Handbook Of Pi And Pid Controller Tuning Rules

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HANDBOOK OF

PI AND PID CONTROLLER TUNING RULES

3rd Edition

P575.TP.indd 1

3/20/09 5:16:12 PM

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HANDBOOK OF

PI AND PID CONTROLLER TUNING RULES 3rd Edition

Aidan O’Dwyer Dublin Institute of Technology, Ireland

ICP P575.TP.indd 2

Imperial College Press 3/20/09 5:16:18 PM

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF PI AND PID CONTROLLER TUNING RULES (3rd Edition) Copyright © 2009 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-1-84816-242-6 ISBN-10 1-84816-242-1

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

Steven - Hdbook of PI & PID (3rd).pmd

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Dedication

Once more, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian.

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Preface

Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for over seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past two decades, when work by K.J. Åström, T. Hägglund and F.G. Shinskey, among others, sparked a revival of interest in the use of this ‘workhorse’ of controller implementation. There is strong evidence that PI and PID controllers remain poorly understood and, in particular, poorly tuned in many applications. It is clear that the many controller tuning rules proposed in the literature are not having an impact on industrial practice. One reason is that the tuning rules are not accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book, now in its third edition, is to bring together and summarise, using a unified notation, tuning rules for PI and PID controllers. The author restricts the work to tuning rules that may be applied to the control of processes with time delays (dead times); in practice, this is not a significant restriction, as most process models have a time delay term. In this edition, the structure of the book has been modified from the previous edition, with controller tuning rules for non-self-regulating process models being summarised in a different chapter from those for self-regulating process models. It is the author’s belief that this book will be useful to control and instrument engineering practitioners and will be a useful reference for students and educators in universities and technical colleges. I wish to thank the School of Electrical Engineering Systems, Dublin Institute of Technology, for providing the facilities needed to complete the book. Finally, I am deeply grateful to my mother, Lillian, and my father, Sean, for their inspiration and support over many years and to my family, Angela, Catherine and Fiona, for their love and understanding. Aidan O’Dwyer

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Contents

Preface

........................................................................................................

vii

1. Introduction ........................................................................................................ 1.1 Preliminary Remarks ...................................................................................... 1.2 Structure of the Book......................................................................................

1 1 2

2. Controller Architecture........................................................................................... 2.1 Introduction .................................................................................................. 2.2 Comments on the PID Controller Structures . ................................................ 2.3 Process Modelling .......................................................................................... 2.3.1 Self-regulating process models . ........................................................ 2.3.2 Non-self-regulating process models................................................... 2.4 Organisation of the Tuning Rules ...................................................................

4 4 11 12 12 14 16

3. Controller Tuning Rules for Self-Regulating Process Models ............................... 3.1 Delay Model ................................................................................................. 3.1.1 Ideal PI controller – Table 2 .............................................................. 3.1.2 Ideal PID controller – Table 3 .......................................................... 3.1.3 Ideal controller in series with a first order lag – Table 4 ................ 3.1.4 Classical controller – Table 5 ........................................................... 3.1.5 Generalised classical controller – Table 6 ........................................ 3.1.6 Two degree of freedom controller 1 – Table 7 .................................. 3.2 Delay Model with a Zero ............................................................................... 3.2.1 Ideal PI controller – Table 8 . ............................................................ 3.3 FOLPD Model ............................................................................................... 3.3.1 Ideal PI controller – Table 9 .............................................................. 3.3.2 Ideal PID controller – Table 10 ........................................................ 3.3.3 Ideal controller in series with a first order lag – Table 11 ................ 3.3.4 Controller with filtered derivative – Table 12.................................... 3.3.5 Classical controller – Table 13 ........................................................ 3.3.6 Generalised classical controller – Table 14 . ..................................... 3.3.7 Two degree of freedom controller 1 – Table 15 ............................. 3.3.8 Two degree of freedom controller 2 – Table 16 ............................. 3.3.9 Two degree of freedom controller 3 – Table 17 .............................

18 18 18 23 24 25 26 27 28 28 30 30 78 118 122 134 149 152 168 170

ix

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Handbook of PI and PID Controller Tuning Rules

3.4 FOLPD Model with a Zero ............................................................................ 3.4.1 Ideal PI controller – Table 18 ............................................................ 3.4.2 Ideal controller in series with a first order lag – Table 19 ................ 3.5 SOSPD Model .............................................................................................. 3.5.1 Ideal PI controller – Table 20 .......................................................... 3.5.2 Ideal PID controller – Table 21 ........................................................ 3.5.3 Ideal controller in series with a first order lag – Table 22 ................ 3.5.4 Controller with filtered derivative – Table 23.................................... 3.5.5 Classical controller – Table 24 ........................................................ 3.5.6 Generalised classical controller – Table 25 . ..................................... 3.5.7 Two degree of freedom controller 1 – Table 26 ............................. 3.5.8 Two degree of freedom controller 3 – Table 27 ............................. 3.6 SOSPD Model with a Zero ............................................................................ 3.6.1 Ideal PI controller – Table 28 ......................................................... 3.6.2 Ideal PID controller – Table 29 ........................................................ 3.6.3 Ideal controller in series with a first order lag – Table 30 ............... 3.6.4 Controller with filtered derivative – Table 31.................................... 3.6.5 Classical controller – Table 32 ......................................................... 3.6.6 Generalised classical controller – Table 33 ....................................... 3.6.7 Two degree of freedom controller 1 – Table 34 .............................. 3.6.8 Two degree of freedom controller 3 – Table 35 .............................. 3.7 TOSPD Model .............................................................................................. 3.7.1 Ideal PI controller – Table 36 . .......................................................... 3.7.2 Ideal PID controller – Table 37 ......................................................... 3.7.3 Ideal controller in series with a first order lag – Table 38 ................ 3.7.4 Controller with filtered derivative – Table 39 .................................... 3.7.5 Two degree of freedom controller 1 – Table 40 ................................. 3.7.6 Two degree of freedom controller 3 – Table 41 . ............................... 3.8 Fifth Order System Plus Delay Model ........................................................... 3.8.1 Ideal PID controller – Table 42 ......................................................... 3.8.2 Controller with filtered derivative – Table 43 .................................... 3.8.3 Two degree of freedom controller 1 – Table 44 . ............................... 3.9 General Model . .............................................................................................. 3.9.1 Ideal PI controller – Table 45 ........................................................... 3.9.2 Ideal PID controller – Table 46 ......................................................... 3.9.3 Ideal controller in series with a first order lag – Table 47 ................ 3.9.4 Controller with filtered derivative – Table 48 .................................. 3.9.5 Two degree of freedom controller 1 – Table 49 . .............................. 3.10 Non-Model Specific ...................................................................................... 3.10.1 Ideal PI controller – Table 50 ............................................................ 3.10.2 Ideal PID controller – Table 51 ......................................................... 3.10.3 Ideal controller in series with a first order lag– Table 52 . ................ 3.10.4 Controller with filtered derivative – Table 53 ................................... 3.10.5 Classical controller – Table 54 ......................................................... 3.10.6 Generalised classical controller – Table 55 ......................................

180 180 182 183 183 206 232 236 238 251 253 264 277 277 279 282 284 286 288 289 292 293 293 296 297 298 299 302 303 303 305 308 310 310 312 315 316 317 318 318 324 332 336 341 343

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Contents

xi

3.10.7 Two degree of freedom controller 1 – Table 56 . .............................. 346 3.10.8 Two degree of freedom controller 3 – Table 57 . .............................. 349 4. Controller Tuning Rules for Non-Self-Regulating Process Models ....................... 4.1 IPD Model ..................................................................................................... 4.1.1 Ideal PI controller – Table 58 . .......................................................... 4.1.2 Ideal PID controller – Table 59 ...................................................... 4.1.3 Ideal controller in series with a first order lag – Table 60 ................ 4.1.4 Controller with filtered derivative – Table 61 .................................. 4.1.5 Classical controller – Table 62 ......................................................... 4.1.6 Generalised classical controller – Table 63 ..................................... 4.1.7 Two degree of freedom controller 1 – Table 64 . .............................. 4.1.8 Two degree of freedom controller 2 – Table 65 . .............................. 4.1.9 Two degree of freedom controller 3 – Table 66 . .............................. 4.2 IPD Model with a Zero ................................................................................. 4.2.1 Ideal PI controller – Table 67 . .......................................................... 4.3 FOLIPD Model .............................................................................................. 4.3.1 Ideal PI controller – Table 68 ......................................................... 4.3.2 Ideal PID controller – Table 69 ........................................................ 4.3.3 Ideal controller in series with a first order lag – Table 70 ................ 4.3.4 Controller with filtered derivative – Table 71 ................................... 4.3.5 Classical controller – Table 72 ......................................................... 4.3.6 Generalised classical controller – Table 73 ...................................... 4.3.7 Two degree of freedom controller 1 – Table 74 .............................. 4.3.8 Two degree of freedom controller 2 – Table 75 .............................. 4.3.9 Two degree of freedom controller 3 – Table 76 .............................. 4.4 FOLIPD Model with a Zero ........................................................................ 4.4.1 Ideal PID controller – Table 77 . ....................................................... 4.4.2 Ideal controller in series with a first order lag – Table 78 ............... 4.4.3 Classical controller – Table 79 ......................................................... 4.5 I 2 PD Model..................................................................................................... 4.5.1 Ideal PID controller – Table 80 . ....................................................... 4.5.2 Classical controller – Table 81 ........................................................ 4.5.3 Two degree of freedom controller 1 – Table 82 .............................. 4.5.4 Two degree of freedom controller 2 – Table 83 ................................ 4.5.5 Two degree of freedom controller 3 – Table 84 .............................. 4.6 SOSIPD Model ............................................................................................. 4.6.1 Ideal PI controller – Table 85 . .......................................................... 4.6.2 Two degree of freedom controller 1 – Table 86 . .............................. 4.7 SOSIPD Model with a Zero............................................................................ 4.7.1 Classical controller – Table 87 ......................................................... 4.8 TOSIPD Model............................................................................................... 4.8.1 Two degree of freedom controller 1 – Table 88 ..............................

350 350 350 359 364 366 368 371 372 378 381 383 383 385 385 388 392 394 395 397 399 416 418 420 420 422 423 424 424 425 426 427 429 430 430 431 436 436 437 437

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4.9 General Model with Integrator ....................................................................... 4.9.1 Ideal PI controller – Table 89 . .......................................................... 4.9.2 Two degree of freedom controller 1 – Table 90 ................................ 4.10 Unstable FOLPD Model ................................................................................ 4.10.1 Ideal PI controller – Table 91 ............................................................ 4.10.2 Ideal PID controller – Table 92 ........................................................ 4.10.3 Ideal controller in series with a first order lag – Table 93 ............... 4.10.4 Classical controller – Table 94 ....................................................... 4.10.5 Generalised classical controller – Table 95 ....................................... 4.10.6 Two degree of freedom controller 1 – Table 96 ................................ 4.10.7 Two degree of freedom controller 2 – Table 97 .............................. 4.10.8 Two degree of freedom controller 3 – Table 98 .............................. 4.11 Unstable FOLPD Model with a Zero ........................................................... 4.11.1 Ideal PI controller – Table 99 ............................................................ 4.11.2 Ideal controller in series with a first order lag – Table 100 ............. 4.11.3 Generalised classical controller – Table 101 .................................... 4.11.4 Two degree of freedom controller 1 – Table 102 ............................ 4.12 Unstable SOSPD Model (one unstable pole) ................................................ 4.12.1 Ideal PI controller – Table 103 ........................................................ 4.12.2 Ideal PID controller – Table 104 . ..................................................... 4.12.3 Ideal controller in series with a first order lag – Table 105 .............. 4.12.4 Classical controller – Table 106 ........................................................ 4.12.5 Two degree of freedom controller 1 – Table 107 .............................. 4.12.6 Two degree of freedom controller 3 – Table 108 .............................. 4.13 Unstable SOSPD Model (two unstable poles) ............................................... 4.13.1 Ideal PID controller – Table 109 ...................................................... 4.13.2 Generalised classical controller – Table 110 . ................................... 4.13.3 Two degree of freedom controller 2 – Table 111 .............................. 4.14 Unstable SOSPD Model with a Zero .............................................................. 4.14.1 Ideal PI controller – Table 112 . ........................................................ 4.14.2 Ideal controller in series with a first order lag – Table 113 .............. 4.14.3 Generalised classical controller – Table 114 ................................... 4.14.4 Two degree of freedom controller 1 – Table 115 .............................. 4.14.5 Two degree of freedom controller 3 – Table 116 ..............................

438 438 439 440 440 447 455 458 462 463 473 475 480 480 481 483 484 486 486 488 490 491 497 503 506 506 508 509 511 511 513 516 518 520

5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers ................................................................... 5.1 Introduction ................................................................................................... 5.2 The Analytical Determination of Gain and Phase Margin ............................. 5.2.1 PI tuning formulae ............................................................................... 5.2.2 PID tuning formulae ........................................................................... 5.3 The Analytical Determination of Maximum Sensitivity ................................ 5.4 Simulation Results ..........................................................................................

521 521 522 522 525 529 529

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Contents

5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for All Values of Delay................................................................................... 5.5.1 PI controller design.............................................................................. 5.5.1.1 Processes modelled in FOLPD form...................................... 5.5.1.2 Processes modelled in IPD form ........................................... 5.5.2 PID controller design .......................................................................... 5.5.2.1 Processes modelled in FOLPD form – classical controller.... 5.5.2.2 Processes modelled in SOSPD form – series controller......... 5.5.2.3 Processes modelled in SOSPD form with a negative zero – classical controller ...................................................... 5.5.3 PD controller design . .......................................................................... 5.6 Conclusions . ..................................................................................................

xiii

534 534 534 536 539 539 541 542 542 543

Appendix 1 Glossary of Symbols and Abbreviations................................................. 544 Appendix 2 Some Further Details on Process Modelling........................................... 551 Bibliography

........................................................................................................ 565

Index

........................................................................................................ 599


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Chapter 1

Introduction

1.1 Preliminary Remarks The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu (1991), for example, suggest that there are perhaps 5–10% of control loops that cannot be controlled by single input, single output (SISO) PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements (Hwang, 1993; Åström and Hägglund, 1995). It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers (Bialkowski, 1996) and that in process control applications, more than 95% of the controllers are of PID type (Åström and Hägglund, 1995). PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often (Isermann, 1989). However, despite decades of development work, surveys indicating the state of the art of control in industrial practice report sobering results. For example, Ender (1993) states that in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned). The situation does not appear to have improved more recently, as Van Overschee and De Moor (2000) report that 80% of PID controllers are badly tuned; 30% of PID controllers operate in manual with another 30% of the controlled loops increasing the short term variability of the process to be controlled (typically due to too strong integral action). The authors state that 25%

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of all PID controller loops use default factory settings, implying that they have not been tuned at all. These and other surveys (well summarised by Yu, 1999, pp. 1–2) show that the determination of PI and PID controller tuning parameters is a vexing problem in many applications. The most direct way to set up controller parameters is the use of tuning rules; obviously, the wealth of information on this topic available in the literature has been poorly communicated to the industrial community. One reason is that this information is scattered in a variety of media, including journal papers, conference papers, websites and books for a period of over seventy years. The purpose of this book is to bring together, in summary form, the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. 1.2 Structure of the Book Tuning rules are set out in the book in tabular form. This form allows the rules to be represented compactly. The tables have four or five columns, according to whether the controller considered is of PI or PID form, respectively. The first column in all cases details the author of the rule and other pertinent information. The final column in all cases is labelled ‘Comment’; this facilitates the inclusion of information about the tuning rule that may be useful in its application. The remaining columns detail the formulae for the controller parameters. Chapter 2 explores the range of PI and PID controller structures proposed in the literature. It is often forgotten that different manufacturers implement different versions of the PID controller algorithm (in particular); therefore, controller tuning rules that work well in one PID architecture may work poorly in another. This chapter also details the process models used to define the controller tuning rules. Chapters 3 and 4 detail, in tabular form, PI and PID controller tuning rules (and their variations), as applied to a wide variety of self-regulating process models and non-self-regulating process models, respectively. One-hundred-andsixteen such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table starts on its own page; each table is preceded by the controller used, together with a block diagram showing the unity feedback closed loop arrangement of the controller and process model. In Chapter 5, analytical calculations of the gain and phase margins of a large sample of PI and PID controller tuning rules are determined, when the process is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios

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of time delay to time constant of the process model. Results are given in graphical form. An important feature of the book is the unified notation used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods used to determine process model parameters; this information is presented in summary form, as this topic could provide data for a book in itself. However, sufficient information, together with references, is provided for the interested reader. Finally, a comprehensive reference list is provided. In particular, the author would like to recommend the contributions by McMillan (1994), Åström and Hägglund (1995), (2006), Shinskey (1994), (1996), Tan et al. (1999a), Yu (1999), Lelic and Gajic (2000) and Ang et al. (2005) to the interested reader, which treat comprehensively the wider perspective of PID controller design and application.

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Chapter 2

Controller Architecture

2.1 Introduction The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows:

U (s ) = G c (s ) E (s )

(2.1)

with G c (s) = K c (1 +

1 + Td s) Ti s

(2.2)

and with K c = proportional gain, Ti = integral time constant and Td = derivative time constant. If Ti = ∞ and Td = 0 (i.e. P control), then it is clear that the closed loop measured value y will always be less than the desired value r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs. Many variations of the PID controller structure have been proposed (indeed, the PI controller structure is itself a subset of the PID controller structure). As Tan et al. (1999a) suggest, one important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. A substantial number of the variations in the controller structures used may be 4

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summarised by controller structure ‘supersets’. In this book, nine such supersets are specified, which allows a sensible restriction on the number of tables that need to be detailed. The controller structures specified are detailed below. 1. The ideal PI controller structure:

 1   G c (s) = K c 1 +  Ti s 

(2.3)

2. The ideal PID controller structure:   1 G c (s) = K c 1 + + Td s   Ti s 

(2.4)

This controller structure has also been labelled the ‘non-interacting’ controller (McMillan, 1994), the ‘ISA’ algorithm (Gerry and Hansen, 1987) or the ‘parallel non-interacting’ controller (Visioli, 2001). A variation of the controller is labelled the ‘parallel’ controller structure (McMillan, 1994). G c (s ) = K c +

1 + Td s Ti s

(2.5)

This variation has also been labelled the ‘ideal parallel’, ‘non-interacting’, ‘independent’ or ‘gain independent’ algorithm. The ideal PID controller structure is used in the following products: (a) Allen Bradley PLC5 product (McMillan, 1994) (b) Bailey FC19 PID algorithm (EZYtune, 2003) (c) Fanuc Series 90–30 and 90–70 Independent Form PID algorithm (EZYtune, 2003) (d) Intellution FIX products (McMillan, 1994) (e) Honeywell TDC3000 Process Manager Type A, non-interactive mode product (ISMC, 1999) (f) Leeds and Northrup Electromax 5 product (Åström and Hägglund, 1988) (g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune, 2003).

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3. Ideal controller in series with a first order lag:   1 1 + Td s  G c (s) = K c 1 +  Ti s  Tf s + 1

(2.6)

4. Controller with filtered derivative:

  Td s 1 G c (s ) = K c  1 + +  Ti s 1 + s Td  N 

     

(2.7)

This structure is used in the following products: (a) Bailey Net 90 PID error input product with N = 10 (McMillan, 1994) and FC156 Independent Form PID algorithm (EZYtune, 2003) (b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003) (c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8 (EZYtune, 2003) (d) Foxboro EXACT I/A series PIDA product (in which it is an option labelled ideal PID) (Foxboro, 1994) (e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune, 2003) (f) Modicon 984 product with 2 ≤ N ≤ 30 (McMillan, 1994; EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10 (ISMC, 1999) and the S7 FB41 CONT_C PID product (EZYtune, 2003).

5. Classical controller: This controller is also labelled the ‘cascade’ controller (Witt and Waggoner, 1990), the ‘interacting’ or ‘series’ controller (Poulin and Pomerleau, 1996), the ‘interactive’ controller (Tsang and Rad, 1995), the ‘rate-before-reset’ controller (Smith and Corripio, 1997), the ‘analog’ controller (St. Clair, 2000) or the ‘commercial’ controller (Luyben, 2001).  1  1 + sTd  G c (s) = K c 1 +  Ti s  1 + s Td N

(2.8)

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7

The structure is used in the following products: (a) Honeywell TDC Basic/Extended/Multifunction Types A and B products with N = 8 (McMillan, 1994) (b) Toshiba TOSDIC 200 product with 3.33 ≤ N ≤ 10 (McMillan, 1994) (c) Foxboro EXACT Model 761 product with N = 10 (McMillan, 1994) (d) Honeywell UDC6000 product with N = 8 (Åström and Hägglund, 1995) (e) Honeywell TDC3000 Process Manager product – Type A, interactive mode with N = 10 (ISMC, 1999) (f) Honeywell TDC3000 Universal, Multifunction and Advanced Multifunction products with N = 8 (ISMC, 1999) (g) Foxboro EXACT I/A Series PIDA product (in which it is an option labelled series PID) (Foxboro, 1994). A subset of the classical PID controller is the so-called ‘series’ controller structure, also labelled the ‘interacting’ controller or the ‘analog algorithm’ (McMillan, 1994) or the ‘dependent’ controller (EZYtune, 2003).  1  (1 + sTd ) G c (s) = K c 1 +  Ti s 

(2.9)

The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (Åström and Hägglund, 1995) (c) Foxboro EXACT 760/761 product (Åström and Hägglund, 1995). A further related structure is one labelled the ‘interacting’ controller (Fertik, 1975).   Td s 1   1 + G c (s) = K c 1 +  Ti s  1 + s Td N 

     

The structure is used in the following products: (a) Bailey FC156 Classical Form PID product (EZYtune, 2003) (b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003).

(2.10)

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6. Generalised classical controller:   Td s 1 + G c (s) = K c 1 + T  Ti s 1+ d  N 

  2  b f 0 + b f 1s + b f 2 s  1 + a f 1s + a f 2 s 2 s  

   

(2.11)

7. Two degree of freedom controller 1:      [1 − β]Td s R (s) − K 1 + 1 + Td s 1 + U (s) = K c  [1 − α] + c Td  T   Ti s Ti s 1 1+ d + s    N  N  

  Y(s) (2.12)  s 

This controller is also labelled the ‘m-PID’ controller (Huang et al., 2000), the ‘ISA-PID’ controller (Leva and Colombo, 2001) and the ‘P-I-PD (only P is DOF) incomplete 2DOF algorithm’ (Mizutani and Hiroi, 1991). This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with α = −1 , β = −1 , N = 10 (McMillan, 1994) (c) Omron E5CK digital controller with β = 1 and N = 3 (ISMC, 1999). Notable subsets of this controller structure are:    Td s 1  1 R (s) − K c 1 + • U (s) = K c 1 + + T  Ti s  Ti s  1+ d  N 

  Y(s)  s 

(2.13)

which is used in the following products: (a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and Logix5550 products (EZYtune, 2003) (b) Modcomp product with N = 10 (McMillan, 1994).

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   1  1 R (s) − K c 1 + • U (s) = K c 1 + + Td s Y (s) Ti s  Ti s   

9

(2.14)

Also labelled the ‘PI+D’ controller structure (Chen, 1996) or the dependent, ideal, non-interacting controller structure (Cooper, 2006a), it is used in the following products: (a) ABB 53SL6000 product (ABB, 2001) (b) Genesis product (McMillan, 1994) (c) Honeywell TDC3000 Process Manager Type B, non-interactive mode product (ISMC, 1999) (d) Square D PIDR PID product (EZYtune, 2003).  1    1 R (s) − K c 1 + • U (s) = K c  + Td s Y(s) Ti s  Ti s   

(2.15)

which is used in the following products: (a) Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al., 1987) (b) Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999). 8. Two degree of freedom controller 2:     T s 1 d  1 + b f 1s (F(s)R (s) − Y(s) ) − K Y(s) , + U(s) = K c 1 + 0  T  1 + a f 1s Ti s 1+ d s  N   1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = (2.16) 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4

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9. Two degree of freedom controller 3: U (s) = F1 (s)R (s) − F2 (s)Y(s) ,     ( ) 1 − β T s 1 + b f 1s [1 − χ] + δ + d  , F1 (s) = K c  [1 − α ] +  Td  1 + a f 1s + a f 2 s 2 Ti s 1 + Ti s 1+ s   N       1+ b f 2s [ 1 − φ] [1 − ϕ]Td s   [ ] + F2 (s) = K c 1 − ε +   T Ti s 1 + a f 3s + a f 4 s 2 1+ d s   N   K (b + b f 4 s ) − 0 f3 (2.17) 1 + a f 5s Notable subsets of this controller structure are:

    Td s 1   R (s) − 1 + • U (s) = K c 1 + Y(s)    Ti s    1 + (Td N )s  

(2.18)

Also labelled the ‘reset-feedback’ controller structure (Huang et al., 1996), it is used in the following products: (a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products (ISMC, 1999) (b) Moore Model 352 Single-Loop Controller product (Wade, 1994).   1 + Td s 1   R (s) − • U(s) = K c 1 + Y(s)  1 + (Td N )s Ti s   

(2.19)

Also labelled the ‘industrial controller’ structure (Kaya and Scheib, 1988), it is used in the following products: (a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999; McMillan, 1994) (b) Foxboro Model 761 product with N = 10 (McMillan, 1994) (c) Fischer-Porter Micro DCI product (McMillan, 1994)

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11

(d) Moore Products Type 352 controller with 1 ≤ N ≤ 30 (McMillan, 1994) (e) SATT Instruments EAC400 product with N = 8.33 (McMillan, 1994) (f) Taylor Mod 30 ESPO product with N = 16.7 (McMillan, 1994) (g) Honeywell TDC3000 Process Manager Type B, interactive mode product with N = 10 (ISMC, 1999).   Kc 1  1 + sTd  R (s) − K c 1 + • U (s) = Tis  Tis  1 + sTd N 

  Y (s)   

(2.20)

which is used in the following products: (a) Honeywell TDC3000 Process Manager Type C, interactive mode product with N = 10 (ISMC, 1999) (b) Honeywell TDC3000 Universal, Multifunction and Advanced Multifunction products with N = 8 (ISMC, 1999). 2.2 Comments on the PID Controller Structures

In some cases, one controller structure may be transformed into another; clearly, the ideal and parallel controller structures (Equations 2.4 and 2.5) are very closely related. It is shown by McMillan (1994), among others, that the parameters of the ideal PID controller may be worked out from the parameters of the series PID controller, and vice versa. The ideal PID controller is given in Equation (2.22) and the series PID controller is given in equation (2.23).   1 G cp (s) = K cp 1 + + Tdp s   Tip s   

(2.22)

 1  (1 + sTds ) G cs (s) = K cs 1 +  Tis s 

(2.23)

Then, it may be shown that  T K cp = 1 + ds Tis 

  Tis  K cs , Tip = (Tis + Tds ) , Tdp =    Tis + Tds

 Tds . 

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Similarly, it may be shown that, provided Tip > 4Tdp ,

  T  , T = 0.5T 1 + 1 − 4 dp is ip   Tip    Tdp  . Tds = 0.5Tdp 1 − 1 − 4  Tip   

 Tdp K cs = 0.5K cp 1 + 1 − 4  Tip 

 ,  

Åström and Hägglund (1996) point out that the ideal controller admits complex zeroes and is thus a more flexible controller structure than the series controller, which has real zeroes; however, in the frequency domain, the series controller has the interesting interpretation that the zeroes of the closed loop transfer function are the inverse values of Tis and Tds . O’Dwyer (2001b) developed a comprehensive set of tuning rules for the series PID controller, based on the ideal PID controller; as these are not strictly original tuning rules, only representative examples are included in the relevant tables. In a similar manner, it is straightforward to show that controller structures (2.6), (2.7) and (2.8) are subsets of a more general controller structure, and controller parameters may be transformed readily from one structure to another. 2.3 Process Modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables (see Chapters 3 and 4). These modelling strategies are outlined in Appendix 2. Process models may be classified as selfregulating or non-self-regulating, and the models that fall into these categories are now detailed. 2.3.1 Self-regulating process models

1. 2.

Delay model: G m (s) = K m e −sτm Delay model with a zero: G m (s) = K m (1 + Tm 3 s)e − sτ m or G m (s) = K m (1 − Tm 4 s)e − sτ m

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13

3.

First order lag plus time delay (FOLPD) model: G m (s) =

4.

FOLPD model with a zero:

K m e −sτ m 1 + sTm

K m (1 − sTm 4 )e − sτ m K m (1 + sTm 3 )e −sτm or G m (s) = G m (s ) = 1 + sTm1 1 + sTm1 5.

Second order system plus time delay (SOSPD) model: K m e − sτ m or G m (s) = G m (s) = 2 (1 + Tm1s )(1 + Tm 2 s ) Tm1 s 2 + 2ξ m Tm1s + 1 K m e − sτ m

6.

SOSPD model with a zero: G m (s ) =

G m (s) = 7.

K m (1 + sTm 3 )e − sτm 2

Tm1 s 2 + 2ξ m Tm1s + 1

or G m (s) =

K m (1 − sTm 4 )e −sτ m K m (1 − sTm 4 )e − sτ m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1

Third order system plus time delay (TOSPD) model: G m (s ) =

K m (1 + b1s + b 2 s 2 + b 3s 3 )e − sτ m or 1 + a 1s + a 2 s 2 + a 3s 3

(

G m (s ) =

)

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) G m (s) =

(T

m1

8.

K m (1 + sTm 3 )e −sτm (1 + sTm1 )(1 + sTm 2 )

K m (Tm 3 s + 1)e −sτ m

)

s + 2ξ m Tm1s + 1 (Tm 2 s + 1)

2 2

Fifth order system plus delay model: G m (s ) =

K m (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm 1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5

(

)

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9. General model. 10. Non-model specific. Corresponding tuning rules may also apply to non-selfregulating processes. 2.3.2 Non-self-regulating process models

1.

Integral plus time delay (IPD) model: G m (s) =

2.

IPD model with a zero: G m (s) =

3.

K m e −sτ m s

K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s s

First order lag plus integral plus time delay (FOLIPD) model: K m e −sτ m G m (s) = s(1 + sTm )

4.

FOLIPD model with a zero: G m (s) =

5. 6.

7. 8.

K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s(1 + sTm1 ) s(1 + sTm1 )

Integral squared plus time delay ( I 2 PD ) model: G m (s) =

K m e − sτ m

s2 Second order system plus integral plus time delay (SOSIPD) model: K m e −sτ m K m e −sτ m G ( s ) or = G m (s ) = m 2 2 s(1 + sTm1 ) s(Tm1 s 2 + 2ξ m Tm1s + 1)

K m (1 + sTm 3 )e − sτ m s(1 + sTm1 )(1 + sTm 2 ) Third order system plus integral plus time delay (TOSIPD) model: SOSIPD model with a zero: G m (s) =

G m (s) =

K m e − sτ m K m e − sτ m or G m (s) = 3 s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) s(1 + sTm1 )

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9.

