physical realizability" important within the context of digital controller implementation?

981

CHAP 26 DESIGN OF DIGITAL CONTROLLERS physic ally realizable? What does it mean for a digita l controller to be

4.

"indir ect" appro ach to the design of What is the funda menta l philos ophy behin d the philos ophy behin d the "direc t" the digita l contro llers? How is it differ ent from appro ach?

5.

ach to the design of

6.

the "indir ect" appro What are the advan tages and disadvantages of digita l contro llers?

7.

the "direc t" appro ach to What are the advan tages and disadv antage s of llers? contro l digita

8.

g on the prope rties What are the main effects of sampl ing and holdin feedback contro l loop?

9.

y affect the closed-loop How does the sampl e-and- hold device quantitativel a sampl ed-dat a control system?

tage 10. What is the single most overw helmi ng advan analog contro l?

is it 13. What is Tutsin 's appro ximat ion, and what

of a sampl ed-dat a

stability of

of digita l contro l over conve ntiona l

contro 11. What is the positi on form of the discrete PID velocity form? 12. How are digita l PID controllers tuned in genera

the design of

ller? How is it differe nt from the

l?

used for?

ion discus sed in this chapte r, which is 14. Of the two altern atives to Tutsin 's appro ximat more useful? Why? differe nce" appro ximat ion to conve rt a 15. What is requir ed in using the "back ward l form? digita to analog controller transfer function from design by direct synthe sis in discrete time? 16. What is involv ed in carryi ng out controller Chapt er 19 for direct synthe sis design in Is it any differe nt from what was discussed in n? the continuous-time domai contro ller? Why is this not a practical 17. What are the requir ement s for a deadb eat contro ller for most applications? menon 18. What is "ringing," and how does this pheno

arise?

thm 19. What strateg y is emplo yed in Dahlin's algori ller? the deadb eat contro

for impro ving on the perfor mance of

of the 20. What are some of the basic characteristics

Dahli n controller?

's algori thm, and in what way does it 21. What is the Vogel-Edgar modification to Dahlin impro ve the Dahlin controller's performance? almos t synon ymou s with that of design ing 22. Why is the issue of multiv ariabl e contro l "digit al" controllers for multiv ariabl e systems?

L COMPUTER PROCESS CONTRO

982

PROBLEMS 26.1

wing l gas furnace consists of the follo The con!Tol syst em for an indus!Tia to: g acco rdin (a) A digi tal cont rolle r oper atin g u(k)

= O.?Su(k- 1)

8£(k - 1) + 0.25 u(k- 2) + O.Ole(k) - 0.00

elements:

(P26.1)

(b) A ZOH element; (c) A valve, with tran sfer function: gv(s)

75.0 +I = 0 Ols

CFMI . pslg)

(S

(P26.2)

function: (d) A tran sfer line, with transfer e-0.2 •

g,(s)

= O.Ols +

(e) The mai n process, with transfer g•ls) 1'

I (SCFM/SCFM)

(P26.3)

function:

= 0.5s2 ·0+

I

(°F/SCFM)

(P26.4)

The unit of time is seconds. and com bine it sfer func tion g(s) for the proc ess, (a) Obt ain the cons olid ated tran give n that g(z) tion func sfer tran e puls the process with the ZOH elem ent to obta in roller. t.t ::: 0.1 sec. Also obta in g,(z) for the cont obta in the Figu re 26.1(b) for this proc ess; in type the of ram diag k (b) Dra w a bloc ate the stig Inve . tion and the char acte risti c equa clos ed-l oop tran sfer func tion stability of the closed-loop system. er mac hine , u, and the the thick stoc k flow rate to a pap 26.2 The rela tion ship betw een win g tran sfer been app roxi mat ed by the follo basi s wei ght of the pap er, y, has function model:

O.Sse-8•

y(s) ::: 7 .5s + I u(s)

(P26.5)

n tuni ng rule s el, alon g with the Coh en and Coo (a) Use this Lap lace dom ain mod resu lting PI the e retiz Disc r. rolle gt~ a PI cont pres ente d in Cha pter 15, to desi us proc ess inuo cont ent the digi tal cont rolle r on the cont rolle r usin g At = 2. Imp lem the basi s in ge chan step unit a em resp onse to and simu late the clos ed-l oop syst · . weig ht set-point. Plot the response plin g rate At= 4, discretize the sam d ease incr the at ated oper (b) If the procesS is to be e new cond ition s. 4 and repe at part (a) und er 'thes cont inuo us PI cont rolle r usin g At= s. Com pare the closed-loop response y in the proc ess first add At/2 to 'the time dela 4, = At with n atio (c) For the situ cont inuo us PI the in Coh en and Coo n rule s to obta tran sfer func tion befo re usin g the at part (b). repe and 4} = M ng (usi cont rolle r cont rolle r. Disc retiz e this new significant any e controller to that in part (b). Is ther Com pare the perf orm ance of this r? rolle cont last d in obta inin g this adva ntag e to the strat egy emp loye tion of etha nol, y, in an tion mod el relates the top com posi 26.3 The following tran sfer func the feed flow rate, d: and u, rate, flow x to the reflu etha nol/ wat er disti llati on colu mn

983

ITAL CONTROUERS CHAP 26 DESIGN OF DIG y(.r)

=

0.14e-12.0s

0.661!-l.SI

+ 1 d(.r) 6.7 .r + 1 u(.r) + 6.2 .r

The uni t of time h; minute

del s. Observe tha t the mo y(.r)

(P26.6)

is of the form:

= g(s) u(s) + gJ.s) d(.r)

rw ard con tro lle er function for a fee dfo and rec all tha t the transf en by: basis of such model is giv

r des ign ed on the

-gJ .s)

g-' s) 11'

= -g(.r)

e ital form wit h sam ple tim be implemented in dig to is ller the tro m con fro rd z), rwa gj. If such a feedfo nsf er fun ctio n, uired digital controller tra At = 0.5 min, obtain the req ng: usi s, for this proces par tic ula r gj.s) obt ain ed e app rox im atio n enc fer dif rd wa for (a) Th e e app rox im atio n enc fer (b) Th e bac kw ard dif tion (c) Th e Tu tsin approxima transfer function: t-order system wi th the 26. 4 Fo r the gen era l firs (P2 6.7 ) K g(s)

=1:s + 1

uired syn the sis con tro ller req t the con tin uou s direct tha 19 ller. er tro apt con Ch PI in a is wn y it wa s sho erence tra jec tor t-order closed-loop ref ain ed firs obt ied ~ cif wa spe ller a e tro iev con ach sis to syn the ital ver sio n of this dir ect In Ex am ple 26.2, the dig n. tio ima rox app nce ere n for the gen era l using the bac kw ard diff (26.39) - the exp res sio Eq. th wi .5) ula r (26 Eq. ing (a) By com par Ke and -r1 for thi s par tic y obt ain the val ues for Wh r At. lle e, tro tim con ple PID sam te dis cre epe nde nt of the tha t these values are ind controller and confirm t dep end on At? tha rs ete am par troller wi th tha t is it des ira ble to hav e con atio n; com par e the res ult im rox usi ng Tu tsin 's app At? on (b) Repeat Ex am ple 26.2 end dep troller par am ete rs no w sam ple tim es, in Eq. (26.53). Do the con time sys tem s wi th lon g tecre dis for t tha d nde me om the model: rec on n d bee tea ins has It ed {c) if the design is bas ved pro im be can nce ma controller perfor (P2 6.8 ) K 1!-a.r g(s)

=1:s + 1

ller suc h as tha t in 1 in the res ult ing con tro tin g D set n the e un it as pa rt and t, DA wi th a= sam pli ng del ay of 1 tim iva len t to int rod uci ng a pre dic tor equ is ith is .Sm (Th d ). fie .69 odi (26 . "m Eq ref er to thi s as the us t Le .) vel oci ty del (in mo ller s tro of the pro ces l res ult in a PI con suc h an exercise wil l stil wh at t on tha ent ow Sh mm H Co ch. ). roa .53 app wi th the on e in Eq. (26 ller tro con s thi e . par form), and com alternative controller on the par am ete rs of this effect increasing At has firs ttely, con sid er the sim ple of Pro ble m 26.4 concre nts poi in ma the ate 26.5 To illu str er function: ord er pro ces s wi th tra nsf (P2 6.9 ) 0.6 6 6.1 s + 1 g(s)

=

=

984

COMPUTER PROCESS CONTROL (a) Obtain a direct synthesis controller for this process for 1'r = 2.0. Implement this in standard digital form on the continuous process, with a sample time At = 1.0, and obtain the closed-loop system response to a unit step change in the process set-point (b) Implement the same controller on the same process for the same set-point change, but now with sample time At =.4.0. Compare the closed-loop response obtained in this case with that obtained in part (a). Comment on the effect of increased At on the closed-loop system performance. (Recall Examples 25.8 and 25.9 of Chapter 25.) (c) Use the "modified Smith predictor approach" outlined in Problem 26.4(c) to obtain the digital controller for the case when At = 4.0. Implement this controller on the process (with At = 4.0) and obtain the closed-loop response for the same set-point change. Compare the response with that obtained in part (b).

26.6

[This problem is open ended] The following model for a CSTR in which an isothermal second-order reaction is taking place, was first given in Problem 19.12:

(P19.18) The manipulated variable is F, the volumetric flowrate of reactant A into the reactor; the disturbance variable is CAQt the inlet concentration of pure reactant in the feed; and the output variable to be controlled is CA• the reactor concentration of A; V is the reactor volume, assumed constant The nominal process operating conditions are given as:

F

= 7.0 X w-3 m3/s

V

= 7.0 m3/s

/c = 1.5 x I0-3 m3/kg moJo-s CAo = 2.5 kg molelm3 CA = 1.0 kg molefm3 Consider the situation in which the gas chromatograph measurement of CA has a delay of 60 seconds, and that CAO is not available for measurement. (a) Linearize the model around the provided steady-state operating condition, obtain the approximate transfer function model, and use it to design a classical PID controller, using any tuning method of your choice. Carefully justify any design dtoices made. (b) Obtain a discrete version of the controller obtained in part (a); implement it on the original nonlinear system using At = 30 sec, and obtain the closed-loop system response to a changeinCAO from25 to20 kg mole/~ (remember, CAO is unavailable ~~~

.

(c) Use the linearized model to design an IMC controller, justifying any design choices made. Discretize this controller using the approximation technique of your choice,~ At= 30 sec, and repeat part (b). Compare the~ with that of the classical controller you designed and implemented in part (b).

26.7

In Problem 24.5, the transfer function model for a thermosiphon reboiler was given as:

() _ 0.9{0.3s-1)u{) Ys

-

s (2.Ss

+ 1)

with the input and output variables defined as:

s

(P24.7)


985

(Fs- F s *) (lb/hr )J. ' F s * (lb/hr )

u

=[

y

= [ (h-h*

h*) (in)] (in)

steady-state operating conditions. Here, h* (ins), and F5 * (lb/hr ), represent nominal The time constants are in minutes. ter to sample the process outpu t It is desired to control this process using the compu and to take control action every 0.1 min. e trajectory: (a) Use the transfer function model and the referec s q()

=

(l-0. 3s) (1

+ 0.3s)( -r,s + I)

ller. What dynamic system can with -r, = 1.0 min, to design a direct synthesis contro ller? Also, what is the closest contro al unusu hat somew be used to imple ment this n? imatio classical controller that can be w;ed as an approx nce approach, imple ment it differe ard backw the using ller contro (b) Discretize the a unit step change in the to se respon -loop closed on the process, and obtain the normalized level set-point. ior might be "missed" if the sample (c) On the chance that the inverse response behav te version of the direct synthesis discre time is increased toM = 1.0 min, rederive the process (using the new sample the on it ment imple ions, condit these controller under unit step chang e in the same the to se respon time), and obtain the closed-loop the longer sample time does s, normalized level set-point. For this particular proces ? antage disadv a or provid e an advan tage 26.8

Dahli n's digita l contro ller design This probl em is conce rned with comp aring que. techni sis synthe direct teclmique with the continuous Given the first-order-plus-time-delay system: K e-ar

g(s) = - -

(P26.1 0)

-rs + I

uous direct synthesis controller and an appro priate reference trajectory, the contin was given in Chapt er 19, in Eq. (19.24). sion for 8c(z) (Keep in mind that (a) Discretize this controller and obtain an expres MM.) a= where the discre tecou nterpa rtofe- as is~ obtain g(z), the corres pondi ng (b)From the transfer function in Eq. (P26.10), first use this to obtain the Dahli n and ZOH; a g oratin pulse transfer functi on incorp are the Dahlin controller Comp ). controller, emplo ying the q(z) given in Eq. (26.79 sis controller. synthe direct uous contin the of n versio with the discretized 26.9

s whose transfer function model (a) Design a Dahlin controller for the paper proces was given in Problem 26.2 as: 0.55e- 85 (P26.5 ) y(s) = 7 .Ss + 1 u(s)

it expression for the controller. for llt =2.0, r= 8.0, and -r,. =3.0. Write out an explic ented by: repres s proces "true" Implement this controller on the y(s)

=

OAOe-lOs S.Os + 1 u(s)

(P26.1 1)


986

unit step change in the basis weigh t and obtain the closed-lOop system response to a set-point. closed-loop system respon se with that (b) Repeat part (a) with -r,. = 6.0. Comp are the of the param eter -r,. on the controller effect the on obtain ed in part (a). Comm ent performance in the face of plant/ mode l mismatch. 26.10 [This problem is open ended] trial heat excha nger that utilizes Figur e P26.1 is a schematic diagra m for an indus was first introd uced in Proble m it ; chilled brine to cool down a hot process stream oq, remai ns constant, that the (in rature tempe brine the , 14.2. Assum e that T8 the following transfer function that measu ring device has negligible dynamics, and

relations:

-(1 - O.Se- 108}

(T-TI' )("C)

(P26.12a)

= (40s + l)(lSs +I) (Fs- Fs*) (kgls)

(P26.1 2b)

are reason ably accurate. ss using a techn ique of your (a) Desig n a classical PID contro ller for this proce choice. Justify any design choices made. ed in part (a) using backw ard (b) Discre tize the classical PID contro ller obtain ller on the process and obtain contro this ment Imple differences, withA l = 5 seconds. e of +SOC in the process feedst ream the closed-loop system response to a step chang

temperature.

ability , determ ine &om it an (c) Obtai n a proce ss reacti on cubes t of your this appro ximat e mode l to Use l. mode y e-dela appro ximat e first-order-plus-tim choices made. Imple ment design obtain a Dahlin controller for Al = 5 sec. Justify the Comp are the closed (a). part in as e rbanc distru same this contro ller for the

responses.

t to perfo rm better ? Did the (d) Which contro ller would you norma lly expec in. simul ations confirm this expectation? Expla

Chilled Brine

Heat Exchanger Process

Feed

Stream

Chilled Brine Out

Figur e P26.1.

T,F

ITAL CONTROLLERS CHAP 26 DESIGN OF DIG r functio 26.11 Giv en the tran sfe

n for an ope n-l oop uns tab

987

le process as: (P26.13)

- 0.5 5

g(s) = 5.0~

for s inc opo rati ng a ZO H, r function for the proces sfe tran se pul the (z) (a) Ob tain gHp ation. esponding difference equ M 1 min. Write out the cor digital PI controller: the ng usi led trol s is to be con (b) Giv en tha t this proces (P26.14) -1)

=

u(k)

= u(k -l) -1. 2K e(k )+K e(k

atio n int o the dif fer enc e equ rod uce this exp res sio n int the K, r ain ete obt am nt, par sta con free is wit h the set -po int y4 (a), and ass um ing tha t the val ues tha t of ge ran the mo del obt ain ed in par t tain Ob atio n rela ting y(k) to Yd· loo p sys tem to be stable. closed-loop difference equ tak e in ord er for the closed can K r ete am the controller par ble ms 85 and process intr odu ced in Pro designed for the pol ym er ler trol con ital dig A 12 26. 25.15 has bee n giv en as: k) 0.375Su(k- 2) + 0.325e( u(k) = 0.6 25u (k- 1) + (P26.15) 2) (k99e 0.1 + -1) (k - O.S08e en as At for imp lem ent atio n giv wit h the sam ple tim e en by: giv is s ces pro n for the composite transfer functio 25.oe-6 s + 1) ) = (1 Os + 1 )(1 2s

=3.0 min.

Recall tha t the

(P26.16)

g(s

. the ZO H, for At= 3.0 min r function inc orp ora ting sfe tran se pul the ), g(z (a) Ob tain din g difference equation. atio n Write out the cor res pon 5) into the difference equ exp res sio n in Eq. (P26.1 ller the tro nge con rra the rea uce and rod (b) Int y is con sta nt, um e tha t the set -po int 4 equ atio n to this use ; y to ) obt ain ed in par t (a); ass 4 y(k ting difference equ atio n rela exp res sio n to obt ain a . ility stab p in yd; -loo check for closed ent a uni t ste p cha nge atio n in par t (b), imp lem in equ e lied enc imp fer is dif ller the tro ng con (c) Usi pon se. Wh at typ e of res tem sys the t plo sim ula te and

Eq. (1'26.15)?

pa rt V §fE CKA L

CON TR0 1L TOPKCS In Parts I-IV we have presented key concepts and many importan t tools for

understan ding the dynamics, developin g a model, and designing a control system for a process of interest. However, this material is only a fundamen tal base from which to latmch further forays into deeper and more advanced aspects of Process Dynamics, Modeling, and Control. Even after the extensive and detailed discussions presented so far, there remain a significant number of importan t subjects yet to be explored. To sample some of these, we provide in Chapters 27-29 an introduction to a broad spectrum of "special topics" and truly advanced process control concepts which the engineer will likely encounter in practice. To avoid undue excursions d(k - I) + e(k)

(28.28)

t-order autoregressive, or ite noise, is kno wn as a firs wh ian uss Ga is ) e(k ere wh (28.29) AR(l), model, while: = e(k )- ee(k - l) d(k)

l model. For reasons we wil mo vin g average, or MA(l), stic cha sto y nar tio is kno wn as a first-order sta of two mo del s are exa mp les no t go int o her e, the se m walk" model: ndo "ra the : are s del mo ary processes. Two nonstation (28.30) d(k) = d(k -1) + e(k) IM A( l)) model: rder mo vin g average," or and the "in teg rat ed first-o ) - Be( k -1) d(k) = d(k -1) + e(k

(28.31)

(p, D,q ) model, the ser ies mo del is the AR IMA e tim ear lin l era gen st Th e mo , wit h pa s the ord er of the d, Mo vin g Average model AutoRegressive, Int egr ate

1052

SPECIAL CONTROL TOPICS

autoregressiv e part q as the order of the moving average part, and D as the· degree of differencing (or integration). It has the form: (28.32)

where 1/J(z-1) and O(z-1 ) are, respectively, pth-order and qth-order polynomials in the backshift operator z-1, and \1° is the Dth difference operator defined as: V 0 d(k) = d(k) -d(k-D)

The interested reader is referred to Box and Jenkins [1] for further details; for our purposes here, we only note an important result due to MacGregor [14] that as the interval between observations increases, the behavior of an ARIMA(p,1, q) model- for any p or q - approaches that of the IMA{l,1) model shown in Eq. (28.31). Titis is the motivation for restricting our attention to the model in Eq. (28.31).

28.5.3 Minimum Variance Control: General Resultc; The overall model used for stochastic process control is of the form: (y(k) - y 0 ) = P(r 1)r"' u(k -l) + d(k)

(28.33a)

(y(k) - y 0 ) = P(z- 1) u(k- m- I) + d(k)

(28.33b)

or

with appropriate form chosen for P(z-1), and for d(k); and an appropriate value chosen for m, the process delay. The typical objective is that control action, u(k), should be determined to minimize the expected squared deviation of y(k) from its desired target value Yd· Since E[(y- yd) 2] is the theoretical definition of the variance of y (given that its mean value is yd), this objective is commonly referred to as the "minimum variance" control objective. To determine the "minimum variance" control action, without loss of generality, we take the nominal value y 0 around which the model was obtained to be the target value yd, and note from Eq. (28.33) that: (y(k +m +I)-

= P(z- 1)u(k) + d(k + m +I)

yd)

(28.34)

It can now be shown (cf. Box and Jenkins [1), MacGregor [15]) that the control action, u(k), that minimizes E([y(k + m + 1) - yd]2) in Eq. (28.34), is given by: 1

•

u(k) "' P(z-•) [-d(k + m + Ilk)]

(28.35)

where d(k + m + 11 k) is the minimum variance forecast of d(k + m + 1) given information up to the present time k. Such a forecast is obtained using procedures discussed in detail in Box and Jenkins [1], and some illustrative examples will be given below. For now, we simply note that the procedure is

CHAP 28 STATISTICAL PROCESS CONTROL

1053

equivalent to applying a "forecast filter" on the currently available disturbance information, d(k), i.e.: d(k + m +Ilk)

= F(z- 1) d(k)

(28.36)

and that the specific form taken by F(z- 1) is determined by the disturbance model. Whenever an observation y(k) becomes available, we may use Eq. (28.33) to obtain the corresponding value of d(k) as: (28.37a)

where (28.37b)

is the dynamic model prediction of the deterministic part of the observed data y(k). Thus, Eq. (28.35) may now be rewritten with the aid of Eqs. (28.36) and (28.37) as: (28.38)

In diagramatic form, the strategy represented by Eq. (28.38) is as shown in Figure 28.6 (cf. MacGregor [15]), and is structurally the same as the IMC form developed apparently under the strictly deterministic, standard control paradigm in Chapter 19. However, it is important to observe that a crucial part of the IMC strategy is the requirement of an estimate of the unknown disturbance d, to be determined from plant output data and the process model. Whenever unknown information must be deduced from data, there is always a statistical underpinning, whether explicit or not.

28.5.4 Minimum Variance Control: Specific Results For certain specific simple forms of dynamic process and disturbance models, the nature of the resulting minimum variance controller can be very instructive. We shall consider four examples.

d(k) +

Forecast Filter Deterministic Dynamic Model

Figure 28.6.

Block diagram of the minimum variance control strategy.


1054 1. Pure gain process with Gaus sian disturbanc

e.

The opera ting mode l is: (y(k) -

Yd)

= Ku(k -1) + e(k)

(28.39)

and the mini mum variance control is given by: u(k)

(28.40)

I"e(k +Ilk) =-K

of e(k + 1) is obtained as E[e(k + 1)], e(k + 11 k), the mini mum variance forecast a zero mean Gaus sian sequence, i.e., its expe cted value. Now , if e(k) is truly then: (28.41) E{e(k + 1)]

=0

and the mini mum variance control is: u(k)

(28.42)

=0

i.e., no control action shou ld be taken. exam ple are preci sely those Obse rve that the cond ition s impl ied in this the Shew art Char t is base d. This impl ied by the assum ption s upon whic h ation to "do noth ing" when ever justifies the tradi tiona l Shew art recom mend = E[e(k + 1}1 = 0). k) 11 + ~(k the process is "on-t arget " (implied by e(k) takes on a "sign ifican t" error the of value If howe ver, the obse rved that there has been a "shif t" or value , say, b, the inter preta tion is usua lly is assum ed to be constant, then shift distu rbanc e of the same amount; and if this r these circumstances, the Unde 8. , value ero E[e(k + 1)] is now equa l to the nonz mini mum variance control is: (28.43) u(k) = - _!. 8 K e. 2. Pure gain process with IMA (l,l) disturbanc The opera ting mode l in this case is: (y(k )-

with

Yd) = Ku(k -1)

+ d(k)

d(k) = d(k- I) + e(k) - () e(k- 1)

(28.44) (28.31)

mum variance control is obtai ned and e(k) is zero mean Gaussian noise. The mini from: 1 (28.45) u(k) = - K d(k + l I k) A

forecast of d(k + 1), is its expe cted Now , J(k + 11 k), the mini mum varia nce s: from Eq. (28.31): value, E[d(k + 1)], which is comp uted as follow d(k+ l)

= d(k)

+e(k +l)- Be(k)

(28.46)

1055

S CONTROL CHA P 28 STA TIST ICA L PROCES

so that: - 6 e(k) d(k + I I k) = E[d( k + 1)] = d(k)

(28.47)

a kno wn ing bee n observed, is con side red ion for since E[e(k + 1)] = 0 and e(k), hav ress exp the in obta 7) from Eq. (28.46) we quantity. By subtracting Eq. (28.4 r: the "one-step-ahead" forecast erro ) e(k+ l) = d(k +l) -d( k+ llk

from which we obtain:

e(k) = d(k )- d(k I k- 1)

(28.48)

7) yields: Introducing this now into Eq. (28.4 - () )d(k) d(k + Ilk) = 6 d(k lk- l) + (l

(28.49)

as an EWMA er filter on d(k), or equivalently, recognizable eith er as a first-ord 1\ plot does of d(k). nt (in the sense that the EWMA Thus, if d(k +11 k) is not significa ssary; nece is on acti 45) implies that no control be not ,Vigger a signal), then Eq. (28. to trol con e ianc var then the min imu m if d(k +ll k) is sign ific ant, 5). implemented is as given in Eq. (28.4 E GAI N IANCE CONTROLLER FOR PUR Exa mpl e 28.1 MIN IMU M VAR .. CE TURBAN PROCESS WIT H IMA (l,l) DIS process who se model imu m variance controller for the Obt ain an expr essio n for the min has been identified as : (28.50a) = 5.0u (k- I) + d(k) (y(k ) - y

d)

with

d(k)

d(k - I) +e(k ) - 0.6 e(k - I)

(28.50b)

Solu tion : in the requ ired para met ers in Eq. (28.50), we obta From Eq. (28.45) and the mod el as: ession min imu m vari ance controller expr u(k) = - 0.2

The min imu m vari ance forecast

d(k

J (k + Ilk)

requ ired in this specific case is obta

ined according to:

+ Ilk)= 0.6J (k I k - I) + 0.4d(k)

rban ce, d(k), whi ch is an EWM A of the distu d(k)

as: itself obta ined from Eq. (28.50a)

= [y(k) - ydl - 5.0u(k - I)


1056

3. Pure gain process with IMA(l,l) disturbance: (Control action to be implemented at every time instant k). Jn this case we wish to write an explicit expression for the minimum variance controller we have just derived in Eq. (28.45). First, the forecast formula in Eq. (28.49) is rewritten as: (I- f)z-1) d(k + 11 k) = (1- fJ)d(k)

which can be written in the transfer function form: d(k + I I k) :::: F(r1)d(k)

(28.5la)

{1-fJ)

(28.51b)

with 1

-

F(r) - (I - (}z-1)

(Note that both the parameter f) and the form of this "forecast filter" F(z-1) arise naturally from the IMA(l,l) disturbance modeL) The minimwn variance control expression now becomes: u(k) =

1(1-fJ)

-K (1- (}z-1) d(k)

or (1 - 9r 1)u(k) :::: -

1 (1 K

(} )[(y(k) - yd)- Ku(k- 1)]

(28.52)

where we have made use of the model in Eq. (28.44) to write d(k) in terms of known process information. If we now note that Yd- y(k) is E(k), the familiar feedback error, then Eq. (28.52) rearranges to give: u(k)-u(k-1) = (1

~ (} ) E(k)

or u(k) :::: (1 - (})

K

(28.53a)

±

E(i)

(28.53b)

i=l

recognizable as a pure integral control algorithm. Example 28.2

MINIMUM VARIANCE CONTROLLER IMPLEMENTED AT EVERY TIME INSTANT k FOR A PURE GAIN PROCESS WITH IMA(l,l) DISTURBANCE.

Obtain an explicit expression for the minimum variance controller derived for the process given in Example 28.1 if the controller is to be implemented at every time instant k. Solution: Recalling from the model in Eq. (28.50) that the process parameters are K = 5.0 and £J = 0.6, then from Eq. (28.53b), the required controller is the pure integral controller: k

u(k) = 0.08

L £(i) i= I

1057

CHAP 28 STATISTICAL PROCESS CONTROL : 4. FiiSt-order process with IMA( l,l) distur bance

(Control action to be imple mente d at every time instan

t k).

Tne mode l in this case is: (28.54)

with (28.31)

d(k) = d(k-1 ) +e(k )- ee(k -l)

and the minim um variance controller is given by: [ u(k) = -

} - 1r I

~~

z- I

J

(l - (} )

(1 _ (}z-') d(k)

(28.55)

This rearra nges to give:

rearrangement, we obtain: upon using the mode l to replace d(k). After furthe r u(k)- u(k-1 ) =

] (l-fJ ( )[ e(k)- n1€(k-l )

(28.56)

1

ssion is recognizable as where e(k) is the feedback error Ya- y(k). This expre rearra nged to give the r the velocity form of a PI controller; it may be furthe familiar positi on form: u(k) =

)[ (l-fJ E(k) +(I- n 1) ,. ':>!

.I E(i)J

k-1

•= I

(28.57)

a secon d-ord er model, with The reade r shoul d show, as an exercise, that for ller is the digita l PID contro IMA( l,l) distur bance , the minim um varian ce controller. Examp le 28.3

MENTED AT MINIM UM VARI ANCE CONT ROLL ER IMPLE R PROCESS -ORDE FIRST A FOR k EVERY TIME INSTA NT CE. RBAN DISTU ,l) IMA(l WITH

The deterministic model for a process has been obtain (y(k) -

ed as:

yd) = 0.5 (y(k- I) - yd) + 0.8u(k - I)

with the accompanying stochastic disturbance model

given by:

(28.58)

SPECIAL CON TROL TOPICS

1058 d(k)

= d(k -I)

+ e(k) - 0.6 e(k- I)

(28.59)

um variance controller for this process if Obta in an explicit expression for the minim instant k. time the controller is to be implemented at every Solut ion:

param eters are {1 =0.8, n1 = 0.5, and Obse rve that for this process, the mode l controller is the PI controller: 8 =0.6. Thus, from Eq. (28.57), the required u(k)

= 0.5 [ e(k) + 0.5

.L E(i) J

k- 1

(28.60)

ao:l

this stochastic process control We may now note the following with in framework: algorithm is: 1. Implicit in the classical PID control

mic model • A first- or second-order deterministic dyna el mod ce rban distu • An IMA(l,l) stochastic tive • A minimum variance performance objec

t control action, then implicit in 2. If the EWMA char t is used to impl emen such a strategy is: • A pure gain deterministic mod el el • An IMA(l,l) stochastic disturbance mod tive objec ce rman • A minimum variance perfo sponds to: 3. The Shewart Charting para digm corre el • A pure gain deterministic mod rbance model • A Gaussian white noise stochastic distu tive • A minimum variance performance objec

28.6

TECHNIQUES MORE ADVANCED MULTIVARIATE

likely to face a situa tion in which Realistically, the process engineer is more are to be mon itore d and controlled seve ral process and qual ity variables ess control, extending the ideas we simultaneously. As with multivariable proc ate case is fraug ht with man y have pres ente d thus far to the mult ivari mult ivari ate equi vale nt of the ient effic complications. Nevertheless, a very ented in Kresta et al. [8]. It is based Shewart plot has been developed and pres l analysis as Principal Com pone nt on such tools of mult ivari ate stati stica Structures (PLS). A full discussion of Analysis (PCA) and Projection to Latent outside the intended scope of this these multivariate SPC monitoring tools lies ed to consult the cited reference. chapter; the interested reader is encourag techniques to the desig n and The application of these mult ivari ate SPC nce controllers has also been varia mum implementation of multivariable mini industrial polymerization process has carried out. A specific application to an h the interested reader is referred. been reported in Roffel et al. [21], to whic