15

General model with integrator: G m (s) =

K m e − sτ m

s(1 + sTm1 )

n

or

K G m (s ) = m s

∏ (T ∏ (T i

i

)∏ (T s + 1)∏ (T

m1i s + 1

m 3i

i

i

m 2i m 4i

2 2

) e s + 1)

s + 2ξ m ni Tm 2i s + 1

2 2

s + 2ξ mdi Tm 4i

−sτ m

K m e −sτm Tm s − 1 11. Unstable FOLPD model with a zero:

10. Unstable FOLPD model: G m (s) =

G m (s) =

K m (1 + sTm 3 )e −sτm K (1 − sTm 4 )e − sτ m or G m (s) = m Tm s − 1 Tm1s − 1

12. Unstable SOSPD model (one unstable pole): G m (s ) =

K m e − sτm (Tm1s − 1)(1 + sTm 2 )

G m (s ) =

K m e − sτ m (Tm1s − 1)(Tm 2 s − 1)

13. Unstable SOSPD model (two unstable poles):

14. Unstable SOSPD model with a zero:

G m (s) =

K m (1 + Tm 3 s )e −sτ m 2

Tm1 s 2 − 2ξ m Tm1s + 1

or

K m (1 + Tm 3 s )e − sτ m 2

− Tm1 s 2 + 2ξ m Tm1s + 1 G m (s ) =

or

K m (1 − sTm 4 )e − sτ m 2

Tm1 s 2 − 2ξ m Tm1s + 1

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16

Table 1 shows the number of tuning rules defined for each PI/PID controller structure and types of process model. The following data is key to the process model type: Model 1: Stable FOLPD model (with or without a zero) Model 2: Stable SOSPD model (with or without a zero) Model 3: Other stable models Model 4: Non-model specific Model 5: Models with an integrator Model 6: Open loop unstable models Table 1: PI/PID controller structure and tuning rules – a summary Process model Controller Equation (2.3) (2.4) (2.6) (2.7) (2.8) (2.11) (2.12) (2.16) (2.17) Total

1

2

3

4

5

6

261 140 20 36 74 7 74 7 28 649

63 82 23 9 53 7 36 3 15 291

58 20 3 5 1 1 14 0 1 103

59 63 9 11 12 4 9 0 2 169

90 66 16 6 26 5 106 16 8 339

32 35 17 0 17 6 37 8 30 182

Total 563 406 88 67 183 30 276 34 84 1731

Table 1 shows that for the tuning rules defined, 60% are based on a selfregulating (or stable) model, 10% are non-model specific, with the remaining 30% based on a non-self-regulating model. 2.4 Organisation of the Tuning Rules

The tuning rules are organised in tabular form in Chapters 3 and 4. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process reaction

curve methods). (ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or optimum servo action.

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17

(iii) Tuning rules that give a specified closed loop response (direct synthesis

tuning rules). Such rules may be defined by specifying the desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin. (iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built into the design process. (v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods). (vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or to achieve magnitude and frequency information at a particular phase lag. Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. All symbols used in the tables are defined in Appendix 1.

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Chapter 3

Controller Tuning Rules for Self-Regulating Process Models

3.1 Delay Model G m (s) = K m e − sτm

 1   3.1.1 Ideal PI controller G c (s) = K c 1 +  Ti s  R(s)

+

U(s)

 1   K c 1 + Ti s  

E(s)

−

K m e −sτ

Y(s) m

Table 2: PI controller tuning rules – G m (s) = K m e −sτ m

Rule 1

Callender et al. (1935/6). Model: Method 1

Kc

Ti

0.568 K m τ m

Process reaction 3.64τ m

2

0.65 K m τ m

2.6τ m

3

0.79 K m τ m

3.95τ m

4

0.95 K m τ m

3.3τ m

1

Decay ratio = 0.015; period of decaying oscillation = 10.5τ m .

2

Decay ratio = 0.1; period of decaying oscillation = 8τ m .

3


4

Decay ratio = 0.33; period of decaying oscillation = 5.56τ m . 18

Comment

Representative results deduced from graphs.

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule

Kc

Ti

Comment

Two constraints method – Wolfe (1951). Model: Method 1

0.2 K mτm

0.3τ m

Decay ratio is as small as possible.

Minimum ISE – Haalman (1965). Model: Method 1 Minimum error: step load change – Gerry (1998). Minimum IAE – Gerry and Hansen (1987). Minimum IAE – Shinskey (1988), p. 123. Minimum IAE – Shinskey (1994). Model: Method 1

Minimum IAE – Shinskey (1988), p. 148. Minimum IAE – Shinskey (1996), p. 167. Minimum IAE – Huang and Jeng (2002). Minimum ISE – Keviczky and Csáki (1973). Model: Method 1

Fertik (1975). Model: Method 1 Åström and Hägglund (2000). Model: Method 1

Minimum error integral (regulator mode). Minimum performance index: regulator tuning undefined 1.5τ m G c (s) = 1 Tis M s = 1.9 ; A m = 2.36 ; φ m = 50 0 .

0.3 Km

0.42τ m

Model: Method 1

0.345 Km

0.455τ m

Model: Method 1

0.43 Km

0.5τ m

Model: Method 1

0. 4 K m

0.5τ m

p. 67.

undefined

1.6K m τ m

G c (s) = 1 Tis ; p. 63.

0.4255 Km

0.25Tu

Model: Method 1

0. 4 K u

0.25Tu

Model: Method 1

Minimum performance index: servo tuning Model: Method 1 0.36 K m 0.47τ m

Minimum IAE = 1.377 τ m . A m = 2 ; φ m = 60 0 . undefined

1.33τ m

G c (s) = 1 Tis ; Minimum ISE = 1.53τ m ; K m = 1 .

0.5

0.635τ m

Km = 1

Minimum performance index: other tuning 0.35 K m 0.4 τ m K c , Ti values deduced from graph. K c Ti maximised 0.25 0.35τ m Km subject to M max = 2 .

19

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20 Rule

Kc

Ti

Comment

Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1

0.16 K m

0.340τ m

M max = 1.4

0.200 K m

0.318τ m

M max = 1.6

0.26 K m

0.306τ m

M max = 2.0

x1 K m

x 2τm

Åström and Hägglund (2004). Model: Method 1

x1

Kc

Ti

undefined

2.70K m τ m

undefined

2.50K m τ m

undefined

x 2τm

1

OS

x2

0.64

105% x1

1.00

x1

OS

x1

5%

0.72

0.68

Kc =

2

Kc =

(

Stochastic disturbance. Critically damped dominant pole. Damping factor (dominant pole) = 0.6. G c (s) = 1 Tis

Coefficient values: OS OS x2

x2

x2

OS

17%

2

4%

Coefficient values: OS OS x1

x1

OS

50% 1.5 2τ m

0.75

Kc

Ti

15% 0.78 20% Model: Method 1

0.223607 K m

0.309017 τm

Model: Method 1

Kuwata (1987).

1

2

)

10%

(

)

m

1 − 0.267 τm − 0.25τm .

e − ωd τ m τ 1 − 0.5e − ωd τ m , Ti = − ωmτ 1 − 0.5e − ωd τ m . Km e d m

(

M max

0.400 1.1 0.211 0.342 1.6 0.389 1.2 0.227 0.334 1.7 0.376 1.3 0.241 0.326 1.8 0.363 1.4 0.254 0.320 1.9 0.352 1.5 0.264 0.314 2.0 Direct synthesis: time domain criteria Step disturbance. 0. 5 K m 0.5τ m

Bryant et al. (1973). Model: Method 1; G c (s) = 1 Tis

Nomura et al. (1993).

x2

0.057 0.103 0.139 0.168 0.191

Van der Grinten (1963). Model: Method 1

Keviczky and Csáki (1973). Model: Method 1; representative coefficient values estimated from graphs; K m = 1 .

Coefficient values: M max x1

x2

0.4τm

K m 1 − 1 − 0.267 τm

) , T = 1.25τ i

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models

Rule Kamimura et al. (1994), Manum (2005). Model: Method 1; G c (s) = 1 Tis .

Åström and Hägglund (1995), pp. 187–189. Model: Method 1

Kc

Ti

Comment

undefined

2K m τ m

OS = 10%

Also given by Schlegel (1998); M s = 1 . undefined

2.6667 K m τ m

undefined

2.3529 K m τ m

0.305 K m

0.279τm

‘Quick’ servo response; ‘negligible’ overshoot. ξ = 0.3

0.273 K m

0.285τm

ξ = 0.4

OS = 0%

0.218 K m

0.288τm

ξ = 0.6

0.174 K m

0.276τm

ξ = 0.8

0.154 K m

0.265τm

ξ = 0.9

Åström and Hägglund (1995), pp. 187–189, Åström and Hägglund (2006), pp. 185–186. Model: Method 1

0.388 K m

0.258τm

ξ = 0.1

0.343 K m

0.270τm

ξ = 0.2

0.244 K m

0.288τm

ξ = 0.5

0.195 K m

0.284τm

ξ = 0.707

0.135 K m

0.250τm

ξ = 1.0

Hansen (2000).

0. 2 K m

0.3τ m

Model: Method 1

Buckley (1964). Model: Method 1; M max = 2dB . Schlegel (1998). Model: Method 1 Hägglund and Åström (2004). Model: Method 1 Åström and Hägglund (2006), p. 243.

Direct synthesis: frequency domain criteria undefined 1.45K m τ m G c (s) = 1 Tis 0.075 K m

0.1τ m

0.53 K m

τm

0.55 K m

1.59 τ m

0.318 K m

0.405τm

0.15 K m

0.35τ m

0.15 K u

0.17Tu

0.1677 K m

0.363τm

Representative results. Ms = 1 M s = 1.4

Model: Method 1; M s = M t = 1. 4 .

Robust

Bequette (2003), p. 300.

τm K m (2λ + τ m )

0.5τ m

Skogestad (2003). Model: Method 1

0.294 K m

0.5τ m

Model: Method 1; λ > τm (Yu, 2006, p. 19). IMC based rule.

21

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22 Rule

Kc

Ti

Comment

1 0.5τm K m 0.5τm + λ

0.5τm

Model: Method 1

undefined

x1K m τm

G c (s) = 1 Tis ;

Skogestad (2003).

0.25 K m

Other tuning 0.333τ m

Model: Method 1

McMillan (2005), p. 35.

0.25 K m

0.25τ m

Model: Method 1

Yu (2006), p. 18.

0.30K u

Ultimate cycle 0.25Tu

Model: Method 1

Åström and Hägglund (2006), p. 189. Urrea et al. (2006). Model: Method 1

3

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3

1.5 ≤ x1 ≤ 2 .

λ = Tm (aggressive tuning). λ = 3Tm (robust tuning).

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  1 + Td s  3.1.2 Ideal PID controller G c (s) = K c 1 +  Ti s  R(s)

E(s)

−

+

  1 K c 1 + + Td s  T s i  

Y(s)

U(s)

K me

− sτ m

Table 3: PID controller tuning rules – delay model G m (s) = K m e − sτ m

Rule Callender et al. (1935/6). Model: Method 1 Ream (1954). Model: Method 1 Nomura et al. (1993). Model: Method 1

Kc

Ti

Td

Comment

0.749 K m τm

Process reaction 2.734 τ m 0.303τ m

0.2269 K m

0.3109τm

0.0264τm

‘Butterworth’.

0.2635 K m

0.3610τm

0.1911τm

0.3548 K m

0.4860τm

0.2735τm

‘Minimum ITAE’. ‘Binomial’.

0.511 K m

0.511τm

0.1247 τm

Model: Method 1

0.40 K m

0.46τm

0.149τm

Regulator.

Representative results deduced 2 1.31τ m 0.303τ m 1.24 K m τm from graphs. Direct synthesis: time domain criteria Decay ratio 0.61 K m 0.491τm 0.015τm ≈ 33% . ‘Kitamori’. 0.230 K m 0.315τm 0.0086τm

1

Robust

Gong et al. (1998). Wang (2003), pp. 15–17. Model: Method 1

Servo; ±10% overshoot. Deduced from graphs; robust to ±25% uncertainty in process model parameters. 0.29 K m

0.40τm

0.096τm

1


2


23

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24

3.1.3 Ideal controller in series with a first order lag

 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s R(s)

+

E(s)

−

  1 K c 1 + + Td s  Ti s  

1 1 + Tf s

Y(s)

U(s)

K m e −sτ

m

Table 4: PID controller tuning rules – delay model G m (s) = K m e −sτ m

Rule

Kc

Ti

Td

Comment

Robust

Bequette (2003), p. 300. Model: Method 1

τm K m (4λ + τ m )

0.5τ m

0.167 τ m

λ > 1.7 τm (Yu, 2006, p. 19).

Tf = 2

2λ − 0.167τ m 4λ + τ m

2

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   1  1 + Td s    3.1.4 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   E(s)

R(s)

−

+

     1  1 + Td s  U(s)  K c 1 + Td  Ti s   s 1 + N  

Y(s) K m e − sτ

m


Rule

Kc

Ti

Td

Comment

Process reaction

Hartree et al. (1937). Model: Method 1

1

2

0.70 K mτm

2.66τ m

τm

0.78 K mτm

2.97 τ m

0.5τ m

1

N = 2. Decay ratio = 0.15; period of decaying oscillation = 4.49 τ m .

2

N = 2. Decay ratio = 0.042; period of decaying oscillation = 5.03τ m .

Representative results.

25

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26

3.1.5 Generalised classical controller

  Td s 1 + G c (s) = K c 1 + T  Ti s 1+ d  N  R(s)

E(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

Rule Yu (2006), p. 18. Model: Method 1

  2  b f 0 + b f 1s + b f 2 s 2  1 + a f 1s + a f 2 s s 

U(s)


K m e − sτ

Y(s) m

Table 6: PID controller tuning rules – delay model K me −sτ m Comment Kc Ti Td 0.30K u

Ultimate cycle 0 0.23Tu

b f 0 = 0 , b f 1 = 0.12Tu , b f 2 = 0 , a f 1 = 0.012Tu , a f 2 = 0 .

   

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3.1.6 Two degree of freedom controller 1   Td s 1 + U (s ) = K c  1 +  Ti s 1 + Td  N 

     βTd s   E (s ) − K c  α +  R (s ) Td    s 1+ s   N   

    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

−

+

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    + s 

−

U(s)

Y(s) K me

− sτ m


Rule

Kc

Kuwata (1987). Model: Method 1 Nomura et al. (1993). Model: Method 1

1

Kc =

(

1

Ti

Kc

0.2 K m

0.278τm

1 Km

1.667 τm − 2.5

0.8τm

K m 1 − 1 − 0.267 τm

).

Comment

Td

Direct synthesis: time domain criteria 0.278τm 0 1.667 τm − 4.167

0.1 K m

27

0

α =1

α = 1 ; also given by Kuwata (1987).

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28

3.2 Delay Model with a Zero

G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4 s)e −sτ m

 1   3.2.1 Ideal PI controller G c (s) = K c 1 +  Ti s  E(s)

R(s)

+

 1   K c 1 +  T is  

−

Y(s)

U(s) Model

Table 8: PI controller tuning rules – delay model with a zero G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4s)e −sτ m

Rule

Kc

Ti

Comment

Direct synthesis: time domain criteria 1 ‘Smaller’ τm Tm 4 . Ti undefined

Chidambaram (2002), p. 157. Model: Method 1; G c (s) = 1 Tis .

2

( 1 + 4.712(T

1

Ti = 0.637 τm 1 + 3.142 Tm 4 τm

2

2

Ti = 0.424τm

2

m4

τm

). ).

Ti

‘Larger’ τm Tm 4 .

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Rule

Jyothi et al. (2001). Model: Method 1; G c (s) = 1 Tis .

Jyothi et al. (2001). Model: Method 1; G c (s) = 1 Tis .

Kc

Ti

Comment

Direct synthesis: frequency domain criteria 3 τm Tm3 < 1 Ti Ti

τm Tm3 > 1

5

Ti

0 < τm Tm 3 < 1

6

Ti

0 < τm Tm 3 < 10

4

undefined

1.5K m (τ m − Tm 3 )

2K m Tm 4

undefined

3

Ti = 0.64K m τm 1 + 9.870(Tm3 τm ) .

4

Ti = 1.27 K m τm 1 + 2.467(Tm3 τm ) .

5

Ti = 2.0309K mTm3e0.419692 τ m

6

Ti =

7

Ti = 1.27 K m τm 1 + 2.467(Tm 4 τm ) .

8

Ti =

9

Ti =

τm Tm 4 < 1

Ti

τm Tm 4 > 1

8

Ti

0 < τm Tm 4 < 1

9

Ti

0 < τm Tm 4 < 10

7

1.5K m (τm + Tm 4 )

2

2

Tm 3

.

K mTm3 . exp[− 0.706636 ln (τm Tm3 ) − 0.962232] 2

K m Tm 4

0.5 + 0.001945(τm Tm 4 ) − 0.03091(τm Tm 4 )2 K m Tm 4

.

0.6178 − 0.1452(τm Tm 4 ) + 0.0141(τm Tm 4 )2 − 0.0004883(τm Tm 4 )3

.

29

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30

3.3 FOLPD Model G m (s) =

K m e −sτm 1 + sTm


E(s)

+

−

U(s)

 1   K c 1 + Ti s  

Y(s)

K m e − sτ 1 + sTm

m

Table 9: PI controller tuning rules – FOLPD model G m (s) = Rule Callender et al. (1935/6). Model: Method 1 Ziegler and Nichols (1942). Model: Method 2 Hazebroek and Van der Waerden (1950). Model: Method 2

K m e − sτ m 1 + sTm

Comment

Kc

Ti

1

0.568 K m τ m


2

0.690 K m τ m

2.45τ m

τm = 0.3 Tm

0.9Tm K mτm

3.33τ m

Quarter decay ratio; τm Tm ≤ 1 .

x1Tm K m τm

x 2τm

τm Tm

x1

x2

Coefficient values: x1 x2 τm Tm

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.84 0.87

7.14 4.76 3.70 3.03 2.50 2.17 1.92 1.75 1.61

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.90 0.93 0.96 0.99 1.02 1.06 1.09 1.13 1.17

1.49 1.41 1.32 1.25 1.19 1.14 1.10 1.06 1.03

1


2


τm Tm

x1

x2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81

1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80

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Kc

Hazebroek and Van Tm der Waerden (1950) – K m τm continued. Oppelt (1951). Model: Method 2 Moros (1999). Model: Method 1

3

4

Kc ≤

1 Km

τm Tm > 3.5

1.66τ m

τ m >> Tm

Kc

3.32τ m

τ m 0.35

1.00Tm K m τ m

3.0τ m

τ m Tm = 0.2

1.04Tm K m τ m

2.25τ m

τ m Tm = 0.5

1.08Tm K m τ m

1.45τ m

τ m Tm = 1

1.39Tm K m τ m

τm

τ m Tm = 2

0.214 − 0.346

τm Tm

 τm    T   m

, 1.977

0.874  TL    Kc = K m  Tm 

− 0.099 + 0.159

τm Tm

 τm    T   m

0.871  TL    Kc = K m  Tm 

− 0.015 + 0.384

τm Tm

 τm    T   m

τm − 0.55 Tm 

0.513  TL  Kc = K m  Tm

   

0.218

τm Tm

 τm  T  m

   

−1.451

,

TL τ ≤ 2.641 m + 0.16 . Tm Tm

, − 4.515

τm Tm

+ 0.067

 τm    T   m

0.876

, 2.641

τm T + 0.16 ≤ L ≤ 1 . Tm Tm

−1.055

, T T  Ti = m  L  0.444  Tm 

9

1.123

τ   m T   m

−1.041

T T  Ti = m  L  0.415  Tm  8

τm ≤ 0.35 Tm

−1.256

T T  Ti = m  L  0.214  Tm  7

Comment

Kc

Minimum IAE – Yu (1988). Model: Method 1. Load disturbance K e −sτ L . model = L 1 + sTL

35

T T , Ti = m  L 0.670  Tm

   

− 0.003

− 0.217

τm Tm

τm − 0.213 Tm 

− 0.084

τ   m T   m

 τm  T  m

   

0.56

.

0.867

, 1
0.35

Ti 0.945

1.346  Tm  K m  τ m

τm

1.157  TL    Kc = K m  Tm 

τm ≤ 0.35 Tm

Kc

0.2 0.4 0.6 0.8

15

Comment

15

1.279  Tm  K m  τ m

1 Km

Minimum ITAE – Frank and Lenz (1969). Model: Method 1

Ti

41

− 0.889

T T , Ti = m  L 0.596  Tm

   

−1.112

0.372

τm Tm

τm − 0.094 Tm 

− 0.44

τ   m T   m

 τm  T  m

   

0.898

, 1
0.35

Ti

0.304

τm − 0.112 Tm 

τ   m T   m

0.196

,

TL τ ≤ 2.385 m + 0.112 . Tm Tm

−1.055

, − 5.809

τm Tm

+ 0.241

 τm    T   m

0.901

, 2.385

τm T + 0.112 ≤ L ≤ 1 . Tm Tm

−1.042

, T T  Ti = m  L  0.431  Tm 

21

x1

2.33 0.47 0.59 4.00 0.42 0.52 9.00 0.375 0.47 Model: Method 1; τm ≤ 0.35 . Tm

Ti

T T  Ti = m  L  0.425  Tm  20

τm Tm

Ti

T T  Ti = m  L  0.805  Tm  19

Comment

Ti

Minimum ITAE with x1 K m OS ≤ 8% – Fertik x1 x2 τm (1975). T m Model: Method 3; 5.0 0.53 x 1 , x 2 deduced from 0.11 0.25 2.6 0.74 graphs. 0.43 1.45 0.80 18 Minimum ITAE – Kc Yu (1988). 19 Kc Load disturbance 20 K L e −sτ L Kc . model = 1 + sTL 21 Kc

18

FA

− 0.909

T T , Ti = m  L 0.794  Tm

   

− 0.148

0.228

τm Tm

τm − 0.365 Tm 

τ   m T   m

− 0.257

 τm  T  m

   

0.901

, 1
0.2 and Martin (1983).  0.3 + 0.3 Tm + 0.4 τ m T Km  τm  m Model: Method 1 Minimum IAE – 1.4 K m 0.72τ m τm Tm = 0.11 Marlin (1995), K c , Ti deduced from graphs; robust to ±25% process model pp. 301–307. parameter changes. Model: Method 1 Minimum IAE – τ 0.1 ≤ m ≤ 1 29 Huang et al. (1996). T Kc i Tm Model: Method 1 Minimum IAE – Model: Method 1; 0.6Tm Smith and Corripio T 0 .1 ≤ τm Tm ≤ 1.5 . m K mτm (1997), p. 345. Minimum IAE – Minimum IAE = 0.59Tm Huang and Jeng T 2.1038τ m ; m Kmτm (2002). τ m Tm < 0.2 . Model: Method 1

29

Kc =

−1.0169 3.5959  τ  τ  1  τ  + 1.1075 m  − 13.0454 − 9.0916 m + 0.3053 m  Km  Tm   Tm   Tm   

+

τ 1  − 2.2927 m Km   Tm 

  

3.6843

τm  + 4.8259e Tm  ,  

2 3 4  τ   τ  τ  τ Ti = Tm 0.9771 − 0.2492 m + 3.4651 m  − 7.4538 m  + 8.2567 m    Tm  Tm    Tm   Tm  

 τ + Tm  − 4.7536 m   Tm 

5

τ   + 1.1496 m  Tm 

  

6

.  

b720_Chapter-03

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Rule Minimum IAE – Arrieta Orozco (2003). Model: Method 10 Minimum IAE – Tavakoli and Fleming (2003). Model: Method 1 Minimum IAE – Tavakoli et al. (2006). Minimum IAE – Harriott (1988). Ti coefficients deduced from graph.

30

1 Kc = Km

Kc 30

Comment

Ti

0.1 ≤

Ti

Kc

τm ≤ 1.2 Tm

Also given by Arrieta Orozco and Alfaro Ruiz (2003). 31

0.1 ≤

Ti

Kc

τm ≤ 10 Tm

Minimum A m = 6dB ; minimum φ m = 60 0 . 32

Kc

0.5K u

Ti

Model: Method 42; A m ≥ 3 ; φ m ≥ 60 0 .

x 2Tu

Model: Method 1

x2

τm Tm

x2

τm Tm

x2

τm Tm

1.61 0.94

0.2 0.5

0.61

1.0

0.48

2.0

1.2099  0.8714        0.2438 + 0.5305 Tm   , Ti = Tm 0.9377 + 0.4337 τm  . τ  T      m  m       

31

Kc =

   1  Tm τ + 0.3047 , Ti = Tm 0.4262 m + 0.9581 . 0.4849 Km  τm Tm   

32

Kc =

1 Km

 Tm  + 0.071 , Ti = min[Tm + 0.143τ m ,9τ m ] . 0.5 τ m  

47

b720_Chapter-03

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48 Rule Small IAE – Hwang (1995). Model: Method 44 Minimum ISE – Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. Minimum ISE – Khan and Lehman (1996). Model: Method 1 Minimum ITAE – Rovira et al. (1969). Model: Method 5

33

17-Mar-2009

µ1 =

(

Ti

Comment

K c µ 2 K u ωu

Decay ratio = 0.1; τ 0.1 ≤ m ≤ 2.0 . Tm

Kc 33

(1 − µ1 )K u

0.980  Tm  K m  τ m

  

0.892

1.072  Tm  K m  τ m

  

0.560

Tm

τ 0.690 − 0.155 m Tm Tm

τ 0.648 − 0.114 m Tm

0.01 ≤ τm Tm ≤ 0.2

35

Kc

Ti

0.2 ≤ τm Tm ≤ 20

0.586  Tm  K m  τ m

  

Tm

0.916

τ 1.030 − 0.165 m Tm

0.1 ≤

τm ≤ 1.0 Tm

)

)

1.11 1 − 0.467ωu τm + 0.0657ω2u τ2m , K u K m (1 + K u K m )

(

τm ≤ 2.0 Tm

Ti

µ2 = µ1 =

1.1 ≤

Kc


(

τm ≤ 1.0 Tm

34

µ2 = µ1 =

0.1 ≤

)

(

)

0.0328 − 1 + 2.21ωu τm − 0.338ω2u τ2m , r = 0.5; K u K m (1 + K u K m )

(

)

0.0477 − 1 + 2.07ωu τm − 0.333ω2u τ2m , r = 0.75; K u K m (1 + K u K m )


(

)

0.0609 − 1 + 1.97ωu τm − 0.323ω2u τ2m , r = 1.0; K u K m (1 + K u K m ) r = parameter related to the position of the dominant real pole. µ2 =

34

35

 0.7388 0.3185  Tm  0.7388Tm + 0.3185τ m   K c =  +  K , Ti = τ m  − 0.0003082T + 0.5291τ  . T τ m m  m m    m  0.808 0.511  0.808Tm + 0.511τm − 0.255 Tm τm  0.255  Tm . Kc =  + − , Ti = τm   τm  Tm  0.095Tm + 0.846τm − 0.381 Tm τm  Tm τ m  K m 

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Rule

Kc

Ti

49

Comment

Minimum ITAE with x1 K m x 2 (τm + Tm ) OS ≤ 8% – Fertik x1 x2 x1 x1 x2 τm τm τm x2 (1975). T T T m m m Model: Method 3; x 1 , x 2 deduced from 0.11 3.05 0.95 0.67 0.71 0.76 2.33 0.42 0.59 0.25 1.70 0.88 1.00 0.55 0.71 4.00 0.39 0.52 graphs. 0.43 1.05 0.82 1.50 0.47 0.65 9.00 0.365 0.47 Minimum ITAE – τ 0.1 ≤ m ≤ 1.2 36 Arrieta Orozco Ti Kc Tm (2003). Also given by Arrieta Orozco and Alfaro Ruiz (2003). Model: Method 10 Minimum ITAE – 2 ≤ Am ≤ 4 ; 37 Arrieta Orozco Ti Kc Model: Method 10 (2006), pp. 90–92. τ Minimum ITAE – − 0.9707 m Tm 38 Model: Method 1 Barberà (2006). 0 . 9894 e Kc

36

37

1 Kc = Km

1.5426  1.0382        . 0.1440 + 0.5131 Tm   , Ti = Tm 0.9953 + 0.2073 τ m  τ  T      m  m       

−1.0871+ 0.0048 A m + 0.0047 A m 2    τm  1  2 ,  Kc = 0.9639 − 0.4436A m + 0.0496A m + x1   Km  Tm     2

x1 = 1.3296 − 0.4355A m + 0.0509A m , 0.1 ≤ τm Tm ≤ 2.0 ; 6.1845 − 3.4585 A m + 0.4764 A m 2    τm  2 ,   Ti = Tm 2.7836 − 1.2511A m + 0.1692A m + x 2    Tm     2

x 2 = 0.8298 − 0.1193A m + 0.0083A m , 0.1 ≤ τm Tm < 1.0 ; 1.6705 − 0.2509 A m − 0.0447 A m 2    τm  2  ,  Ti = Tm 2.0174 − 0.5089A m + 0.0311A m + x 2    Tm     2

x 2 = 1.0065 − 0.4874A m + 0.0889A m , 1.0 ≤ τm Tm ≤ 2.0 . 38

Kc =

1 . 2    τm  τ  + 2.0214 m + 0.00911 K m 1.5853 Tm  Tm     

b720_Chapter-03

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50 Rule

Kc

Ti

Comment

Smith (2002).

0.586Tm K m τm

Tm

Model: Method 1

Approximately minimum ITAE. Minimum ISTSE – Model: Method 1;  1  Tm + 0.2 Fukura and Tamura 0.56 Tm + 0.36τm 0.05 ≤ τm Tm ≤ 1 . Km  τm  (1983). 0.921 Minimum ISTSE – τ Tm 0.712  Tm  0.1 ≤ m ≤ 1.0   Zhuang (1992), T 0 . 968 − 0 . 247 τ T m m m K m  τm  Zhuang and Atherton (1993). 0.559 τ Tm 0.786  Tm  1.1 ≤ m ≤ 2.0 Model: Method 1 or   T 0 . 883 − 0 . 158 τ T m m m K m  τm  Method 5 or Method 6 or Method 15. 39 Minimum ISTSE – Model: Method 1; Ti Kc Zhuang (1992). 0.1 ≤ τm Tm ≤ 2.0 . Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Minimum ISTSE – Majhi (2005). Model: Method 52

39

40

41

0.361K u 41

40

Ti

Ti

Kc

Model: Method 1 Model: Method 37

0.712  Tm  K m  τ m

  

0.921

0.712Tm 0.694 − 0.182 τm Tm

0.1 ≤

τm ≤ 1.0 Tm

0.780  Tm  K m  τ m

  

0.543

0.780Tm 0.683 − 0.120 τm Tm

1.1 ≤

τm ≤ 2.0 Tm

0.673  Tm  K m  τ m

  

0.331

0.673Tm 0.511 − 0.066 τm Tm

2.1 ≤

τm ≤ 3.5 Tm

 4.264 − 0.148K m K u   K u , Ti = 0.083(1.935K m K u + 1)Tu . K c =    12.119 − 0.432K m K u  τ Ti = 0.083(1.935K m K u + 1)Tu , 0.1 ≤ m ≤ 2.0 . Tm ∧   ∧ τm  ∧  1.506K m K u − 0.177  ∧ Kc =   K u , Ti = 0.055 3.616K m K u + 1 T u , 0.1 ≤ T ≤ 2.0 . ∧  3.341K K u + 0.606    m m  

b720_Chapter-03

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FA


Rule Minimum ISTES – Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15.