CHAP 28 STATISTICAL PROCESS CONTROL

28.7

1059

SUMMARY

ess and quality control in the face of The indu stria lly impo rtant issue of proc focus of this chap ter. We have inhe rent varia bilit y has been the main (QC) meth ods - the Shew art, trol revie wed the tradi tiona l Qual ity Con ing the salie nt poin ts and the basic CUSUM, and EWMA charts - high light The role of stan dard proc ess control assu mpti ons unde rlyin g each technique. tical process cont rol (SPC) was also with in the fram ewor k of traditional statis in all situa tions , and neith er is cable appli discu ssed . Trad ition al SPC is not unde r which each appr oach is most stan dard process control; the circumstances appr opria te have been carefully note d. d in this chap ter as a technique in Stochastic process control was pres ente mic mod els are con1bined with whic h appr opri ate dete rmin istic dyna els and appr opri ate perfo rman ce appr opri ate stoch astic distu rban ce mod control decisions. It was show n nal objectives for the purp ose of mak ing ratio sed tradi tiona l char ting meth ods and that the appa rentl y diametrically oppo y inter preta tion with in this unify ing the classical PID cont rolle rs find read ivari ate tech niqu es were brief ly fram ewo rk. High er dime nsio nal, mult the read er inter ested in purs uing this men tione d with references prov ided for rapid ly deve lopin g topic further. REFERENCES AND SUGGESTED FUR

THER REA DING

Analysis: Forecasting and Control, HoldenBox, G. E. P. and G. M. Jenkins, Time Series Day, San Francisco (1976) Statu s and Futur e Needs: The View From 2. Dow ns, J. J. and J. E. Doss, "Pres ent CPC IV, (Ed. Y. Arku n and W. H. Ray), trolCon ss Indus try," in Chemical Proce AIChE, New York (1991) s T. Sastri, "Qua lity Monitoring of Cont inuou 3. English, J. R., M. Krishnamurthi, and ) (1991 251 20, , eering Engin Flow Processes," Computers and Industrial elate d "Stat istica l Proce ss Cont rol for Corr 4. Harr is, T. J. and W. H. Ross, ) (1991 48 Observations," Canadian f. Chern. Eng., 69, , 18, ng Average," Journal of Quality Technology 5. Hunt er, J. S., "The Exponential Movi 203 (1986) ric alent to the Shewart Char t with Western Elect 6. Hunt er, J. S., "A One-point Plot Equiv Rules," Quality Engineering, 2, 13 (1989) Statistical Process Control in Automated 7. Keats, J. B. and N. F. Hube le, Eds. ) Manufacturing, Marcel Dekker, New York (1989 "Multivariate Statistical Monitoring n, Marli E. T. and , 8. Kresta, J. V., J. F. MacGregor . Eng, 69, 35 (1991) Chern f. ian of Process Oper ating Performance," Canad Technometrics, 15, 833 (1973) e," Schem rol Cont sk V-Ma 9. Lucas, J. M., "A Modified Schemes," Journal of Quality rol Cont sk of V-Ma 10. Lucas, J. M., "The Design and Use Technology, 8, 1 (1976) l of USUM Qual ity Cont rol Schemes," fourna 11. Lucas, J. M., "Com bined Shewart-C Quality Technology, 14, 51 (1982) rol Initial Response for CUSUM Quality Cont 12. Lucas, J. M. and R. B. Crosier, "Fast 24, 199 (1982a) ics, ometr Techn ," -start Head a M Schemes: Give Your CUSU A CUSUM," Communications in Statistics, Part 13. Lucas, J. M. and R. B. Crosier, "Robust - Theory and Methods, 11, 2669 (1982b) rol," of Samp ling Inter val for Process Cont 14. MacGregor, J. F., "Opt imal Choice ) Technometrics, 18, 151 (1976 ) Process Control," Chern. Eng. Prog., 21 (1988 15. MacGregor, J. F., "On-line Statistical

1.

SPECIAL CONTROl. TOPICS

1060 16. MacGregor,

J.

F., "A Different View of the Funnel Experiment," Journal of Quality

Technology, 22, 255 (1990)

17. MacGregor, J. F., J. S. Hunter, and T. J. Harris, SPC Interfaces, unpublished course notes (1993) 18. Page, E. S., "Continuous Inspection Schemes," Biometrika, 41,100 (1954) 19. Page, E. 5., "Cumulative Sum Charts," Technometrics, 3,1 (1961) 20. Roberts, S. W., "Control Chart Tests Based on Geometric Moving Averages,"

Technometrics, 1, 239 (1959) 21. Roffe!, J. J., J. F. MacGregor, and T. W. Hoffman, "The Design and Implementatio n of a Multivariable Internal Model Controller for a Continuous Polybutadiene Polymerization Train," Proceedings DYCORD+ 89, IFAC Maastricht, 9-15 (1989) 22. Shewart, W. A., "The Application of Statistics in Maintaining Quality of a · Manufactured Product," JASA, 546 (1925) 23. Shewart, W. A., Economic Control of Quality, Van Nostrand, New York (1931) 24. Wetherill, G. B. and D. W. Brown, Statistical Process Control: Theory and Practice, Chapman and Hall, London (1990)

REVIEW QUESTIONS 1.

What is Statistical Process Control (SPC) in its broadest sense?

2.

Within the context of maintaining a process variable steady in the face of unavoidable, inherent variability, what is the analysis problem and what is the control strategy problem?

3.

What is ''common cause" variation, and how is it different from "special cause" variation?

4.

Why do the traditional Quality Control (QC) methods pay more attention to the analysis of process variability and not so much to control strategy?

5.

Why does standard process control pay more attention to control strategy and not so much to the analysis of process variability?

6.

How is the Gaussian probability density function used in the traditional QC methods?

7.

In hypothesis testing, what is the difference between a Type I error and a Type II error?

B.

What is an a risk and what is a jJ risk?

9.

What is a classical Shewart Chart, and on what premise is it based?

10. When does a "signal" occur in a ShewartChart?

11. What is the recommended plan of action when a "signal" occur in a Shewart Chart?

12. In employing the standard "3-sigma" limits on a Shewart Chart, what is the risk of raising a false alarm? 13. What is the motivation behind augmenting the Shewart Chart with the Western Electric rules? 14. What is the average run length, (ARL), and how does this concept help in assessing the usefulness of a QC method?

1061

CHAP 28 STATISTICAL PROCESS CONTROL 15. What process variable is plotted on a CUSUM chart?

chart is likely to be more 16. Other things being equal, why do you think the CUSUM shifts? process smaller up picking in Chart t Shewar the than sensitive how 17. Theoretically, on what test is the CUSUM method based, and

does the test work?

data, how does it stand in 18. What is the EWMA, and in terms of how it uses process s? method CUSUM and t relation to the Shewar and the EWMA parameter, 19. Given the inheren t standar d deviation of raw process data chart? EWMA how are the "3-sigma" limits set for the ent to the Shewar t Chart 20. When does the EWMA chart become virtuall y equival incorporating the Western Electric rules? ility of traditional QC 21. What is serial correlation and how does it affect the applicab methods? and the "stocha stic" 22. What are the typical characteristics of the "determ inistic" ment? measure variable portions of a typical industrial process y aim" of standar d 23. As summa rized by Downs and Doss [2], what is the "primar process control? es should be investigated in 24. The status of what three basic process and control attn'but situation? given a in le deciding whethe r or not traditional SPC is applicab 25. When is traditional SPC appropriate? 26. When is standar d process contrbl appropriate? , and what are the three 27. What is the basic premise of Stochastic Process Control m? paradig this for essential elements required stochastic process control 28. What models are typical used for the deterministic part of models? stochastic process control? 29. What models are typical used for the disturbance models in control? 30. What is the performance objective of "minim um variance" represe ntation of the 31. What are the primary elemen ts in the block diagram matic the primary elements of minimu m variance controller, and how do they compare with the IMC strategy discussed in Chapter 19? "perform 32. Within the context of stochastic process control what is the ? method t the Shewar

ance objective" of

er become 33. Under what condition does the minimum variance controll ? method Shewart

equivalent to the

er become 34. Under what condition does the minimum variance controll EWMA method?

equivalent to the

er become 35. Under what condition does the minimu m variance controll er? controll classical PI

equivalent to the

CHAPT1EJR

29 SELECTED TOPICS IN

NT RO L AD VA NC ED PROCESS CO trol and cha pter s to Model Predictive Con y of Even after dev otin g ind ivid ual arra t vas a still is e ther in Par t V, d. Statistical Process Control her e usse disc trol topics that we hav e not yet d ippe imp orta nt adv anc ed process con equ are s, s, and many existing processe data However, almost all new processe er put com ital dig ems (DCS) so that As a wit h Distributed Computer Syst par t of the plan t env iron men t. gral inte an are d logg ing and control ente lem trol methods are now bein g imp e thes consequence, Advanced Process Con ng igni des rs stries. While the enginee , .'ling broadly throughout the process indu trai. ial spec and rees hav e advanced deg ts advanced control systems usually cep con c basi the and erst und t rations mus all engineers involved in plan t ope on and chapter, we will introduce the jarg this in s Thu s. hod met trol beh ind these con cess pro h some of the adv anc ed qua lita tive idea s associated wit plants. methods being installed in today's the use advanced process control involve of s hod met the Essentially all of er to ord in (e.g., a detailed process model) dge of quantitative process knowledge wle kno ess lem. For example, this proc the solve a difficult process control prob with l dea to ts, men of on-line measure to , can be use d to overcome the lack time h wit nge cha tics eris ami c charact mal pro blem of processes whose dyn opti w esses with spatial profiles, to allo provide control systems for proc of failure ntain safe process operation in case economic performance, and to mai each of In or). sens e, valv system (e.g., pump, trol of a key component of the control con ed anc adv t eren diff a ribe shall desc a the sections of this chapter, we with e clud it is designed to solve. We con met hod and the special problem . process control technologies brief discussion of some emerging

29.1

OF GO OD ON-LINE CONTROL IN THE ABSENCE ESTIMATION ME AS UR EM EN TS - STATE

key process process indu stry that there are It is very often the case in the It is eve n s. rval inte d on-line at frequent time basis. variables that can not be measure tine rou a on le ilab ava at all are possible that no mea sure men ts

1064


Sometimes these measurements can be taken only very infrequently (as in the case of on-line gas chromatography with long elution times) or perhaps samples must be sent to the lab for analysis. In both these cases the measurement result is known only after a significant delay. For these very common situations, the process time constants may be short relative to the time between measurements, and in addition, the measurement result is delayed, indicating the state of the process .some time in the past. Hence, direct feedback control based on these measurements is impractical. In such situations, a State Estimator may be designed to estimate the values of these process variables between measurements. This state estimator is used to "observe" the values of unmeasured variables and is often called an "observer." Another type of problem that arises less frequently in process control is a situation where the process variable is measured on-line, but there is a high level of noise in the measurement signal. If the frequency of the noise is widely separated from the frequency of the process measurement itself (as in the case of 60 cycle noise superimposed on slowly varying temperature measurements}, then simple filters such as described in Chapter 2 may be used. However, if there is significant overlap of frequencies in the noise signal and in the process measurement, then more sophisticated model-based filtering may be used as part of the state estimator. The most common of these filters is based on the work of Kalman in 1960 [cf. Refs. 1-3] and is termed a Kalman Filter.

29.1.1 State Estimator Structure The general structure of a state estimator is shown in Figure 29.1. components of the estimator are

The

1. A dynamic model for the states of the system x having dimension n:

dx

dt = f(x, u, a) + ~(t)

(29.1)

where ~(t) is some model error. The model depends on a knowledge of the process inputs u and model parameters a. 2. Initial conditions: x(O) = x 0

+

~O

(29.2)

where ~ represents initial condition error. 3. Measurement devices producing an l vector of signals y: y = h(x, u, j3) + 11

(29.3)

where 11 represents measurement error. Usually there are fewer measurements than states (l < n). Here the model for the measurement devices h(x, u, j3,) depends on a set of parameters j3. The actual data y and the model h(x, u, j3) differ by the error term 11·

1065

CHAP 29 SELECTED TOPICS IN ADVANCE D PROCESS CONTROL

When on-line measurem ents are available continuou sly (as in the case of a temperatu re, level, or pressure measurem ent), these componen ts may combine into a state estimator of the form:

dx dt

f(x, u, a) + K(t)[y(t) - h(x, u, ~)]

(29.4) (29.5)

x

where K(t) is ann xI dimension al correction gain matrix and denotes an estimate of the true state x. Note that the last term in the state estimator equation provides a correction to the process model based on the difference between the actual measurem ent, y(t), and the value of the measurem ent signal predicted by the model, h(x, u, ~)- Figure 29.2(a) illustrates a typical state estimator trajectory in the case when y(t) is continuou s and is the measured value of x(t). When some of the measurem ents are available only at discrete time intervals tl' t 2, t 3, ••• , tk and may arrive with some significant delay, then the state estimator consists of two parts. The first part has the form: (29.6)

State Model

:k =f(:r.:,u,o:) + ~(t)

State Estimator

Initial Condition

i

x(O) =Xo +~

= f(i,u,o:) + K(t) [)(t)- h(i,u,j3))

Estimate

x(O} = Xo

Output data y =h(:r.:,u,j3) + 11

Known:

To be estimated optimally:

1. Model with uncertain

1. Process states x(t).

initial condition. 2. Mean and covariance of model errors (~(t), ~ ). 3. Mean and covariance of output errors (ll(t)). 4. Inputs, u(t), and outputs, y(t). Figure 29.1.

The structure of a state estimator.

i{t)


1066

0

(a) Continuous data

~(t)

tl

ts

ts

(b)

Figure 29.2.

t,

-tDiscrete

ts

te

data

s data, and (b) discrete

nuou Typical state estimator trajectories for (a) conti data.

een discrete samples tk-l < t < tk. Here and prov ides estimates for the time betw are continuous. The second part of the y 1(t) represents those measurements that sure men t resul ts beco mes know n at estim ator is activ ated whe n discr ete mea according to: t =tk to prov ide an instantaneous correction (29.7) u, ~>J + ~- h ==

x(tt> x

2(x,

ents available at t =tr The correction Here y1 (t.t} deno tes discrete meas urem rding to the type of estim ator one gain s K 1(t), K 2(tk) may be defin ed acco state estim ation trajectory for the case employs. Figure 29.2(b) show s a typical discrete data whe n y(tJ is a measured of no continuous meas urem ents and only valu e of x(t).

29.1.2 Applications of the State Estimator are two princ ipal functions that can be As indic ated in the last section, there perfo rmed by a state estimator:

s, and 1. Observation of unm easu red process state measurements. noisy from s state ess proc the of 2. Extraction ervability cond ition s" be satisfied in The first task requ ires that certa in "obs ble of observation. For example, if we orde r for the unm easu red state to be capa measure only the temp eratu re history have a closed pres sure vessel, and we can estim ate the pres sure histo ry inside of the gas insid e the vessel, can we also use of an equa tion of state such as: the vessel? This ough t to be possible beca

PV = nRT

(29.8)

1067

ANCED PROCESS CONTROL CHAP 29 SELECTED TOPICS IN ADV

fixed, and quan tity of gas in the vessel are Thus, by kno win g that the volu me this In . sure pres the rve obse we may also and by mea suri ng the temp erat ure, wed gas allo also we if , ever How . fied case the observability condition is satis d not , then the pres sure in the vessel coul flow into the vessel in an unk now n way ility rvab obse the use beca t mea sure men be estim ated base d on a tem pera ture of an ion rvat obse ul essf succ for that is here cond ition is not satisfied. The lesson e ain enou gh information to uniquely defin cont t mus el mod the , state red easu unm rs, Filte an Kalm as well s of obser-vers, as that unm easu red state. Man y type to easu red states. The read er is referred unm of ion rvat obse for have been used Refs. [1-3] for further discussion. y ator to extract process state s from nois To be successful in using a state estim For ess. proc the of stics abo ut the stati data requ ires deta iled info rma tion tities and covariance for all rele vant quan n mea the ire requ ld wou example, one men t sure mea initial condition error, ~ and such as process mod elin g error, ~(t)i n utio evol l stica stati ld be prov ided to erro r, 11 (t). This info rma tion wou ces choi mal opti ugh Thro ). (29.4 K(t) in Eq. equa tion s used in the calculation of The can be extr acte d from the data y(t). x(t) of ates estim mal opti of K(t) , the of form e som this task usua lly invo lves part icul ar state estim ator used for Kalm an Filter (see Refs. [1-3]). REACTOR (ADAYfED A CONTINUOUS STIRRED TANK FRO M REF. [1]) reactions: us stirred tank reactor (CSTR) for the Let us consider an isothermal continuo

Exam ple 29.1

The rates for each of the reactions are reactor:

c in the given in terms of the concentrations cA, 8

eling equations for the reactor where kl' k2 are rate constants. The mod de;. V

= F(cA f- c;.) - Vk 1c;. c;.(O) = c;.0

-;g des

V

take the form:

dt

= F(c81 - c8 ) + V(k 1cA

- ~c8 )

c8 (0)

= c80

are the of A and B respectively and cAo- c80 Here CAf, c8! are the feed concentrations ible to poss is it that ose supp Now at start-up. initial concentrations in the reactor sure mea ot cann we but B continuously on-line, mea sure the concentration of species to wish we Thus e. valu its know to is important the concentration of A even though it using a state estimator. estimate the concentration of species A F, cAf' noting that c81 = 0 and the quantities V, by tions equa the lify simp First let us rs: mete para less nsion dime the e defin can k1, k2 are all constant. Then we 2V

k 1V

Da 1

=F ; CA

xl

CAref

Da2

k =r:

l=[ i

en

u; ~

X2=-

CArtf

t'V

CAref

1068


where cAre[ is the nominal initial charge of species A to the reactor. With these definitions the model becomes: dxl

dt

= -(1 +Da 1)x1 +u

If we have a continuous measurement of species B, then our sensor model is: y(t) = xp) + 77(1)

where 77(t) is the measurement error. Then going to Eqs. (29.4) and (29.5) for our state estimator equation, we obtain:

To illustrate the performance of this state estimator, numerical computations were carried out for the parameters Da 1 = 3.0, Da 2 = 1.0, tt= 2.0, true initial conditions x 10 = 1.0, x20 = 0; and inlet feed concentration u = 0. The state estimator was given erroneous initial conditions 1 = 0.85, ; 2 = 0.15. The values of k 11 • k 21 were determined from Kalman filter equations [1). Gaussian measurement errors having zero mean and standard deviation 0' = 0.05 were simulated by a random number generator and added to the x 2 values to produce the sensor signals, y(t). The state estimatesx1, 2 are shown in Figure 29.3 along with the "true" values x 1, Xz- Note how the unmeasured and measured states both converge to the "true" values with time.

x

x

LO

_ "true" states - - state estimates 0.5

LO

t__..