Kc 0.569  Tm  K m  τ m

  

0.951

0.628  Tm  K m  τ m

  

0.583

Nearly minimum IAE and ITAE – Hwang and Fang (1995).

43

44

τ 1.023 − 0.179 m Tm Tm

τ 1.007 − 0.167 m Tm

0.1 ≤

τm ≤ 1.0 Tm

1.1 ≤

τm ≤ 2.0 Tm

Ti

Kc

x1 K m

44

42

Tm

Model: Method 30; 0.1 ≤ τ m Tm ≤ 2.0 ; decay ratio = 0.03. Minimum performance index: other tuning Model: Method 30; 43 Ti 0.1 ≤ τ m Tm ≤ 2.0 ; Kc decay ratio = 0.05. 42

Simultaneous servo/regulator tuning – Hwang and Fang (1995). Minimum performance index – Wilton (1999). Model: Method 1; coefficients of K c estimated from graphs.

Comment

Ti

Tm

Coefficient values: b τ m Tm

τ m Tm

x1

x1

b

0.1 0.1 0.1 0.1

750.0 17.487 4.8990 2.1213

0.0001 0.04 0.2 1

0.2 0.2 0.2 0.2 0.2

3.3675 2.2946 1.7146 1.1313 0.8764

0.008 0.04 0.2 1 5

0.4 0.4 0.4 0.4 0.4

1.6837 1.5297 1.2247 0.9192 0.7668

0.008 0.04 0.2 1 5

0.5 0.5 0.5 0.5

150.01 7.1386 2.6944 1.1314

0.0001 0.04 0.2 1

2  τ τ  (K c / K u ωu ) K c =  0.438 − 0.110 m + 0.0376 m 2 K u , Ti = . 2    T  Tm  m   0.0388 + 0.108 τm − 0.0154 τm  2  Tm Tm   2  τ τ  (K c / K u ωu ) K c =  0.46 − 0.0835 m + 0.0305 m 2 K u , Ti = . 2    T  T m m    0.0644 + 0.0759 τm − 0.0111 τm  2  Tm Tm  

Performance index =

∫ [e (t) + bK ∞

0

2

2 m

(y(t ) − y∞ )2 ]dt .

51

b720_Chapter-03

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FA


52 Rule Wilton (1999) – continued.

Minimum ITAE – ABB (2001). Model: Method 7 Ream (1954). Model: Method 1; decay ratio ≈ 33% . Bryant et al. (1973). Model: Method 1

Kc

Ti

x1 K m

Tm

τ m Tm

x1

b

τ m Tm

x1

b

0.6 0.6 0.6 0.6 0.6

1.1225 1.1218 0.9798 0.7778 0.6572

0.008 0.04 0.2 1 5

0.8 0.8 0.8 0.8 0.8

0.8980 0.9178 0.7348 0.7071 0.6025

0.008 0.04 0.2 1 5

1.0 1.0

72.004 1.6657

0.0001 0.2

1.0 1.0

3.6713 0.04 0.8485 1 τm < 0.5 Tm

0.68

 τ   1.4837Tm  m    Tm  Direct synthesis: time domain criteria 1.56 K m 0.368(τm + Tm ) τm Tm = 1.12 0.85 K m

0.454(τm + Tm )

τm Tm = 2.86

0.37Tm K m τm

Tm

0.40Tm K m τm

Tm

undefined

x 2K mτm

Critically damped dominant pole. Damping ratio (dominant pole) = 0.6. G c = 1 (Tis )

Coefficient values (deduced from graph): x2 τ m Tm x2

0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67

Mollenkamp et al. (1973). Model: Method 1

0.977

0.8591  Tm  K m  τ m

τ m Tm

Chiu et al. (1973). Model: Method 1

Comment

12.5 11.1 11.1 9.1 7.1 6.3 5.6 4.3

1.0 2.0 10.0

6.7 4.2 3.0

Critically damped dominant pole.

1.0 2.0 10.0

3.2 2.1 1.6

Damping ratio (dominant pole) = 0.6.

λ variable; suggested values: 0.2, 0.4, 0.6, 1.0. λ = 1.0 , OS < 10%, Model: Method 11 (Pomerleau and Poulin, 2004). Appropriate for Tm T ‘small τ m ’. m K (T + τ ) λTm K m (1 + λτ m )

m

CL

m

Tm

b720_Chapter-03

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FA


Rule

Kc

Ti

Comment

0.44Tm Kmτm

Tm

0.04 ≤ τ m Tm ≤ 1.4 ; deduced from graph.

0.43Tm K mτm

Tm

1% overshoot – servo; τm Tm > 0.2 .

1 Km

Tm

53

undefined 1 (x 2 τm ) G c = 1 Tis Keviczky and Csáki x 2 coefficient values (estimated from graph): (1973); 0% 5% 10% 15% 20% OS values in first τ m Tm OS row; τ m Tm values 0.1 30 22 16 12 10 0.2 18 11 9 7.5 6 in first column; 0.5 8 6 5 4 3.5 Km = 1. 1.0 5 4 3.5 3 2.5 Model: Method 1 2.0 4 3 2.5 2.2 2 5.0 3 2.4 2 1.8 1.6 10.0 2.5 2.1 1.8 1.7 1.5 5% overshoot – servo 0.52Tm K m τm Tm 0.04 ≤ τm Tm ≤ 1.4 – Smith et al. (1975). Model: Method 1. Also given by Bain and Martin (1983). 1% overshoot – servo – Smith et al. (1975). Model: Method 1 Bain and Martin (1983). Model: Method 1

Ti

‘Very conservative tuning for small delay’. Model: Method 1

0.5Tm K m τm

Tm

OS = 10%

0.5Tm K m τm

Tm

Model: Method 1

Kuwata (1987). Suyama (1992). Model: Method 1 5% overshoot – servo – Smith and Corripio (1997), p. 346. Vítečková (1999), Vítečková et al. (2000a) (Model: Method 9 – Vítečková et al., 2000b).

45

Kc =

45

Kc

φ m = 60 0 ; 0.25 ≤ τm Tm ≤ 1 (Voda and Landau, 1995). x1Tm K m τm x1

OS (%)

0.368 0.514 0.581 0.641

0 5 10 15

Tm

Coefficient values: OS (%) x1 0.696 0.748 0.801 0.853

20 25 30 35

x1

OS (%)

0.906 0.957 1.008

40 45 50

0.4(Tm + τm ) 0.5 − , Ti = 1.25x1 - 0.25(Tm + τm ) , with K m (Tm + τm − x1 ) K m x1 =

(Tm + τm )2 − 0.267τm 2 (3Tm + τm ) .

b720_Chapter-03

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54 Rule Okada et al. (2006). Model: Method 1

x1

Kc

Ti

x1Tm K m τm

Tm

ξ

Comment

Coefficient values: ξ x1 ξ x1

x1

ξ

x1

ξ

0.368 1.0 0.454 0.8 0.578 0.6 0.765 0.4 1.06 0.2 0.407 0.9 0.510 0.7 0.661 0.5 0.896 0.3 1.29 0.1 Mann et al. (2001). 0.51Tm K m τm Tm 0 < τm Tm < 2 Model: Method 37 or 46 Ti Kc Method 41. Closed loop response overshoot (step input) < 5%. 47 undefined G c = 1 (Tis ) Ti Vítečková and 48 Ti Kc Víteček (2002). Model: Method 1 For 0.05 < τ T < 0.8 , T = T ; overshoot is under 2% m

m

i

m

(Vítečková and Víteček, 2003). Pinnella et al. (1986). Model: Method 24

46

undefined

49

Gc = Ki s

Ki

Critically damped servo response.

    0.56 + 1.02 Tm − Tm 1.0404 Tm − x1  + x 2  ,   τm τm  τm      0.56 + 1.02(Tm τm ) − (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2 τ Ti =  m − 1  T   0.04 − 1.02(Tm τm ) + (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2  m x 1 = 0.4, x 2 = 3.68, 1 < τ m Tm < 2 ; x 1 = 0.8, x 2 = 3.28, 2 < τ m Tm < 4 ;

Kc =

1 Km

  Tm ; 

x 1 = 1.2, x 2 = 2.88, 4 < τ m Tm . 47

Km

Ti = −

2    1 0.25 1 0.5    Tm  Tm e   + − − + − + 0 . 25 0 . 5   2 2 τ    τ τ T τ T m m  m m  m    m   1 48 τm Tm x12 + (2Tm + τm )x1 + 1 e τ m x1 , Kc = − Km

[

Ki =

1 4K m τ m 2

.

]

x1 = −

49

2

 τ  0.5 τ m 1+ 0.25 m  − −1 Tm  Tm 

2−

τm  τm   + Tm  Tm 

2

2 0.5 − + τ m Tm

2 τm

2

+

0.25 Tm

2

, Ti = − 

τm

τm Tm x12 + (2Tm + τm )x1 + 1 x12 (τm Tm x1 + Tm + τm )

2  τm     

2  0.5  2 − Tm +  Tm   τ  τ   Tm  2 − m +  m  + 1 e    Tm  Tm   

.

.

b720_Chapter-03

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FA


Rule

Kc

Ti

Comment

Schneider (1988), Bekker et al. (1991), Khan and Lehman (1996). Model: Method 1 Bekker et al. (1991). Model: Method 1; ξ = 1 ; Tv = valve opening time.

0.368Tm K m τm

Tm

ξ =1

0.403Tm K m τm

Tm

ξ = 0.6

1.571Tm K m τm

Tm

ξ = 0.0

0.368Tm K m τm

Tm

Regulator – Gorecki et al. (1989). Model: Method 1 Hang et al. (1991). Model: Method 1

50

Kc =

τm > 0.5Tm or T v < 20Tm . τm < 0.5Tm or

50

Kc

51

Kc

Tv > 20Tm . Ti

undefined 53

52

Ti

Ti

Kc

Pole is real and has maximum attainable multiplicity. 0.16 ≤ τm Tm < 0.96

Servo response: 10% overshoot, 3% undershoot.

0.311Tm  τ  0.1092 ln  m  Tm τm   

 K m τm    Tm 

 T  − 0.0760 ln  v  T   m

2    τ  2 Tm  51 Kc = 2 +  m  − 1 e  K m τm   2Tm   

   − 0.2685     

 τ 2 +  m  2 Tm

  Tv  0.0121 ln   Tm Tv   

 τ   m

.

   + 0.1026    

2

 τ  − 2− m  2 Tm 

, 2

52

G c = 1 (T i s ) , Ti =

55

 τ  1 +  m   2Tm  . Ti = τm 2 3 2 2  τm   τ m   τm    τ m    τm  +  +  − 2 +    2+  3 +          2T   2Tm   Tm   2Tm    2Tm    m 0.5K m τm . 2  

 1 + (τ 2T )2 − 1 e m m  

 1+  τ m  2T   m 

 τ  −1− m   2 Tm   

     τm    12 + 2  11(τm Tm ) + 13    11 + 13     ( ) τ − 37 T 4 4 5 T  m m   K , T = 0.2 53 m   + 1Tu . Kc =   u i  15  τm  6  11(τm Tm ) + 13   − 4     37 + 14 15         Tm  37(τm Tm ) − 4   

b720_Chapter-03

Rule

Kc

McAnany (1993). Model: Method 29 Sklaroff (1992), Åström and Hägglund (1995).

54

0.5Tm K m τm 55

56

Kamimura et al. (1994). Model: Method 1; servo tuning. Zhang (1994), Smith (2002). Model: Method 1

Kc =

Kc

Ti

Comment

Ti

TCL = 1.67Tm

Model: Method 32 or 3 Method 33. T m K m (1 + 3τ m Tm ) pp. 250–251, 253. Honeywell UDC 6000 controller. 0.5556Tm K m τm 5τ m τm Tm < 0.33

Fruehauf et al. (1993). Model: Method 2 Nomura et al. (1993). Model: Method 1 Gonzalez (1994) – pp. 114–115. Model: Method 1

55

FA


56

54

17-Mar-2009

Kc

Tm

τm Tm ≥ 0.33

Ti

0.01 ≤ τm Tm ≤ 10

0.5τm

δ0 = 0.21, 0.5, 0.7 (pp. 142–143).

2

x 2 Tm τm 2K m 57

Kc

Ti

10% overshoot.

58

Kc

Ti

0% overshoot.

59

Kc

Ti

‘Quick’ response: ‘negligible’ overshoot.

Tm K m τm

Tm

‘Good’ servo response; τ m is ‘small’.

(1.44Tm + 0.72Tm τm − 0.43τm − 2.14) , T

(

2

K m 0.72τm + 0.36τm + 2

)

i

=

(

)

K m 5.56 + 2τm + τm 2 . 4Tm + 1.28τm − 2.4

0.5x1 − 0.1(Tm + τm ) Kc = , Ti = 1.25x1 − 0.25(Tm + τm ) , with K m (Tm + τm − x1 )

2

2

x1 = 0.2τm + 0.4τmTm + Tm . 56

x2 = −

2x1 + τm

4 x12 τm

2

+

2 1 , x1 = , Tm τm 1 + (2π loge [b a ])2 b a = desired closed loop response decay ratio.

57

Kc =

58

Kc =

59

Kc =

0.5(Tm + τm ) 0.5τm (2Tm + τm ) + 0.10 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.10] Tm + τm

0.375(Tm + τm ) 0.5τm (2Tm + τm ) + 0.0781 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.0781] Tm + τm 0.425(Tm + τm ) 0.5τm (2Tm + τm ) + 0.083 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.083] Tm + τm

b720_Chapter-03

17-Mar-2009

FA


Rule Davydov et al. (1995). Model: Method 6 Kuhn (1995). Model: Method 17 Khan and Lehman (1996). Model: Method 1 Abbas (1997). Model: Method 1

Ti

Comment

Ti

ξ = 0.9; 0.2 ≤ τ m Tm ≤ 1 .

0.5 K m

0.5(Tm + τ m )

‘Normal tuning’.

1 Km

0.7(Tm + τ m )

‘Rapid tuning’.

Kc 60

Kc

61

Kc

Ti

0.01 ≤ τm Tm ≤ 0.2

62

Kc

Ti

0.2 ≤ τm Tm ≤ 20

63

Kc

Tm + 0.5τ m

0 ≤ V ≤ 0.2 ; 0.1 ≤ τm Tm ≤ 5.0 .

Valentine and Chidambaram (1997a). Bi et al. (1999).

τm  1  0.43 − 0.97  Km  Tm 

0.5064Tm K m τm

Tm

Model: Method 16

Bi et al. (2000).

0.5064Tm K m τm

1.9747 K m Tm

Model: Method 16

Ettaleb and Roche (2000).

1 Km

Tm + τ m

Model: Method 34; TCL = Tm + τ m .

Ti

x1 > 0

Prokop et al. (2000). Model: Method 1

60

Kc =

61

Kc =

65

Kc

0.413 + 0.8

τm Tm

Model: Method 1; ξ = 0.707 64.

  τ 1 , Ti =  0.153 m + 0.362 Tm .   T τm m   K m 1.905 + 0.826  Tm   τ  Tm  0.3852 0.723 + − 0.404 m2  , K m  τm Tm Tm 

Ti = τm

− 0.404τm 2 + 0.723Tm τm − 0.3852Tm 2 − 0.525τm 2 + 0.4104Tm τm − 0.00024Tm 2

62

Kc =

0.404Tm + 0.256τm − 0.1275 Tm τm Tm  0.404 0.256 0.1275  + − , T = τm .  i K m  τm Tm τ T 0.0808Tm + 0.719τm − 0.324 Tm τm m m  

63

Kc =

0.148 + 0.186(Tm τm ) . K m (0.497 − 0.464V 0.590 )

64

±1% TS = 2.5Tm , 0.1 ≤ τ m Tm ≤ 0.4 ; ±1% TS = 5Tm , 0.5 ≤ τ m Tm ≤ 0.9 .

65

57

1.045

 K m 2 τ m 2  K m 2 x1 + + Tm  x1 − 1 2   Tm T τ T x + 2Tm x1 − 1  Tm  Kc = m m m 1 , Ti = . K m (τm + Tm )(1 + x1τm ) x1 (Tm + τ m )

(

)

.

b720_Chapter-03

FA


58 Rule

Kc

Chen and Seborg (2002). Chidambaram (2002), p. 155. Skogestad (2003). Model: Method 56

69

Kc =

Comment

Kc

Ti

Model: Method 1

67

Kc

0.413Tm + 0.8τm

Model: Method 1

min (Tm ,8τ m )

Ti = 8τm (Faanes and Skogestad, 2004). 0 ≤ τm Tm ≤ ∞

Kc

Ti

Tm (x1K m τm )

Tm

68

Gorez (2003). Model: Method 1

Ti

66

0.5Tm K m τm

Klán (2000). Model: Method 18

66

17-Mar-2009

70

(1 − x 2 )τm + Tm

Tm x1K m τm

τm ≤

τm ≤1 τ m + Tm

T τ + 2Tm TCL − TCL 2 1 Tm τm + 2Tm TCL − TCL 2 , , Ti = m m 2 Tm + τm Km (TCL + τm ) −τ s

Tise m  y 0 < TCL < Tm + Tm 2 + Tm τ m ,   . = 2  d desired K c (TCLs + 1) 67

1 Kc = Km

68

Kc =

69

x1 =

2     3.113 − 5.73 τm + 3.24 τm   . T   Tm   m  

 1  2τm (Tm + τm ) 1− K m  1 + 1 + 2τ 2 (τ + T )2 m m m 

[

]

  , T = τm + Tm  i 2 

2 1 1.5M s − 2  τm    + 2 M s (M s − 1) 0.32M s (M s − 1)  τm + Tm 

2

2  1 + 1 + 2 τm   − τ . τ +T   m  m   m  

  5τ m  3 −   ( ) T M τ + m m s  

  2.5τ m 0.5 1 −  −  M s (M s − 1)  ( ) T M τ + m m s  

(

2 M s 2 −1

) ; 0≤

2 2   τm    τm   − 0.4 M s    − 0.4 M s   0.1M s   τ + Tm    τ m + Tm  . 70 x1 = 7.5 −  m   , x 2 = 1 − 0.5  Ms −1  1 − 0.4 M s 1 − 0.4 M s             

τm ≤ τm . τ m + Tm

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Comment

Klán and Gorez (2005a). Klán and Gorez (2005b).

71

Kc

Ti

Model: Method 1

72

Kc

Ti

Model: Method 1; 0 ≤ τm Tm ≤ ∞ .

Skoczowski (2004).

73

Kc

Tm

Model: Method 1

Sree et al. (2004), Chidambaram et al. (2005). Foley et al. (2005).

0.9179  Tm    K m  τm 

Haeri (2005). Model: Method 1 McMillan (2005), pp. 60–61. Model: Method 13

71

72

73

74

0.14Tm Kc = e K m τm

Model: Method 1; 0.1 ≤ τm Tm ≤ 1 .

0.8915 74

Ti

75

Kc

Ti

Model: Method 1

76

Kc

0.96Tm + 0.46τ m

0 ≤ τm Tm ≤ 4

Tm + 0.25τ m

τm Tm > 1

0.25 K m

  τm τm 1.96 +1.65  τ m + Tm  τ m + Tm 

   

2

   

, Ti = 5.38τ m e

59

  τm τm  − 3.94 +1.80  τ m + Tm  τ m + Tm 

   

2

   

.

2 2 2     1 + 1 − τm   , Ti = (τm + Tm ) + Tm . 2(τm + Tm )   τm + Tm     2T ln (1 OS) K c ≤ m 1 + 2x12 − 2 x1 1 + x12  , with x1 ≈ . 2 2 τm   π + [ln (1 OS)]

0.5 Kc = Km

2    τm  τ τm   , 0.1 ≤ m ≤ 0.4 ;   − + Ti = Tm 10.59 2 . 3588 0 . 8985  T T T   m m  m   4 3 2   τ  τ  τ  τ Ti = Tm 0.7719 m  − 3.6608 m  + 6.5791 m  − 5.1652 m + 2.8059 , Tm    Tm   Tm   Tm   

0.4 ≤ 75

76

Kc = Kc =

(1 + 2T

m

λK mω90 0 1 Km

) 1+ T [T + τ (1 + T

2

ω90 0 m

2

2

m

m

m

ω90 0

2

2

ω90 0

2

2

)]

, Ti =

  6.282 . 0.407 + 1.195  1 + 14.956(τm Tm )  

2

[

1 + 2Tm ω90 0

(

2 2

ω90 0 Tm + τm 1 + Tm ω90 0

2

)] .

τm ≤ 1.5 . Tm

b720_Chapter-03

17-Mar-2009

FA


60 Rule Trybus (2005). Model: Method 1

x1

Kc

Ti

Comment

0.6Tm K m τm

0.8τm + 0.5Tm

x1Tm K m τm

Tm

Servo response: small overshoot, short settling time. 77 τm Tm ≤ 2

OS

x1

OS

x1

OS

OS

x1

x1

OS

x1

OS

0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25% x1 K m 0.5τm τm Tm ≥ 2 x1

Xu et al. (2005).

OS

78

Cuesta et al. (2006). Model: Method 1

77

78

x1

OS

x1

OS

OS

x1

x1

OS

x1

OS

0.17 0% 0.27 5% 0.32 10% 0.37 15% 0.41 20% 0.45 25% Model: Method 1 (0.3K m τm ) Tm 3.33τm

0.4 K m

79

Ti

Kukal (2006). Model: Method 1

undefined

80

Ti

Mesa et al. (2006). Model: Method 1

Tm + 0.5(1 − x1 )τm K m τm (x1 + 1)

81

40% overshoot; unit step servo response. 0% overshoot; unit step servo response. 0.01 ≤ τm Tm ≤ ∞

Ti

Kc

Kc

Ti

0.01 ≤ τm Tm ≤ ∞

Tm + 0.5(1 − x1 )τm

Sample x1 values: 0.3, 0.5, 0.9.

The author quotes the following source for this rule: Findeisen, W. (Ed.), Control Engineering Handbook (2nd Edition). WNT, Warsaw, 1974 (in Polish). 1 100τm + 6.25Tm T (100τm + 6.25Tm )(1000τm + 463Tm ) Kc = , Ti = m . (100τm + 46.25Tm )(5.2τm + 684Tm ) K m 100τm + 46.25Tm 0.4Tm (1000τm + 463Tm ) . (5.2τm + 684Tm )

79

Ti =

80

Gc =

2   T  T  1 , Ti = K m 2 m  + 4 m  + 1  e − x1 , x1 =  τ  Tis  τm    m  

81

Kc =

2   T  1   Tm   + 1 − 2 m   e x 1 , x1 = 8  τ  K m   τm   m   

4(Tm τm )2 + 1 − 1 2(Tm τm )

8(Tm τm )2 + 1 − 1 2(Tm τm ) 3

−1 .

−2; 2

1.5

T  T  T   T  24 m  + 4 m  + 8 m  + 1 +  8 m + 1 τ  τm   τm   τm  Ti = τm  m  3 T  8 m + 1 τ  m 

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Skogestad (2006a). Skogestad (2006b). Model: Method 1 Visioli (2006), p. 157. Vítečková (2006). Hougen (1979), p. 335. Model: Method 1; coefficients of K c estimated from graphs. Latzel (1988). Gp = Gm ; Model: Method 1 Latzel (1988).

Gp =

K pe

− sτ p

(1 + sTp1 ) 2

;

Model: Method 2 Latzel (1988).

Gp =

K pe

− sτ p

(1 + sTp1 )3

;

Model: Method 2

82

Kc ≥

∆d 0 ∆y max

82

Kc

1

61

Comment

Ti

min[Tm ,4(TCL + τm )]

Model: Method 1

min[Tm , 4Tm K m ]

Slow control; ‘acceptable’ disturbance rejection performance. Model: Method 1 0.3Tm K m τm min[Tm ,8τm ] ≥ 0.368Tm K m τm

τm

Model: Method 9

Tm

Direct synthesis: frequency domain criteria Maximise crossover x1 K m Tm frequency. Tm x1 τ m Tm x1 τ m Tm x1

0.1 7.0 0.2 3.5 0.3 2.2 0.4 1.8 0.785Tm K m τm

0.5 0.6 0.7

1.4 1.1 1.0

0.8 0.9 1.0

0.8 0.75 0.7

Tm

φm = 450

0.524Tm K m τm

Tm

φm = 600

0.45Tm K m (τm + 0.03Tm )

0.57Tm

φm = 450

0.30Tm K m (τm + 0.03Tm )

0.57Tm

φm = 600

0.42Tm K m (τm + 0.10Tm )

0.53Tm

φm = 450

0.28Tm K m (τm + 0.10Tm )

0.53Tm

φm = 600 ; also given by Klein et al. (1992).

, τm ≤ TCL ≤

Tm ∆y max − τm ; ∆d 0 = maximum magnitude of a K m ∆d 0

sinusoidal disturbance over all frequencies, ∆y max = maximum controlled variable change, corresponding to the sinusoidal disturbance.

b720_Chapter-03

Rule

Kc

Regulator – Gorecki et al. (1989). Model: Method 1 Regulator – Gorecki et al. (1989). Model: Method 1 Somani et al. (1992). Model: Method 1; suggested A m : 1.75 ≤ A m ≤ 3.0 .

83

Kc =

Comment

Ti

2K m (τm + Tm )

Low frequency part of the magnitude Bode diagram is flat. x 2τm

x 1 K m A m 84 τ m Tm

x1

0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449

36.599 7.754 4.361 3.055 2.369 1.950 1.672

Coefficient values: x2 τ m Tm 3.57 3.33 3.13 2.94 2.78 2.63 2.50

[

]

τm 0.9806 < 1 , Kc ≈ Tm K mAm

Kc =

1.476 2.38 1.333 2.27 1.226 2.17 1.146 2.08 1.086 2.00 1.042 1.92 1.011 1.85 0.1 ≤ τm Tm ≤ 1

Ti

Kc

x2

1 1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3 , Km 4 1 + 3(Tm τm ) + 3(Tm τm )2

Kc ≈ 85

x1

1.869 2.392 3.067 3.968 5.263 7.246 10.870

Ti = τm For

Low frequency part of magnitude Bode diagram is flat. G c = 1 (Tis )

Ti

Kc

undefined

85

84

FA


62

83

17-Mar-2009

( 1.886 + (4τ

4 m

Tm ) − 1.373

T 0.9806 1 + 1.886 m KmAm  τm

1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3

[

3 1 + 2(Tm τm ) + 2(Tm τ m )2

)

2

]

.

+ 1 or

2

T  τ 0.9806  ; for m > 1 , K c ≈ 1 + 8.669 m Tm KmAm  τm 

5Tm 1  Tm  . 0.80 + 1.33  , Ti = K mAm  τm  1.04[0.80 + 1.33(Tm τm )]2 − 1

2

  . 

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc 86

Lee et al. (1992). Model: Method 1

x1 = 0.5 + 0.207

Hang et al. (1993a), (1993b), pp. 61–65. O’Dwyer (2001a). Model: Method 1; τm > 0.3 Tm (Yang and Clarke, 1996). O’Dwyer (2001a). Model: Method 1

2.42 86

Kc =

88

Comment

Ti

M s = 1.25

τm , τm Tm < 2.42 ; x1 = 1 , τm Tm ≥ 2.42 . Tm

ωp Tm

87

Model: Method 37 or Method 41.

Ti

KmAm x 1Tm K mτm

A m = π 2x 1 ;

Tm

φ m = π 2 − x 1 .88

Representative results φm x1

x1

Am

Am

φm

1.048

1.5

30

0

0.524

3

60 0

0.7854

2

450

0.393

4

x1 τm

Tm

Tm K u Tu 2

2

Tu + 4 π Tm

2

Am

67.50 = π 2x 1 ;

φm = π 2 − x 1 .

 τ  τm τ T  τ − 4 + 1.21 m + 4.4x1 m  m + 2 − 2.2 x1 m   τm  Tm Tm Tm Tm   ,  τm τm   + 1.21 K m 4 + 4.4 x1  Tm Tm  

 τ τm τ T  τ − 4 + 1.21 m + 4.4 x1 m  m + 2 − 2.2 x1 m   Tm τ Tm T T m m m   Ti = τ  T τ 0.96 + 2.42 m  m + 2 − 2.2 x1 m  τm  Tm Tm  τ 1 Ti = ; m ≤ 0.3 (Yang and Clarke, 1996). 2 Tm 4ωp τ m 1 2ωp − + π Tm

2.42

87

Kc

Ti

63

   .

Given A m , ISE is minimised when φ m = 68.8884 − 34.3534 A m + 9.1606

τm Tm

for servo tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 ; Given A m , ISE is minimised when φ m = 45.9848A m 0.2677 (τ m Tm )0.2755 for regulator tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 .

b720_Chapter-03

17-Mar-2009


64 Rule

Kc

x1

Am

0.50

3.14

0.61

2.58

0.67

2.34

0.70

2.24

Comment

Ti

x1Tm K m τm

Chen et al. (1999b). Model: Method 1

Tm Coefficient values: Ms x1 Am

φm 61.4

0.72

2.18

48.7

0

1.30

55.0

0

1.10

0.76

2.07

46.50

1.40

51.6

0

1.20

0.80

1.96

0

1.50

50.0

0

1.26

Symmetrical optimum principle – Voda and Landau (1995). Model: Method 1

3.5 G p ( jω1350 )

4.6 ω1350

1

4

2.828 G p ( jω1350 )

ω1350

90

Ti

44.1

Model: Method 1

τm ≤ 0.1 Tm 0.1
2 ; A m > 3. Tm

G p ( jω1350 )

1

ωr − 1.273τm ωr + (1 Tm )

Kc =

89

1

90

Ti =

Ms

1.00

r1ωr Tm K m

Friman and Waller (1997). Model: Method 1

φm

0

Chen et al. (1999a).

89

FA

2

1 4.6 G p ( jω1350 ) − 0.6K m

, ωr = , Ti =

π 2 r1 A m − 1 . 4 τ m (r1 A m )2 − 1 1.15 G p ( jω1350 ) + 0.75K m

[

ω135 0 2.3 G p ( jω1350 ) − 0.3K m

].

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Schaedel (1997), Henry and Schaedel (2005) – optimal servo response; T1 , T2 given in footnote 91.

91

0.5Ti K m (T1 − Ti )

2

T1 − 2T2

0.375Ti K m (T1 − Ti )

2

T1 =

Tm 1 + 3.45

T1 =

τ am 2

Tm − 0.7 τ am

4T2 T1

2

Ti

2   1 − 2 T2  2  T1  

2  3.86T2 2  1.46T2 2   Ti  1  1 −   0 . 69 − 1   T1   T12  Km  T  2   93

Kc

94

Kc

Tm

+ τ m , T2 2 = 1.25Tm τ am +

1 + 3.45 2

+ τ m , T2 = Tm τ am +

0.7Tm

2

Tm − 0.7 τ am

τ am

93

94

Minimum ITAE. Minimum φ m = 50 0 .

Tm

τ m + 0.5τ m 2 , 0
0.104 . Tm

65

b720_Chapter-03

17-Mar-2009

FA


66 Rule

Kc

Potočnik et al. (2001). Model: Method 48

Cluett and Wang (1997), Wang and Cluett (2000), p. 177. Model: Method 1; τ 0.01 < m ≤ 10 . Tm

Ti

Comment

95

Kc

Ti

τm Tm ≤ 0.1

96

Kc

Ti

0.1 < τm Tm < 0.15

97

Kc

Ti

0.1 < τm Tm < 1

98

Kc

Ti

99

Kc

Ti

100

Kc

Ti

7.50 ≤ A m ≤ 8.28 ,

78.50 ≤ φm ≤ 79.70 . 4.35 ≤ A m ≤ 5.02 ,

70.7 0 ≤ φm ≤ 74.00 . TCL = 1.33τ m , 3.29 ≤ A m ≤ 3.89 ,

64.40 ≤ φm ≤ 70.60 . 101

95

Kc =

(

G p jω135 0

)

Kc

0.5  G ( jω 0 ) 1.14 + 13.24 p 135 Km  

97

98

99

(

K m − 2 G p jω1350

[

Kc =

)

2K m K m − 2 G p ( jω135 0 )

(

]

, Ti =

) 

 G jω 0 0.75 + 1.15 p 135  Km  0.019952τm + 0.20042Tm Kc = , Ti K m τm

[K

m

57.80 ≤ φm ≤ 68.50 .

,

   

Ti =

96

2.74 ≤ A m ≤ 3.30 ,

Ti

(

4 ω135 0

(

+ 2 G p jω135 0

(

)

(

 1 + 2.48 G p jω1350 + 17.40 G p jω1350 2  Km Km 

ω135 0 G p jω135 0

) ][2 G p (jω135 ) − K m ] 0

) [ G p ( jω135 ) − K m ] G p (jω135 )  

.

0

 0 0.75 + 1.15 .   Km ω135 0     0.099508τm + 0.99956Tm = τm , TCL = 4τ m . 0.99747 τm − 8.7425.10−5 Tm

Kc =

2K m ω135 0

Kc =

0.16440τm + 0.99558Tm 0.055548τm + 0.33639Tm τm , TCL = 2τ m . , Ti = K m τm 0.98607 τm − 1.5032.10− 4 Tm

, Ti =

1

100

Kc =

0.20926τm + 0.98518Tm 0.092654τm + 0.43620Tm τm . , Ti = K m τm 0.96515τm + 4.2550.10− 3 Tm

101

Kc =

0.24145τm + 0.96751Tm 0.12786τm + 0.51235Tm τm , TCL = τ m . , Ti = K m τm 0.93566τm + 2.2988.10− 2 Tm

) 2   

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Comment 2.39 ≤ A m ≤ 2.93 ,

Cluett and Wang (1997), Wang and Cluett (2000) – continued.

102

Kc

Ti

103

Kc

Ti

38.10 ≤ φm ≤ 66.30 .

Modulus optimum principle – Cox et al. (1997). Model: Method 20 Smith (1998). Model: Method 1

104

Kc

Ti

τm Tm ≤ 1

0.5Tm K m τm

Tm

τm Tm > 1

0.35 Km

0.42τ m

6 dB gain margin – dominant delay process.

Tm

1.3 ≤ M s ≤ 2

Wang et al. (2000a). Model: Method 37 or 38.

102

103

104

105

Kc

67

49.60 ≤ φm ≤ 67.10 . 2.11 ≤ A m ≤ 2.68 ,

0.26502τ m + 0.94291Tm 0.16051τm + 0.57109Tm τ m , TCL = 0.8τ m . , Ti = K m τm 0.89868τ m + 6.9355.10 − 2 Tm 0.19067 τm + 0.61593Tm 0.28242τm + 0.91231Tm Kc = , Ti = τm , TCL = 0.67τ m . K m τm 0.85491τm + 0.15937Tm Kc =

Kc =

0.5  Tm 3 + Tm 2 τ m + 0.5Tm τ m 2 + 0.167 τm 3  , 2 2 3  K m  Tm τm + Tm τm + 0.667 τm   T 3 + Tm 2 τm + 0.5Tm τm 2 + 0.167 τm 3  . Ti =  m 2 2   Tm + Tm τm + 0.5τm  

105

 1.508  Tm K c = 1.451 − .  Ms  K m τm 

b720_Chapter-03

FA


68 Rule

Kc

Wang and Shao (2000a). Model: Method 1 Chen and Yang (2000).

106

0.7Tm K m τm

(1 + ω

1.5 ≤ x1 ≤ 2.5

Model: Method 8

Tm 0.8Tm

0.2 ≤ τm Tm < 1

0.1Tm 0.15 + K m τm K m

0.3τ m + 0.5Tm

τm >1 Tm

0.15 K m

0.3τ m

τm Tm → ∞

107

Kc

Ti

108

Kc

Tm

Km 2 90

0

Tm

)

2 1.5

[(T

m

1 2

1 + ω900 Tm

2

[(T

m

M s = 1.4 ;

τm ≥ 0.5 . Tm

 ,  + {1 + ω900 2 Tm 2 }τ m ) sin( − ω900 τ m − tan −1 ω900 Tm ) −

f 2 ( ω900 ) = −

Ti

0.25Tm K m τm

 1 1 f 2 (ω90 0 ) − λf1 (ω90 0 )  ω90 0

f1 ( ω900 ) =

Comment

M s = 1.26 ; A m = 2.24 ; φ m = 50 0 .

Hägglund and Åström (2002). Model: Method 1 Leva et al. (2003). Model: Method 1

Kc =

Kc

Ti

A m > x1 ; φm > cos −1 (1 x1 ) (Wang and Shao, 1999b).

Hägglund and Åström (2002). Model: Method 5; M s = 1.4 .

106

17-Mar-2009

Km

(1 + ω

900

2

2

2

Tm 2

)

1.5

[ω

900

]

Tm 2 cos( − ω900 τ m − tan −1 ω900 Tm ) ,

+ {1 + ω900 Tm }τ m ) cot( − ω900 τ m − tan −1 ω900 Tm )

]

− Ti =

[

2

2

]

2

2

2

2

ω900 Tm 2 1 + ω900 2 Tm 2

;

2

ω90 0 Tm + (1 + ω90 0 Tm )τm cos θ + (1 + 2ω90 0 Tm ) sin θ 3

]

2

− ω90 0 Tm cos θ + ω90 0 [Tm + (1 + ω90 0 Tm )τm ] sin θ

, θ = − ω900 τ m − tan −1 ω900 Tm .

107

108

Kc =

1 Km

 T  0.14 + 0.28 m  τm 

 6.8τ m Tm  , Ti = 0.33τ m + .  10τ m + Tm 

 T (0.5π − φm ) 5Tm  K c = minimum  m ,  ; minimum φ m , maximum ωc . K m τm K m TCL  

b720_Chapter-03

17-Mar-2009

FA


Rule Leva et al. (2003) – continued. Mataušek and Kvaščev (2003). Model: Method 1 Kristiansson and Lennartson (2002b). Model: Method 1

110

Comment

Kc

Tm

minimum φ m = 50 0 .

x 1Tm K mτm

Tm

x 1 = 0.5π − φ m .110

109

Kristiansson (2003). Model: Method 12 Kristiansson and Lennartson (2006).

109

Ti

Kc

111

Kc

0.57Tm + 0.29τm

0.2 ≤ τm Tm ≤ 5

112

Kc

Ti

τm Tm < 1

113

Kc

0.60Tm + 0.28τm

τm Tm ≥ 1

114

Kc

1.25 Tm

1.6 ≤ M s ≤ 1.9

115

Kc

1.4 τ m + 0.8Tm

τm Tm < 0.2

69

Model: Method 12; 1.65 ≤ M max ≤ 1.80 .

 0.698Tm  τm + 5Tm 5x1Tm K c = minimum  , , 4 ≤ x1 ≤ 10 .  , TCL = x1  K m τm K m (τm + 5Tm )  ‘Acceptable’

values

of

0

x1 :

0.5 ≤ x 1 ≤ 3 .

‘Recommended’

values

of

x1 :

0

0.32 ≤ x 1 ≤ 0.54 ( 59 ≤ φ m ≤ 72 ). Choose x 1 = 0.32 when large variations in process

parameters are expected; choose x 1 = 0.54 to give 6% ≤ OS < 13% and 1.4 < M s ≤ 2 . 111

112

113

114

115

Kc =

0.57(0.57 + 0.29[τm Tm ]) , 1.65 ≤ M max ≤ 1.75 . K m ([τm Tm ] − 0.055)

2   τm   0.57 τm  ,   Kc = 0.37 + 0.94 − 0.42 K m ([τm Tm ] − 0.055)  Tm Tm     

Kc =

0.57(0.60 + 0.28[τm Tm ]) . K m ([τm Tm ] − 0.055)

 ω150 0  0.075 .  0.2 + K m  K150 0 K m + 0.05  0.57(0.8Tm + 1.4τm ) Kc = . K m Tm ([τm Tm ] − 0.055) Kc =

[

]

2  τ   τ Ti = Tm 0.37 + 0.94 m − 0.42 m   . Tm   Tm   

b720_Chapter-03

Rule

Kc

Kristiansson and Lennartson (2006) – continued. Bai and Zhang (2007). Model: Method 59 Leva et al. (1994).

Hägglund and Åström (2004). Model: Method 1

116

Kc =

117

Kc = Kc =

Ti

Comment

116

Kc

τ m + Tm

0.2 ≤ τm Tm ≤ 1

117

Kc

0.6τ m + 1.2Tm

τ m Tm ≥ 1

0.526Tm K m τm

Tan et al. (1996).

Kc =

A m = 2.98 ,

Tm

φm = 59.80 .

118

Kc

Ti

Model: Method 54

119

Kc

Ti

Model: Method 36

120

Kc

Ti

1.920 < φ < 2.356 ; M s = 1.2 .

121

Kc

Ti

φ = 2.09 radians (good choice).

0.57(Tm + τm ) . K m Tm ([τm Tm ] − 0.055)

0.57(1.2Tm + 0.6τ m ) . K m Tm ([τ m Tm ] − 0.055)

ωcn Ti Km

1 + ωcn 2Tm 2 2

1 + ωcn Ti

, Ti =

2

[

]

tan φm − 1.571 + τm ωcn + tan −1 (ωcn Tm ) ωcn

with ωcn = 119

FA


70

118

17-Mar-2009

(

x1Ti ωφ 1 + x1Tmωφ

(

A m 1 + x1Ti ωφ

)

2

)2

, Ti =

2.82 − φm − tan −1[(Tm τm )(1.571 − φm )] . τm

1 , ωφ < ω u . x1ωφ tan − tan x1Tm ωφ − x1τm ωφ − φ

[

]

−1

x1 = 0.8, τm Tm < 0.5 ; x1 = 0.5, τm Tm > 0.5 . 120

121

Kc =

Kc =

2.01 − 0.80φ  G jω 1 + (15.45 − 6.30φ) p φ Km  

( )

0.33

( ) 

 G jω 1 + 2.26 p φ Km  

 

( )

G p jωφ

  G p jωφ  

6.283(1.72φ − 2.55)

, Ti =

( ) 

 G jω 1 + (3.44φ − 5.20 ) p φ Km   6.61 , Ti = . 2  G p jωφ  1 + 2.01  ωφ Km    

( )

( )

 

.

2

ωφ

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Hägglund and Åström (2004) – continued.

Hägglund and Åström (2004). Xu et al. (2004). Model: Method 37

Ti

Comment

122

Kc

Ti

2.094 < φ < 2.443 ; M s = 1.4 .

123

Kc

Ti

φ = 2.269 radians (reasonable choice).

124

Kc

Ti

2.356 < φ < 2.705 ; M s = 2.0 .

125

Kc

Ti

126

Kc

Ti

φ = 2.53 radians (good choice). Model: Method 5; ‘AMIGO’ tuning rule. 1.5 ≤ x1 ≤ 2.5

Tm x1K m τm

Tm

x1 = 2 implies A m > 2, φm > 600 (Xu and Shao, 2004b).

Huang et al. (2005a). Model: Method 51

0.50Tm + 0.22 τ m Kmτm

0.9Tm + 0.4τ m

Huang and Jeng (2005).

0.495Tm + 0.22τm K m τm

0.9Tm + 0.4 τ m

122

123

124

125

126

Kc =

2.50 − 0.92φ  G jω 1 + (10.75 − 4.01φ) p φ Km  

( )

  G p jωφ  

Model: Method 1

6.283(1.72φ − 3.05)

, Ti =

( ) 

2

( ) 

2

 G jω 1 + (3.44φ − 6.10 ) p φ Km   5.36 0.41 , Ti = . Kc = 2  G p jωφ   G p jωφ  1 + 1.65  G p jωφ 1 + 1.71  ωφ Km   Km       6.283(0.86φ − 1.50) 3.43 − 1.15φ , Ti = Kc =  G p jωφ   G jω 1 + (11.30 − 4.01φ)  G p jωφ 1 + (1.72φ − 2.90) p φ Km   Km     4.25 0.52 , Ti = . Kc = 2  G p jωφ   G p jωφ  1 + 1.15  G p jωφ 1 + 1.45  ωφ Km   Km      

( )

( )

( )

( )

. ωφ

 

( )

( )

( )

Kc =

71

( )

( )

0.15  τm Tm  Tm 6.7 τm Tm 2 + 0.35 − , Ti = 0.35τm + 2 . 2 K m  (τm + Tm )  K m τm Tm + 2Tm τm + 10τm 2

 

. ωφ

b720_Chapter-03

Rule

Kc

Åström and Hägglund (2006), pp. 228–229. Åström and Hägglund (2006), pp. 258–260. Model: Method 5; detuned ‘AMIGO’ tuning rule. Clarke (2006). Model: Method 27

K c (1) =

Comment

Ti (1)

Model: Method 5; ‘AMIGO’ tuning rule; M s = M t = 1.4 .

Ti

( 2)

τm Tm > 0.11

( 3)

Ti

( 3)

( 4)

Ti

( 4)

τm < 0.11 Tm

(1)

Ti

( 2)

127

Kc

128

Kc

129

Kc

130

Kc

Tm τm

0.5 T 1+ m Km τm

τ m Tm  Tm 13τ m Tm 2 0.15  (1) + 0.35 − , T = 0 . 35 τ + .  i m K m  (τ m + Tm )2  K m τ m Tm 2 + 12Tm τ m + 7 τ m 2 K c (1)

128

129

FA


72

127

17-Mar-2009

Kc

( 2)

= x1K c

K c ( 3) ≥

(1)

, x1 = 0.5,0.1 or 0.01; Ti

( 2)

= Ti

(1)

K c ( 2) Ti (1)

K m + 1 − M s −1

K c (1) K c ( 2) K m + 1 − M s −1

2K c (1) (τm + Tm ) 1 + −1 , 2 (1) (1) −1 M   s M s  M s + M s − 1 Ti 1 − M s + K c K m  

(

)

Ti (3) = Ti (1) 130

K c ( 4)
1 ;       ∧ 2 ∧ 2    1 + ωu Tm 2 1 + ωu TCL 2  , τm < 1 ;      

Ti =

133

x1 =

2φm + π(A m − 1)

(

)

2 Am2 − 1

∧   tan − τm ωu + π − φm  , φm measured in radians.   ωu

1 ∧

Tm x1 K m τm . ; Ti = A m x1  A m x1  1 + 1 −  τm  π  Tm

b720_Chapter-03

17-Mar-2009

FA


74 Rule

Kc

Ti

Comment

Robust

Tm λK m

Tm

2Tm + τ m 2λ K m

Tm + 0.5τ m

Tm K m (τ m + λ)

Tm

Tm + 0.5τ m K m (λ + τ m )

Tm + 0.5τ m

λ ≥ 1.7 τ m ,

λ > 0.1Tm . Rivera et al. (1986). λ ≥ 1 . 7 τ , λ > 0 . 2 T (Luyben, 2001); λ = 2τ (minimum ISE – Model: Method 1 m m m de Oliveira et al., 1995); λ = max[2.3τm ,0.15Tm ] (Visioli, 2005).

Chien (1988). Model: Method 1 Brambilla et al. (1990). Model: Method 1

134

λ ≥ 1.7 τ m , λ > 0.1Tm . 134

No model uncertainty: λ = τ m , 0.1 ≤ τ m Tm ≤ 1 ; λ = [1 − 0.5 log10 (τ m Tm )]τ m , 1 < τ m Tm ≤ 10 .

λ = Tm (Chien, 1988); (‘aggressive’ tuning) (Åström and Hägglund, 2006). λ = 3Tm (‘robust’ tuning) (Åström and Hägglund, 2006). λ = 2τ m (Bialkowski, 1996); (‘aggressive, less robust’ tuning) (Gerry, 1999). λ > Tm + τ m (Thomasson, 1997). 0.45τm ≤ λ ≤ 0.8τm (Huang et al., 1998). λ = 2(Tm + τ m ) (‘more robust’ tuning) (Gerry, 1999). λ = a. max(Tm , τ m ) : a > 3 (‘slow’ design), 2 ≤ a ≤ 3 (‘normal’ design),

1 ≤ a < 2 (‘fast’ design), a < 1 (‘faster’ design) (Andersson (2000), p. 11). Model: Method 10. 0.1Tm ≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ; λ = 1.5(τm + Tm ),0.25 < τm Tm ≤ 0.75 ; λ = 3(τm + Tm ), τm Tm > 0.75 (Leva, 2001).

Tm ≤ λ ≤ τm or τm ≤ λ ≤ Tm (Smith, 2002). λ = 1.7Tm (Faanes and Skogestad, 2004). λ = max[0.25τ m ,0.2T m

] (Leva, 2007).

λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ] (‘moderate’ tuning), λ > [10Tm ,80τm ] (‘conservative’ tuning) (Arbogast et al., 2006).

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Pulkkinen et al. (1993).

0.7Tm K m τm

λ (τ m + Tm )

x1 K m

x 2 Tm

Ogawa (1995). τm Model: Method 1; coefficients of K c , Ti Tm deduced from graphs. 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 Thomasson (1997). Model: Method 1 2K Chen et al. (1997). Model: Method 31

Comment

Ti

x1

x2

0.9 1.3 0.6 1.6 0.7 1.3 0.47 1.7 0.47 1.3 0.36 1.8 0.4 1.3 0.33 1.8 τm m (τ m + λ)

0.5Tm K m max( τ m ,0.5)

Model: Method 1

τm Tm

x1

x2

2.0 10.0 2.0 10.0 2.0 10.0 2.0 10.0

0.45 0.4 0.4 0.35 0.32 0.3 0.3 0.29

2.0 7.0 2.2 7.5 2.4 8.5 2.4 9.0

0.5τ m

λ = TCL .

minimum (Tm ,6τ m )

Isaksson and Graebe (1999). Model: Method 1 Chun et al. (1999). Model: Method 43

τm 2(λ + τ m ) K m (λ + τ m )

Tm +

τ m < 0.5

Desired closed loop Tm +

τm2 2(λ + τ m )

response =

e − τ ms . (λs + 1)

Tm + 0.25τ m K mλ

Tm + 0.25τ m

Tm > τ m ; λ not specified.

Tm (τ m + 2λ ) − λ2 K m (τ m + λ )

Tm (τ m + 2λ ) − λ2 τ m + Tm

Representative λ = 0.4Tm .

Tm K m (λ + τm )

Tm

x1 K m

x 2 (τm + Tm )

2

Zhong and Li (2002). Model: Method 1 Wang (2003), p. 14. Model: Method 1. Regulator; x1 , x 2 deduced from graphs; robust to ±25% uncertainty in process model parameters.

τ m > 0.5

3

2


20% uncertainty in process parameters. 33% uncertainty in process parameters. 50% uncertainty in process parameters. 60% uncertainty in process parameters. τ m >> Tm ;

∆τm ≤ λ ≤ 1.4∆τm ;

x1

x2

τ

x1

x2

τ

5.1 4.5 3.6 2.9 2.3 1.8 1.4

0.23 0.34 0.44 0.50 0.57 0.60 0.63

0.05 0.11 0.18 0.25 0.33 0.43 0.54

1.1 0.9 0.8 0.8 0.7 0.6 0.5

0.65 0.64 0.63 0.62 0.61 0.58 0.56

0.67 0.82 1.0 1.22 1.5 1.9 2.3

∆τ m = time delay uncertainty. τ = τm Tm τ x1 x2

0.4 0.4 0.4 0.4 0.3

0.50 0.50 0.46 0.44 0.42

3.0 4.0 5.7 9.0 19.0

75

b720_Chapter-03

17-Mar-2009

FA


76 Rule

Kc

Wang (2003), p. 13. Model: Method 1. Servo; ±10% overshoot, step input. x1 , x 2 deduced from graphs; robust to ±25% uncertainty in process model parameters

x1 K m

Wang (2003), pp. 50–54. Model: Method 1. Servo; ±10% overshoot, step input. Wang (2003), pp. 50–54. Model: Method 1. Regulator; x1 deduced from graph; robust to ±25% uncertainty in process model parameters. Åström and Hägglund (2006), p. 187.

x1

Comment

Ti

x 2 (τm + Tm )

τm Tm

x2

2.94 2.73 2.20 1.65 1.29 1.04 0.85

0.79 0.05 0.82 0.11 0.81 0.18 0.70 0.25 0.75 0.33 0.72 0.43 0.73 0.54 Tm K m (τ m + λ)

x1

x2

τm Tm

x1

x2

τm Tm

0.81 0.68 0.65 0.59 0.50 0.43 0.38

0.67 0.63 0.63 0.60 0.55 0.51 0.47 Tm

0.67 0.82 1.0 1.22 1.5 1.9 2.3

0.34 0.31 0.29 0.29 0.27

0.43 0.43 0.39 0.39 0.38

3.0 4.0 5.7 9.0 19.0

λ = 0.0132(τm + Tm ) , τm Tm ≤ 0.05 ;

λ = 0.286τm + 0.0034(τm + Tm ) , 0.05 ≤ τm Tm ≤ 19.0 ; λ = 0.27(τm + Tm ) , τm Tm ≥ 19.0 ;

robust to ±25% uncertainty in process model parameters. Tm λ = x1 (τm + Tm ) Tm K m (τ m + λ) x1

τm Tm

0.01 0.03 0.04 0.06 0.07

0.05 0.11 0.18 0.25 0.33 Tm K m (τ m + λ) λ = Tm

x1

τm Tm

x1

0.08 0.11 0.12 0.13 0.15

0.43 0.54 0.67 0.82 1.00

0.16 0.18 0.19 0.21 0.22

τm Tm

x1

τm Tm

1.22 0.25 4.0 1.5 0.26 5.7 1.9 0.27 9.0 2.3 0.27 19.0 3.0 Model: Method 1; delay dominated max [Tm ,3τm ] processes. (aggressive tuning); λ = 3Tm (robust tuning). Ultimate cycle

McMillan (1984). Model: Method 2 or Method 55.

135

135

Kc

Ti

Tuning rules developed from K u , Tu .

    0.65   T    1.881 Tm  1   m  = τ + T 1 . 67 1 Kc = ,    i m 0.65   . K m τm   T     Tm + τm   m   1 +     Tm + τm  

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

77

Comment

Ti

x 1K u x 2 Tu Kuwata (1987). x1 x2 x1 x2 x1 x2 τm τm τm Model: Method 1; T T Tu x 1 , x 2 deduced from u u 0.29 0.84 0.28 0.24 0.24 0.35 0.225 0.17 0.43 graphs. 0.27 0.46 0.30 0.23 0.20 0.38 0.225 0.155 0.45 0.25 0.31 0.33 0.225 0.18 0.40 0.225 0.115 0.48 0.654 Boe and Chang τ  (1988). Quarter decay ratio. 1.935Tm  m  0.54 K u  Tm  Model: Method 1 τ 0.876 Claimed equivalent to Hill and Adams 1.36 − 0.245 m 0.961  Tm  Tm Ziegler–Nichols (1988).   0 . 83 τ e m K m  τm  ultimate cycle Model: Method 1 method. Geng and Geary 0.71Tm τm < 0.167 (1993). 3.33τ m K mτm Tm Model: Method 1 Luyben (1993). 0.225K u 0.415Tu τm Tm = 0.05 Model: Method 1 ^

^

Perić et al. (1997). Model: Method 49 Matsuba et al. (1998). Model: Method 1 Huang et al. (2005a). Model: Method 51

136

K u tan 2 φ m

0.1592 Tu tan φ m

1 + tan 2 φ m 0.99  Tm  K m  τ m 136

  

0.9

Kc

τ 2.63τ m  m  Tm Ti

  

0.097

0.1 ≤

τm ≤ 1.0 Tm

  2    ^    0.9  K u  − 1 0.55     + Kc = 0 . 4 , 2   K m  ^    −1  π − tan   K u  − 1               2 2    ^   ^    −1     Ti = 0.159 Tu 0.9  K u  − 1 + 0.4π − tan  K u  − 1   .            ^

b720_Chapter-03

17-Mar-2009

FA


78

  1 + Td s  3.3.2 Ideal PID controller G c (s) = K c 1 +  Ti s  E(s)

R(s)

+

  U(s) 1 K c 1 + + Td s   Ti s 

−

Y(s)


m

Table 10: PID controller tuning rules – FOLPD model G m (s) = Rule Callender et al. (1935/6). Model: Method 1 Ziegler and Nichols (1942). Model: Method 2 Chien et al. (1952) – regulator. Model: Method 2

Chien et al. (1952) – servo. Model: Method 2 Cohen and Coon (1953). Model: Method 2

Kc 1

1.066 K mτm

Ti


Comment

Td

Process reaction 0.353τ m or 1.418τ m 0.47τ m

τm = 0.3 Tm

x1Tm K m τm

2τ m

0.5τ m

0.95Tm K mτm

2.38τ m

0.42τ m

1.2 ≤ x1 ≤ 2 ; quarter decay ratio. 0% overshoot; 0.1 < τm Tm < 1 .

1.2Tm K mτm

2τ m

0.42τ m

20% overshoot; 0.1 < τm Tm < 1 .

0.6Tm K mτm

Tm

0.5τ m


0.95Tm K mτm

1.36Tm

0.47τ m


Ti

0.37 τm 1 + 0.19(τm Tm )

2

Kc

1


2

Kc =

 2.5(τm Tm ) + 0.46(τm Tm ) 2   1  T . 1.35 m + 0.25  , Ti = Tm      Km  τm 1 + 0.61(τm Tm )   

Quarter decay ratio; 0.1 < τm Tm ≤ 1 .

b720_Chapter-03

17-Mar-2009

FA


Kc

Comment

Td

Ti

0.950 0.738 Three constraints K c K m Td Tm  τ m  1.370  Tm  = 0.5 ; 3     method – Murrill T Tm d 1.351  Tm  (1967), p. 356. K m  τ m  0.1 < τm Tm < 1 . Model: Method 5 Quarter decay ratio; minimum integral error (servo mode). Åström and Model: 3 Km T90% 0.5τ m Hägglund (1988), Method 28 pp. 120–126. T90% = 90% closed loop step response time (Leeds and Northrup Electromax V controller). Parr (1989), Model: Method 2 1.25Tm K m τm 2.5τ m 0.4 τ m p. 194. Borresen and Model: Method 2 Tm K m τm 3τ m 0.5τ m Grindal (1990). 4 Sain and Özgen Model: Ti Td Kc (1992). Method 29 Connell (1996), Approximate 1.6Tm p. 214. quarter decay 1.6667 τ m 0.4 τ m K mτm Model: Method 2 ratio. x1 K m x 2τm x 3τ m Hay (1998), pp. Coefficient values – servo tuning: 197,198. τ T x x2 x3 τm Tm x1 x2 x3 m m 1 Model: 0.125 6.5 9.0 0.47 0.125 5.0 9.8 0.34 Method 1; 5.0 6.5 0.45 0.167 3.7 7.0 0.32 coefficients of 0.167 0.25 3.5 4.2 0.40 0.25 2.5 4.8 0.30 K c , Ti , Td 0.5 1.8 2.2 0.35 0.5 1.1 2.5 0.27 deduced from Coefficient values – regulator tuning: graphs. τm Tm x1 x2 x3 τm Tm x1 x2 x3

0.125 0.167 0.25 0.5

3

4

τ  Td = 0.365Tm  m   Tm  Kc =

9.8 7.2 4.5 2.2

2.0 1.8 1.6 1.4

0.52 0.50 0.45 0.35

0.1 0.125 0.167 0.25 0.5

10.0 8.0 6.0 4.0 2.0

2.2 2.2 2 1.8 1.4

0.35 0.35 0.34 0.32 0.28

0.950

.

 0.8647Tm + 0.226τm 0.0565Tm 1  T  0.6939 m + 0.1814  , Ti = , Td = .   T T Km  τm m  + 0.8647 0.8647 m + 0.226 τm τm

79

b720_Chapter-03

FA


80 Rule Hay (1998), p. 200. Model: Method 2; coefficients of K c , Ti , Td deduced from graphs.

Lipták (2001). Chidambaram (2002), p. 154. Alfaro Ruiz (2005a). ControlSoft Inc. (2005). PMA (2006). Model: Method 2

Kc =

1 Km

Comment

Kc

Ti

Td

x1 K m

x 2 τm

x 3τm

x1

4.7 2.4 2.8 1.5 1.8 0.5 1.2 0.8 1.0 0.7 1.2Tm K m τm

Moros (1999). Model: Method 1 1.2Tm K m τm

5

17-Mar-2009

0.85Tm K m τm 5

Kc

x2

x3

0.87 1.87 0.95 1.55 1.00 1.33 1.00 1.15 0.95 1.00

1.32 1.28 1.07 1.0 0.90 0.78 0.78 0.64 0.72 0.54

τm Tm

0.3 0.4 0.5 0.6 0.7

2τ m

0.42τ m

2τ m

0.44τ m

1.6τ m

0.6τ m

2.4τm

0.38τm

Regulator. Servo. Regulator. Servo. Regulator. Servo. Regulator. Servo. Regulator. Servo. Attributed to Oppelt. Attributed to Rosenberg. Model: Method 2

Model: Method 1. ‘Improved’ Ziegler–Nichols method. x1Tm K m τm 2.5τm 0.4τm 2.0 ≤ x1 ≤ 3.33 ; Model: Method 2 6 Ti Td 1.0 ≤ x1 ≤ 1.67 ; Kc Model: Method 1 7 Model: 2 Km τ m + Tm Td Method 11 2τm 0.59Tm 2τm K m τm Alternative. τm

  Tm + 0.45 . 1.8   τm

 τ  τ  Tm  0.40 + 0.153 m Tm  2.50 + 0.957 m  Tm Tm   x  T   6 , Td = K c = 1 1.56 m + 0.68 , Ti = ( ) 1 + 0 . 787 τ T Km  τm ( 1 0 . 787 T + τ m m m m)  τ T  τ T  7 Td = max  m , m  (slow loop); Td = min  m , m  (fast loop). 3 6    3 6 

   

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Td

Comment

Minimum performance index: regulator tuning 0.921 0.749 Minimum IAE – Tm  τ m  1.435  Tm  8     Model: Method 5 Murrill (1967), Td K m  τm  0.878  Tm  pp. 358–363. Minimum IAE – 12.52 K m 0.20Tm 0.04Tm τm Tm = 0.10 Lopez (1968), 2.68 K m 0.69Tm 0.22Tm τm Tm = 0.50 p. 50. 1.47 K m 1.04Tm 0.42Tm τm Tm = 1.00

Minimum IAE – Marlin (1995), pp. 301–307. Model: Method 8

Minimum IAE – Peng and Wu (2000). Model: Method 21

1.3 K m

0.7 τ m

0.01τ m

τ m Tm = 0.43

1.0 K m

0.68τ m

0.05τ m

τ m Tm = 0.67

0.85 K m

0.65τ m

0.08τ m

τ m Tm = 1.0

0.65 K m

0.65τ m

0.10τ m

τ m Tm = 1.5

0.45 K m

0.55τ m

0.14τ m

τ m Tm = 2.33

0.4 K m

0.50τ m

0.21τ m

τ m Tm = 4.0

Coefficients of K c , Ti , Td estimated from graphs; robust to ±25% process model parameter changes. 6.558K m 0.925Tm 0.061Tm τ m Tm = 0.1 2.311K m

0.677Tm

0.189Tm

τ m Tm = 0.2

1.479 K m

0.633Tm

0.233Tm

τ m Tm = 0.3

1.117 K m

0.647Tm

0.250Tm

τ m Tm = 0.4

0.947 K m

0.705Tm

0.268Tm

τ m Tm = 0.5

0.825K m

0.752Tm

0.289Tm

τ m Tm = 0.6

0.731K m

0.792Tm

0.312Tm

τ m Tm = 0.7

0.695K m

0.871Tm

0.316Tm

τ m Tm = 0.8

0.630K m

0.895Tm

0.326Tm

τ m Tm = 0.9

0.651K m

1.030Tm

0.306Tm

τ m Tm = 1.0

1.137

8

τ  Td = 0.482Tm  m   Tm 

, 0.1 ≤

τm ≤1. Tm

81

b720_Chapter-03

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FA


82 Rule

9

Minimum IAE – Kosinsani (1985), pp. 46–48. Model: Method 1

Kc

Ti

Td

Comment

Kc

Ti

Td

Load disturbance model = 1 . 1 + sTL

0.05 ≤

τm T ≤ 1.0 ; 0.05 ≤ L ≤ 7.0 . Tm Tm

Values of x1 to x 15 for varying τm Tm provided. The controller tuning formula is not recommended when τm Tm = 0.4, TL Tm > 6.5 ; τm Tm = 0.5, TL Tm > 5.5 ; τm Tm = 0.75, TL Tm > 5.0 ; τm Tm = 1.0, TL Tm > 4.5 .