Figure 29.3.

Comparison of "true" states and state estimator estimates for continuous data. 77(1) is the actual measurement error used (0'= 0.05).

CHAP 29 SELECTED TOPICS IN ADVANCED PROCESS CONTROL

Figure 29.4.

1069

Feedback control loop in which a state estimator is used to provide "measurements" to the controller.

29.1.3 Use of State Estimation in Feedback Control Now that we understand how to generate state estimates, we may use these estimates in a feedback control loop. For example, suppose that we have the chemical reactor of the last section and we wish to configure a control loop to keep the concentration of species A at its set-point. However, since we cannot measure species A directly, we must use a state estimator to provide these "measurements" to the controller. The control loop is shown in Figure 29.4. Obviously one must have great confidence in the state estimator in order to implement such a control scheme. However, plant experience has shown that with good models, careful instrument calibration and regular control system maintenance, state estimators can be successfully used to improve product quality and control system performance.

29.2

CONTROL IN mE FACE OF PROCESS VARIABILITY AND PLANT/MODEL MISMATCH -ROBUST CONTROLLER DESIGN

One of the issues that has always arisen in control system design is how sensitive the controlled system is to expected variations in the process parameters. For example, if we have tuned a PI controller for a particular loop, how will this control loop behave if the process gain changes by 20% or the time delay changes by 30%? This is a question of process sensitivity and the robustness of the control system design. New design methods have been developed for addressing this issue, and we shall discuss some of them in this section.

29.2.1

SISO Sensitivity and Robust Controller Design

Let us begin our discussion with SISO systems because we can readily illustrate the important concepts. To have a specific example in mind, suppose that we consider the conventional control loop shown in Figure 29.5. If we were to apply classical tuning of a proportional controller to the loop, we would construct a Bode plot of g(s) and determine the process gain at the crossover frequency me (i.e., when the phase lag is 180°), as shown in Figure 29.6. The critical process gain takes the value A, and a proportional controller gain of Kcu = 1/A, would bring the control loop to the verge of instability. Thus, depending on the


1070

Figure 29.5.

A convent ional SISO control loop.

phase margin and particu lar controller tuning strategy selected, we use various e, Ziegler-Nichols exampl For gain. tional propor the select to gain margin rules tional gain of propor a that tuning would call for a gain margin of 2, so rules involve tuning our of many that Note K, = 1/2A is recomm ended. le and then unstab go to process the causes which gain ler control detel'll'Uning the timeThese safety. of reduci ng the gain by some factor to provid e a margin ff trade-o a always is there that fact simple the tested design rules recognize (usually lity instabi to ess robustn and ance, perform system l betwee n contro contro l system termed · robust stability). Higher contro ller gains to favor that robustness so y stabilit for safety of margin perform ance always reduce the proper balance the seek always must one 29.7, Figure in shown As decreases. betwee n these two factors for the proble m at hand. one would like In order to determine the best margin of safety for robustness, se the true Suppo g(s). in lity variabi ed expect and ainty uncert to know the process can be represented exactly by: g(s)

Kctzs

= 1:S +

1

(29.9)

a, which can then we have a process gain K, a time consta nt 1', and a time delay I in Figure g(s) I curve entire the vary. Variations in process gain K will cause consta nt 't time the in ns variatio factor; same the 29.6 to move up or down by mediu m at vary to g(s) L and cies frequen high at vary to I will cause I g(s) I g(s) I on effect no have will frequencies; finally, variations in the time delay a tes illustra This cies. frequen higher at vary to and will cause the curve L g(s) Note ters. parame s proces the in ons variati to plot Bode the the sensiti vity of frequency around that loop stabilit y is threate ned only by changes at higher ined by process determ be to tends y stabilit robust the crossover frequency. Thus r function in transfe the for that means This cies. frequen variabi lity at higher , while critical most are gain Eq. (29.9), variations in the time delay and process ant import so variations in the time constant are not ms with There are two practic al situati ons that lead to proble arise as a and tch misma robustness. Both could be said to lead to plant/m odel consequence of the following situations: no longer 1. Proces s Variab ility: in which an otherwise adequa te model itself has process the e matche s the observ ed plant behavi or becaus les examp two are fouling ger exchan heat and change d. Catalyst decay lity. variabi process such ying underl nisms of mecha an inadeq uate 2. Inade quate Mode ling: in which the model is e the model becaus be may This or. behavi plant true of represe ntation case, for the is (as n imatio approx poor entally fundam a is re structu ar nonline y severel a nt represe example, when a linear model is used to ters parame model the of es estimat poor of e becaus process), or it may be in an otherwise adequa te model structure.

1071

L ADV ANC ED PRO CES S CON TRO CHA P 29 SELECTED TOPICS IN

Lg(s )

~~~

~~~~~~~~~~

-180~~~~~~

.. Figu re 29.6.

Bode plot for g(s).

Process Var iab ility model, a r e>.:periment, or the use of a goo d Sup pose that thro ugh trial and erro service. into put and ility stab st robu of gin loop is tune d wel l with a certain mar y. In bilit insta rs in the process to thre aten Then, ove r time, some chan ge occu hea t y, deca lyst cata ., (e.g the process gain y this case, the chan ges cou ld be in dela time the in or t, stan con in the time me exch ang er surf ace fouling, etc.), assu us let flowrate, etc.) In this case (change in feed pipe diameter, feed are t but that the vari atio ns AK, Aa ican gnif insi that vari atio ns in -r are nds bou the by n give on regi y fuzz g(s) lies in a significant so that the true process on the transfer function (K + AK)e- (a+ t.a).< (29.10) g(s) =

-rs + 1

er sign, er variations 11K, Aa can be of eith We will assu me that the para met and vary between: $;IlK~

-IMmaxl

-l11amaxl ~

Figu re 29.7.

IMmaxl

11a $;

jl1amaxl

control syste m perf orm ance and The ever-present balance between stability in control system design.

robu st


1072

The bounds of this fuzzy reqion can be seen in Figure 29.6. The curve for I g(s) I while the curve for L g(s) has possible has possible variabilit y± I t.Kmax Note the large potential variabilit y in the variabilit y ± ro t.amax crossover frequency roc and correspon ding critical gain, 1/ A, caused by the variability in K and a. It would be useful to have a transfer function for the bounds on this region of variability. Let us call this transfer function the variability factor, lf/(s). Then we could say that the true process, g(s), varies from the nominal process, gm(s),

I

I.

by:

I,

g(s) = gm(s)~s)

(29.11)

-- K +Kt.K e -!J.as "''s) .,..~

(29.12)

Hence for our example here:

Thus, the nominal process when LlK =t.a = 0 has a variability factor of unity. We mar view the Bode plot of this variability factor for b.K = ± t.Kmax ,, Aa = ± b.amax to see how it influences the frequency response of the nominal process, gm(s) (cf. Figure 29.8). If one were to convert the Bode plot in Figure 29.6 into a Nyquist diagram (see Figure 29.9), then we can see the uncertainty in gain and phase angle in the same graph. Notice that the variability factor lf/(s) causes the possible domain of g(s) to appear as wedge-sh aped regions of uncertain ty around gm(jro) when there is variability in only K and a. For example, see the wedge formed around the point g(jro1) as a result of the uncertain ty in the actual values of L g(jro1)

I

I

lljl (s) I

A.K,.,

1.0

-&.,..,.

I (II

(II

Figure 29.8.

Bode plot for the variability factor V'(s) given by Eq. (29.12); liK lia= ±2, K = r= a= 5.

=± 1,


-3

-2

1073

-1

Real Axis

Figure 29.9.

Nyquist diagram for g(s), gm(s) given by Eqs. (29.11) and (29.12).

by and I g(jOJ1) I. The envelop e of all these regions is the band given phase g involvin rules K ± AKmax fihO~ in the figure. Notice that tuning margin and gain II,l'argin are designed to make sure that no part of the possible domain of g(s) encompa sses the point -1 in the Nyquist diagram . The gain to margin rules ensure that the gain variations K ± AKmax do not cause g(s) that ensure hand other the on rules margin phase the OJ,; include -1 when OJ = not variation s in phase lag due to time-delay variabili ty (a± Aamax )OJ do how see can one Thus figure. the in cause g(s) to encompa ss -1 for some m < m, of PID controller tuning rules address the question of robust stability in the face process variations.

I

I

I

I

I

I

Inadequate Modeling of Suppose that a linear process model, gm(s), was develop ed from a set are there r, Howeve data. test pulse or test step experim ental data such as to always inadequ acies in such experim ental data sets. This could be due noise ment measure or unmode led nonlinea rities, poor experim ental design, problem s that prevent the model from represen ting all features of the process. part of In normal process identification experim ents it is the high-fre quency /model process the is This the model which is usually in greatest error. mismatc h problem . In this situation, the robust stability of a control system depends on how sensitive the control system design is to model inadequacy. In practice, one finds that the variability factor lfl(s) resulting from process identification experim ents has the general shape shown in Figure 29.10. Note is that the variabili ty factor is close to unity until the failure frequency m1 to ate inadequ are data ental reached. Above this frequency, the experim identify reliable model paramete rs. If this failure frequenc y is well beyond for the crossover frequency m1 >>m,, then obviously the model will be adequate to dose is y frequenc failure the if , conventi onal control system design; however stability serious to lead may model the using designed OJc, then control systems problems. We shall discuss some possible remedies for this problem below.


1074 .

l'l'(s)l

1.0

/ ~====::::---1 \

Ill--

L'l'(s)

variability factor, ~s), resulti ng from Figure 29.10. Typical frequency depen dence of the . Here m1 repres ents the failure ments experi n ficatio proces s identi inadequate. frequency beyon d which the identification experiments are

imes useful to neglect For the analysis of more complex systems, it is somet Figut e 29.9 and to in the specific shape of the variability region g(jof) d by: define ((J)) I radi11S with 4 approximate them with a circular disk (29.13)

f.
= .,.........::..:......;....,. -

(29.14)

lcmU(J)>I

Q)

-

passing disk size I 4 { ro) as a fraction of Figure 29.11. Frequency dependence of the encom the nominal process lgm(jm} I.

1075

TROL IN ADVANCED PROCESS CON CHAP 29 SELECTED TOPICS

ure 29.11. Note endent behavior sho wn in Fig factor IJI(jm), should have the frequency-dep ility iab var tha t in terms of the 14) (29. and 13) (29. . Eqs from · Tm(m) mu st satisfy: (29.15)

Ro bu st Controller Design and of mo del effects of process var iab ility s a controller With this bac kgr oun d on the ign des one how r position to conside ure 29.7, we uncertainty, we are now in a Fig m ance and robust stability. Fro rov e the imp THAT provides good perform to n gai pro por tion al con trol ler factor one is recall· tha t inc rea sin g the re the r, eve How ity. ess to instabil trol con d goo performance reduces the robustn lly, trade-off somewhat. Genera y. onl cies tha t allows us to improve this uen freq low at hig h controller gains near gain ler sys tem performance requires trol con the by d ine ility is determ Conversely, the question of stab response of our we could shape the frequency if s, Thu cy. uen and a sharp the crossover freq cies uen h controller gains at low freq controller so as to provide hig d, we could che roa app is cy the crossover frequen as gain ler trol con in ion uct red our controller. ance and robust stability from the n guarantee both high perform 2. 29.1 curved balance beam in Figure This would correspond to the ler for the system trol con PI ust rob a ign des To illustrate this po~t, let us (29.12). Let us Eq. by en giv iability factor 'l'(s) var ·the h wit 9) (29. Eq. by en giv the controller so ent weighting function W(s) in intr odu ce a frequency-depend tha t: (29.16)

uency in order to the controller gain at high freq uce red to ) W(s like uld wo We !J.a. This means wo rst parameter variations !J.K, gua ran tee stability und er the in Figure 29.13, so a Bode plot of the type sho wn tha t ideally W(s) should have frequencies. h phase angle increases at hig tha t the gain drops off and the ~

II ~

~ ~ ~

Po.

s

s

~ "ll

.!!

8

f.

~

~

j

$

il

5.

!

em performance and rob ust balance between control syst Fig ure 29.12. Altering the hav e the pro per freq uen cy to ler con trol stab ility by des igni ng the dependence.

1076


·~"I 1.0

.,

'"· · I

.,

~I

j

Figur e 29.13. Ideal form of contr oller weighting to achieve high perfo rman ce at low frequencies and robus t stability at high frequ encies.

Ther e are diffi culti es in impl emen ting this ideal cont rolle r weig hting because simple forms such as:

efi.,s W(s) = (-r,.s

+ l)"

(29.17) whos e Bode plots have the desir ed shap e, are not causal. How ever , there are convenient appr oxim ation s such as: (29.1&a)

or W(s) =

( -r s + 1)" + K (I - e P..s) w w

(29.18b)

whic h may be used. Let us illustrate with a particular numerical exam 10

1o-•+ -or-rT rnrnt- --r-rn rnrd- r-r.,.. ,.mnt 10-·

--r-rT Jrrm!

10

to•

Figur e 29.14. Bode plots for the control weightings W(s) given by Eq. (29.19).

ple.

l

CHAP 29 SELECTED TOPICS IN ADVANCED PROCESS CONTROL Example 29.2

1077

ROBUST CONTROLLER DESIGN FOR A PROCESS WITH TIME DELAYS.