τm Tm

0.075

0.1

0.15

0.2

0.3

0.4

0.5

0.75

1.0

x1

27.567 18.817 13.984 9.3318 7.0210 4.7110 3.6156 2.9398 2.0164

1.5805

x2

3.2174 1.2714 7.6166 4.9233 3.9864 2.9383 2.2643 1.8975 2.0290

0.6154

x3

-2.3336 -1.8823 6.8896 4.5267 3.2537 2.0947 1.5546 1.2108 0.7548

0.5345

x4

-31.416 -31.416 -1.9E-6 -3.4E-7 -1.0E-5 -5.6E-8 -9.0E-9 -9.5E-9 0.4277

0.3603

x5

-7.2662 -6.2990 -9.9212 -0.0978 1.9228 1.6537 2.3143 1.5861 1.5845 -0.6779

x6

7.8891 3.7533 1.1676 0.9631 1.1989 0.5403 2.5070 2.6381 0.6400

1.1821

x7

8.1491 8.0893 8.5751 6.3996 4.9687 4.0498 7.8855 0.7385 2.5128

2.6522

x8

-0.4706 -0.5880 -0.6459 -0.6104 -0.4955 -0.5196 -0.0233 -0.2809 -0.1774 -0.0445

x9

14.020 11.353 5.2382 1.7098 0.6395 0.1405 1.7941 3.5610 0.0825

x 10

9

0.05

0.6528

-3.6987 -3.4514 -1.7079 -1.3466 -1.3080 -0.2366 -0.5171 -0.5509 -0.0100 -0.4917

x 11

4.6405 3.5079 3.1794 3.5195 7.5219 1.8090 0.5537 0.1952 1.3773

x 12

-0.8986 -0.4818 -0.2947 -0.4508 -2.7213 -4.4931 -1.8157 -2.5837 -4.7205 -2.3611

x 13

0.0270 -1491.0 -4829.2 -1224.6 -13635 -31129 -31539 -40901 -52408

x 14

-0.0090 1491.1 4829.3 1224.7

52408

59325

x 15

-4.1759 6.2E-7 3.8E-7 3.2E-6 5.0E-7 3.9E-7 5.6E-7 5.7E-7 6.8E-7

7.5E-7

Kc = Ti =

1   x1 − e − x 2 TL Km  

(

x6 + x7 1 − e

13635

31130

31539

40901

  T   T   x 3 cosx 4 L  + x 5 sin x 4 L   ,    Tm   Tm    Tm , Td = Tm x13 + x14e x15 TL   TL x 10 TL Tm sin  x11 + x 9e + x12  Tm  

0.5675

-59324

Tm 

x 8 Tl Tm

)

(

Tm

).

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Comment

Td

Ti

Load disturbance Minimum IAE – x1 K m x 2Tm x 3Tm 1 model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (this page), x 2 values (p. 84) and x 3 values (p. 85) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm TL Tm = 0.05

19.0

11.8

8.6

TL Tm = 0.1

21.2

13.6

9.8

TL Tm = 0.3

22.0

14.6

10.9

TL Tm = 0.5

22.2

14.7

11.0

TL Tm = 0.7

22.1

14.7

TL Tm = 1.0

22.0

14.8

TL Tm = 1.2

22.2

TL Tm = 1.5

5.5

4.2

2.88

2.20

1.80

1.32

1.08

6.1

4.5

3.00

2.26

1.83

1.33

1.09

7.1

5.3

3.44

2.58

2.06

1.40

1.12

7.2

5.5

3.66

2.76

2.22

1.51

1.19

11.0

7.3

5.6

3.76

2.85

2.33

1.59

1.24

11.1

7.4

5.7

3.81

2.94

2.39

1.66

1.32

14.9

11.2

7.4

5.7

3.84

2.95

2.41

1.70

1.35

22.4

15.0

11.3

7.5

5.8

3.88

2.98

2.43

1.72

1.37

TL Tm = 2.0

22.5

15.0

11.2

7.5

5.8

3.91

3.00

2.45

1.74

1.40

TL Tm = 2.5

22.6

15.2

11.2

7.6

5.9

3.93

3.01

2.46

1.75

1.40

TL Tm = 3.0

22.8

15.0

11.2

7.7

5.9

3.95

3.02

2.48

1.76

1.40

TL Tm = 3.5

22.7

15.1

11.0

7.5

5.9

3.94

3.02

2.48

1.75

1.40

TL Tm = 4.0

22.7

15.0

10.9

7.6

5.9

3.94

3.02

2.48

1.76

1.40

TL Tm = 4.5

22.7

15.1

11.2

7.6

5.9

3.95

3.03

2.48

1.77

1.41

TL Tm = 5.0

22.6

15.1

11.2

7.5

5.9

3.94

3.03

2.50

1.78

1.42

TL Tm = 5.5

22.7

15.1

11.3

7.6

5.9

3.94

3.03

2.51

1.79

1.42

TL Tm = 6.0

22.6

15.0

11.5

7.6

5.9

3.94

3.03

2.51

1.80

1.44

TL Tm = 6.5

22.6

15.0

11.3

7.5

5.8

3.94

3.04

2.52

1.80

1.45

TL Tm = 7.0

22.7

14.9

11.5

7.6

5.9

3.95

3.05

2.54

1.81

1.46

83

b720_Chapter-03

17-Mar-2009

FA


84 Rule

Kc

Comment

Td

Ti

Load disturbance Minimum IAE – 1 x1 K m x 2Tm x 3Tm model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (p. 83), x 2 values (this page) and x 3 values (p. 85) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm 1.0

1.0

1.0

1.0

1.0

0.92

0.90

0.90

0.82

0.78

TL Tm = 0.1

1.0

1.0

1.0

TL Tm = 0.3

10.5

6.6

4.6

1.0

1.0

0.92

0.90

0.90

0.82

0.78

2.3

1.1

1.00

0.96

0.92

0.82

0.79

TL Tm = 0.5

10.7

7.0

5.1

3.2

2.3

1.48

1.18

1.04

0.86

0.80

TL Tm = 0.7

10.4

TL Tm = 1.0

9.7

7.0

5.2

3.4

2.7

1.80

1.40

1.20

0.95

0.82

6.7

5.1

3.5

2.8

2.00

1.62

1.40

1.06

0.92

TL Tm = 1.2

10.3

7.0

5.4

3.8

2.9

2.14

1.75

1.48

1.13

0.96

TL Tm = 1.5

11.0

7.6

5.8

4.0

3.1

2.32

1.86

1.59

1.20

1.00

TL Tm = 2.0

12.3

8.4

6.5

4.6

3.6

2.60

2.10

1.79

1.30

1.06

TL Tm = 2.5

13.1

9.0

7.0

4.9

3.8

2.80

2.26

1.90

1.40

1.15

TL Tm = 3.0

13.7

9.6

7.3

5.2

4.1

3.00

2.39

2.01

1.50

1.22

TL Tm = 3.5

14.1

9.8

7.8

5.4

4.2

3.12

2.50

2.13

1.57

1.26

TL Tm = 4.0

14.6

10.2

8.0

5.6

4.4

3.21

2.59

2.20

1.61

1.30

TL Tm = 4.5

15.0

10.5

8.0

5.7

4.5

3.32

2.66

2.23

1.63

1.33

TL Tm = 5.0

15.4

10.8

8.2

5.9

4.7

3.40

2.72

2.30

1.65

1.35

TL Tm = 5.5

15.6

10.9

8.3

6.0

4.8

3.45

2.76

2.33

1.67

1.37

TL Tm = 6.0

15.9

11.1

8.3

6.1

4.9

3.50

2.79

2.36

1.70

1.39

TL Tm = 6.5

16.1

11.2

8.6

6.2

5.0

3.54

2.80

2.38

1.70

1.40

TL Tm = 7.0

16.4

11.5

8.5

6.2

5.0

3.57

2.83

2.40

1.72

1.40

TL Tm = 0.05

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Comment

Td

Load disturbance Minimum IAE – 1 x1 K m x 2Tm x 3Tm model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (p. 83), x 2 values (p. 84) and x 3 values (this page) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm TL Tm = 0.05

0.022 0.034 0.045 0.067 0.089 0.13

0.17

0.21

0.30

0.39

TL Tm = 0.1

0.020 0.030 0.042 0.064 0.088 0.13

0.17

0.21

0.30

0.40

TL Tm = 0.3

0.027 0.038 0.050 0.068 0.081 0.13

0.17

0.21

0.31

0.41

TL Tm = 0.5

0.027 0.039 0.051 0.074 0.094 0.13

0.17

0.22

0.32

0.41

TL Tm = 0.7

0.026 0.038 0.050 0.075 0.097 0.14

0.18

0.22

0.32

0.41

TL Tm = 1.0

0.024 0.037 0.050 0.073 0.097 0.14

0.19

0.23

0.33

0.42

TL Tm = 1.2

0.026 0.038 0.051 0.076 0.100 0.14

0.19

0.23

0.34

0.43

TL Tm = 1.5

0.027 0.040 0.052 0.079 0.104 0.15

0.20

0.24

0.34

0.44

TL Tm = 2.0

0.028 0.041 0.057 0.083 0.110 0.16

0.21

0.26

0.37

0.46

TL Tm = 2.5

0.028 0.042 0.058 0.085 0.113 0.17

0.22

0.27

0.38

0.48

TL Tm = 3.0

0.029 0.045 0.059 0.088 0.118 0.17

0.23

0.28

0.40

0.50

TL Tm = 3.5

0.029 0.045 0.062 0.090 0.120 0.18

0.23

0.28

0.41

0.52

TL Tm = 4.0

0.029 0.046 0.064 0.090 0.122 0.18

0.24

0.29

0.42

0.52

TL Tm = 4.5

0.030 0.045 0.062 0.092 0.123 0.18

0.28

0.30

0.42

0.52

TL Tm = 5.0

0.030 0.044 0.062 0.095 0.125 0.18

0.24

0.30

0.42

0.52

TL Tm = 5.5

0.030 0.047 0.062 0.094 0.126 0.19

0.24

0.30

0.42

0.52

TL Tm = 6.0

0.031 0.048 0.061 0.094 0.128 0.19

0.25

0.30

0.42

0.52

TL Tm = 6.5

0.031 0.048 0.063 0.095 0.133 0.19

0.25

0.30

0.42

0.52

TL Tm = 7.0

0.031 0.049 0.062 0.095 0.130 0.19

0.25

0.30

0.42

0.52

85

b720_Chapter-03

17-Mar-2009

FA


86 Rule

Kc

Minimum IAE – Arrieta Orozco (2006).

10

Kc

Ti

Td

Ti

Td

Comment 0.1 ≤

τm ≤ 2.0 Tm

pp. 23–24. Model: Method 10 Minimum IAE – τ 0.1 ≤ m ≤ 1.0 Pessen (1994). 0.7 K u 0.4Tu 0.149Tu Tm Model: Method 1 0.749 0.921 Modified Tm  τ m  3  Tm  11   minimum IAE – Model:   Td 0.878  Tm  Method 14 Cheng and Hung K m  τ m  (1985). 1.006 0.1 ≤ τ 0.945 0.771 Minimum ISE – m Tm ≤ 1 ; τ  Tm  τ m  1.495  Tm  0.56Tm  m      Murrill (1967), Model: Method 5 K m  τm  1.101  Tm   Tm  pp. 358–363. Minimum ISE – 13.41 K m 0.14Tm 0.05Tm τm Tm = 0.10 Lopez (1968), 2.84 K m 0.55Tm 0.28Tm τm Tm = 0.50 p. 50. 1.52 K m 0.86Tm 0.56Tm τm Tm = 1.00 0.970 0.753 τ 1.473  Tm  Tm  τ m  0.1 ≤ m ≤ 1.0 12   Minimum ISE –   T T d m 1115 . Zhuang (1992), K m  τ m   Tm  Zhuang and 0.735 0.641 τ Tm  τ m  Atherton (1993). 1.524  Tm  1.1 ≤ m ≤ 2.0 13     T T d m K m  τm  1.130  Tm  Model: Method 1 or Method 5 or Method 6 or Method 15.

10

1 Kc = Km

1.0158  0.5022         , Ti = Tm − 0.2228 + 1.3009 τm  , 0.2068 + 1.1597 Tm  τ  T      m  m       

τ  Td = 0.3953Tm  m   Tm  11

12

13

τ Td = 0.482Tm  m  Tm

1.137

   

τ  Td = 0.550Tm  m   Tm 

0.948

τ  Td = 0.552Tm  m   Tm 

0.851

.

.

.

0.8469

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum ISE – 14 Ho et al. (1998). Ti Kc Model: Method 1 0.947 0.738 Minimum ITAE Tm  τ m  1.357  Tm      – Murrill (1967), K m  τm  0.842  Tm  pp. 358–363. Minimum ITAE 11.80 K m 0.22Tm – Lopez (1968), 2.59 K m 0.75Tm p. 50. 1.39 K m 1.15Tm Minimum ITAE – Arrieta Orozco (2006), pp. 90–92.

14

16

0.116  τm  1.0722 φm −   Kc = 0.8432   K m Am  Tm 

−0.908

15

1.0082

, Ti =

1.2497Tm φm

A m 0.2099 Td =

2 ≤ Am ≤ 5 ,

Td

300 ≤ φm ≤ 600 . 0.1 ≤ τm Tm ≤ 1 ; Model: Method 5

Td

0.03Tm

τm Tm = 0.10

0.19Tm

τm Tm = 0.50

0.37Tm

τm Tm = 1.00

Td

2 ≤ Am ≤ 4 ; Model: Method 10

Ti

Kc

Comment

Td

Ti

 τm    T   m

0.3678

,

0.4763Tm φm −

Given A m , ISE is minimised when φ m = 46.5489A m

Am 0.2035

0.328

0.0961

(τ m

87

Tm )

1.0317

 τm    T   m

0.3693

, 0. 1 ≤

τm ≤ 1.0 ; Tm

;

2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1.0 (Ho et al., 1999). 15

16

τ  Td = 0.381Tm  m   Tm 

0.995

.

Tuning rule continued onto p. 88. −0.8757 − 0.0851A m + 0.0120 A m 2    τm  1  2 ,  Kc = 0.6791 − 0.2272A m + 0.0253A m + x1   Km  Tm     2

x1 = 1.9742 − 0.7126A m + 0.0813A m , 0.1 ≤ τm Tm ≤ 2.0 ; −5.1852 + 4.4910 A m − 0.7891A m 2    τm  2  ,  Ti = Tm 7.1461 − 4.8748A m + 0.6843A m + x 2    Tm     2

x 2 = −5.6764 + 4.7562A m − 0.6850A m , 0.1 ≤ τm Tm < 1.0 ; 0.6969 + 0.0948 A m − 0.0231A m 2    τm  2  ,  Ti = Tm 0.2931 + 0.2270A m − 0.0427 A m + x 2    Tm     2

x 2 = 0.9924 − 0.2202A m + 0.0217 A m , 1.0 ≤ τm Tm ≤ 2.0 .

b720_Chapter-03

17-Mar-2009

FA


88 Rule

Kc

Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993).

Ti

Td

17

Kc

Ti

Td

18

Kc

Ti

Td

Comment

Model: Method 1 or Method 5 or Method 6 or Method 15. Model: Method 1 Ti 0.144Tu K

19 20

c

0.960

Model: Method 37 τ 0.1 ≤ m ≤ 1.0 Tm

∧

Ti

Kc

0.15 T u 0.746

Tm  τ m  1.531  Tm      Minimum ISTES K τ 0 .971  Tm  m  m  – Zhuang (1992), 0.705 0.597 Zhuang and Tm  τ m  Atherton (1993). 1.592  Tm    K m  τ m  0.957  Tm 

23

Td

24

Td

1.1 ≤

τm ≤ 2. 0 Tm

Model: Method 1 or Method 5 or Method 6 or Method 15.

(

Td = Tm 0.4718 − 0.1251A m + 0.0202A m

(

2

)

1.8232 − 0.5092 A m − 0.0721A m 2

)

1.4576 − 0.4119 A m + 0.0813A m 2

 τm    T   m

τ  Td = Tm 0.5312 − 0.1662A m + 0.0271A m 2  m   Tm  17

Kc =

1.468  Tm    K m  τm 

0.970

, Ti = 0.730

Tm  τm    0.942  Tm 

0.725

τ  , Td = 0.443Tm  m   Tm 

, 0.1 ≤

τm < 1.0 ; Tm

, 1.0 ≤

τm ≤ 2.0 . Tm

, 0. 1 ≤

τm ≤ 1.0 . Tm

0.939

0.847

0.598

τ  τ , 1.1 ≤ m ≤ 2.0 , Td = 0.444Tm  m  Tm  Tm  τ 4.434K m K u − 0.966 1.751K m K u − 0.612 19 Kc = K u , Ti = Tu , 0.1 ≤ m ≤ 2.0 . Tm 5.12K m K u + 1.734 3.776K m K u + 1.388 18

Kc =

20

Kc =

1.515  Tm    K m  τm 

, Ti =

∧

23

24

6.068K m K u − 4.273 ∧

5.758K m K u − 1.058

τ  Td = 0.413Tm  m   Tm 

0.933

τ  Td = 0.414Tm  m   Tm 

. 0.850

.

∧

Tm  τm    0.957  Tm 

K u , Ti =

∧

τ 1.1622K m K u − 0.748 ∧ T u , 0. 1 ≤ m ≤ 2. 0 . ∧ Tm 2.516K m K u − 0.505

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Td

Comment

Minimum error – step load change – Gerry (1998). Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 44; decay ratio = 0.15.

0.3 Km

0.5τ m

Tm

τm Tm > 5 ; Model: Method 1

Hwang and Fang (1995).

25

26

Kc

Ti

26

Kc

Ti

27

Kc

Ti

28

Kc

Ti

29

Kc

25

3 ≤ ε1 < 20

Nearly minimum IAE, ITAE. Model: Method 30; decay ratio = 0.12. Td

[

]

2 2  0.674 1 − 0.447ωH1τm + 0.0607ωH1 τm  , K c =  K H1 −   K m (1 + K H1K m )   τ K c (1 + K H1K m ) Ti = , 0. 1 ≤ m ≤ 2. 0 . 2 2 Tm ωH1K m 0.0607 1 + 1.05ωH1τm − 0.233ωH1 τm

 K c =  K H1 −  

[

(

2 2 0.778 1 − 0.467ωH1τm + 0.0609ωH1 τm

K m (1 + K H1K m )

[

[

)

] ,  

(

K c (1 + K H1K m )

ωH1K m 0.0309 1 + 2.84ωH1τm − 0.532ωH12 τm 2

2 ω τ  1.31(0.519 ) H1 m 1 − 1.03 ε1 + 0.514 ε1 K c =  K H1 −  K m (1 + K H1K m ) 

] ,  

K c (1 + K H1K m )

(

ωH1K m 0.0603(1 + 0.929 ln[ωH1τm ]) 1 + 2.01 ε1 − 1.2 ε12

2 2  1.14 1 − 0.482ωH1τm + 0.068ωH1 τm K c =  K H1 −  K m (1 + K H1K m ) 

Ti = 29

2.4 ≤ ε1 < 3 ε1 ≥ 20

Ti

Ti = 28

ε1 < 2.4

0.471K u K m ωu

Ti = 27

89

] ,  

(

K c (1 + K H1K m )

ωH1K m 0.0694 − 1 + 2.1ωH1τm − 0.367ωH12 τm 2

2  τ   τ K c = 0.802 − 0.154 m + 0.0460 m   K u , Tm   Tm    Kc Ti = , K u ωu 0.190 + 0.0532(τm Tm ) − 0.00509(τm Tm )2

[

]

2

 τ  τ Ku τ Td = 0.421 + 0.00915 m − 0.00152 m   , 0.1 ≤ m ≤ 2.0 . T Tm T ω K   m  m  u c 

). ). ).

b720_Chapter-03

17-Mar-2009

FA


90 Rule

Kc

O’Connor and Denn (1972). Model: Method 1; K m = Tm .

Comment

Td

Ti ∞

∫[

Minimum 0.5 x 2 y( t ) + (du ( t ) dt ) 2

2

] dt .

0

30

Syrcos and Kookos (2005). Model: Method 1 Minimum IE – Devanathan (1991). Model: Method 1; coefficients of K c , Ti provided are deduced from graphs.

2

31

τm Tm < 0.2

x1τm

2

x 2 τ m + 2 x1 ( 2 + x1 ) τ m

x 2 τm + 2 x1 2τ m x 2

τm x 2 + 2 x1

Ti

Td

Kc

2

2 ≤ τm Tm ≤ 5

Minimum performance index with constraints. 32

0.32Tu

0.08Tu

x 2 Tu

x 3Tu

Kc

τ m Tm

Coefficient values: x3 τ m Tm

x2

x2

0.2 0.4 0.6 0.8 1.0

0.31 0.08 1.2 0.26 0.29 0.07 1.4 0.26 0.29 0.07 1.6 0.24 0.29 0.07 1.8 0.24 0.28 0.07 Minimum performance index: servo tuning x1 K m x 2 (Tm + τ m ) x 3 (Tm + τ m )

x3

0.07 0.07 0.06 0.06

Minimum IAE – Representative coefficient values (deduced from graphs): Gallier and Otto τ T x1 x2 x3 τ m Tm x1 x2 x3 (1968). m m Model: 0.053 12.3 0.95 0.0215 0.67 1.45 0.75 0.155 Method 1 0.11 6.8 0.93 0.041 1.50 0.85 0.63 0.21 0.25 3.4 0.88 0.083 4.0 0.57 0.53 0.19 0.869 Minimum IAE – T m 1.086  Tm  33   Model: Method 5 Rovira et al. τ Td K m  τm  0.740 − 0.13 m (1969). Tm

30

x1 =

x 2 τm 2 − 2(τm Tm ) + 2 (τm Tm )2 + 2τm 2 x 2 2 + (τm Tm )

0. 6 τm

2

≤ x2 ≤

1 τm 2

.

2.2     Tm   Tm  τm      T T 0 . 777 0 . 45 , 0 . 31 0 . 6 , + = + T T 0 . 44 0 . 56 = −   i  d m m τ  . Tm  τm      m  

31

Kc =

1 Km

32

Kc =

ξ cos tan −1 (1.57 D R − (0.16 D R )) . K m cos tan −1 (6.28Tm Tu )

33

; recommended

[

[

τ  Td = 0.348Tm  m   Tm 

]

0.914

, 0.1
1.  T m 

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum ISTSE – Zhuang and Atherton (1993). Model: Method 1 Chidambaram (1995a).

98

98

99

mK u cos(φ m )

Ti

Td

Comment

Ti

Ti x1

A m = 2, φ m = 60 0 .

Ti

Td

Model: Method 1

mK u cos(φ m ) 100

Kc

(

tan (φm ) + x1 = 0.413(3.302K m K u + 1) , Ti = x1 99

100

)

m = 0.614(1 − 0.233e −0.347 K m K u ) , φ m = 33.8 0 1 − 0.97e −0.45K m K u , 0.1 ≤ τm Tm ≤ 2.0 , 4 + tan 2 (φm ) x1 2ωu

, Model: Method 1.

∧ ∧ ∧   m = 0.613(1 − 0.262e −0.44 K m K u ) , φ m = 33.2 0 1 − 1.38e −0.68 K m K u  , x1 = 1.687 K m K u ,   0.1 ≤ τm Tm ≤ 2.0 , Model: Method 37.

Kc ≈ Ti ≈

Td ≈

 1  K mAm    1.17   

(

 1.17   

(

2   or 0.85 1 + 4.45 Tm  ; + 0 . 85 2 τ   KmAm  m 4.45 + 4(τm Tm ) − 2.11  5Tm 5Tm or ; 2 2   Tm  3.42  − 0.15 4.45 + 0.85 − 1 2  τm   4.45 + 4(τm Tm ) − 2.11  0.8Tm 0.8Tm or . 2 2   Tm  3.42  − 0.15 4.45 + 0.85 − 1 2  τm   4.45 + 4(τm Tm ) − 2.11 

(

3.42

)

)

)

107

b720_Chapter-03

17-Mar-2009

FA


108 Rule Chidambaram (1995a) – continued.

Kc

Ti

Td

Comment

x1 K m A m

x2τm

x 3τ m

Alternative.

τm Tm

x1

0.062 0.083 0.150 0.196 0.290 0.339 1.20Tm

29.770 22.100 12.470 9.670 6.686 5.784 K m τm

x2

2.34 0.37 2.33 0.37 2.29 0.37 2.27 0.36 2.23 0.36 2.21 0.35 2.4 τ m

Chidambaram (1995a). 1.03Tm K m τm Model: Method 1 0.90Tm K m τm 101 Åström and Hägglund (1995), pp. 208–210. τ 0.14 ≤ m ≤ 5.5. Tm

102

Coefficient values: x3 τm Tm

Kc

x2

x3

2.19 2.14 2.00 1.72 1.67

0.35 0.34 0.32 0.28 0.27

A m = 1.5

2.4 τ m

0.38τ m

A m = 1.75

2.4 τ m

0.38τ m

A m = 2.0

Ti

Td

M s = 1.4

Td

Ti

Ti Kc

x1

5.104 3.781 2.262 1.209 1.105

0.38τ m

103

104

0.389 0.543 1.03 2.92 3.69

105

Td

M s = 2.0

Td

Ti

Model: Method 5 or Method 17.

Ku cos φ m Am

Tan et al. (1996). Model: Method 36

Kφ

107

Am

101

x1Td

106

Td

Ti

x1 chosen arbitrarily.

Td

Arbitrary A m , φ m at ωφ .

  1 Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] + + Tds  . T s i   2

2 2 1.52e −8.22 τ +10.1τ Tm , Ti = 2.08τme −2.32 τ +1.4 τ , Td = 2.23τm e −0.55τ − 6.9 τ . K m τm

102

Kc =

103

Ti = 0.184Tm e 2.98τ + 0.7 τ , Td = 0.193Tm e 4.82 τ − 7.6 τ .

104

Kc =

105

Ti = 0.06Tme 4.45τ −1.549 τ , Td = 0.345Tm e −2.75τ −1.151τ .

106

Td = Tu

2

2

2

107

2 2 1.85e −8.95 τ + 9.851τ Tm , Ti = 0.704τm e −0.85τ − 0.879 τ , Td = 3.91τme −2.55τ − 0.491τ . K m τm 2

Ti =

tan φm +

(4 x1 ) + tan 2 φm 4π

(

r2 K φ ωu 2 − ωφ 2 ωu ωφ

2

2

2

2

)

K u − r2 K φ

2

, Td =

; recommended A m = 2 , φ m = 450 . ωu K u 2 − r2 2 K φ 2

(

r2 K φ ωu 2 − ωφ 2

)

, r2 = 0.1 + 0.9

Ku . Kφ

b720_Chapter-03

17-Mar-2009

FA


Friman and Waller (1997). Model: Method 1; Am > 2

Comment

Kc

Ti

Td

0.25K u

1.1879Tu

0.2970Tu

2.6064 ω1500

0.6516 ω1500

0.4830

(

G p jω1500

)

109

τ m Tm < 0.25 ,

φ m > 60 0 . τm ≤ 2.0, Tm

0.25 ≤

φ m > 450 . Branica et al. (2002). Model: Method 1; M s = 1.8

108

Kc

Ti

Td

109

Kc

Ti

Td

Kc

Ti

Td

110 111

Gorez (2003). Model: Method 1 Åström and Hägglund (2006), pp. 261–262.

108

109

110

111

112

Kc =

x 2 − x1 x 2K m 113

Kc

(1)

Ti

(x 2 − x1 )

(1)

Td

T 1  T   0.021 + 0.788 m  , Ti = 1.534τ m  m τ τm  K m   m

   

0.326

T  1  T   0.292 + 0.671 m  , Ti = 1.239τm  m  Kc = τ  τm  K m   m

0.554

T  1  T   0.338 + 0.623 m  , Ti = 1.228τm  m    τ  τm  Km   m

0.480

Kc =

τm ≤ 0.5 Tm

0.5
0.75 (Leva, 2001).

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc


Ti

Ti

119

K m (λ + τ m )

Comment

Td

Td

Ti

TCL = λ ; recommended TCL = 0.5τm (Lee et al., 2006). 120

Kc

Tm K m (λ + 0.5τ m )

Gerry (1999). Model: Method 11

113

Ti

Td

Tm

0.5τ m

λ = 2τ m (aggressive, less robust tuning); λ = 2(Tm + τ m ) (more robust tuning).