Let us suppose that K = 5, t = 5, and a = 5 in the process given by Eq. (29 .9). If we allow AK = ±1 and Lia = ±2, then the process variability has already been plotted in Figures 29.6, 29.8, and 29.9. It can be shown that if we choose W(s) to be of the form: W: · 5s+l (s) = 5 s + I + 2(1 - e- 5 •)

(29.19)

then W(s) will have the frequency response shown in Figure 29.14. This is similar to the ideal form in Figure 29.13 for frequencies w < 0.5. The overall control scheme for the robust controller is represented in Figure 29.15. The actual process is given by the model g (s) combined with the variability factor as in Eq. (29.11). In a similar way, the controller is a combination of a PI controller and the input weighting function W(s) as in Eq. (29.16). For the W(s) chosen (Eq. (29.19)), the Bode plot shown in Figure 29.14 is not as effective in decreasing amplitude and increasing phase lag at high frequencies as the idealized form in Figure 29.13. Nevertheless, as we shall see, it does decrease the process gain and phase lag in the region close to the critical frequency that aids both performance and stability. To see the loop shaping effect, the nominal (lfl(s) = 1) ueffective" process Km(s) W(s) seen by the PI controller has the Bode plot shown in Figure 29.16 for W(s) =1 and for W(s) give11 by Eq. (29.17). Note how much the amplitude ratio has decreased at llJc. In fact; 'the gain margin has gone from 0.45 for the uncompensated plant to 1.18 for the compensated,plant. This should allow the PI controller gain to be much lower for the same performance or the gain to be increased to improve performance with no cieterioration in robust stability. To illustrate the control system performance the closed-loop step response of the nominal plant, Eq. (29.9), with uncompensated PI controller (W(s) = 1) tuned optimally to minimize Integral Time Squared Error (ITSE) for Km<s) is shown in Figure 29.17(a). The corresponding step responses are shown in Figure 29.17(b) for the PI controller tuned "manually" when the plant is Km(s) W(s) with W(s) given by Eq. (29.19). Note that for this "nominal" plant with no parameter variations, compensation with W(s) given by Eq. (29.19) gives much better performance as expected. However, one must test the effect of parameter variations on the control system.

- -

r

--

r

Process g(s) -- - -

-

I

Nominal

I

Process y

L

- -

- -

-

-

_]

L

-

-

-

_I

Figure 29.15. Block diagram for a robust controller Kc(s) consisting of a PI controller and input weighting, W(s). The process consists of the nominal process, Km(s), and the variability factor 'l'(s).


1078 10

:

~

IW(s )gm( s)l

W (s)g0

lgm (s)l

1

gm (s)