Gong (2000). Model: Method 1

121

Kc

Wang (2003), x1 K m pp. 16–17. τm Tm x1 Model: 0.05 13.2 Method 1. 0.11 5.3 Regulator; 0.18 4.3 x1 , x 2 , x 3 0.25 3.3 deduced from 2.6 graphs; robust to 0.33 0.43 2.0 ±25% 0.54 1.6 uncertainty in 0.67 1.4 process model 0.82 1.3 parameters. 1.0 1.1

119

Ti = Tm +

120

Kc =

x 2 (τm + Tm )

0.3866τm τ 1 + 0.3866 m Tm x 3 (τm + Tm )

x2

x3

τm Tm

x1

x2

x3

0.18 0.33 0.43 0.49 0.51 0.56 0.59 0.61 0.61 0.61

0 0 0 0 0.044 0.074 0.091 0.106 0.120 0.124

1.2 1.5 1.9 2.3 3.0 4.0 5.7 9.0 19.0

0.9 0.8 0.7 0.6 0.5 0.5 0.4 0.4 0.4

0.60 0.59 0.57 0.56 0.51 0.50 0.47 0.46 0.46

0.136 0.139 0.144 0.146 0.147 0.147 0.148 0.149 0.149

τm 2 τm 2  τ  1 − m  . , Td =  2(λ + τm ) 2(λ + τm )  3Ti 

Ti T 2 + x1τm − 0.5τm 2 , Ti = Tm + x1 − CL , K m (2TCL + τm − x1 ) 2TCL + τm − x1 Tm x1 −

Td =

Tm + 0.3866τm

0.167 τm 3 − 0.5x1τm 2 T 2 + x1τm − 0.5τm 2 2TCL + τm − x1 − CL , Ti 2TCL + τm − x1 2

τm

 T  − x1 = Tm − Tm 1 − CL  e Tm . Tm   121

Kc =

τ  0.7556Tm  1 + 0.3866 m  . K m τm  Tm 

b720_Chapter-03

17-Mar-2009


114 Rule

Kc

Ti

x 2 (τm + Tm )

Wang (2003), x1 K m pp. 15–16. τm Tm x1 Model: 0.05 2.89 Method 1. 0.11 2.61 Servo; ±10% 2.28 overshoot, step 0.18 0.25 1.96 input; x1 , x 2 , x 3 0.33 1.58 deduced from 1.40 graphs; robust to 0.43 0.54 1.19 ±25% 0.67 1.04 uncertainty in 0.82 0.90 process model 1.0 0.85 parameters. Shamsuzzoha et 122 al. (2006c). Kc Model: Method 1

122

FA

Kc =

x 3 (τm + Tm )

x2

x3

τm Tm

x1

x2

x3

0.81 0.86 0.86 0.87 0.85 0.84 0.78 0.77 0.73 0.74

0 0 0 0.034 0.043 0.066 0.075 0.094 0.100 0.120

1.2 1.5 1.9 2.3 3.0 4.0 5.7 9.0 19.0

0.72 0.68 0.56 0.50 0.43 0.38 0.34 0.32 0.29

0.68 0.65 0.60 0.55 0.52 0.48 0.44 0.44 0.40

0.130 0.142 0.149 0.140 0.137 0.136 0.122 0.116 0.096

Ti

Td

Ti λ2 + x1τm − 0.5τm 2 , Ti = Tm + x1 − , K m (2λξ + τm − x1 ) 2λξ + τm − x1 Tm x1 −

Td =

Comment

Td

0.167 τm 3 − 0.5x1τm 2 λ2 + x1τm − 0.5τm 2 2λξ + τm − x1 , Ti 2λξ + τm − x1

(

)

 λ2 − 2λξTm + Tm 2 e − τ m x1 = Tm 1 − 2 Tm 

Tm

 . 

λ , ξ not defined.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Comment

Td

λ = x1Tm

123 Ti Td Kc Shamsuzzoha Coefficients of x1 (deduced from graph): and Lee (2007a). Model: Method 1 M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0

123

Kc =

0.11

0.10

0.08

0.07

0.06

0.20

0.17

0.15

0.14

0.12

τm = 0.1Tm

0.35

0.31

0.27

0.25

0.22

τm = 0.2Tm

0.46

0.41

0.37

0.34

0.30

τm = 0.3Tm

0.55

0.49

0.44

0.40

0.36

τm = 0.4Tm

0.62

0.55

0.50

0.46

0.42

τm = 0.5Tm

0.68

0.60

0.55

0.51

0.47

τm = 0.6Tm

0.72

0.65

0.59

0.55

0.50

τm = 0.7Tm

0.77

0.69

0.63

0.59

0.53

τm = 0.8Tm

0.81

0.72

0.66

0.62

0.56

τm = 0.9Tm

0.84

0.76

0.69

0.65

0.59

τm = 1.0Tm

0.91

0.82

0.75

0.70

0.64

τm = 1.25Tm

τm = 0.05Tm

0.97

0.87

0.80

0.75

0.70

τm = 1.5Tm

undefined

0.91

0.84

0.78

0.73

τm = 1.75Tm

undefined

0.96

0.87

0.81

0.76

τm = 2.0Tm

Ti 3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2 , Ti = Tm + 2 x 2 − , K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2 2Tm x 2 + x 2 2 −

Td =  x 2 = Tm 1 − 

λ3 + 0.167 τm3 − x 2 τm 2 + x 2 2 τm 3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2 3λ + τm − 2 x 2 , Ti 3λ + τm − 2 x 2

 λ 1 −  T m 

3

 −τm  e  

Tm

 .  

115

b720_Chapter-03

17-Mar-2009

FA


116 Rule

Kc

Td

Comment

Ti

0.25Ti


x 2 Tu

x 3 Tu

Ti

Ultimate cycle

McMillan (1984). Model: Method 2 or Method 55.

124

Kc

x 1K u

Kuwata (1987). x 1 , x 2 deduced from graphs.

x1

x2

x3

τm Tu

x1

x2

x3

τm Tu

x1

x2

x3

τm Tu

0.44 0.80 0.08 0.28 0.37 0.26 0.06 0.39 0.28 0.18 0.03 0.50 0.42 0.43 0.08 0.32 0.33 0.22 0.05 0.43 0.40 0.32 0.07 0.35 0.30 0.18 0.04 0.47 Geng and Geary τm 0.94Tm < 0.167 (1993). 2τ m 0.5τ m Tm Kmτm Model: Method 1 Wojsznis and 1.3 ≤ x1 ≤ 1.6 ; Tm τ m Tm + 0.5τm Blevins (1995). Tm + 0.5τ m 2Tm + τ m Model: x1K m τm Method 45 Perić et al. ^ Recommended (1997). x1Td Td 125 K u cos φ m Model: x1 = 4 . Method 49 Matsuba et al. τ 0.1 ≤ m ≤ 1.0 126 (1998). Ti Td Kc Tm Model: Method 1 127 Wojsznis et al. Model: Method 1 x1Ti K u cos φ m Ti (1999). A m

124

125

126

127

    0.65   T    1.415 Tm  1   m  = τ + T 1 Kc = ,  i m 0.65    . K m τm   T     Tm + τm   m   1 +     Tm + τm    Td =  tan φm +  

1.31  Tm    K m  τm 

0.9

^

T 4 + tan 2 φm  u .  4π x1 

2.19  Tm    ≤ Kc ≤ K m  τm 

0.9

T  , Ti = 1.59τm  m   τm  T Ti =  tan φ m + 4 x1 + tan 2 φ m  u .   4πx1

0.097

T  , Td = 0.40τm  m   τm 

0.097

.

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Rule

Td

Comment

Ti

0.15Ti

Default values.

0.38K u

1.2Tu

0.18Tu

0.27 K u

0.87Tu

0.13Tu

Kc

Wojsznis et al. (1999) – continued.

Ti

0.5K u cos φ m

128

φ m = 450 ; τ m Tm < 0.2 . Am = 3 ,

φ m = 330 ; τ m Tm ≈ 0.2 5 .

Wojsznis et al. (1999). Model: Method 45 Gu et al. (2003).

128

129

130

129

Ti

Kc

0.125Ti

0.3 ≤ x1 ≤ 0.4 ; 0.25 ≤ x 2 ≤ 0.4 ;

0.2 ≤ 130

Ti

Kc

τm ≤ 0.7 . Tm

Model: Method 1

Td

Ti = 0.53Tu  tan φm + 0.6 + tan 2 φm  .  

          0.6 0.6    . K c = K u x 2 + 0.251.0 − x1 − , T T x = +  i u 1 T   T    −  u − 7.0   −  u − 7.0     τ τ       1+ e  m 1+ e  m     Representative tuning rule: 0.6K u ≤ K c ≤ K u , Ti = 0.5Tu , Td = 0.125Tu , with K u ≈ Tu ≈ 2π

τm 3 + 5τm 2Tm + 12τm Tm 2 + 12Tm3

[

(

K m τm τm 2 + 5τm Tm + 8Tm 2

(

)

,

)

τm K m K u τm τm 2 + 6τm Tm + 12Tm 2 + τm 3 + 6τm 2Tm + 18τm Tm 2 + 24Tm 3

(

12(K m K u + 1) τm 2 + 4τm Tm + 6Tm 2

)

].

117

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118



E(s)

−

+

  1 K c 1 + + Td s  Ti s  

U(s)

1 1 + Tf s

Y(s)


m

Table 11: PID controller tuning rules – FOLPD model G m (s) = Rule

Kc

Lim et al. (1985). Schaedel (1997). Model: Method 25

1

2

Kc Kc

Ti


Comment

Td

Direct synthesis: time domain criteria 0 Model: Method 1 Tm Direct synthesis: frequency domain criteria 2 2 Tf = Td N ; T1 − T2 T2 2 T33 − 2 5 ≤ N ≤ 20 . T1 − Td T1 T2

2

1

Kc =

TCL 2 Tm . , Tf = 2ξTCL 2 + τ m K m (2ξTCL 2 + τm )

2

Kc =

0.375Ti Tm , T1 = + τm , K m (T1 + (Td N ) − Ti ) 1 + 3.45 τam Tm

(

2

T2 = 1.25Tm τ am + T1 =

0.7Tm

2

Tm − 0.7 τ am

T3 3 =

(

τam Tm 2 τ + 0 . 5 τ , T = 0 , 0 < ≤ 0.104 ; 3 m m Tm 1 + 3.45 τam Tm

(

0.7Tm

2

+ τ m , T2 = Tm τ am +

)

Tm − 0.7 τ am

)+T

0.952Tm 2 τ am − 0.1 Tm −

2

0.7 τ am

)

+ 0.5τ m 2 ,

a mτmτm

+

0.35Tm 2 τ m 2 Tm −

0.7 τ am

+ 0.167τ m 3 ,

τ am > 0.104 . Tm

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kristiansson and Lennartson (2002b).

Kc =

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

3

0.2 ≤ τm Tm ≤ 5 , 1.65 ≤ M max ≤ 1.85 .

Kristiansson and Lennartson (2006).

3

Kc

4

Kc

Ti

Td

Model: Method 1

5

Kc

Ti

Td

Model: Method 12

0.05 ≤ τm Tm ≤ 2 , 1.65 ≤ M max ≤ 1.85 .

[

]

1.8 0.13 + 0.5(τm Tm ) − 0.04(τm Tm )2 , τ   0.12 K m  m − 0.05 0.7 + (τm Tm ) + 0.05   Tm 

2    τ   τ 0.12 Ti = 2Tm 0.13 + 0.5 m − 0.04 m   0.7 + (τm Tm ) + 0.05  Tm   Tm     2

2 2    τm    τm   τm τm         0.5Tm 0.13 + 0.5 − 0.04 0.9Tm 0.13 + 0.5 − 0.04 Tm Tm   Tm Tm         ,T =   . Td = f     τm   0.12 Tm ,25 − 0.05 min 5 + 2 0.7 +  (τm Tm ) + 0.05  τm     Tm 

        0 . 54 T ( ) 0 . 12 + τ 0 . 12 4 m m 0.7 +  , Ti = 0.6(τm + Tm )0.7 + , Kc = τm τm τ    + 0.05  + 0 . 05 K m τm  m − 0.05    T T m m T      m 

Td =

5

Kc = Ti =

0.15(τm + Tm )

  0.12 0.7 +  ( ) τ T + 0 . 05 m m  

[

, Tf =

0.081(τm + Tm )2 τ , 0.7 ≤ m ≤ 4 .  τm  Tm   T − 0.05 min 5 + 2 m ,25  T τ m  m   

2[1.1(Tm τm ) − 0.1] 0.06 + 0.68(τm Tm ) − 0.12(τm Tm )2

[

K m 0.68 + 0.84(τm Tm ) − 0.25(τm Tm )

[ [0.68 + 0.84(τ

2

2Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) m

Tm ) − 0.25(τm Tm )2

]

2

],

]

],

2 2   τ   τ   τ τ Td = 0.5Tm 0.06 + 0.68 m − 0.12 m   0.68 + 0.84 m − 0.25 m   ,  Tm Tm  Tm     Tm   

[

]

T 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1] Tf = m . min{5 + 2(Tm τm ),25} 2 2

119

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120 Rule Kristiansson and Lennartson (2006).

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 12

6

0.7 < τm Tm < 2 ; 1.65 ≤ M max ≤ 1.8 .

Robust

Tm + 0.5τ m K m (λ + τ m )

Horn et al. (1996). Model: Method 1

Tm + 0.5τ m

Tm τ m 2Tm + τ m

λ = max (0.25τ m ,0.2Tm ) (Luyben, 2001);

Tf =

λτ m , 2(λ + τ m )

λ ∈ [τ m , Tm ]

λ > 0.25τ m (Bequette,

2003); λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ]

(‘moderate’ tuning), λ > [10Tm ,80τm ] (‘conservative’ tuning) (Cooper, 2006b). 7 H ∞ optimal – Tm + 0.5τ m τ m Tm Kc τ m + 2Tm Tan et al. (1998b). λ = 2 : ‘fast’ response; λ = 1 : ‘robust’ tuning; λ = 1.5 : recommended. Model: Method 1 Rivera and Jun Tm τmλ Tf = ; (2000). 0 Tm K m (2 τ m + λ ) 2τ m + λ Model: Method 1 λ not specified.

 T  0.6667(Tm + τm )1.1 m − 0.1 0.6667(Tm + τm )  τm  6 Kc = , Ti = , 2 2       τ τ τ τ 0.68 + 0.84 m − 0.25 m   K m τm 0.68 + 0.84 m − 0.25 m   T   Tm Tm   Tm     m    



 − 0.1 2   τm  , T = 0.1667(τ + T )0.68 + 0.84 τm − 0.25 τm   . d m m T   Tm   T   m   9Tm  5 + 2 m  τm  

(Tm + τ m )2 1.1 Tm Tf =

7

Kc =

 τm 0.265λ + 0.307  Tm    τ + 0.5  , Tf = 5.314λ + 0.951 . Km  m 

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Rule

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1 Tf = Td .

Tm + 0.5τ m K m (λ + τ m )

Tm + 0.5τ m

Tm 2Tm + τ m

Lee and Edgar (2002). Huang et al. (2002). Model: Method 1

8

Tf =

λτ m 2(λ + τ m )

λ = 0.65τ m : minimum ISE; λ = 0.5τ m : overshoot to a step input

≤ 5% , A m = 3 . In general, A m = 2(λ τ m + 1) .

Zhang et al. (2002b).

9

Vilanova (2006). Model: Method 1

10

Tang et al. (2007). Model: Method 1

121

Kc

Tm + 0.5τ m

Tm τ m 2Tm + τ m

Kc

Tm + 1.75τm

1.72τm Tm Tm + 1.75τm

τm + 2Tm K m (4λ + τm )

0.5τm + Tm

Tm τm τm + 2Tm

Model: Method 1

φm ≥ 450 Tf =

2λ2 4λ + τm

Labelled the H ∞ Smith predictor; 0.3 ≤ λ τm ≤ 0.8 . τm + 2Tm 2K m (λ + τm )

0.5τm + Tm

Tm τm τm + 2Tm

Tf =

λτ m 2(λ + τm )

Labelled the H 2 Smith predictor or the Dahlin controller; 0.3 ≤ λ τm ≤ 0.8 .

2Tm + τm 3K m τm

8

Kc =

2  τm 2 1 3Tm − τm Tm + τm + τm + K m (λ + τm )  2(λ + τm ) 6(λ + τm ) 4(λ + τm )2  2

Ti = Tm + 9

Kc =

10

Kc =

Tm τm τm + 2Tm

Tm + 0.5τm

Recommended λ = 0.5 ; Tf = 0.167 τm .

 ,  

2

2

τm 3Tm − τm τm 3Tm − τm τm + τm + , Td = τm + . 2 2(λ + τm ) 6(λ + τm ) 4(λ + τm ) 6(λ + τm ) 4(λ + τm )2

0.1Tm τ m 1  Tm + 0.5τm  λ2   , Tf = . or Tf = 2λ + 0.5τ m τ m + 2Tm K m  2λ + 0.5τm 

0.377(Tm + 1.75τm ) , Tf = 1.72τm . K m τm

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122

  Td s 1 + 3.3.4 Controller with filtered derivative G c (s) = K c 1 +  Ti s 1 + s Td  N  R(s)

E(s)

−

+

  Td s 1 + K c 1 + T  Ti s 1+ s d  N 

     

U(s)

Rule

Kc

Ti

Y(s)


m

Table 12: PID controller tuning rules – FOLPD model G m (s) =

     


Comment

Td

Minimum performance index: regulator tuning Ti Td 1 Kc

Minimum IAE – Alfaro Ruiz Model: Method 10; N = 10. (2005a). Minimum IAE – Ti Td 2 Kc Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003).

1

Kc =

1.0158  0.5022    T   1   , Ti = Tm  − 0.2228 + 1.3009 τm  , 0.2068 + 1.1597 m  T  Km      τm   m    

τ  Td = 0.3953Tm  m   Tm  2

Kc =

1 Km

0.8469

, 0.05 ≤

τm ≤ 2.0 . Tm

0.9946  0.4796        0.1050 + 1.2432 Tm   , Ti = Tm − 0.2512 + 1.3581 τm  , τ  T      m  m       

 τ  Td = Tm  − 0.0003 + 0.3838 m    Tm  

0.9479 

 , 0.1 ≤ τm ≤ 1.2 . Tm  

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Rule

Kc

Ti

Td

Comment

Ti Td 3 τ Kc 0.1 ≤ m ≤ 1.2 Minimum ITAE Tm – Arrieta Orozco Model: Method 10 ; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). Minimum performance index: servo tuning Tavakoli and Minimum IAE. Ti 4 Kc Tavakoli (2003). 0.0111Tm Minimum ISE. Ti 5 Model: Kc Method 1;     τ     0.1 ≤ m ≤ 2 ;   T m 1 0 . 8  τ m  0.3 +  Tm  τ  0.06  Minimum ITAE.  m  τm   τm K m  τm  N = 10 + 0.1  + 0.04      Tm  Tm 

3

1 Kc = Km

1.0191  0.4440        0.1230 + 1.1891 Tm   , Ti = Tm  − 0.3173 + 1.4489 τm  , τ  T      m  m       

0.9286   τ   (Arrieta Orozco, 2003); Td = Tm 0.0053 + 0.3695 m  Tm      

 τ  Td = Tm − 0.0053 + 0.3695 m    Tm   4

5

0.9286 

 (Arrieta Orozco and Alfaro Ruiz, 2003).  

τ          0.3 m + 1.2    1  1 0.06  Tm 2    Kc = , Ti = τm .  τm   τm  K m  τm + 0.2  + 0 . 08 + 0 . 04        Tm   Tm   Tm  τm     + 0.75   0.3   1 1  Tm   . , Ti = 2.4τm Kc =  τm  K m  τm + 0.05  + 0 . 4     T T m    m 

123

b720_Chapter-03

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124 Rule

Kc

Ti

Comment

Td

Minimum IAE – Ti Td 6 Kc Alfaro Ruiz Model: Method 10; N = 10. (2005a). Minimum IAE – Ti Td 7 Kc Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). Minimum ITAE T Td 8 i Kc – Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003).

6

Kc =

0.9971  0.8456   T    1   , Ti = Tm 0.9781 + 0.3723 τm  , 0.3295 + 0.7182 m  T  τm  Km     m       

τ  Td = 0.3416Tm  m   Tm  7

Kc =

1 Km

0.9414

Kc =

τm ≤ 2.0 . Tm

0.9597  1.0092        0.2268 + 0.8051 Tm   , Ti = Tm 1.0068 + 0.3658 τm  , τ  T      m  m       

 τ  Td = Tm − 0.0146 + 0.3500 m    Tm   8

, 0.05 ≤

0.8100 

 , 0.1 ≤ τm ≤ 1.2 . Tm  

0.9462  0.6884   T    1   , Ti = Tm 0.9581 + 0.3987 τm  , 0.1749 + 0.8355 m  T  τm  Km     m        0.7417    τm    , 0.1 ≤ τm ≤ 1.2 .   Td = Tm − 0.0169 + 0.3126  Tm T    m  

b720_Chapter-03

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FA


Rule

Kc

K c = x1

Comment

9

12 13

1.473  Tm    K m  τm 

T τ  Ti = x1 m  m  1.115  Tm 

Td

Minimum performance index: servo/regulator tuning Td τ Kc 0.1 ≤ m ≤ 1.0 Ti T 10 11 m Kc Td

Minimum ISE – Arrieta and Vilanova (2007). Model: Method 10; N = 10.

9

Ti

125

Kc Kc

0.970

0.753

τ  Td = 0.550x1Tm  m   Tm 

Td

Ti

14

τm ≤ 2.0 Tm

0.897

+ (1 − x1 )

1.048  Tm  K m  τ m

+ (1 − x1 )

Tm ; 1.195 − 0.368(τm Tm )

0.948

  

Td

1.1 ≤

;

τ  + 0.489(1 − x1 )Tm  m   Tm 

0.888

;

x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2

10

11

12

Kc =

1.473x1 + 1.048(1 − x1 )  Tm    τ  Km  m

K c = x1

1.524  Tm    K m  τm 

Tm  τm    1.130  Tm 

0.735

0.641

τ  Td = 0.552x1Tm  m   Tm 

14

.

τ  Td = [0.550x1 + 0.489(1 − x1 )]Tm  m   Tm 

Ti = x1

13

0.970 x 1 + 0.897 (1− x 1 )

Kc =

+ (1 − x1 )

+ (1 − x1 )

0.851

0.948 x 1 + 0.888(1− x 1 )

.

1.154  Tm  K m  τ m

  

0.567

Tm 1.047 − 0.220

;

τ  + 0.490(1 − x1 )Tm  m   Tm 

1.524 x1 + 1.154((1 − x1 )  Tm    τ  Km  m

;

τm Tm

0.708

T  ; x1 = 0.2382 + 0.2313 m   τm 

0.735 x 1 + 0.567 (1− x 1 )

τ  Td = [0.552 x1 + 0.490(1 − x1 )]Tm  m   Tm 

. 0.851x 1 + 0.708(1− x 1 )

.

7.1208

.

b720_Chapter-03

FA


126 Rule

Kc

Minimum ISTSE – Arrieta and Vilanova (2007). Model: Method 10; N = 10.

15

17-Mar-2009

K c = x1

1.468  Tm    K m  τm 

T τ  Ti = x1 m  m  0.942  Tm 

15

Kc

16

Kc

18

Kc

19

Kc

0.970

0.725

τ  Td = 0.443x1Tm  m   Tm 

Ti

Td

Ti

Td 17

Ti

τm ≤ 1.0 Tm

1.1 ≤

τm ≤ 2.0 Tm

Td

0.897

+ (1 − x1 )

1.042  Tm  K m  τ m

+ (1 − x1 )

Tm ; 0.987 − 0.238(τm Tm )

0.939

  

0.1 ≤

Td

Td 20

Comment

;

τ  + 0.385(1 − x1 )Tm  m   Tm 

0.906

;

x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2

16

17

18

Kc =

1.468x1 + 1.042(1 − x1 )  Tm    τ  Km  m

K c = x1

1.515  Tm    K m  τm 

Tm  τm    0.957  Tm 

0.730

0.598

τ  Td = 0.444x1Tm  m   Tm 

20

.

τ  Td = [0.443x1 + 0.385(1 − x1 )]Tm  m   Tm 

Ti = x1

19

0.970 x 1 + 0.897 (1− x 1 )

Kc =

+ (1 − x1 ) + (1 − x1 )

0.847

0.939 x 1 + 0.906 (1− x 1 )

.

1.142  Tm  K m  τ m

  

0.579

Tm 0.919 − 0.172

;

τm Tm

τ  + 0.384(1 − x1 )Tm  m   Tm 

1.515x1 + 1.142(1 − x1 )  Tm  τ Km  m

   

;

0.839

T  ; x1 = 0.2382 + 0.2313 m   τm 

0.730 x 1 + 0.579 (1− x 1 )

τ Td = [0.440x1 + 0.384(1 − x1 )]Tm  m  Tm

.    

0.847 x 1 + 0.839 (1− x 1 )

.

7.1208

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum ISTES – Arrieta and Vilanova (2007). Model: Method 10; N = 10.

21

K c = x1

1.531  Tm    K m  τm 

T τ  Ti = x1 m  m  0.971  Tm 

21

Kc

22

Kc

24

Kc

25

Kc

0.960

0.746

τ  Td = 0.413x1Tm  m   Tm 

Ti

Td

Ti

Td 23

Ti

26

0.1 ≤

τm ≤ 1.0 Tm

1.1 ≤

τm ≤ 2.0 Tm

Td

0.904

+ (1 − x1 )

0.968  Tm  K m  τ m

+ (1 − x1 )

Tm ; 0.977 − 0.253(τm Tm )

0.933

  

Comment

Td

Td

127

;

τ  + 0.316(1 − x1 )Tm  m   Tm 

0.892

;

x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2

22

23

24

Kc =

1.531x1 + 0.968(1 − x1 )  Tm    τ  Km  m

K c = x1

1.592  Tm    K m  τm 

Tm  τm    0.957  Tm 

0.705

0.597

τ  Td = 0.414x1Tm  m   Tm 

26

.

τ  Td = [0.413x1 + 0.316(1 − x1 )]Tm  m   Tm 

Ti = x1

25

0.960 x 1 + 0.904 (1− x 1 )

Kc =

0.850

.

1.061  Tm  K m  τ m

+ (1 − x1 )

+ (1 − x1 )

0.933 x 1 + 0.892 (1− x 1 )

  

0.583

Tm 0.892 − 0.165

;

τm Tm

τ  + 0.315(1 − x1 )Tm  m   Tm 

1.592 x1 + 1.061(1 − x1 )  Tm    τ  Km  m

T , x1 = 0.2382 + 0.2313 m  τm

0.832

.

0.705 x 1 + 0.583(1− x 1 )

τ  Td = [0.414 x1 + 0.315(1 − x1 )]Tm  m   Tm 

. 0.850 x 1 + 0.832 (1− x 1 )

.

   

7.1208

;

b720_Chapter-03

FA


128 Rule

Kc

Davydov et al. (1995). Model: Method 6 Kuhn (1995). Model: Method 17; N = 5. Schaedel (1997).

27

17-Mar-2009

Kc =

Ti

Comment

Td

27

Kc

28

Kc

Direct synthesis: time domain criteria Ti 0.25Ti ξ =0.9; 0 . 2 ≤ τm Tm ≤ 1. T 0.4T i

i

1 Km

0.66(τ m + Tm )

0.167(τ m + Tm )

‘Normal tuning’.

2 Km

0.8(τ m + Tm )

0.194(τ m + Tm )

‘Rapid tuning’.

T2 2 T33 − T1 T2 2

Model: Method 25

29

2 T1

− T2 T1 − Td

Kc

2

  1 τ , Ti =  0.186 m + 0.532 Tm , N = K m . T K m (1.552(τm Tm ) + 0.078) m  

  1 τ , Ti =  0.382 m + 0.338 Tm , N = K m . T K m (1.209(τm Tm ) + 0.103) m   Tm 0.375Ti 29 , T1 = + τm , Kc = K m (T1 + (Td N ) − Ti ) 1 + 3.45 τam Tm 28

Kc =

(

2

T2 = 1.25Tm τ am + T1 =

0.7Tm

2

Tm − 0.7 τ am

T3 3 =

N=

(

Tm τa τm + 0.5τm 2 , T3 = 0 , 0 < m ≤ 0.104 ; a Tm 1 + 3.45 τm Tm

(

0.7Tm

2

+ τ m , T2 = Tm τ am +

)

2

Tm − 0.7 τ am

)+T

0.952Tm 2 τ am − 0.1 Tm − 0.7 τ am

(a + τm − Ti )2 + 0.375 TiTd 4

+ 0.5τ m 2 ,

a m τm τm +

Td  a + τm − Ti  − + 2  

)

0.35Tm 2 τ m 2 Tm − 0.7τ am

+ 0.167τ m 3 ,

τ am > 0.104 . Tm

, 0.05 ≤ a ≤ 0.2 , K v = prescribed

Kv

overshoot in the manipulated variable for a step response in the command variable.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Goodwin et al. (2001).

Latzel (1988). Model: Method 1 Latzel (1988). Model: Method 2

30

Kc = Ti =

Ti

Comment

p. 426; Model: Method 1 Direct synthesis: frequency domain criteria 0.84Tm 0.067Tm φm = 450 Ti

Td

Kc

0.84Tm

0.067Tm

φm = 600

33

Kc

0.76Tm

0.13Tm

φm = 450

34

Kc

0.76Tm

0.13Tm

φm = 600

30

Kc

31

Kc

32

(4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2

(

K m 16ξ2TCL 2 2 + 8ξTCL 2 τm + τm 2

),

(4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2

(

(

2 16ξ 2TCL 2 2 + 8ξTCL 2 τm + τm 2 2

Td = 16ξ 2TCL 2 + 8ξTCL 2 τm + τm

2

)

[(4ξT

)

CL 2

N=

Td

129

4ξTCL 2 + τm 2TCL 2 2

Td with G CL (s) =

(4ξTCL 2 + τm ) ,

+ τ m )2 τ m Tm − 2TCL 2 2 (4ξTCL 2 + τ m )(τ m + 2Tm ) + 8TCL 2 4

(4ξTCL2 + τ m )3 [(4ξTCL2 + τ m )(τ m + 2Tm ) − 4TCL2 2 ] 1

TCL 2 2 s 2 + 2ξTCL 2 s + 1

31

Kc =

0.66Tm , N = 8.38; G p = G m K m (τm − 0.02Tm )

32

Kc =

0.44Tm , N = 8.38; G p = G m K m (τm − 0.02Tm )

33

Kc =

K pe p 0.60Tm , N = 6.19; G p = . K m (τm − 0.07Tm ) (1 + sTp1 ) 2

34

Kc =

K pe p 0.40Tm , N = 6.19; G p = . K m (τm − 0.07Tm ) (1 + sTp1 ) 2

− sτ

− sτ

.

],

b720_Chapter-03

17-Mar-2009

FA


130 Rule Latzel (1988). Model: Method 2

Kc

Ti

Td

Comment

35

Kc

0.74Tm

0.14Tm

φm = 450

36

Kc

0.74Tm

0.14Tm

φm = 600

Tm τ m 2Tm + τ m

N = 10.

Robust

Chien (1988). Model: Method 1

Tm + 0.5τ m K m (λ + 0.5τ m )

Tm + 0.5τ m

λ ∈ [τm , Tm ] , Chien and Fruehauf (1990); λ > [0.8τm ,0.2Tm ] , Yu (2006), p. 19.

Morari and Zafiriou (1989). Model: Method 1 Gong et al. (1996). Maffezzoni and Rocco (1997).

37

Kc

Tm + 0.5τ m

38

Kc

Tm + 0.3866τ m

39

Kc

Ti

λ > 0.25τ m ,

Tm τ m 2Tm + τ m

λ > 0.2Tm .

3 ≤ N ≤ 10 ; 0.3866Tm τ m Tm + 0.3866τ m Model: Method 1 Model: Method 14

Td

− sτ p

35

Kc =

K pe 0.58Tm , N = 5.83; G p = K m (τm − 0.08Tm ) (1 + sTp1 )3

36

Kc =

K pe p 0.39Tm , N = 5.83; G p = K m (τm − 0.08Tm ) (1 + sTp1 )3

37

Kc =

2T (λ + τ m ) 1  Tm + 0.5τm   , N= m . λ (2Tm + τ m ) K m  λ + τm 

38

Kc =

39

Kc =

− sτ

N=

(0.1388 + 0.1247 N )Tm + 0.0482 Nτ m τ . Tm + 0.3866τm , λ= m 0.3866( N − 1)Tm + 0.1495Nτ m K m (λ + 1.0009τm ) 2Tm (τm + x1 ) + τm 2 2K m (τm + x1 )2

2Tm (τ m + x 1 ) − τ m x 1

[

τm τm 2 [2Tm (τm + x1 ) − τm x1 ] , Td = , 2(τm + x1 ) 2(τm + x1 ) 2Tm (τm + x1 ) + τm 2 2

, Ti = Tm +

]

[

]

τ . x 1 = 0.25τ m , uncertainty on τ m ; x1 > 0.1Tm , m < 0.4 , Tm

x 1 2Tm (τ m + x 1 ) + τ m 2 uncertainty on the minimum phase dynamics within the control band.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc 40

Gong et al. (1998). Model: Method 1; λ = x1τm ; x1 values deduced from graph.

Kc =

41

Comment

Ti

Td

N = 3, 5 or 10.

x1

τm Tm

x1

τm Tm

x1

τm Tm

1 2 3 4 1 2 3 4 1 2 3 4

0.46 0.44 0.43 0.41 0.41 0.40 0.39 0.38 0.37 0.36 0.36 0.35

5 6 7 8 5 6 7 8 5 6 7 8

0.41 0.40

9 10

N = 3.

0.38 0.37

9 10

N = 5.

0.35 0.35

9 10

N = 10.

Kc

Ti

Td

42

Kc

Ti

Td

43

Kc

Ti

Td

41

(Tm + 0.3866τm )(λ + τm ) − (0.3866λ − 0.1247τm )τm K m (λ + τm )2

Ti = Tm + 0.3866τm − Td =

Td

0.63 0.57 0.51 0.48 0.52 0.48 0.45 0.43 0.42 0.40 0.38 0.38

Kasahara et al. (1999). Model: Method 1; N = 10; 0 < τm Tm < 1 .

40

Kc

Ti

131

Robust to 20% change in plant parameters. Robust to 30% change in plant parameters. Robust to 40% change in plant parameters.

,

0.3866λ − 0.1247 τm τm , λ + τm

0.3866Tm τ m 0.3866λ − 0.1247 τ m − τm . Tm + 0.3866τ m λ + τm

K c = 0.6K u [0.718 − 0.057(τm Tm )] , Ti = 0.5Tu [0.296 + 0.838(τm Tm )] ,

Td = 0.125Tu [0.635 − 1.091(τm Tm )] .

42

K c = 0.6K u [0.690 − 0.076(τm Tm )] , Ti = 0.5Tu [0.274 + 0.788(τm Tm )] ,

Td = 0.125Tu [0.526 − 0.057(τm Tm )] .

43

K c = 0.6K u [0.331 + 0.149(τm Tm )] , Ti = 0.5Tu [0.041 + 0.857(τm Tm )] ,

Td = 0.125Tu [0.495 − 0.064(τm Tm )] .

b720_Chapter-03

FA


132 Rule

Kc

Kasahara et al. (1999) – continued. Leva and Colombo (2000).

44

Kc

45

Kc

Leva (2001). Model: Method 1

46

Kc

Leva et al. (2003), (2006).

47

Kc

48

Kc

Ti

Td

Ti

Td

Comment Robust to 50% change in plant parameters. λ not specified; Model: Method 1

2

Zhang et al. (2002b). Model: Method 1

44

17-Mar-2009

Tm +

τm 2(λ + τ m )

Td

Tm +

τm2 2(λ + τ m )

Td

Tm +

τm2 2(λ + τ m )

Td

Model: Method 1

τm Tm Td − 2Ti N

λ > 0.0735τm for stability (Ou et al., 2005c).

Ti

K c = 0.6K u [0.195 + 0.266(τm Tm )] , Ti = 0.5Tu [− 0.013 + 0.731(τm Tm )] ,

Td = 0.125Tu [0.489 − 0.065(τm Tm )] .

45

46

Kc =

2Tm (λ + τ m ) Tm (λ + τm ) 1 2(λ + τm )Tm + τm ,N= . , Td = 2 Km 2( λ + τ m ) 2 2(λ + τm )Tm + τm λ 2(λ + τ m )Tm + τ m 2

Kc =

2Tm (λ + τ m ) 1 2(λ + τm )Tm + τm , N= −1, Km 2( λ + τ m ) 2 λ 2(λ + τ m )Tm + τ m 2

2

[

]

2

2

Td = τm 2

2

[

2λTm + (2Tm − λ )τm

[

2(λ + τm ) 2Tm (λ + τm ) + τm 2

[

]

] . 0.1T

m

≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ;

λ = 1.5(τ m + Tm ) , 0.25 < τ m Tm ≤ 0.75 ; λ = 3(τ m + Tm ) , τm Tm > 0.75 . 47

Kc =

2 τm Tm (λ + τm ) 1 2(λ + τm )Tm + τm , Td = , Km 2( λ + τ m ) 2 2(λ + τm )Tm + τm 2

2Tm (λ + τ m )

[

]

2

N=

[

λ 2(λ + τ m )Tm + τ m 2

] . 0.1T

m

≤ λ ≤ 0.5Tm , τm Tm ≤ 0.33 ;

λ = 1.5(τ m + Tm ) , 0.33 < τm Tm ≤ 3 ; λ = 3(τ m + Tm ) , τm Tm > 3 . 48

Kc =

λ2 0.5τm + Tm − (Td N ) T T or N = 10. , Ti = 0.5τm + Tm − d , d = K m (2λ + 0.5τm ) N N 2λ + 0.5τm

]

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Vilanova and Balaguer (2006).

Ti K m (x1 + x 2 ) 50

Tang et al. (2007). Model: Method 1; Td N = Tf .

49

Kc

Ti 49

Ti

Ti

Td

x2

x1 + x 3 N x1 + x 2

133

Comment Model: Method 1

Td

Labelled the H ∞ Smith predictor. 51

Kc

Ti

Td

Labelled the H 2 Smith predictor or the Dahlin controller.

2(3Tm + τm ) 9K m τm

Tm + 0.333τm

(6Tm − τm )τm 6(3Tm + τm )


 x + x3  T x x + x2 , N= m 1 1 Ti = Tm + τm + x 3 − x1 − x 2  1 −1 .  Ti τm x1 + x 3  x1 + x 2  For minimum ISE, servo response: x 3 = τm ;

for minimum IE, regulator response: x 2 = τm ; x1 = real constant. 50

51

(τm + 2Tm )(4λ + τm ) − 4λ2 , T = 0.5τ + T − 2λ2 , i m m 4λ + τ m K m (4λ + τm )2 2 Tm τm (4λ + τm ) 2λ 2λ2 λ Td = − T = , ; 0.3 ≤ ≤ 0.8 . f 2 4λ + τm τm (τm + 2Tm )(4λ + τm ) − 4λ 4λ + τm (τ + 2Tm )(λ + τm ) − λτm , T = 0.5τ + T − λτm , Kc = m i m m 2(λ + τm ) 2K m (λ + τm )2 λτ m Tm τ m (λ + τ m ) λτ m λ ,T = − Td = ≤ 0.8 . ; 0.3 ≤ (τ m + 2Tm )(λ + τ m ) − λτ m 2(λ + τm ) f 2(λ + τm ) τm

Kc =

b720_Chapter-03

17-Mar-2009

FA


134

   1  1 + Td s    3.3.5 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   E(s) R(s)

−

+

    U(s)  1  1 + Td s    K c 1 + Td  Ti s   s 1 + N  

Y(s)


m


Kc

Kraus (1986). Model: Method 23

0.833Tm K m τm

Witt and Waggoner (1990).

Comment

Td

Process reaction 1.5τ m 0.25τm

N=∞

Foxboro EXACT controller pre-tuning. Also given by Hang et al. (1993b), p. 76. x1Tm K m τm τm τm 10 ≤ N ≤ 20 ; 0.6 ≤ x1 ≤ 1.0 . Model: Method 2. ‘Equivalent to’ Ziegler and Nichols (1942). 1 Ti Td 10 ≤ N ≤ 20 Kc

Witt and Waggoner (1990).

1

Ti


Model: Method 2. ‘Preferred’ tuning.

Tuning ‘equivalent to’ Cohen and Coon (1953).

1.350 Kc =

τ  Tm T τ + 0.25 + m 0.7425 + 0.0150 m + 0.0625 m  τm τm Tm  Tm  2K m Ti =

2

,

Tm τ  τ T T 1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m  τm τm Tm  Tm 

2

,

Tm

Td = T T 1.350 m + 0.25 + m τm τm

τ  τ 0.7425 + 0.0150 m + 0.0625 m  Tm  Tm 

2

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

2 Witt and Kc Waggoner (1990) – continued. St. Clair (1997), Tm K m τm p. 21. Model: Method 0.5Tm K m τm 2; N ≥ 1 . Harrold (1999). 1 Km Model: 0.5 K m Method 11; 0.25 K m N not specified. Tan et al. 0.6Tm (1999a), p. 25. K mτm

Shinskey (2000). Model: Method 2

Ti

Td

Ti

Td

135

Comment

‘Alternate’ tuning. 10 ≤ N ≤ 20 . 5τ m

0.5τ m

5τ m

0.5τ m

Tm

≤ 0.25Tm

‘Aggressive’ tuning. ‘Conservative’ tuning. τ m Tm ≤ 0.25 τ m Tm ≈ 0.5 τ m Tm ≥ 1

0.889Tm K mτm

τm

τm

N =∞; Model: Method 2

1.75τ m

0.70τ m

τm Tm = 0.167 ; N not specified.

x1Tm K m τm x 2τm x 3τ m N=∞ O’Dwyer Representative results: equivalent to Chien et al. (1952) – regulator. (2001b). x2 x3 Model: Method 2 x 1 0.7236 1.8353 0.5447 0% overshoot; 0.1 < τ m Tm < 1 .

0.84

1.350 2

Kc =

1.4

0.6

20% overshoot; 0.1 < τ m Tm < 1 .

τ  Tm T τ + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m  τm τm Tm  Tm  2K m Ti =

2

,

Tm τ  τ T T 1.350 m + 0.25 + m 0.7425 + 0.0150 m + 0.0625 m  τm τm Tm  Tm  Td =

2

,

Tm τ  τ T T 1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m  τm τm Tm  Tm 

2

.

b720_Chapter-03

17-Mar-2009

FA


136 Rule O’Dwyer (2001b). Model: Method 2; N =∞.

Ti

Td

Comment

3

Kc

Ti

Td

0% overshoot; τ m Tm < 0.5 .

4

Kc

Ti

Td

20% overshoot; τ m Tm < 0.7234

Representative results: equivalent to Chien et al. (1952) – servo. Model: 0.47Tm Method 1; τ τ m m K m τm N = 10. Minimum performance index: regulator tuning 8.1 K m 0.29(τm + Tm ) 0.13(τm + Tm ) τm Tm = 0.11

Faanes and Skogestad (2004).

Minimum ITAE with OS ≤ 8% – Fertik (1975). Model: Method 3; N = 11.

Minimum IAE – Huang and Chao (1982). Minimum IAE – Kaya and Scheib (1988).

Kc

3.8 K m

0.46(τm + Tm )

0.23(τm + Tm )

0.46(τm + Tm )

0.40(τm + Tm )

0.50(τm + Tm )

1.85 K m 0.93 K m

0.33(τm + Tm )

0.40(τm + Tm )

0.56 K m

0.44(τm + Tm )

τm Tm = 0.25 τm Tm = 0.43 τm Tm = 0.67 τm Tm = 1.00

Coefficients of K c , Ti , Td deduced from graphs and converted to this equivalent controller architecture. 0.982 N = 8; 0.817  Tm  5 Model: Method 1   Td Ti K m  τm  Model: Method 6 5; N = 10; Ti Td Kc 0 < τm Tm ≤ 1 .

3

Kc =

0.3Tm  2τ 1 − 1 − m K m τm  Tm

4

Kc =

0.475Tm K m τm

 T  2τ  Ti = m 1 + 1 − m 2 Tm   ,

 τ 1 + 1 − 1.3824 m Tm 

 T  2τ   Td = m 1 − 1 − m  2 Tm    ,

  τ  Ti = 0.68Tm 1 + 1 − 1.3824 m Tm  , 

   ,

 τ  Td = 0.68Tm 1 − 1 − 1.3824 m  T m    5

6

τ  Ti = 0.903Tm  m   Tm 

Kc =

0.780

0.98089  Tm    K m  τm 

τ , Td = 0.602Tm  m  Tm

0.76167

   

0.954

. 1.05211

, Ti =

Tm  τm    0.91032  Tm 

τ  , Td = 0.59974Tm  m   Tm 

0.89819

.

b720_Chapter-03

17-Mar-2009

FA


Rule Minimum IAE – Witt and Waggoner (1990).

Kc

Ti

Td

7

Kc

Ti

Td

8

Kc

Ti

Td

Comment

0.1 < τm Tm ≤ 0.258 ; 10 ≤ N ≤ 20 ; Model: Method 2. 0.95Tm K m τ m

1.43τ m

0.52τ m

τ m Tm = 0.2

1.17 τ m

0.48τ m

τ m Tm = 0.5

1.03τ m

0.40τ m

τ m Tm = 1

0.77τ m

0.35τ m

τ m Tm = 2

Minimum IAE – Shinskey (1996), 0.96Tm K m τ m p. 119.

1.43τ m

0.52τ m

Model: Method 1; τ m Tm = 0.2 .

Minimum IAE – Edgar et al. 0.94Tm K m τ m (1997), pp. 8-14, 8-15.

1.5τ m

0.55τ m

Minimum IAE – Shinskey (1988), 0.95Tm K m τ m p. 143. 1.14Tm K m τ m Model: Method 1 1.39Tm K m τ m

7

‘Preferred’ tuning. K c =

0.718  Tm    K m  τm 

0.921 

1 + 1 − 1.693 τm  T    m 

−1.886

Model: Method 1; Time constant dominant; N = 10.

 ,  

1.137

Ti =

8

137

‘Alternate’ tuning. K c =

0.718  Tm    K m  τm 

τ  0.964Tm  m   Tm 

1.886

τ  1 − 1 − 1.693 m   Tm 

0.921 

Ti =

1 − 1 − 1.693 τm  T    m 

τ 0.964Tm  m  Tm

−1.886

1.137

, Td =

τ 1 − 1 + 1.693 m  Tm

1.886

   

1.886

.

τ  1 + 1 + 1.693 m   Tm 

 ,  

1.137

   

τ  0.964Tm  m   Tm 

1.137

, Td =

τ  0.964Tm  m   Tm 

1.886

τ  1 + 1 − 1.693 m   Tm 

.

b720_Chapter-03

Rule

Kc

Minimum IAE – Edgar et al. 0.88Tm K m τ m (1997), p. 8–15. Minimum IAE – Smith and Tm Corripio (1997), K mτm p. 345. Minimum IAE – 0.89Tm K m τm Shinskey (2003). 0.56K u Minimum IAE – 0.49 K u Shinskey (1988), 0.51K u p. 148. Model: Method 1 0.46K u

Minimum IAE – K u τm Shinskey (1994), 3τ − 0.32T m u p. 167. Minimum ISE – 1.101  T  0.881  m Huang and Chao K m  τ m  (1982). Minimum ISE – Kaya and Scheib (1988).

11

11

12

  

Comment

Ti

Td

1.8τ m

0.70τ m

Tm

0.5τ m

1.75τm

0.70τ m

Model: Method 2

0.39Tu

0.14Tu

τ m Tm = 0.2

0.34Tu

0.14Tu

τ m Tm = 0.5

0.33Tu

0.13Tu

τ m Tm = 1

0.28Tu

0.13Tu

τ m Tm = 2

9

Ti

0.14Tu

10

Ti

Td

Ti

Kc

Minimum ITAE 0.745  T  m – Huang and K m  τm Chao (1982).

10

FA


138

9

17-Mar-2009

Td

1.036

12

Td

Ti

Model: Method 2; N = 10. Model: Method 1; τ 0.1 ≤ m ≤ 1.5 . Tm

Model: Method 1; N not specified. Model: Method 1; N = 8. Model: Method 5; 0 < τm Tm ≤ 1 ; N = 10. Model: Method 1; N = 8.

  T Ti = Tu  0.15 u − 0.05  . τ m   τ  Ti = 1.134Tm  m   Tm  Kc =

0.883

1.11907  Tm    K m  τm 

τ  Ti = 0.771Tm  m   Tm 

τ  , Td = 0.563Tm  m   Tm 

0.89711

0.595

, Ti =

0.881

.

Tm  τm    0.7987  Tm 

0.9548

1.006

τ  , Td = 0.597Tm  m   Tm 

.

τ , Td = 0.54766Tm  m  Tm

   

0.87798

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum ITAE – Kaya and Scheib (1988). Minimum ITAE – Witt and Waggoner (1990). Minimum ITAE – Ou and Chen (1995). Minimum ITSE – Huang and Chao (1982).

13

Kc

Ti

Td

Comment

Ti

Td

0 < τm Tm ≤ 1

N = 10; Model: Method 5. 14

Kc

Ti

Td

Preferred tuning.

15

Kc

Ti

Td

Alternate tuning.

Model: Method 2; 10 ≤ N ≤ 20 . 1.357  Tm  K m  τ m

  

0.947

0.994  Tm    Km  τm 

Kc =

0.77902  Tm    K m  τm 

14

Kc =

0.679  Tm    K m  τm 

Tm  τ m  0.842  Tm

, Ti =

17

0.947 

1 + 1 − 1.283 τm  T    m 

0.947 

1 − 1 − 1.283 τm  T    m 

Ti =

16

17

τ  Td = 0.381Tm  m   Tm  τ  Ti = 1.032Tm  m   Tm 

0.995

0.869

Model: Method 57

0.738

16

Td

Model: Method 1; N = 8.

, N=±

Td

Ti

Tm  τm    1.14311  Tm 

Ti =

0.679  Tm    Kc = K m  τm 

  

0.907

1.06401

13

15

139

−1.733

0.70949

1.03826

τ  , Td = 0.57137Tm  m   Tm 

  , 0.1 < τ m < 0.379 ;  Tm 

τ  0.762Tm  m   Tm 

0.995

1.733

τ  1 − 1 − 1.283 m   Tm  −1.733

, Td =

τ  0.762Tm  m   Tm 

0.995

1.733

.

τ  1 + 1 + 1.283 m   Tm 

 ,  

τ 0.762Tm  m  Tm

   

0.995

τ 1 − 1 + 1.283 m  Tm

1.733

   

, Td =

0.75K m K cTd , N>0. Ti + K m K c (Td + Ti − τm − Tm )

τ  , Td = 0.570Tm  m   Tm 

.

0.884

.

τ 0.762Tm  m  Tm

   

0.995

τ 1 + 1 − 1.283 m  Tm

1.733

   

.

b720_Chapter-03

17-Mar-2009

FA


140 Rule

Kc

Minimum ITAE with OS ≤ 8% – Fertik (1975). Model: Method 3; N = 11.

Ti

5.0 K m

Td

Comment

Minimum performance index: servo tuning 0.63(τm + Tm ) 0.13(τm + Tm ) τm Tm = 0.11

2.4 K m 1.2 K m 0.69 K m 0.51 K m

0.57(τm + Tm )

0.23(τm + Tm )

0.51(τm + Tm )

0.33(τm + Tm )

0.46(τm + Tm )

0.40(τm + Tm )

0.40(τm + Tm )

0.44(τm + Tm )

τm Tm = 0.25 τm Tm = 0.43 τm Tm = 0.67 τm Tm = 1.00

Coefficients of K c , Ti , Td deduced from graphs and converted to this equivalent controller architecture. 1.00 Minimum IAE – Model:   0.673 Tm 18   Huang and Chao Method 1; T Ti d K m  τm  (1982). N = 8.

Minimum IAE – 0.65  Tm Kaya and Scheib  K m  τ m (1988).

Minimum IAE – Smith and Corripio (1997), p. 345.

  

1.04432

0.83Tm K mτm

19

Td

Ti

0.5τ m

Tm

1.032

18

Ti =

19

Ti =

τ  Tm , Td = 0.502Tm  m  τ   Tm  0.148 m  + 0.979 T  m Tm 0.9895 + 0.09539

τm Tm

τ , Td = 0.50814Tm  m  Tm

.

1.08433

   

.

Model: Method 5; 0 < τm Tm ≤ 1 ; N = 10. Model: Method 1; τ 0.1 ≤ m ≤ 1.5 ; Tm N not specified.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum IAE – Witt and Waggoner (1990).

Kc =

1.086  Tm    K m  τm 

Td

Comment

20

Kc

Ti

Td

Preferred tuning.

21

Kc

Ti

Td

Alternate tuning.

0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2.

Minimum ISE – 0.787  T  m Huang and Chao K m  τ m (1982).

20

Ti

  

22

1 + 1 − 1.392 τm  T    m 

Td

Ti

0.914

  τm   0.74 − 0.13   , Tm   

τ  0.696Tm  m   Tm  τ  1 − 1 − 1.392 m   Tm 

0.869

1.086  Tm    K m  τm 

0.869 

1 − 1 − 1.392 τm  T    m 

Ti =

0.914

1.028

T  Ti = 1.678Tm  m   τm 

τ  0.696Tm  m   Tm  τ  1 + 1 − 1.392 m   Tm 

τ  , Td = 0.594Tm  m   Tm 

τ  0.696Tm  m   Tm  τ  1 + 1 − 1.392 m   Tm 

0.869

0.914

.

 τm  0.74 − 0.13  Tm  

  τm   0.74 − 0.13   , Tm   

0.869

0.971

.

0.914

,

 τm  0.74 − 0.13  T m 

Td =

22

0.914

 τm  0.74 − 0.13  T m 

Td =

Kc =


0.964

0.869 

Ti =

21

141

τ  0.696Tm  m   Tm  τ  1 − 1 − 1.392 m   Tm 

0.869

0.914

 τm  0.74 − 0.13  Tm  

.

b720_Chapter-03

FA


142 Rule

Kc

Minimum ISE – 23 Kaya and Scheib Kc (1988). Model: Method 5 0.995 Minimum ITAE 0.684  Tm  – Huang and   K m  τm  Chao (1982). Minimum ITAE 25 – Kaya and Kc Scheib (1988). Model: Method 5 0.85 Minimum ITAE 0.965  Tm    – Ou and Chen K m  τm  (1995).

1.03092

23

17-Mar-2009

Kc =

0.71959  Tm    K m  τm 

, Ti =

Ti

Td

Ti

Td

24

Td

Ti

Ti

26

Td

Td

Ti

Tm

τm Tm

1.12666 − 0.18145

Comment τm ≤ 1; Tm N = 10.

0
0. Ti + K m K c (Td + Ti − τm − Tm )

b720_Chapter-03

17-Mar-2009

FA


Rule Minimum ITAE – Witt and Waggoner (1990).

Kc

0.965  Tm    Kc = K m  τm 

Td

Comment

27

Kc

Ti

Td

Preferred tuning.

28

Kc

Ti

Td

Alternate tuning.

0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2.

Minimum ITSE – 0.718  Tm  Huang and Chao K m  τ m (1982).

27

Ti

  

29

1 + 1 − 1.232 τm  T    m 

Td

Ti

0.929

  τm   0.796 − 0.1465   , Tm   

τ  0.616Tm  m   Tm  τ  1 − 1 − 1.232 m   Tm 

0.85

0.85 

1 − 1 − 1.232 τm  T    m 

Ti =

0.929

Ti =

τ  0.616Tm  m   Tm  τ  1 + 1 − 1.232 m   Tm 

τ  0.616Tm  m   Tm  τ  1 + 1 − 1.232 m   Tm 

τ Tm , Td = 0.596Tm  m τ   Tm 0.063 m  + 1.061 T  m

,

0.85

0.929

.

 τm  0.796 − 0.1465  T m 

  τm   0.796 − 0.1465   , Tm   

0.85

1.028

   

0.929

,

 τm  0.796 − 0.1465  T m 

Td =

29

0.929

 τm  0.796 − 0.1465  T m 

Td =

0.965  Tm    Kc = K m  τm 


1.00

0.85 

Ti =

28

143

.

τ  0.616Tm  m   Tm  τ  1 − 1 − 1.232 m   Tm 

0.85

0.929

 τm  0.796 − 0.1465  T m 

.

b720_Chapter-03

17-Mar-2009

FA


144 Rule

Kc x1Tm K m τm

Tsang et al. (1993). Model: Method 19

x1

ξ

1.6818 1.3829 1.1610

0.0 0.1 0.2

x1

ξ

Comment

Td

Ti

Direct synthesis: time domain criteria Tm 0.25τ m

Coefficient values: N = 2.5. x1 ξ x1 ξ 0.9916 0.3 0.6693 0.6 0.8594 0.4 0.6000 0.7 0.7542 0.5 0.5429 0.8 Coefficient values: N = ∞ . x1 ξ x1 ξ

x1

ξ

0.4957 0.4569

0.9 1.0

x1

1.8194 0.0 1.0894 0.3 0.7482 0.6 0.5709 1.5039 0.1 0.9492 0.4 0.6756 0.7 0.5413 1.2690 0.2 0.8378 0.5 0.6170 0.8 Tsang and Rad 0.809Tm K m τm Tm 0.5τ m (1995). Overshoot = 16%; N = 5; Model: Method 15.

ξ 0.9 1.0

Smith and 0.5τ m 0.5Tm K m τm Tm Corripio (1997), Servo – 5% overshoot; N not specified; Model: Method 1. p. 346. 2 Schaedel (1997). T T1 2 − 2T2 2 T1 − 2 30 Model: N not defined. Kc T1 Method 25 Wang et al. Tm 1 (2002). Labelled ‘IMC’. 0.5τ m    τ τ  Model: K m 1 + m  K m 1 + m  2Tm  2Tm    Method 1; N = 10.

30

Kc =

T1 =

 T2 0.375 T1 − 2   T1    T 2 − 2T 2  2 T  2 2  2 Km  1 + 2 − T1 − 2T2  N T1    

Tm 1 + 3.45

T1 =

0.7Tm

τ am Tm 2

Tm − 0.7 τ am

+ τ m , T2 2 = 1.25Tm τ am +

,

Tm 1 + 3.45

2

+ τ m , T2 = Tm τ am +

0.7Tm 2 Tm − 0.7 τ am

τ am Tm

τ m + 0.5τ m 2 , 0
0.104 . Tm

≤ 0.104 ;

b720_Chapter-03

17-Mar-2009

FA


Rule Wang et al. (2002) – continued.

Kc 0.65  Tm  K m  τ m

Ti

  

Td

Comment

Td

Labelled ‘IAE setpoint’.

1.04432

31

Ti

145

32

Kc

Ti

Td

33

Kc

Ti

Td

34

Kc

Ti

Td

35

Kc

Ti

Td

36

Kc

Ti

Td

Labelled ‘IAE load’. Labelled ‘ISE setpoint’. Labelled ‘ISE load’. Labelled ‘ITAE setpoint’. Labelled ‘ITAE load’.

Åström and 1 Tm Hägglund (2006), K 0.5τ + λ Model: Method 1 Tm 0.5τm m m pp. 188–189. λ = Tm (aggressive tuning), λ = 3Tm (robust tuning). N = ∞ .

31

32

33

34

35

36

Ti =

1.08433

Tm 0.9895 + 0.09539

τm Tm

0.98089  Tm  K m  τ m

  

0.76167

0.71959  Tm  Kc = K m  τ m

  

1.03092

1.11907  Tm  K m  τ m

  

0.89711

1.12762  Tm  Kc = K m  τ m

  

0.80368

0.77902  Tm  K m  τ m

  

Kc =

Kc =

Kc =

τ  , Td = 0.50814 m   Tm 

1.05211

τ  , Ti = 1.09851Tm  m   Tm 

, Ti =

Tm 1.12666 − 0.18145

τ  , Ti = 1.2520Tm  m   Tm 

, Ti = 1.06401

.

τm Tm

0.95480

τ  , Td = 0.54568 m   Tm 

   

0.89819

.

0.86411

τ  , Td = 0.54766Tm  m   Tm 

.

0.87798

.

1.0081

Tm 0.9978 + 0.02860

τ  , Ti = 0.87481Tm  m   Tm 

τ , Td = 0.59974Tm  m  Tm

τm Tm

0.70949

τ  , Td = 0.42844 m   Tm 

τ , Td = 0.57137Tm  m  Tm

.

1.03826

   

.

b720_Chapter-03

17-Mar-2009

FA


146 Rule Hougen (1979) – p. 335. O’Dwyer (2001a). Model: Method 1


Kc 37

Kc

Td

Ti

Comment

Direct synthesis: frequency domain criteria Tm 0.45τm

Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . x1Tm K m τm Tm N Tm A m = π 2 x 1 N ; φ m = 0.5π − x 1 N ; representative result below.

0.079Tm K mτm

0.1Tm

Tm

0.65Tm K m τm

Tm

0.4 τ m

A m = 2.0; φ m = 450 .

N = 20, A m = 2.7 , φ m = 650 .

The above is a representative tuning rule (also given by Huang et al. (2003), N not defined, Model: Method 37). Other rules may be deduced from a graph provided by the authors for other A m , φ m values.

Huang and Jeng (2005).

0.65Tm K m τm

Harris and Tyreus (1987). Model: Method 1

Tm K m (τ m + λ )

τm

N = 40; Model: Method 1

0.5τ m

N = (τ m + λ ) λ

Tm

Robust


Tm

λ ≥ maximum of [0.45τ m ,0.1Tm ]

Tm K m (λ + 0.5τ m )

Tm

0.5τ m

0.5τ m K m (λ + 0.5τ m )

0.5τ m

Tm

Tm

0.55τm

0.5τ m

1.1Tm

Tm Chien (1988). K (λ + 0.5τ ) m m Model: Method 1 0.5τ m K m (λ + 0.5τ m )

λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 10. λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 11.

Tuning rules converted to this equivalent controller architecture.

37

K c = 9.0 K m , τm Tm = 0.1 ; K c = 4.2 K m , τm Tm = 0.2 ; K c = 2.8 K m , τm Tm = 0.3 ; K c = 2.1 K m , τm Tm = 0.4 ; K c = 1.7 K m , τm Tm = 0.5 ; K c = 1.45 K m , τm Tm = 0.6 ; K c = 1.2 K m , τm Tm = 0.7 ; K c = 1.1 K m , τm Tm = 0.8 ; K c = 0.95 K m , τm Tm = 0.9 ; K c = 0.85 K m , τm Tm = 1.0 ; values deduced from graph.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Zhang et al. 38 (1996b). Kc Model: 39 Kc Method 47 40 Zhang et al. Kc (2002b). Shi and Lee 41 (2002). Kc Model: Method 1 Lee and Shi 42 (2002). Kc Model: Method 1 Faanes and 0.77Tm K m τm Skogestad 0.5Tm K m τm (2004). N = 10. Tm Tang et al. K m (2λ + 0.5τm ) (2007). τm Model: Method 1 K m (4λ + τm )

Comment

Ti

Td

0.5τ m

Tm

Tm

0.5τ m

≤ 1. 2 τ m .