---- - ••••

~ ··········z;_ .....____.,

: 'I

1

10-3

~~~---------.

0~--------~~~~== LW( s)g111 (s)

~

------r----~~~

Lgm (s)

~00~--------~r----

~

J 1

-loop frequency pensated and uncompensated open Figu re 29.16. Bode plots of the com ). gm(s W(s) response of the nominal plot,

Nominal

1.25 •

1.00

0.75 y

0.50

0.26

0.00 0

10

20

30

40

50

er PI control onse for the process in Eq. (29.9) und Figu re 29.17. Closed~loop step resp tuned to be r rolle cont ed mpensat with no parameter variations: (a) unco pensated com (b) 6); 0.12 = 1/r 4, 1 ITSE optimal for W(s) = l(K, = 0.18 given by is W(s) n response whe controller tuned "manually" for good . Eq. (29.19)(Kc = 0.45, l/r1 = 0.06)

1079

CONTROL IN ADVANCED PROCESS CHAP 29 SELECTED TOPICS AK= l,ll a.= 2

2.0

1.5

y

1.0

0.5

50

40

30

20

10

0

t

control cess in Eq. (29.9) und er PI p step response for the pro ures -loo Fig sed in Clo as 8. ing 29.1 tun 2; ure = Fig ons AK = 1, Aa wit h par am ete r var iati 29.17(a) and (b).

AK a-l ,lla =-2

y

0

10

20

t

30

40

50

9) und er PI control wit h onse of the process in Eq. (29. resp step p -loo sed Clo 9. ing as in Fig ure s Figure 29.1 AK = -1, Aa = -2; tun par am ete r var iati ons 29.17(a) and (b).

1080

SPECIAL CONTROL TOPICS 10 0

:a

I W (s)gm (s) I

P2

"""'

~ ~"" 10-1 10--1

0 LW(s)gm (s)

...."' ~

. g::

"'.,

-200

-400 10--1 (I)

Figure 29.20. Open-loop frequency response of the compensated plant for various process parameter variations. For parameter variations that cause the plant to be less stable (M< = 1, 11a =2), the two closed-loop step responses are shown in Figure 29.18. By contrast, parameter variations that cause the plant to be more sluggish (M< =-1, 11a =-2) are applied and the two closed-loop responses shown in Figure 29.19. Observe that the performance of the compensated PI controller is noticeably better for all parameter variations. Thus the compensator chosen has indeed improved both performance and robust stability when the process gain varies by 20% and the process time delay changes by 40%. The reason can be seen from the Bode plots of the compensated process shown in Figure 29.20. In addition to the fact that the amplitude ratio is reduced by the compensator close to the critical frequency me there is another valuable feature. Close to the critical frequency me there is some significant variation in phase lag with parameter variations, but the amplitude ratio is relatively flat. Thus there is robust stability for reasonably large process parameter variations. Although there are many possible choices for weighting functions W(s) and optimization methods can be used to find good ones, the choice we made in Eq. (29.19) was found by using the equations for a Smith Predictor linked to a proportional controller. This turns out to have the very nice properties noted above.

There is a broad literature describing various approaches for synthesis of robust controllers for specific types of set-point changes or disturbances and for specific forms of model nncertainty, Ym( m). Those wishing more details can begin with Refs. [4-7].

29.2.2 MIMO Robust Controller Design When we consider the design of multiple input/multiple output (MIMO) systems, it is much more difficult to characterize the uncertainty in the process


1081

and to design robust controllers. To understand the reasons for this, consider the 2 x 2 MIMO system with process model: y = G(s)u

where

y

= [~:]'

G(s)

=

u =

e-a,.,s]

K II e-«u•

K 12

[ -r11 s + I

'rl2S

K2Je-«:.•• -r21 s + I

[:J

+ I (29.20)

K22e-«tt' 'rzzs + I

As compared with the SISO system in Eq. (29.9), where everything could be represented by a Bode or Nyquist plot, this MIMO system has four elements gr (s), each one with a different frequency response. Thus, in order to design a controller G c (as shown in Figure 29.21) that will achieve good performance and robust stability for this system, more complex methods must be applied. In keeping with the concept of shaping the frequency response of the control Gc- Doyle and Stein [5] proposed the use of the singular values to construct a Bode-like plot for a MIMO system. Recall from Chapter 22 and Appendix D that the singular values of a matrix may be defined as the positive square root of the eigenvalues of the product G 8 G where G 8 is the complex conjugate transpose of G -termed the ~ermitian of G. That is:

MIMO

a; (s)

= --J .it;(GH(s)G(s)); i = I, 2, •••, N

(29.21)

Thus there are N singular values for an N x N matrix, and the largest and smallest of the singular values provide norms for the matrix G(s): a,_(s) = II G(s) 11 2

(29.22) (29.23)

-~I Figure 29.21.

.. y

A conventional block diagram for a MIMO control system.


1082

Figure 29.22.

Use of O"mJn. O"max to construct a fuzz

y "Bode plot " for IIYII/IIuU 2.

riable "Amplitude Ratios" for this multiva It can therefore be shown that the system may be bounded by: (29.24) s of the vectors y and u. Here IIYlb, llulh denote the Lz norm Figure e the top half of a Bode plot (see wer it if as 4) If we plot Eq. (29.2 een betw Ratio" of the MIMO system lies 29.22), we see that the "Am plit ude the as b /llul view the frequency response of llylh a O"max and O'min so that we can out ked wor e hav [5] . Doyle and Stein shad ed region between O'max and amin the of onse resp cy uen freq the that shapes design procedure for MIMO systems s at uct GG c has sufficiently high gain prod the that ee rant gua y controller Gc to uenc freq high and sufficiently low gains at low frequency for good performance do To d. ecte exP. range of model uncertainty to guarantee robust stability for the Figure ular value plot for II(GG,))II (see sing a ct stru this, Doyle and Stein con t be mus cies uerl freq low at amin er design, 29.23). To achieve the proper controll ll. sma ntly frequencies mus t be sufficie st sufficiently large, and ama x at high robu for nt icie that this desi gn is suff Alth oug h Doyle and Stein prov e

y "Bode plot" for valu es of GG , to cons truc t a fuzz Figu re 29.23. Use of the sing ular IIG G,D2 •

NTROL IN ADVANCED PROCESS CO CHAP 29 SELECTED TOPICS

1083

roximations tha t are vative because of all the app stability, it is often too conser poo rer performance e giv n designs tha t ofte ler trol con in ults res this ma de blem the larg er the usly, this is even more of a pro tha n one would want. Obvio e of this, the ratio of ama x ami n for G(s). Becaus fuzzy region between ama x and and amin for G(s), i.e.: x

(29.25)

O"ma

I(=--

(Jmin

of the sensitivity of , is often used as a measure know.n as the condition number control structures le sib fact, when evaluating pos the process to uncertainty. In eliminate highly to le iab var use d as a screening be can K s, tem sys O MIM for ue analysis that sensitive structures. servatism of the singular val To overcome some of the con for example, en, (ev ) over all elements of G(s y poorly), ver s averages the model uncertainty hap per (s) g very precisely and 12 wn kno be ht mig (s) g en wh 11 " and the concept of of the "structured nncertainty Doyle [6, 7] proposed the use as oth er ideas are Designs using these as well "str uct ure d singular values." ately, the cur ren t tun and Zafiriou [4]. Unfor discussed in detail by Morari l pro duc e rather stil s tem design for MIMO sys ler trol con ust rob for s che approa mance is not the best. cases so that controller perfor ny ma in ults res e ativ serv con ng sha rpe r analysis ains to be don e in ord er to bri Thus, important research rem of stru ctu re can ails lag information and more det to the problem so that phase . be used in the controller design

29.3

OF IL ES - DISTRIBUTED CONTROL OF SPATIAL PR S ER PARAMETER CONTROLL

we have assumed control problems to this point, of ion uss disc our t hou oug Thr ns of time. However, asurements were only fnnctio that all states, inputs, and me iables are functions ms for which the process var there are process control proble urgical processing tall as· ~e. For example, in me of position in space as well s and sheets there slab tal or oth er types of me ; um min alu l, stee for ons operati in the metal. Figure pro per temperature profile are furnaces for creating the ire d lon git udi nal one furnace in wh ich a des 29. 24 sho ws a typical 3-z tion of the firing ula imposed on the slab by manip ons the slab is temperature profile is to be rati ope e the furnace. In som of es zon ual ivid ind the in rates g through the furnace it is a continuous strip movin motionless, and in some cases opt ica l pyr om ete r t the re are con tin uou s wit h vel oci ty v. No te tha ace. Figure 29.25 furn at five fixed positions in the measurements of temperature ual tem per atu re act an and temperature profile nt poi setired des the tes illustra d controller. e through application of a goo profile history tha t could aris s wo uld be in the ces pro the for del riate mo For this problem, the approp ations of the form: form of partial differential equ ox(z. t)

iJt

= a

o2x(z. t) + v ox(z az. t) + {Jx(z, t) + u(z. t) ; 0 az2

s: z :;; L

(29.26)

with boundary conditions: 0

(29.27)


1084 3 Furnace Heating Zones

11111111111111.1111111111111.11111111111111

~ 5 Optical Temperature Sensors

Figure 29.24. Heating of a metal slab in a 3-zone furnace having 5 temperature sensors.

Here a is a thermal conduction parameter, u.is the velocity of the slab through the furnace, f3 represents the convective cooling heat transfer coefficient, and u(z,t) is the radiant heat flux from the three zones of the furnace. The boundary conditions Eq. (29.27) indicate that there is negligible heat transfer from the ends of the slab. The measurements are taken at five points Z; where the optical sensors are located:

= x(z;*• t)

Y,
~:

(These processes are all located on the "back face" of the main cube.) 5.

CATEGORY V PROCESSES (Bottom, Left half, Back face Prima ry Attrib utes:

cube)

Significantly nonlinear processes, with no difficult dynam ics, and with little or no interac tions among the proces s variables. Main Problems: Signif icant Nonlin earity. Sugge sted Techniques: • Single-loop nonlinear model -based techniques, such as Generic Model Contro l {d. Chapt er 19). • Custom -desig ned single-loop PID controllers with process knowl edge incorporated, such as gain schedu ling (cf. Chapt er 18). Typic al Examp les: • Single-point tempe rature (or compo sition) control of very high-p urity colum ns withou t significant time delays . • Single -tempe rature (or composition) contro l of most exothermic chemical reactors.

(I, ..... · '·:.

CHAP 30 CONTROL SYSTEM CASE STUDIES

1137

6. CATEGORY VI PROCESSES (Bottom, Right half, Back face cube) Primary Attributes:

Significantly nonlinear processes, with difficult dynamics, but with little or no interactions among the process variables. Main Problems: Combination of: • Significant Nonlinearity, and • Difficult dynamics. Suggested Techniques: • Single-loop nonlinear model-based techniques, such as Generic Model Control (cf. Chapter 17) or nonlinear MPC (d. Chapter 27). • Custom-designed single-loop PID controllers with process knowledge incorporated (d. Chapter 18), and compensation for the difficult dynamics (e.g., time-delay compensation (cf. Chapter 17)). Typical Examples: • Any Category V process with significantly delayed measurements or other delays. • Exit temperature control of a highly exothermic reactors having inverse response. 7. CATEGORY VII PROCESSES (Top, Left half, Back face cube) Primary Attributes:

Significantly nonlinear processes, with no difficult dynamics, but with significant interactions among the process variables. Main Problems: Combination of: • Significant Nonlinearity, and • Significant Process Interactions. Suggested Techniques: • Full-scale multivariable nonlinear model-based control techniques such as Generic Model Control (cf. Chapter 19); or nonlinear MPC (cf. Chapter 27). Typical Examples: • Two-point temperature (or composition) control of single, or coupled, very high-purity columns without significant time delays. • Simultaneous conversion and composition control of very nonlinear chemical reactors. 8. CATEGORY VIII PROCESSES (Top, Right half, Back face cube) Primary Attributes:

Significantly nonlinear processes, with difficult dynamics, in addition to significant interactions among the process variables. Main Problems: Combination of: • Significant Nonlinearity, and • Difficult Dynamics, and • Significant Process Interactions. Suggested Techniques: • Full-scale multivariable, nonlinear, model-based control techniques with capabilities for handling difficult dynamics, such as Generic Model Control, and nonlinear MPC (d. Chapter 27).

1138


Typical Examples:

• Two-point composition control of single, or coupled, very high-purity columns with delayed measurements or other delays. • Simultaneous conversion and composition control of very nonlinear chemical reactors with delayed composition measurements. • Some nonlinear polymer reactors with multiple control objectives.

Additional Considerations There are many situations of practical importance for which it might be necessary to take other factors into consideration in addition to the three attributes employed above. Some of these factors include: 1. Constraints: requiring controller design techniques such as MPC with explicit and intelligent constraints handling capabilities (cf. Chapter 27). 2. Frequent Disturbances: requiring special controller design techniques specifically for disturbance rejection, such as cascade or feedforward control (d. Chapter 16). 3. Ill-Conditioning: requiring specialized robust multivariable controller design techniques (d. Chapter 29), since ill-conditioning makes standard multivariable model-based techniques particularly sensitive to the effect of plant/model mismatch, thereby placing limitations on the achievable performance of such schemes. 4. Significant Inherent Variability in Process Outputs: requiring controller design techniques specifically for dealing explicitly with the stochastic nature of the measurements, such as the statistical process control schemes discussed in Chapter 28.

30.5.3 Illustrative Example To illustrate the issues raised in this section consider the following example. The process is a CSTR used to manufacture Cyclopentanol (which we shall represent by B) from Cyclopentadiene (represented by A). The kinetic scheme for this reaction is as follows:

A 2A

where C represents Cyclopentanediol, and D represents Dicyclopentadiene. The objective is to make the desired product, Cydopentanol (B), from pure feed of A, maintaining constant reactor temperature, and a constant concentration c8 of the desired product in the reactor. The feedrate and the reactor jacket temperature are available as manipulated variables. The modeling equations for this process obtained by material and energy balances are given as follows:


1139

along with the outputs given by:

Here, x 10 is the feed concentration of A, x 1 is the concentration of A in the reactor, x2 is the concentration of B, x 3 is the reactor temperature, V R is the reactor volume, u 1 is the dimensionless feedrate, and u 2 is the reactor jacket temperature. Using the process parameters reported in Ref. [7] the steady-state behavior of the reactor may be investigated as a function of the feedrate u1• The reactor jacket temperature u 2 is fixed at 130"C, and the feed concentration of A, x10, is fixed at 5.1 mole/liter. Figure 30.33 shows the steady-state behavior of x2, the concentration of B, as a .function of the feedrate. The RGA parameter for this 2 x 2 system, computed as a function of the feedrate, is shown in Figure 30.34. The corresponding steady behavior of x1, the reactor concentration of A, and x31 the reactor temperature, are shown in Figures 30.35 and 30.36. Ll5 . 11 .

.

~vm :

:I

.

.

:

···········-r·············:···· ·····r·········· .·············r·············r··········-,············

1 . . . I········.I···········c····]········] ········~······•r:······ .

.

.

• • •

oooooooooooo}o

.. • • 0

.

.. 0 • •

.

. • 0 •

.

. • 0 •

ooooooooon1oooooooouoooo!oooooo . . ooouo?ooooooooouoo~uoooooooooooo{•••nooooooo

1 1 1 l l ·········· r············T···········r··········T············r············r···········-~·-········ · :IV

··r············r············r··········T············l·············r···········-~·-·········· 0.8'---"----"-------''------'-----J.-----J.-----J.--_J 0 20 40 60 80 100 120 140 160

Feedrate [llh]

Figure 30.33. Steady-state concentration of Cyclopentanol as a function of reactor feedrate.

1140


Characterizing Various Operating Regimes By exam ining Figu res 30.33 and 30.34 (and analy zing the linea r trans fer funct ions obtai ned by linea rizin g the dyna mic mode l aroun d the indic ated stead y-sta te oper ating regim e) we see that this simp le proce ss show s six distin ct oper ating regim es, listed below , each exhib iting radic ally diffe rent dyna mic beha vior in terms of the three attrib utes of Nonl inear ity, Dyna mic Char acter , and Degree of Interaction. Thus depe ndin g on the oper ating cond ition s, this react or may be place d in six of the possi ble eight categ ories discu ssed above.

Operating Regime A: (u1 > 80 hr -I; y (or x 1 2) Prim ary Attri butes :

Char acter izati on: Impl icatio n:

Fairly const ant gain; Figure 30.33 indicates a fairly straig ht line in this regio n ( - Linear Beha vior); Low-order, mini mum phas e (Simple Dynamics) ; RGA const ant at - 1.1 (Virtually noninterac ting). Category I Process. 2 singl e-loo p PI contr oller s shou ld perfo rm reaso nably well in this opera ting regime.

Operating Regime B: (u 1 - 45 hr -l; y (or x 1 2) Prim ary Attri butes :

1.1 mole s/lite r)

-

1.1 mole s/lite r)

-

Mild gain varia tions ; Figu re 30.33 show s only a sligh t curve in this regio n ( - Mildly nonlinear) ; Nonm inim um phas e (Difficult Dynamics); RGA - 0.9 (Virtually noninteracting).

2r-----------------~----

--------------------~

1.5 ......... ... : ......... ....~·-···········;·· ········7········· ···~--

.

1 ......... ..., ......... ...., ............ ~ ........ .. .,. .........

~

.........

0 ........ ..

~- 5

......... :.. ............ .:............

... ; ............ ~......... .. ) ......... .. . ~

: ·r ............!".......... t . . . . . ,. . . . . . r. . . . . .r:. . . . . . ,:. . . . . . +. . . . . . l...........+. . . . . .; . . . . . . 7............ ~.............;............ ~

o.5

. i

.

~

........ ..., ......... ... , ......... ...

~

t·· .........t. . . . . . t. ······"··t······""""j············ ~

~

.

,,,l'' """'"' 'l'"""'"' '"' :•••••••OO••••;•••••OoooOO••:•••••••••"• 1.............,............ -r- .........,............ r............,............. r........ .

-1 .. l,,,,,,,,,,,,,,,,,,,,,.,,,.,,... ,.. -1.5 ......... ...

Feedrate [lJh]

Figur e 30.34.

Relative gain values for the process as a functi

on of reactor feedrate.

~

I

'


1141

3.5.- ----- ----- ----- ----- ----. 3

~

.g "'a "..., :a

2.5

"'a

".,. -§

~

2

>.

0

1.5

20

J

40

60

80 100 Feedrate [1/h)

120

140

160

Figure 30.35. Steady-state concentratio n of Cyclopentad iene as a function of reactor feedrate.

140

135

§

130

~

.a .,f! s"".,

125

.

E-

part V I APPENDKCJES APPENDIX A.

Co ntr ol Sy ste m Sy mb ols Us ed in Process and Ins tru me nta tio n Dia gra ms

APPENDIX B.

Co mp lex Va ria ble s, Dif fer ent ial Equations, and Dif fer enc e Eq uat ion s

APPENDIX C.

Laplace and z-T ran sfo rm s

APPENDIX D.

Re vie w of Matrix Alg ebr a

APPENDIX E.

Co mp ute r-A ide d Co ntr ol Sy

ste m De sig n

'Jr. man sfiouU ~ep liis Cittk 6ra in attic s wit fi a£( tfie furniture tfiat fie is ~(g to am£ tfie rest fie can put a in tfie (um6er room of liis ({6 wfiere fie can get it if lie wants.

'i!

:) I

1148

AJPJPENDKX

A CON TRO L SYST EM SYM BOL S USED IN PRO CESS AND INST RUM ENT ATIO N DIAG RAM S In reading Process and Instrume ntation (P&I) Diagram s, the process sensors, controllers, and actuators are indicated by special symbols. It is importan t to be able to recogniz e these symbols in order to work with design engineer s, operators, et al. Some of the most commonly employed symbols are indicated in the tables below. In many P&I diagram s, only abbrevia ted treatmen t of the control loop is given so that only the measure ment, the controller, and the actuator are shown. The standard s of the Instrume nt Society of America (ISA) are followed in this section. Ref. [1] presents a much more compreh ensive descripti on of the ISA standard s.

[1] InstruJrt entation Symbols and Identific ation, ISA-SS-1 (ANSI/I SA- 1975, R1981) Instrumentation Soc. of America, in Standard s and Practices for Instrume ntation. 7th Edition (1983).

1149

f ·.!I' ji

1150

APPENDIX A

Table A.l. Some Line Symbols

Line

-----------------------------------------------------------

#

#

#

Meaning Connection to process or mechanical link or instrument supply Electrical signal

Pneumatic signal or undefined signal

Table A.2. Some Identification Letters

lJA \.'·

nrL I

Svmbol A

c D E F I

L M p T

v

First Letter Analvsis Conductivity Density

Succeeding Letter Alarm Control

Volta~te

Flowrate Current Level Moisture lhumiditv) Pressure or Vacuum Temperature Viscosity

Indicate

Transmit Valve

1151

CONTROL SYSTEM SYMBOLS USED IN P & I DIAGRAMS

Table A.3. Some Actuator Symbols

Actuator

Svmbol

--

t*l

Manual Control Valve

~

Automatic Control Valve

-~

Motor

--~

Solenoid

~

Pressure control valve, self-contained

~

Pressure control valve, external pressure measurement

~

Pressure relief or safety valve, angular pattern

~ ~

Pressure relief or safety valve, straight through

...

Level control valve with mechanical linkage

w

!I

1152

APPENDIX A Table A.4. Some Sensor Symbol s

Svmbol

Sensor

t~

-

--

Flow sensor I transmitter using orifice meter

Flow sensorI transmitter using turbine meter

Flow sensorI transmitter, undefined type

~ L_

--

~---

Level sensor/t ransmit ter, undefined type

f

---

Pressure sensor I transmitter, undefined type

t

---

Temperature sensor/t ransmit ter, undefined type

t

---

Viscosity sensor I transmitter, undefined type

~

cr

Pressure indicator, undefin ed type

Temperature indicator, undefin ed type

J

~

l

CONTROL SYSTEM SYMBOLS USED IN P & I DIAGRAMS

1153

Table A.S. Some Example Control Loops

S

bol

Meanin

Level measurement/ transmitter, signal sent electronically to controller and actuator signal sent to valve

Cascade controller: Temperature measurement/ transmitter signal sent electronically as set-point to coolant flow controller, flow measurement/ transmitter sent electronically to flow controller, actuator signal sent to valve

Pressure measurement/ transmitter with pressure indicator, signal sent electronically to pressure controller, actuator signal sent to inlet gas control valve

·.·~.!~i'r ' '

l

I

B CO MP LE X VARIABLES, DIFFER ENTIAL EQUATIONS, AN D DIFFEREN CE EQ UA TI ON S Classical process control is kno wn to rest on such trad ition al fund ame ntal elements as linear ordinary different ial equations, the Laplace transform, and the closely related topic of complex vari ables. The corr espo ndin g fund ame ntal elements for computer control are linear difference equations, z-transfo rms, and, aga in, complex variables. Since Laplace and z-tr ansf orm s are disc usse d elsewhere in this book (see Chapter s 3 and 24 for the respective introdu ctions and Appendix C for a more complet e review) this appendix is therefore devoted to reviewing first the algebra of com plex linear ordinary dtfJerential and difference variables, then some basic elements of equations that are most relevant to proc ess dynamics and control studies. Brie f summaries of pert inen t issues rega rdin g nonlinear ordinary differential equa tions are also included.

B.l

COMPLEX VARIABLES

B.l .l Def init ion of a Complex Nu mber A complex num ber z is defined in term s of an ordered pair of real numbers (x,y) and the imaginary number j = {:1 as follows: z=(x ,y)

= x+ jy

(B. I)

The real numbers x and y are calle d, respectively, the real and the ima ginary parts of z; and it is customary to expr ess this as: Rez =x; lmz == y

1155

(B.2)

'II !

1156

APPENDIXB

iJ

p : (x,y)

•

Imag inary Axis

r

'• ''•

~

',y •

I

•

. ' ''

.•• Real Axis

Figure 8.1.

~

X

" ~

The complex number.

B.1.2 Representations of Comple x Numbers Since a complex num ber is defi ned in term s of an orde red pair of real num bers (x, y), it is natu ral to associate each complex num ber with a poin t Pin a plan e who se Car tesi an coo rdin ates are x and y (see Figu re B.l) . Wit h such repr esen tatio n, each complex num ber corr espo nds to just one poin t in the plane, and con vers ely, each poin t in the plan e is associated with just one com plex num ber. In this context, the Cart esia n x-y plan e is referred to as the complex plane; the Car tesi an x-axis beco mes kno wn as the "rea l axis " and the y-ax is as the "im agin ary axis." It is cust oma ry to then refer to diag ram s such as in Figure B.l as Arga nd diagrams. Obs erve now that the same poin t P show n in Figu re B.l can be loca ted in the complex plan e as the end poin t of a vect or of leng th r unit s disp lace d at an ang le 6 (rad ians ) from the posi tive real axis. Thu s we may equ ally well identify the complex num ber z by the ordered pair (r, 9). In gen eral it is possible to repr esen t a complex num ber in any of thes e two enti rely equ ival ent form s (kno wn resp ecti vely as the Olrtesian and the polar forms) and , as will be seen below, one representation is usually mor e convenient than the othe r for perf orm ing cert ain algebraic operations.

Cartesian (or Rectangular) Represe ntation The Cart esia n repr esen tatio n of a com

plex num ber is as given in Eq. (B.l ), i.e.:

z = x+ jy

(B.l)

Abs olut e Val ue (or Mod ulus ) The absolute value (or modulus) of the com plex num ber in Eq. (B.l ) is the non nega tive real num ber den oted I z I and defined by: lzl;::

..Jx2 +l

(B.3)

From Figure B.l it is obvious that this represents the distance from the poin t P to the origin. From Eq. (B.2} we now note that: lzl 2 = (Re d+ (Im z/

(B.4)

I

COMPLEX ALGEBRA, DIFFERENTIA L AND DIFFERENCE EQNS Com plex Con juga te

1157

Associated with ever y complex num ber z with Cartesian representation given in Eq. (B.l) is the num ber its complex conjugate defined as:

z,

z = x-jy

(B.5)

Tha t is, z is a num ber repr esen ted by the poin t (x, -y), a mirr or ima ge of the num ber z reflected in the real axis. The following are imp orta nt relation s to note abo ut complex num bers and their conj ugates. 1.

Their moduli:

IZl = lzl 2.

z

z+Rez = 3.

(B.6)

Their sum and their difference:

2

Imz =

zz =

ld

z- z Zj

(B.7)

Their pro duc t

(B.8)

Relation Eq. (B.8) is particularly usef ul; it indicates that mul tiply ing a complex num ber by its conjugate produces a (nonnegative) real number.

Pol ar Rep rese nta tion The poin t (x, y) may be represented instead by its pola r coordinates ( r, e); in this case, the corr espo ndin g complex num ber may be represented as: z

= rL9

(B.9)

whe re the symbol L is used to indi cate implicitly that 9 is an angle. The explicit pola r representation correspondin g to Eq. (B.l) is obtained by noti ng that x=r cos 9;

and

y= r sin 9

so that z may now be written as:

z=

r(cos 9+ jsin 9)

(B.lO)

The Mod ulus and the Arg ume nt Not e that the mod ulus of the complex

num ber in this form is given by: (B.l l)

~·.··~

l'

APPENDIXB

j

The angle 6 is called the argument of z, and we may write: e = arg z. If the complex num ber z is interpreted as a direc ted line segm ent in the complex plane connecting the poin t (x, y) and the origin, then, as stated earlier, r is the length (or magnitude) of this line and 6 the angle (in radians) that this line makes with the positive real axis. Observe, howe ver, that 6 is not uniq ue beca use for any positive integer k, the angles given by (9 + 2k1r) equally qualify as arguments of the same complex number. Thus we defin e the principal value of arg z (sometimes denoted by Arg z ) as that unique value of (} for which: (B.l2 )

The Exponential Pola r Form It is possible to emp loy Euler's formula: ej8

= cos 6+ jsin B

(B.l3 )

in Eq. (B.10) to obtain straightforwardly: (B.14)

nus is sometimes referred to as the exponentia

l pola r form for obvious reasons.

Transformations The following is a sum mary of the relat ionships used for trans form ing one complex num ber representation to the other . L

From Cart esian to Pola r

Given: z =x + jy, the required r and 0 are obtained from:

r=.VXl+l

(B.l5a)

mn-{;)

(B.15b)

and

o=

In using Eq. (B.15b) care mus t be taken to ensure that the quad rant in which the poin t (x, y) falls is taken into consideration. The pote ntial pitfall lies in the fact that the indiv idua l signs associated with x and with y - the mos t critical infor mati on need ed to deter mine e accu rately - are lost in the process of evalu ating the ratio y/x. The informati on abou t the quad rant in which z is located is not "automatically" contained in Eq. {B.l5b); it mus t be "man ually " supp lied. Observe therefore that as it stand s, Eq. (B.15b) will alwa ys give 0 values in the range [-n"/2, n/2]; this will be valid as long as x is positive. When x is negative, the appropriate value of 9 is obtai ned by addi ng 1r.

';,,i~'li~

.: j.

COMPLEX ALGEBRA, DIFFERENTIAL AND DIFFERENCE EQNS 2. From Pola r to Cart esian

=

Given: z r L 6 eithe r in the trigonometric form in Eq. (8.10}, or in the expo nent ial form in Eq. (B.14), the required Cart esian x and y num bers are obtained from:

x= rcos 6

(B.16a)

= rsin 6

(B.l6b)

and

y

B.1.3 Algebraic Operations The sam e basi c alge braic oper ation s of addi tion , subt racti on, divis ion, and mult iplic ation poss ible with real num bers can also be carri ed out with com plex num bers . Whi le it is true, in princ iple, that all the alge braic oper ation s can be perf orm ed usin g both the Cart esian and· the pola r forms, it is also true that for each algebraic oper ation , there is a pref erred com plex num ber repr esen tatio n whic h cons idera bly simplifies the requ ired man ipula tion. In wha t follo ws, it is with out loss of gene ralit y that we illus trate alge braic operations usin g two complex num bers Zt and Zz given in the Cart esian form by:

=XI +jyl

(B.l7 a)

t2=1 tz+i Y2

(B.l7b)

Zt

and by: Zt

= '• elfl,

(B.l8 a)

z2

= r2ei~

(B.18b)

in (exponential) pola r form; the exte nsio n to three or mor e com plex num bers is strai ghtfo rwar d. L

Add ition and Subt racti on: (Cartesian Form Preferred) Zt

±~

= (XI + jyl) ± ( Xz + jy2) = (x• ±Xz) +i(Yt ±y2)

2.

(B.19)

Mul tipli catio n: (Exponential Polar Form Preferred)

(B.20)

Usin g the alter nativ e Cart esian form, we

obtain:

~~~!!

1160

APPENDIXB

which, upon expanding and rearranging becomes: (B.2I)

3.

Division: (Expone ntial Polar Form Preferred)

(rl eiB•) (rz ei82) ~ e j(0,-11,) (B.22)

r2

It is instructive to examine how division is perform ed using the Cartesia

~ Zz

=

(x• + iYt) (.xz +iYz)

n form:

(B.23)

We must now eliminate the complex number in the denomi nator and replace it with a real number ; this is achieve d by multipl ying both numera tor and denomi nator in Eq. (B.23) by the complex conjugate of .;, a procedure kno·wn as rationalization, i.e.:

(XI + jyl) (Xz - h'2) (.xz+jy2) (.xz-.b'l) with the result that

(xtXz +Y1Y2) +i( -x1Y2 +YtXz) Xz 2 + Yl The following importa nt relations regardi ng the moduli and argume nts of the produc t and quotien t of two complex number s follow directly from the results given above. L

Products:

Ifz =Z 1Zz then:

lzl = lztllz21

(B.24)

and

arg (z) = arg (z1) + arg (z,z) 1'

L ,,,

2

zt Quotien ts: Ifz =-then : Zz

lzl

'j; i:

.,:'I

li

lz.l lz2l

r.:

(B.25)

(B.26)

and

arg (z)

arg (z1) - arg (z2 )

(B.27)

COMPLEX ALGEBRA, DIFFERENTIAL AND DIFFERENCE EQNS

1161

B.1.4 Powe rs and Roots of Complex Num bers The expon ential polar form is usefu l for repres enting integr al powe rs of a complex numb er. H the complex numb er represented by Eq. (8.14): (B.l4)

is multip lied by itself n times, we obtain the result that: (B.28)

Similarly, we have: (B.29)

ln particular, if r = 1 in Eq. (B.28), we obtain:

(B.30)

and upon introd ucing Euler's formula Eq. (8.13) for both sides of Eq. (B.30), we have: (cos 8+ jsin 0)"

= cos nO+ jsin nO

(B.31)

This result is know n as de Moivre's theorem; it is partic ularly usefu l in finding the roots of equat ions involv ing the complex numb er z. To illustrate, we consid er the classic proble m of finding the n complex roots of the equation: z"-1 = 0

(B.32)

where n is a positive integer. This is first rearranged as: z"=l =te.i O

(B.33)

where we have writte n 1 in the exponential polar form. Observe now that in this form, the modu lus and argum ent are given by: r= l and

8 = 0+211k

havin g recalled the fact that e, along with any other angle equal to 8 plus integral multiples of 27r, are equally valid argum ents of the same complex number. Thus, in the trigonometric form Eq. (B.33) becomes:

z"

= 1 = cos 211k +jsin 211k; (k=O, I, 2, ... , n-1)

(B.34)

so that:

z

= 111" = (cos21Zk+jsin2nfc) 11"; (k=O ,l,2, ... ,n-l)

We may now apply de Moivre's theorem to the RHS

z

= cos

e: } e: ) jsin

and obtain finally:

(k = 0, l, 2, ... , n - 1)

(B.35)

1162

APPENDIXB

as the expression for the n roots of the Eq. (B.32). The references provided at the end of the Appendix (particularly Ref. [6]) are recommended to the reader interested in further information on complex variables.

B.2

LINEAR DIFFERENTIAL EQUATIONS

In general, any equation of the following type:

~

~ + ... + an-1

1Jo(t) dt" + a1 (t) dt•-1

t!1.

(t) dt + a.(t)y

:;

= q(t)

(B.36)

involving functions of t (the independent variable), along with y and its derivatives (with respect to t) up to the nth derivative, is known as a linear, differential equation of order n. It is said to be linear because so long as the coefficients a;(t), i = 0, 1, 2, ... , n, and q(t) are functions oft alone (or are constant), the equation is in fact linear in the dependent variable y and its n derivatives; it is of order n because the nth derivative dny/dt" is the highest differential involved. H in particular q(t) = 0, the equation is further classified as homogeneous; otherwise it is nonhomogeneous. The following is a summary of some key methods for solving the most important classes of these linear differential equations.

B.2.1 First-Order Equations Consider the linear first-order homogeneous differential equation:

~+ t.zy

= 0

(B.37)

where a is a constant; it is easily solved by rearranging and separating the variables to give: y(t) = e-at y(O)

(B.38)

Of course not all first-order equations are this simple; but the concepts here illustrated are useful in solving the more general equations. The general nonhomogeneous linear first-order equation is given by:

~+ a(t)y

= q(t)

(B.39)

It is usually solved by the method of integrating factors and has the more general solution:

'

I ~


1163

In the special case when the coefficient a is a constant, the first-order differential Eq. (B. 39) becomes: f!1ffi dt

+ qy(t) = q(t)

(B.41)

and the solution will then be obtained immediately from Eq. (8.40) as: y(t) = e-m y(O) + e-at rea" q(d)da 0

or y(t) = e-m y(O) +

J'

e-a{t-u) q(d)da

(B.42)

0

Note that to avoid the potential confusion arising from using the same variable t in the limit of the integral and as the integration variable, it is customary to introduce a dummy time argument (say, a) under the integral sign, and retain t for current time (and hence in the upper limit). It is straightforward to show (and the reader is encouraged to do so as an exercise) that if we wish to solve the standard state-space model (Eq. (4.28) from Chapter 4) which includes both control action u(t) and disturbance d(t), ie.:

~/) = ax(t) + y

bu(t) + ')d(t)

(B.43)

= cx(t)

then the solution is: y(t) = e 01 y(o)+ c r ea(!-u) bu(d) da+c r ea(!-a)')d(d) da 0

(B.44)

0

B.2.2 Higher Order Equations The nth-order linear equation with constant coefficients: ~

llodtn

~

dn-1y

+ a1dtn-1 + ... + an-1 dt + any

= q(t)

(B.45)

will have a solution of the form: y(t) =

Ao +

n

LA f!m;l r=1

(B.46)

where the m; are the n roots of the characteristic equation: (B.47)

1164

APPENDIXB The specific form of the solu tion is mos t easily foun d from the meth od of Lapl ace trans form s (see App endi x C); it is also possible to solve these equations by matr ix meth ods as show n below.

Ma trix Methods Con side r the situa tion in whic h a new set of variables, z 1 (t), z 2 (t), ... , zn(t) are intro duce d to repr esen t the variable y and its first n-1 derivatives as follows: z1(t)

y

~(t)

4l dt

= ~ dP

z3(t)

(B.48)

z._, (t)

=

z.(t)

=

dn-2 y dt"- 2 dn-1 y dtn-1

From here we easily obtain: dz (t)

-dt1- = z2 (t)

(B.49)

dz 11 _ 1 (t)

dt

= z.(t)

and

whe re the RHS expr essio n has been obta ined by solv ing Eq. (B.45) for dny/d tn and intro duci ng Eq. (B.48) appr opri ately . This set of n linea r first -ord er equa tions now cons titut e an alter nativ e repr esen tatio n of the linea r nth- orde r equa tion. Writ ten in the mor e com pact vector-matrix form (see App endi x D) Eq. (B.49) becomes:

dz}/) = Az(t) + f(t)

(B.50)

whe re then -dim ensi onal vectors z(t), f(t), and then x n matrix A are give n by:

I


=

z(t)

['~] z2(t)

1165

0

0

;

f(t)

= gill_

zn(t)

Go (B.51)

A=

0 0

0

0

0

0

--an

--an-1

--an- 2

--a!

Go

Go

Go

Go

0

0 0

It is impo rtant to note the similarity between Eqs. (B.SO) and (B.41); instead of the scala r functions and coefficients we now have vector functions and matr ix coefficients. The mor e com pact matrix Eq. (B.50) may now be solv ed by the meth ods discussed in App endi x 0; the solution is:

(B.52)

where Zo is the value of the vector z at the poin t t = fo. The matr ix exponentials, using the relation Eq. (0.87 ) in Appendi x D. Note that the general state-space representa tion, first pres ente d in Chapter 4 (Eq. (4.30)), is a special form of Eq. (B.50 ): ~~- 1ol,eA(t-a>, may be evaluated

~~)

= Ax(t )+ Bu(t )+ rd(t)

(B.53)

y = Cx(t)

This has the general solution (from Eq.

(B.52)):

x(t) = eACt -to)"o + ( t eAk-t [Bu(i -1) + rd (i- 1)]

NONLINEAR DIFFERENTIAL EQUA

(B.69)

TIONS

As state d earlier, any equa tion that expresses a relation betw een a function y(t) and its deri vati ves is a diffe rent ial equa tion ; in part icul ar, if the indi cate d relationship is nonlinear, we have a nonl inear differential equation. For exam ple, the equation:

~

= f(t,y )

(B.70)

is a first-order nonl inea r differential equation if f{t,y) is a nonlinear function of y. The general nonl inea r nth-order diffe rential equation may be represented as:

!f2 - 1 Y . . f!r .. )! ( dt" .d" ' dtn- l ' ... ' dt 'Y' t - O where f(·) is som e nonlinear function of the indicated arguments.

(B.7l)

APPENDIXB

1170

For linear equations, there are well-defined, systematic procedures for finding solutions in all cases, some of which were reviewed in Section 8.2. Unfortunately there are no corresponding general procedures for generating solutions to all nonlinear equations. In fact, the determination of analytical solutions for nonlinear equations of any order is not always possible in general; only in a few very special cases are there some specialized methods for obtaining analytical solutions. In the vast majority of cases, one must resort to numerical techniques for constructing the approximate solutions. Thus, this section will be devoted to describing the basic approaches to numerical solution of nonlinear differential equations.

Numerical Methods The basic premise of numerical methods may be summarized as follows: 1.

The differential equation in question has an unknown solution which may be represented as: y

=

Tj(t}

(B.72)

2

This continuous function J}(t) cannot always be obtained in its "true" functional form (explicitly) by analytical techniques so an attempt is made at generating instead the discrete sequence '1(t0 ), 17(~), 17(~), ... , 17(tk), •.. , representing the numerical values it takes at specific discrete points t 0 < t1 < t2 < ... < t" < . .. separated by an interval of size M.

3.

The continuous variables in the differential equation are therefore replaced by discrete variables and the derivatives by appropriate finite difference quotients. The differential equation is thereby converted to a difference equation (in the form of a recurrence relation) which is easily solved especially when programmed on a high-speed digital computer.

4.

Since this process involves some approximation, the discrete sequence generated from the "surrogate" difference equation, which we shall refer to as y(O), y(l), y(2), ... , y(k), ..., will not match the true but unknown values 17(t 0), 17(t1), 17(t2), ••• , 17(t"), •••,precisely; however, approximation errors associated with a good numerical method are usually small.

5.

What differentiates one numerical method from another is the technique adopted for deriving the "surrogate" difference equation from the original differential equation.

Before reviewing specific numerical techniques, we note the following important issues and questions raised by numerical methods: 1.

The numerical solution y(O), y(l), y(2), ... , y(k), ... constitutes only an approximation to the exact but unknown solution because of unavoidable errors introduced in the course of its computation. These errors arise from two separate sources: (a) from the numerical methods formula itseU (which must be obtained by introducing some approximations), and (b) from the fact that in any computation, only a finite number of significant


1171

digits can be retained. The former is referred to as discretization (or

formula) error; the latter as round-off error. 2

Is it always possible to estimate the magnitude of these errors? This question is important from the point of view of evaluating the integrity of the solutions generated by various numerical methods.

3.

As Llt the interval between the discrete points ~~ t1 , t 2, ••. , t1 , ••• tends to zero, will the values taken by the numerical approximation also tend to the values of the exact, but unknown, solution? This question addresses the issue of convergence.

These and other related issues not mentioned here are critical to the successful generation of numerical solutions and are therefore discussed in great detail in texts devoted exclusively to numerical methods. The supplied references [15,16] are recommended to the reader interested in pursuing these aspects further. For now we will review a few of the key techniques used to obtain numerical solutions to differential equations.

The Euler Method In solving the nonlinear first-order equation:

~ given the initial condition y( t0 )

= f(t,y)

(B.73)

=y(O): = t k and therefore replace the

1.

If we let y(k) represent y( t) at the point t continuous J(t,y) withj{t1 , y(k)), and

2.

If we replace the derivative with the finite difference approximation:

(B.74)

then the differential equation becomes:

(B.75)

which is easily rearranged to give: (B.76)

or, simply: (B.77)

W! !/

,j'

1172

APPENDIXB Observe now that starting with the initial value y(O), Eq. (8.77) can be used recursively to generate the sequence y(l), y(2), y(3), . .. as the required approximate solution to Eq. (8.73). The formula in Eq. (8.75) is known as Euler's method; it represents one of the simplest schemes available for solving initial-valu e problems. It has the advantage that it is very straightforward to use, and very easy to program on the computer; its main disadvantag e is that it is not very accurate. It can be shown that the local formula error for Euler's method is proportiona l to (M)2 and to the second derivative of 71(t ), the true, but unknown solution. While it is true, in general, that the accuracy of Euler's method may be improved by reducing the interval size At, unfortunate ly there is a limit beyond which any further reduction in interval size actually worsens the accuracy. Observe that as At is reduced, many more steps are required in going from to to the final point tN; but round-off error is introduced at each step of the calculation so that the accumulate d round-off error actually increases as the interval size At is reduced. Clearly it is possible to cross a point where the increased accuracy achieved by the interval size reduction is offset by the increase in the accumulated round-off error.

The Improved Euler Method The simple Euler method may be improved upon as follows: Let us once again consider the equation: y'(t) =

~ = fl.t,y)

(8.73)

with its initial condition y( fo) = y(O) whose unknown solution is given by tk+I we obtain:

y = 7J(f). By integrating from the point tk to

(8.78)

Observe therefore that in relation to Eq. (8.78) the Euler formula: (8.77)

is equivalent to replacingj[t,7](t)J over the entire interval [t k• tk+ 1] by f(tk,y(k)), the approximat e value taken at the left-hand boundary of the interval. A better approximat ion may be obtained if Jtt, 71( t)) in Eq. (8.78) is replaced by an average of its value at the two boundaries of the interval, i.e:

and upon replacing 1](tk) and 17(tk+l) with their respective approximate values y(k) and y(k + 1) we obtain the improved formula:

COMPLEX ALGEBRA, DIFFERENTIAL AND DIFFERENCE EQNS y(k + 1)

= y(k)

+ t.t {

f(tk ,y(k)) + f(tk+!•y(k + ~

1))}

L,

1173 (B.79)

The only complication now is the appearance of the unknown y(k + 1) in the RHS of Eq. (B.79)i this is resolved by introducing the simpler Euler formula Eq. (B.77) in its place. Simplifying the notation further by using y'(k) to represent f(t k, y(k)) we obtain the final result y(k + 1) = y(k) +

t.t {

y'(k) + f(tk+! ,y(k)+ llty'(k) )} 2

(B.80)

It can be shown that the local formula error for the improved Euler method is proportiona l to (M and since the only reasonable values for M are usually such that IM I
(kl:J.t)" -1

1 s+a

e-at

e-akt>t

1 1- e-a~>tz-1

-

pk

1 1- pz-1

3.

4a.

4b.

1

-

J = z {f(k)}

lim

1

a-+0 ( - )

~~ z)

n-1

5.

1 (s + a) 2

te-at

kMe-akAt

6.

s (s + a) 2

(1- at)e-at

(1 - akllt)e-ak4t

1 - (1 + aM)e-a~>tz- 1 {1 _ e-at>tz-1)2

7a.

a s(s +a)

1-e-t

{1 - e-a~>~~z- 1 (1- z-1)(1- e-aMz- 1)

-

1-pc

(1 -E)z-1 (1- z- 1)(1-pz-1)

e-n • More equations than unknowns (overdefined system). • No "regular" solutions exist (in the sense that no set of values for the unknowns will exactly satisfy all the equations simultaneously). CASE3: m < n • Fewer equations than unknowns ( underdefined system). • An infinite set of solutions exist (in the sense that the excess n - m unknowns can be assigned any arbitrary set of values and the remaining m unknowns solved for in terms of these arbitrary values).

1228

APPENDIXD

For the Case 1 prob lem, as disc usse d in Section D.7, whe neve r I A I"# O, the requ ired solu tion is obtained as given in Eq. (D.35): X

= A- 1b

(D.35)

(It is easy to establish that whe neve r the n x n matrix A is singular, the prob lem redu ces to a Case 3 prob lem with m -no w less than n- being equal to the rank of the rank-deficient A.) In each of the othe r two cases, solutions of the form given in Eq. (D.35) can be obta ined but only after impo sing addi tiona l conditions as we now show.

Overdefined Systems The prob lem here is that there is no vector x for whic h Ax exactly equa ls b. How ever , it is possible to obta in a solu tion in whic h Ax is "closest" tob in the "leas t squa re" sense by finding a vector Q that minimizes: ¢J

= (Ax -bl( Ax -b)

(D.38)

By appl ying the usua l tech niqu es of calculus, it is easy to show that provi ded (ATA) is nonsingular, this "least squa res solution" vector is given by: (D.39a)

or (D.39b)

whic h is of the form in Eq. (D.35) with AL

AL

defined by:

= (ATAriAT

(0.40 )

now play ing the role of A-1. . Note the following abou t AL defined by Eq. (D.40): 1.

ALi sann xmm atrix whil e Ais mxn

2.

ALA = In, the n x n identity matrix, but AA L "#I

3.

AA L = ~ an m x m idempotent matr ix for which I,_ I,_ = I,_

4.

Premultiplying in (2) above by A gives : AALA =A

(D.41)

while post mult iplyi ng by A L gives: (0.42 )

Because only whe n A is multiplied on the ieft by If do we get an identity matr ix, A L is called a left inverse.

REVIEW OF MATRIX ALGEBRA

1229

Underdefined Syst ems The prob lem here is that there is an infinite set of vectors x for whic h Ax exactly equa ls b. How ever, it is possible to obtain a solut ion vector x of minl lnum norm by posin g the following constrained optimizatio n problem:

Min

cp =

t

(XI"x)

subject to: Ax = b

By the usua l meth ods of solvi ng such probl ems (for exam ple, using Lagrange multi plier s) we find that provided (AAT) is nonsingular the requi red "min imum norm solut ion" is: (D.43a) or X

= Ai1>

whic h, once again , is of the form in E"q. (0.35

(D.43b)

) with A R defin ed by:

AR = AT(A A!tl

(D.44)

now playi ng the role of A-1. Again, we note the following abou t AR defin ed by Eq. (0.44): 1.

A R is ann x m matri x while A is m x n

2

AA R =In, them x m identity matrix, but A RA ;i;l

3.

AR A = L:z anoth er n x n idempotent matr ix for whic h, as usual ,

4.

Post multi plyin g the expression in item 2 abov e

Lz Lz =Lz

by A gives: (D.45)

while pre multi plyin g by AR gives: (D.46)

In this case, we obtai n an ident ity matri x only when A is mult iplie d on the right by K; therefore AR is called a right inverse. Noti ng the simil aritie s betw een Eqs. (0.41 ) and (D.45), and also the simil aritie s betw een Eqs. (D.42) and (D.46 ), we can now state the following results: The gene ral syste m of linear algebraic equat ions in Eq. (0.33): Ax= b

(0.33 )

with A as a gene ral m x n matri x (for whic h m is not necessarily equa l to n) has the solution:

1230

APPENDIXD (0.47)

Here A+ is referred to as a generalized (or pseudo) inverse of A, and it satisfies the following conditions:

3.

J

(for the left inverse) (for the right inverse)

Either: A+A= I AA+=I or:

Observe that A-t, the regular inverse of a square, nonsingular matrix satisfies all these conditions. Thus A-t is actually a special case of A+, and t."l)e above is therefore the most general description of the inverse of any matrix. Note, finally, that only A"" 1 is both a left and a right inverse; if Eq. (D.33) is a..1. overdefined system of equations, then a least squares solution can be found only via a left inverse; for an underdefined system of equations, a minimum norm solution can be found only via a right inverse.

ExampleD.S

SOLVING AN OVERDEFINED SYSTEM OF EQUATIONS BY GENERALIZED INVERSES.

To illustrate the use of generalized inverses in the solution of a system of linear algebraic equations, let us consider the set of equations: x 1 + 2x 2

24 - x 2 -2x 1 - x 2

25 -25 -25

a set of three equations in two unknowns. Observe right away that the system is overdefined by one equation, and there is no single set of values for xt and x 2 that will simultaneously satisfy all three equations. These equations may be written in the form Eq. (0.33): Ax = b

(D.33)

In this case:

(D.48)

and from Eq. (0.47), the "least squares" solution is obtained from: (D.47)

where, recalling Eq. (0.40):

*t t '

1231


Now, ATA is obtained by direct multiplication as:

a nonsingular matrix whose determinant is 50; thus:

I [ 6 50 -2

=

(ATArl

~2 ]

so that A+ is given by: 2 A+

14

;0 [ I6 -I3

-10

-5

]

(0.49)

and

14

2 I [ + x=Ab= 50

16 -13

-10 -5

J[:J

which, upon evaluation, gives:

=

X

[:J = [~!]

(050)

We now note that the solution in Eq. (0.50) "satisfies" the system of equations only in the least squares sense; i.e., of all possible x choices, this particular x given in Eq. (0.50) provides an Ax that comes closest to b in the least squares sense. We may also observe that, as asserted earlier:

+

1 [

A A = 50

2

14

16 -13

a 2 x 2 identity matrix; also:

-10 ]

-5

1232

[

rI

APPENDIXD

0~ ~.N ~4]

--0.24

0.82

--0.3

--0.4

--0.3

0.5

(D.5I)

It will be a particularly worthwhile exercise for the reader to carefully multiply this matrix given in Eq. (0.51) by itself and therefore confirm that it is indeed an idempotent matrix.

Example D.9

SOLVIN G AN UNDERDEFINED SYSTEM OF EQUAT IONS BY GENERALIZED INVERSES.

Let us now consider the problem of solving the following set of equation

s:

2

a set of two equations in four unknowns, clearly underdefined. Observe right away that we can arbitrarily assign any values to, say, x , and ll• and solve 1 the resulting equations for Xg and x4 • There is thus an infinite number of solutions. H these equations are written in the form Eq. (033):

Ax

=

b

¥

ti

(D.33)

we would have: 2 A= [

II

3

;J

·=[]

b= [ 1: J

(D.52)

i'

and from Eq. (0.47), the "minimu m norm" solution is obtained from: (D.47)

By direct multiplication, we obtain AAr as:

and hence:

1 [ 4 20 -10

I f

where we now recall from Eq. (0.44) that:

(M1)-1

'[i

-10 ] 30


1233

Thus, we have that A+ is given by:

-10 ] 30

or -{),3

A+ =

[

-{).1

0.1

(D.53)

0.3

and therefore:

X= A+b = [

~:~ 0.1

0.3

00.5 ]

[ 120

J

-{).5

which, upon evaluation, gives:

(0.54)

The reader may now wish to verify that Eq. (0.54) does in fact satisfy the system of equations exactly, and that for the A given in Eq. (0.52) and A+ given in Eq. (0.53), AA + results in the 2 x 2 identity matrix, while A+ A results in the 4 X 4 idemp otent matrix:

[~:: ~:~ ~:~ :: ] 0.1

-{),2

D.9

0.2 0.3 0.1 0.4

0.4 0.7

EIGENVALUES, EIGENVECTORS, AND MATRIX DIAGONALIZATION

The eigen value / eigenvector problem arises in the determ inatio n of the values of a const ant A. for which the following set of n linear algebraic equat ions has nontri vial solutions:

APPENDI XD

1234

1 '

(0.55)

'This can be reexpressed as:

or

=0

(A-AI)x

(0.56)

Being a system of linear homogene ous equations, the solution to Eq. (0.55) or equivalently Eq. (0.56) will be nontrivial (i.e., x 0) if and only if:

"*

=o

IA-A.II

(0.57)

Expandin g this determina nt results in a polynomial of order n in A.:

Do"'111 +a 1....111-1

111-2

+~"'

1 ... +a11 _ 1 .... +a, =

0

(0.58)

This equation (or its determina nt form in Eq. (0.57)) is called the characteristic equation of then x n matrix A; its n roots, A. 11 A. 2, A. 3, •••,A.,.- which maybe real or imaginary, and may or may not be distinct- are the eigenvalues of the matrix. EIGENVALUES OF A 2 x 2 MATRIX.

Example 0.10

Let us illustrate the computatio n of eigenvalues by considering the matrix:

In this case, the matrix (A- AI) is obtained as: (A-A.I)

=[

1 2 3 -4

J_[ A

]. 2 0 ] = [ (1-l) (-4- A) 3 0 A

from which the characteristic equation is obtained as:

lA-AII= -(l-A)(4+ A)-6 = A2 + 3A -10=0 a second-ord er polynomia l (quadratic) with the solution: At.~

= -5,2

1235


Thus in this case the eigenvalues are both real. Note that A- 1 + A. 2 = -3, and that ~~ = -10; also note that Tr(A) = 1 - 4 = -3 and I A I = -10. Thus, we see that ~ + ~ = Tr(A) and A- 1 ~ =I AI. If we change A slightly to:

then the characteristic equation becomes:

lA-AII =

I

I- It

2

-3

4 -.?..

=A? - 5A. + 10

0

whose roots are:

which are complex conjugates. Note again that Tr(A) = 5 = A,. + lt 2 and I A I = 10

=

~~·

Finally if we make another small change in A, i.e.:

then the characteristic equation is: .?..2 -

I +6

=0

which has solutions:

so that the eigenvalues are purely imaginary in this case. Once again Tr(A) ~and IAI =5= ~~·

=0 =.:1. 1 +

Returning now to the original problem in Eqs. (D.55) and (D.56), we note that every distinct eigenvalue It when substituted into Eq. (D.56) gives rise to a corresponding nontrivial solution x; for which: (D.59)

These nontrivial solution vectors are called the eigenvectors of A. Thus every distinct eigenvalue ~._of A has associated with it a corresponding eigenvector x;, j = 1, 2, ... , n. When some eigenvalues are repeated, some specialized considerations - which we will not discuss here - are required. (See the bibliography at the end of the appendix for more details.) We shall restrict ourselves here to the case of distinct eigenvalues. The following are some useful properties of eigenvectors:

1236 1.

2

APPENDIXD Eigenvectors associated with separate, distinct eigenvalues are linearly independent.

The eigenvectors xi are solutio ns of a set of homog eneous linear equatio ns Eq. (0.59) and are thus determ ined only up to a scalar multiplier; i.e., xi and kx.. will both be a solution to Eq. (0.59), if k is a scalar constant. We obtain a normalized eigenv ector X; when the eigenvector xi is scaled by the vector norm xi i.e.:

II II,

_2._

xi= llx.;ll 3.

The normal ized eigenvector

~

is a valid eigenvector since it is a scalar

m~ltiple of ~i; it ~jo~s the sometimes desirab le proper ty that it has

II II- 1.

uruty norm, 1.e., xi

4..

The eigenvectors xi may be calculated by resorti ng to method s of solution of homogeneous linear equations.

It can be shown that the eigenvector xi associated with the eigenvalue A.j is given by any nonzero column of adj(A- A.jl); it can also be shown that there will always

be one and only one independent, nonzero column of adj (AExample D.ll

..:til).

EIGENVECTORS OF A 2 x 2 MATRIX.

Let us illustrate the computation of eigenvectOrs by considering

the 2 x 2 matrix:

for which eigenvalues, A.1 =-5, A.2 =2 were comput ed in Exampl

e 0.10.

For~:

and the adjoint is: adj(A-A - 11)=[ I

-3

-

2

6

]

We may now choose either column of the adjoint as the eigenve ctor 'f, since one is a scalar multipl e of the other; let us choose:

as the eigenvector for ~ =-5. Since the norm of this vector is --J 12 + (-3) 2 = normalized eigenvector corresponding to ~ =-5 will be:

,fTif, the

l I


1237

XI

[~] {10

Similarly for A. 2

= 2:

(A-A. 21)

[~I

2

]

--{j

and

adj (A -..:1.21) =

[~

-2 -1

]

Thus: x2

=[; J

can be chosen as the eigenvect or for eigenvalue this case is:

~

= 2. The normalized eigenvector in

Let us return again to the original eigenval ue/ eigenvector problem , and choose the calculated n linearly indepen dent eigenvectors xi, j = 1, 2, ... , n to be the columns of a matrix M, ;.e.: (0.60)

M is called a modal matrix. Obviously, if the eigenvalues lti ~re distinct, then the eigenvectors xi are all independ ent, and the modal matrix M will be nonsingu lar, and M""1 will a.tways exist. This modal matrix is very useful in converting a matrix A to diagonal form. Recall the eigenval ue/ eigenvector problem: to determin e the values of It i for which nontrivial solutions vectors xi can be found for: j = 1, 2, ... , n

(D.61)

For each j, the complete set of equations may be written more compactly as: (D.62)

If we define the diagonal matrix A as:

APPENDIXD

1238 0

0 0

i]

(D.63)

a matrix whose diagonal elements are the n eigenvalu es of A, and observe that the square bracket on the left-hand side in Eq. (D.62) is just the modal matrix M, then we see immediate ly that the square bracket on the right-han d side is just MA. Thus Eq. (D.62) may be written as: AM= MA

(D.64)

and premultip lying both sides by M-1 thus yields: M-1AM =A

(D.65)

Then x n matrix A is thus reduced to what is known as the diagonal canonical fonn by using the transform ation Eq. (D.65). By pre- and postrnulti plying both sides of Eq. (D.65) by M and M-1 respectively, one obtains the companion relation: (D.66)

These two relations are very useful in the solution of linear algebraic and differenti al equations as shall be seen later. Note that the latter relation in Eq. (D.66) indicates that a square matrix A can be decompo sed into three matrices involving only its eigenvalu es and eigenvecto rs. Example 0.12

MATRIX DIAGONALIZATION.

To illustrate matrix diagonaliz ation,let us consider the 2 x 2 matrix:

discussed in Examples D.lO and D.ll. Recall that the eigenvalue s and eigenvecto rs were determined to be:

Thus the modal matrix M is:

1239


while

[ ]

M""l

2

1 7

--::;

3 7

1 7

and the diagonal eigenvalue matrix is:

Evaluation of the LHS of Eq. (0.65) yields:

M"IAM

[; ;][: [ t -~] [ ~

1

7

7

2 -4 ] [

~3

2

] 0

4 -5 15

2

] =[ :

2

]

which verifies Eq. (0.65). Similarly, evaluation of the RHS of Eq. (0.66) gives:

verifying the equation.

The following are some general properties of eigenvalues worth noting: 1.

The sum of the eigenvalues of a matrix is equal to the trace of that matrix, i.e.: II

Tr(A)

= •L )...J J=l

2.

The product of the eigenvalues of a matrix is equal to the determinant of that matrix, i.e.: n

IAI =

I1 A..

j=l J

1240

APPENDIX D

3.

A singular matrix has at least one zero eigenvalue.

4.

If the eigenvalues of A are~, .:lz, \, ... , A.n, then the eigenv alues of A -1 are 1/ Av 1/ A.2 , 1/ A.3 , •••, 1/ An.

5.

The eigenvalues of a diagonal or a triangular matrix are identical to the elements on the main diagonal.

6.

Given any nonsi ngula r matrix T, the matrices A, and A =T AT-1 have identical eigenvalues. In this case, such matrices A and A are called similar matrices.

D.lO SINGULAR VALUE DECOMPOSITION The concept of matrix diagonalization indicated in Eq. (0.65) and the comp anion relation in Eq. (0.66) can be exten ded to all matrices, squar e and nonsq uare, by singul ar value decomposition (SVD). The main SVD result may be summ arized as follows: For any real m x n matrix A, it is always possible to find orthog onal (i.e., unitary) matrices Wand V such that: (D.67)

Here,

r is the m X n matrix: r

= [ : :J:wi iliS

·[:

0

0

a2

0

0

0

_:

]

(D.68)

o;.

where, for p = min(m,n), the diagonal elements of S, <Ji ~ a2 ~ ••• ~a,

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