Tm

0.5τ m

Model: Method 1

Tm

 ωg τ m 1 tan ωg  2

Tm 43

Ti

0.2τm ≤ λ

  

Suggested ωg = 0.6 τm .

0.5τ m

Tm

0.5τ m

8τm

0.5τ m

Tm

0.5τm

0.5τm

Tm

Suggested ωg = 0.6 τm .

Model: Method 1 Tf =

2λ2 , 4λ + τm

0.3 ≤

λ ≤ 0.8 . τm

Tf =

λτ m , 2(λ + τm )

0.3 ≤

λ ≤ 0.8 . τm

Labelled the H ∞ Smith predictor. Tm K m (λ + τm )

Tm

0.5τm

τm 2K m (λ + τm )

0.5τm

Tm

Labelled the H 2 Smith predictor or the Dahlin controller.

T (2λ + 0.5τ m ) 0.5τm , N= m . K m (0.5τm + 2λ ) λ2

38

Kc =

39

Kc =

0.5τ m (2λ + 0.5τ m ) Tm , N= . K m (0.5τm + 2λ ) λ2

40

Kc =

Tm λ2 . , N = 10 or N = 0.5τ m (2λ + 0.5τ m ) K m (2λ + 0.5τm )

41

42

43

 ωg τ m 2ω bw ωbw Tm tan , N = 1+ ωg K m 1 + 2ωbw ωg tan 0.5ωg τm  2  ωg τ m 2ω bw ωbw Tm tan Kc = , N = 1+ ωg K m 1 + 2ωbw ωg tan 0.5ωg τm  2 T 1 T . Ti = m , 1 < x1 < 1 + m + x1 τm ωbw τm

Kc =

[ (

) (

)]

 .  

[ (

) (

)]

 .  

147

b720_Chapter-03

17-Mar-2009

FA


148 Rule

Kc

Ti

Td

Comment

Tang et al. (2007) – continued.

0.667Tm K m τm

Tm

0.5τm

0.333 K m

0.5τm

Tm


Pessen (1994). Model: Method 1

0.35K u 44


44

Kc

Ultimate cycle 0.25Tu 0.25Tu Ti

N =∞; 0.1 ≤ τm Tm ≤ 1 .

Td

N = 20, A m = 2.7 , φ m = 650 . The above is a representative tuning rule; other rules may be deduced from a graph provided by the authors for other A m , φ m values.

  2    ^     K u  − 1 2 ^ 0.65   ^     , Ti = 0.159 Tu  K u  − 1 , Kc =   2   Km    ^  −1    π −   tan K  u  −1            ^  Td = 0.064 Tu π − tan −1  

2   ^   K u  − 1  .    

b720_Chapter-03

17-Mar-2009

FA


149


  Td s 1 + G c (s) = K c 1 + T  Ti s 1+ d  N  R(s) E(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

  2  b f 0 + b f 1s + b f 2 s 2  1 + a f 1s + a f 2 s s 

U(s)



m



Kristiansson (2003). Model: Method 12

1

Kc x1Tm

Ti

Y(s)


Comment

Td

Direct synthesis: time domain criteria 0 K m τm Tm 2

2

b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2 τ m , a f 2 = 0.01τ m .

ξ

x1

1.851 1.552 1.329

1

ξ

x1

ξ

x1

x1

0.0 1.160 0.3 0.841 0.6 0.622 0.1 1.028 0.4 0.768 0.7 0.553 0.2 0.925 0.5 0.695 0.8 Direct synthesis: frequency domain criteria Kc

Ti

Equations continued into the footnote on the next page; 2  T  τ   τ 21.1 m − 0.1 0.06 + 0.68 m − 0.12 m   Tm  τm    Tm   Kc = , 2    τ τ K m 0.68 + 0.84 m − 0.25 m   Tm   Tm   

Td

0.05 ≤

ξ 0.9 1.0

τm ≤2; Tm

1.6 ≤ M s ≤ 1.9 .

   

b720_Chapter-03

17-Mar-2009

FA


150 Rule Kristiansson (2003) – continued. Shamsuzzoha and Lee (2007b).

Kc

Ti

Td

2

Kc

Ti

Td

3

Kc

0.5τm

0.167 τm

Comment

Model: Method 1

2 2   τ   τ   τ τ 2Tm 0.06 + 0.68 m − 0.12 m   0.5Tm 0.06 + 0.68 m − 0.12 m   Tm Tm    Tm    Tm     Ti = , Td = , 2 2    τm    τm   τ τ m m 0.68 + 0.84 0.68 + 0.84     − 0.25 − 0.25 Tm Tm   Tm Tm          

N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf , a f 2 = Tf 2 with

Tf = 2

[

2 2

0.6667(Tm + τm )[1.1(Tm τm ) − 0.1] 0.6667(Tm + τm ) , Ti = , 2 2    τm   τm   τ τ m m 0.68 + 0.84     K m τm 0.68 + 0.84 − 0.25 − 0.25 Tm Tm   Tm Tm           0.1667(Tm + τm ) Td = , N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf , 2   τm   τ m 0.68 + 0.84   − 0.25 Tm Tm      

Kc =

a f 2 = Tf 2 with Tf = 3

]

Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1] . min{5 + 2(Tm τm ),25}

Kc =

(Tm + τm )2 [1.1(Tm τm ) − 0.1] ; 9Tm (5 + 2(Tm τm ))

0. 7
τ m , λ < Tm .

151

Robust

Horn et al. (1996). Lee and Shi (2002). Model: Method 1 Shamsuzzoha et al. (2004). Model: Method 1 Shamsuzzoha and Lee (2006a). Model: Method 1

4

Kc

Tm + 0.5τ m

τ m Tm τ m + 2Tm

5

Kc

Tm x1

0

6

Kc

Tm + 0.5τm

Tm τm 2Tm + τm

0.5τm K m (λ + τm )

0.5τm

0.167 τm

af1 =

1 < x1
0.2Tm ; α = 1 .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Chien et al. (1999). Chidambaram (2000a). Kasahara et al. (2001). Model: Method 1

19

Kc

Ti

Td

Comment

Ti

Td

Model: Method 1

161

α = 1, β = 1 , N = ∞ . Tm Ti

20

TCL 2 + Ti τm

Ti

0

Model: Method 1; choose TCL , ξ .

21

Kc

Ti

Td

OS = 0%

22

Kc

Ti

Td

OS = 10%

α = 1, β = 1 , N = ∞ .

19

Underdamped system response: ξ = 0.707 . τ m > 0.2Tm . Kc =

1.414TCL 2Tm + τmTm + 0.25τm 2 − TCL 2 2

(

K m TCL 2 2 + 0.707TCL 2 τm + 0.25τm 2 2

Ti =

)

,

2

1.414TCL 2Tm + τmTm + 0.25τm − TCL 2 , Tm + 0.5τm

0.707TmTCL 2 τm + 0.25Tm τm 2 − 0.5τmTCL 2 2

Td =

Tm τm + 0.25τm 2 + 1.414TCL 2Tm − TCL 2 2

.

2

Tm τm + 2TCL ξ − TCL T − ξTCL K K [T + τm ] − Ti , α= i or α = c m i . Tm + τm Ti 2K c K m Ti

20

Ti =

21

Kc =

0.281(τm + 2Tm )3

K m τm (τm + 3Tm )

2

−

3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 5.33τm m − 18.98τm , Km τm + 2Tm (τm + 2Tm )4

Td = τm (τm + 3Tm ) 22

Kc =

0.562(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 0.281(τm + 2Tm )3 − τm (τm + 3Tm )2

.

1.066Tm − 0.467 τm 4.695τm (1.066Tm − 0.467 τm ) , Ti = , K m τm τm + 2Tm  0.331Tm − 0.334τm  . Td = τm    1.066Tm − 0.467 τm 

b720_Chapter-03

Rule Ogawa and Katayama (2001). Srinivas and Chidambaram (2001). Srinivas and Chidambaram (2001). Chidambaram (2002), pp. 157–158.

Kc 23

Kc

Ti

Td

Comment

Ti

Td

N = 10

Minimum ISE: servo. α = 1 , β = 1 ; Model: Method 1. 24

0.9Tm K mτm

3.33τ m

0

Kc

Ti

Td

1.2Tm K m τm

2τ m

0.5τ m

25

26

Desired closed loop response =

e −sτCL

. Kc =

2

 τ  + 0.6974 CL   Tm

2

0.01 ≤

TCL ≤ 1.00 ; Tm

Sample K c , Ti taken by the authors correspond to the process reaction curve τm 1 τ − = 0.30 − 1.11 m for Ti K c K m Tm

this tuning rule. Sample K c , Ti , Td taken by the authors correspond to the process reaction tuning rule of Ziegler and Nichols (1942); α = β=

26

Model: Method 1; N=∞. Model: Method 2; β =1, N = ∞ .

 τ  + 0.007393 , 0.05 ≤ CL ≤ 1.00 . Tm 

tuning rule of Ziegler and Nichols (1942); α = 25

Model: Method 1

1  (τCL Tm ) − 2(TCL Tm ) + 4   , K m  (τCL Tm ) + 2(TCL Tm ) 

(1 + sTCL )  ((τ T ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 )  , Ti = Tm  CL m  2(τCL Tm ) + 4    (τCL Tm )((τCL Tm ) + 4(TCL Tm ) − 2(τCL Tm )2 )  ; Td = Tm   ((τCL Tm ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 )    τ TCL = −0.1902 CL Tm  Tm

24

FA


162

23

17-Mar-2009

1 Td

τm 1 τ − = 0.5 − 0.83 m , Ti K c K m Tm

2  Tm τ  − m  = −0.16 for this tuning rule. τm − K c K m 2Ti  

α = 0.6762 − 0.4332(τm Tm ) , τm Tm ≤ 0.8 ;

or α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm )2 + 8.431(τm Tm )3 , τm Tm < 0.5 ;

α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc 27

Kc

Haeri (2002).

27

Kc =

1 Km

Td

Comment

Ti

Td

Model: Method 1

α = 0 , β =1, N = ∞ .

Ozawa et al. (2003).

28

Kim et al. (2004). Model: Method 1

Ti

163

Kc

x1 K m τ m Tm

Ti

Td

x 2 Tm

x 3Tm

Model: Method 1; 0% ≤ OS ≤ 10%.

Coefficient values: x2 x3

x1

α

β

0.1 12.5 0.22 0.04 0.68 0.75 0.2 6.1 0.41 0.08 0.63 0.70 0.3 4.1 0.57 0.11 0.62 0.70 0.4 3.1 0.71 0.15 0.59 0.69 0.5 2.5 0.83 0.18 0.58 0.69 0.6 2.11 0.94 0.21 0.56 0.69 0.7 1.82 1.05 0.24 0.54 0.69 0.8 1.61 1.13 0.28 0.51 0.68 0.9 1.44 1.22 0.31 0.48 0.69 1.0 1.33 1.26 0.34 0.49 0.66 OS < 20% (servo step); settling time ≤ that of corresponding rule of Chien et al. (1952); a cost function is minimised.

  6.84 + 0.64 ,  0.7 3.5 1 + 2.33(τm Tm ) + 7.82(τm Tm ) 

[ [2.15 − 0.76(τ

] T ) ] , 0.29 < τ

Ti = τm 0.95 + 2.58(τm Tm ) + 3.57(τm Tm ) , 0 < τm Tm ≤ 0.29 , Ti = τm

2

Tm ) + 0.33(τm

m

2

m

m

Tm < 4 ;

2.8   τm    τm   τm  τ   , 0 < m ≤ 1.79 ;  + 3.94  − 4.65 Td = τm 0.29  T +τ   T +τ  T T + τ  m m  m  m   m  m  m   2

[

Td = τm 0.87(τm Tm ) − 0.49(τm Tm ) + 0.09(τm Tm ) 28

2

1.6193(τm + 2Tm )

3

Kc =

K m τm (τm + 3Tm )2

−

2.8

] , 1.79 < τ

m

Tm < 4 .

3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 2.7051τm m − 1.6706τm , Km τm + 2Tm (τm + 2Tm )4

Td = τm (τm + 3Tm )

1.7793(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 1.6193(τm + 2Tm )3 − τm (τm + 3Tm )2

, α = 1, β = 1 , N = ∞ .

b720_Chapter-03

Rule Kotaki et al. (2005a). Model: Method 1; α = 1, β = 1 , N =∞.

Kc =

Kc

Ti

x1 K m

x 2Tm

Td x 3Tm

Closed loop transfer function is 4 order; p = % allowed perturbation on individual model parameters. p x1 x2 x3 τm Tm 6.8016 3.2091 1.4202 0.8298 0.5383 0.3661 29

Kc

30

Kc

31

0.3482 0.0243 0.1 0.6001 0.0453 0.2 0.9000 0.0754 0.4 1.0194 0.0850 0.6 1.0293 0.0656 0.8 0.9685 0.0042 1.0 Direct synthesis: frequency domain criteria 0 Ti Ms Ti

0

0.7Tm

0

Kc

0.4Tm K mτm

= 1.25

0.1 ≤

τm ≤2 Tm

α = 0.5; Model: Method 5 or 17.

(

K m 4 + 4 x1x 2 (τm Tm ) + x 2 2 (τm Tm )2

)

,

2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4x1 )((τm Tm ) + 2 − 2 x1x 2 (τm Tm )) 2 x 2 2 (τm Tm ) − x 2 4 (τm Tm )2 + 2 x 2 2 ((τm Tm ) + 2 − 2 x1x 2 (τm Tm ))

x1 = 0.5 +

0.25 x2

2

,

0.486 + 0.675(Tm τm ) τm τ ; < 1.91 ; x1 = 1 , m ≥ 1.91 ; x 2 = 1 + 0.0972(τm Tm ) Tm Tm

,

α = 0.488 − 0.985(τm Tm ) , τm Tm ≤ 0.495 ; α = 0 , τm Tm > 0.495 . 2

2 2 0.29e −2.7 τ + 3.7 τ Tm , Ti = 8.9τm e−6.6 τ + 3.0 τ or 0.79Tm e −1.4 τ+ 2.4 τ , K m τm 2

α = 1 − 0.81e 0.73τ+1.9 τ , M s = 1.4. 2

31

32 36 46 59 77 100

Model: Method 5 or 17; τ 0.14 ≤ m ≤ 5.5. Tm

2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4 x1 )((τm Tm ) + 2 − 2x1x 2 (τm Tm ))

Ti = Tm

Kc =

Comment

th

Lee et al. (1992). Model: Method 1 Dominant pole design – Åström and Hägglund (1995), pp. 204– 208. Modified Ziegler–Nichols – Åström and Hägglund (1995), p. 208.

30

FA


164

29

17-Mar-2009

2 0.78e −4.1τ + 5.7 τ Tm , α = 1 − 0.44e 0.78 τ−0.45τ , M s = 2.0. Kc = K m τm

b720_Chapter-03

17-Mar-2009

FA


Rule Huang et al. (2000). Model: Method 1

Kc

λ (Tm + 0.4τm ) K m τm α=

Ti

Td

Comment

Tm + 0.4 τ m

0.4Tm τ m Tm + 0.4τ m

For λ = 0.80 , A m = 2.13 .

Tm − 1 , β = 0 , N = min[20,20Td ] . Tm + ατ m

Hägglund and Åström (2002). Model: Method 5; M s = 1.4;

0.35Tm 0.6 − K mτm Km

7τ m

0.35Tm 0.6 − K mτm Km

0.8Tm

0 < α < 0.5 .

0.25Tm K mτm

0.8Tm

Hägglund and Åström (2002).

32

Leva and Colombo (2004).

33

τm < 0.11 Tm 0

0

Ti

Kc

0.11
[Tm ,8τm ] (‘moderate’

tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning).

32

Kc =

6.8τm Tm 1  T   0.14 + 0.28 m  , Ti = 0.33τm + . τm  10τm + Tm K m 

33

Kc =

2 2   1 τm τm ,  , Ti = Tm + Tm + K m (τm + λ )  2(τm + λ )  2(τm + λ )

Td =

τm Tm (τm + λ )

2Tm (τm + λ ) + τm 2

−

2Tm (τ m + λ )2 λτ m ,N= −1 . 2(τm + λ ) λ 2Tm (τ m + λ ) + τ m 2

[

]

3∆τ m , ∆τ m = maximum variation in the time delay, π with only variation in the time delay considered (Leva and Colombo, 2004); λ

α not specified; β = 1 . λ ≥

^

chosen so that nominal cut-off frequency = ωu (Leva, 2005).

165

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166 Rule

Kc

Ti

Comment

Td

34 Ti Td Kc Shamsuzzoha and Lee (2007a). M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0 Model: 0.11 0.10 0.08 0.07 0.06 Method 1; 0.20 0.17 0.15 0.14 0.12 N =∞, 0.35 0.31 0.27 0.25 0.22 α = 0.6 , β = 0 ; 0.41 0.37 0.34 0.30 coefficients of x1 0.46 0.55 0.49 0.44 0.40 0.36 (deduced from graph) indicated 0.62 0.55 0.50 0.46 0.42 in columns. 0.68 0.60 0.55 0.51 0.47

Kc =

λ = x1Tm τm = 0.05Tm τm = 0.1Tm τm = 0.2Tm τm = 0.3Tm τm = 0.4Tm τm = 0.5Tm τm = 0.6Tm

0.72

0.65

0.59

0.55

0.50

0.77

0.69

0.63

0.59

0.53

τm = 0.8Tm

0.81

0.72

0.66

0.62

0.56

τm = 0.9Tm

0.84

0.76

0.69

0.65

0.59

τm = 1.0Tm

0.91

0.82

0.75

0.70

0.64

τm = 1.25Tm

0.97

0.87

0.80

0.75

0.70

τm = 1.5Tm

τm = 0.7Tm

-

0.91

0.84

0.78

0.73

τm = 1.75Tm

-

0.96

0.87

0.81

0.76

τm = 2.0Tm

Ultimate cycle 0 x 2 Tu

x 1K u

Kuwata (1987); x 1 , x 2 deduced from graphs; α = 1.

34

FA

x1

x2

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.37 0.32 0.29 0.26

0.03 0.07 0.08 0.09

0.25 0.28 0.30 0.33

0.24 0.22 0.21 0.20

0.10 0.11 0.12 0.13

0.35 0.38 0.40 0.43

0.20 0.20

0.14 0.14

0.45 0.48

2 2 3λ2 + 2x1τm − 0.5τm − x1 Ti , , Ti = Tm + 2 x1 − K m (3λ + τm − 2x1 ) 3λ + τm − 2x1

2

2Tm x1 + x1 − Td = x1 = Tm 1 − 

λ3 + 0.167 τm 3 − x1τm 2 + x12 τm 2 2 3λ2 + 2x1τm − 0.5τm − x1 3λ + τm − 2 x1 , Ti 3λ + τm − 2x1

(1 − (λ Tm ))3 e− τ

m

Tm

. 

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Rule

Kc

Ti

Kuwata (1987); x 1K u x 1 , x 2 deduced x1 x 2 x 3 from graphs; α = 1, β = 1 , 0.69 0.63 0.08 N =∞. 0.57 0.48 0.08 0.47 0.37 0.08 35 Hang and 0. 6 K u Åström (1988a). 36 0. 6 K u Model: Method 2; β = 1 , N = 10 . 0. 6 K u Hang et al. (1991) – servo.

37 38

Hang and Cao (1996). Model: Method 37

35

36

37

x1

τm Tu

39

x 3 Tu x2

x3

τm Tu

x1

x2

x3

τm Tu

0.25 0.38 0.28 0.07 0.36 0.20 0.14 0.01 0.48 0.28 0.31 0.20 0.05 0.40 0.20 0.14 0 0.50 0.32 0.26 0.18 0.04 0.42 0.5Tu 0.125Tu Ti

0.125Tu

0.335Tu

0.125Tu

0. 6 K u

0.5Tu

0.125Tu

0. 6 K u

'

0.125Tu

0.222 K Tu

0. 6 K u

39

Ti

Td

τm Tm > 1.0 ;

α = 0.2 . Model: Method 2; β = 1 , N = 10 . 0.1 ≤

τm < 0.5 ; Tm

N = 10; β = 1 .

τ τ 1.66τm τ m , < 0.3 ; α = 1 − 2OS − m , 0.3 ≤ m < 0.6 ; Tm Tm Tm Tm OS = % overshoot.  τm τm τm τm  Tu . α = Ti = 0.51.5 − 0.83 − 0.6 , 0.6 ≤ < 0.8 ; α = 0.2 , 0.8 ≤ < 1.0 . Tm Tm Tm Tm   α = 1 − 2(OS − 0.1) −

0.16 ≤

τm 15 − K ' < 0.57 ; α = 1 − , 10% overshoot; Tm 15 + K ' α = 1−

38

Comment

Td

x 2 Tu

167

0.57 ≤

 11[τ m Tm ] + 13  36  . , 20% overshoot with K ' = 2 ' 27 + 5K  37[τ m Tm ] − 4 

τm 8 4  < 0.96 ; α = 1 −  K ' + 1 , 20% overshoot, 10% undershoot. 17  9 Tm 

  τ  τ T Ti =  0.53 − 0.22 m Tu , Td =  0.53 − 0.22 m  u . α equals 1.0, 0.2 and Tm  4 Tm    0.40(τ m Tm ) − 0.05(τ m Tm ) + 0.58 in different circumstances. 2

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168

3.3.8 Two degree of freedom controller 2     T s 1 d  1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0  Ti s T  1 + a f 1s 1+ d s   N   1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) F(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

   1 + b f 1s  1 + a f 1s s 

+ U(s)

−


m

K0


Kc

Ti

Td

Y(s)


Comment

Direct synthesis: time domain criteria 0.375(τ m + 2Tm ) N=∞ Tm τ m Normey-Rico et Tm + 0.5τ m K mτm 2Tm + τ m al. (2000). a = 0 . 13 τ , b = 0 ; a = 0. 5τ m , b f 2 = 0. 2 τ m ; f1 m f1 f2 Model: Method 1 a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . Robust Lee and Edgar (2002). Model: Method 1

1

Kc

0

Ti

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .

1

2

Kc =

1 + 0.25K u K mTi T − 0.25K u K m τm τm . , Ti = m + K m (λ + τm ) 1 + 0.25K u K m 2(λ + τm )

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Lee and Edgar (2002). Model: Method 1

2

Kc

Ti

Td

Comment

Kc

Ti

Td

N=∞

169

a f 1 = Td , b f 1 = 0 ; a f 2 = 0 , b f 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .

Zhang et al. (2002c). 0.1τm ≤ λ1 ≤ τm , 0.1τm ≤ λ 2 ≤ τm .

Tm K m (λ 2 + τm )

0

Tm

Model: Method 1

a f 1 = 0.5τm λ 2 (λ 2 + τm ) , b f 1 = 0.5τm ; a f 2 = λ1 , b f 2 = λ 2 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .

0.5τm 0.167 τm 0 ≤ x1 ≤ 1 Kc Shamsuzzoha and Lee (2007c). a f 2 = b f 1 , b f 2 = x1bf 1 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , Model: Method 1 b f 5 = 0 , K 0 = 0 . λ is given as a multiple of Tm (from a graph, for 3

0.01 ≤ τm Tm ≤ 5 ), for M s = 1.4,1.5,1.6,1.8 and 1.9.

Zhang et al. (2003). Model: Method 1

4

Ultimate cycle 0.5Tu 0.125Tu

0.6K u

N=∞

a f 1 = 0 , b f 1 = 0 ; a f 2 = Ti , b f 2 = x 2Ti ; a f 3 = Ti Td , b f 3 = x1Ti Td , a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .

2

Kc =

2  1 + 0.25K u K m  Tm − 0.25K u K m τm τm + + Td  ,  K m (λ + τm )  1 + 0.25K u K m 2(λ + τm ) 

Ti =

Tm − 0.25K u K m τm τm + + Td , 1 + 0.25K u K m 2(λ + τm )

2

2  0.25K u K m τm τm (T − 0.25K u K m τm )τm 2  Td = τm  − + + m  2 2(1 + 0.25K u K m )(λ + τm )   2(1 + 0.25K u K m ) 6(λ + τm ) 4(λ + τm ) 3

Kc =

τm    λ  − Tm  τm e , b f 1 = Tm 1 − 1 − ,   Tm   2K m (2λ − b f 1 + τm )  

af1 = 4

x1 =

0.5

0.5b f 1τm + λτ m + λ2 − Tm , N = ∞ . 2λ − b f 1 + τ m

2 T + K m K c [Ti − τm ] 2Tm Ti + 2K m K cTi [Td − τm ] + K m τm K c . , x2 = i 2K m K cTi Td K m K c Ti

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170

3.3.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s)     ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d  , F1 (s) = K c  [1 − α ] + + +   T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s   N       [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d  F2 (s) = K c  [1 − ε] + + − 0 f3 2   1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s   N  

R(s)

+

F1 (s)

U(s)

Y(s)

Model

−

F2 (s) Table 17: PID controller tuning rules – FOLPD model G m (s) = Rule

Kc

Hiroi and Terauchi (1986) – regulator. Model: Method 2; τ 0.1 < m < 1 . Tm

0.95Tm K m τm 1.2Tm K m τm

Ti

Td


Comment

Process reaction 2.38τm 0.42τm

0% overshoot

0.42τm

20% overshoot

2τm

α = 0.6 ; β = 1 or β = −1.48 ; χ = 0 ; δ = 0.15 ; N not specified; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; af 4 = 0 , K0 = 0 .

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Rule Kaya and Scheib (1988). Model: Method 5; τ 0 < m ≤ 1. Tm

Kc

Comment

Td

Ti

Minimum performance index: regulator tuning Minimum IAE. x1K c Kc 0 2 Minimum ISE. x1K c K 1

c

3

Minimum ITAE.

x1K c

Kc

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 .

τ  Tm  Tm  τ m  0 < m ≤1     0 T m x 3  Tm   τm  α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; x2

x1 Km

Kaya and Scheib (1988). Model: Method 5

x4

φ = 0 ; ϕ = 0 ; b f 2 = x 5Tm (τm Tm )x 6 ; a f 3 = 0.1bf 3 ; a f 4 = 0 ; K 0 = 0. Coefficient values: x1 x2 x3 x4 x5 x6 Minimum IAE Minimum ISE Minimum ITAE

0.91 1.1147 0.7058

0.7938 0.8992 0.8872

0.8826

1

Kc =

1.31509  Tm    K m  τm 

Kc =

1.3466  Tm    K m  τm 

0.9308

2

Kc =

1.3176  Tm    K m  τm 

0.7937

3

1.01495 0.9324 1.03326

1.3756

, x1 =

Tm  τm    1.2587  Tm 

1.25738

, x1 =

Tm  τm    1.6585  Tm 

, x1 =

Tm  τm    1.12499  Tm 

1.00403 0.8753 0.99138

0.5414 0.56508 0.60006

τ  , x 2 = 0.5655Tm  m   Tm 

0.4576

τ  , x 2 = 0.79715Tm  m   Tm 

1.42603

0.7848 0.91107 0.971

. 0.41941

τ , x 2 = 0.79715Tm  m  Tm

.    

0.41941

.

171

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172 Rule

Kc

Kaya and Scheib (1988). Model: Method 5; τ 0 < m ≤1 Tm

4

Kc

5

Kc

6

Kc

Ti

Comment

Td

Minimum performance index: servo tuning Minimum IAE. x1K c 0 Minimum ISE. x1K c

Minimum ITAE.

x1K c

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 . x1 Km

Kaya and Scheib (1988). Model: Method 5

 Tm   τm

  

Tm

x2

τ x3 − x4 m Tm

0
1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ; K 0 = 0 . 16 Åström and Ti Td Kc Hägglund (2004). α = 0 , β = 1 ; χ = 0 ; δ = 0 ; 4 ≤ N ≤ 10 ; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ; Model: Method 56 ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 ; a f 3 = 2Td N ; a f 4 = Td 2 N 2 ; K 0 = 0 .

15

Kc =

0.4τm + 0.8Tm 0.5τm Tm 1  Tm  τm ; Td = . 0.2 + 0.45  , Ti = Km  τm  τm + 0.1Tm 0.3τm + Tm 2

a f 3 = 0.2τm , a f 4 = 0.01τm (Åström and Hägglund, 2004); a f 3 = 0.17 τm , a f 4 = 0 (Eriksson and Johansson, 2007); 2

a f 3 = 0.333δmax , a f 4 = 0.028δmax , τm < Tm , δmax = jitter margin (Eriksson and Johansson, 2007); 2 a f 3 = 4.6δmax − 6τm (with a f 3 ≥ 1 ), a f 4 = (2.3δ max − 3τm ) , τm > Tm (Eriksson and Johansson, 2007).   [ ] 1 T 0 . 5 T N + 16 m d Kc = 0.2 + 0.45 , Km  τm + 0.5[Td N ]  0.4τm + 0.8Tm + 0.6[Td N ] Ti = (τm + 0.5[Td N ]) , τm + 0.1Tm + 0.55[Td N ]

Td =

0.5(τm + 0.5[Td N ])(Tm + 0.5[Td N ]) τ m >1. ; Tm 0.3τm + Tm + 0.65[Td N ]

b720_Chapter-03

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Rule

Majhi and Atherton (1999), Atherton and Boz (1998). Model: Method 1

Kc

m m m

α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ;  (2Tm + τ m )3 x  , b = 1 , b = x 5 , a = 0 . − 1 − 3 f3 f5 f4 Tm 2Tm 2 τ m   Coefficient values (deduced from graphs). Model: Method 1 x1 x2 x3 x4 x1 x2 x3 x4

1.2 2.6 4.5 6.4 Minimum ISTE – 1.8 servo – Atherton 4.2 and Boz (1998). 6.9 10.0 Minimum ITSE – 1.8 servo – Majhi 4.2 and Atherton 6.9 (1999). 10.0 Minimum ISTSE 2.3 – servo – 5.2 Atherton and Boz 8.4 (1998). 12.4 Minimum ISTSE 2.3 – servo – Majhi 5.2 and Atherton 8.7 (1999). 12.4

17

Comment

Td

Direct synthesis: time domain criteria τm (2Tm + τm )3 x1 τmTm 17 ≤ 0.8 x2 0 2 T 2 T + τ m m m 2K τ T

K0 =

Minimum ISE – servo – Atherton and Boz (1998).

Ti

1 Km

0.58 0.91 0.89 0.98 0.17 0.22 0.25 0.26 0.17 0.22 0.25 0.26 0.08 0.11 0.14 0.13 0.08 0.11 0.12 0.13

-0.81 -0.55 -0.30 -0.22 -0.31 -0.16 -0.11 -0.08 -0.29 -0.16 -0.11 -0.08 -0.16 -0.10 -0.07 -0.04 -0.16 -0.10 -0.06 -0.04

2.36 2.49 2.18 2.23 1.08 0.96 0.95 0.92 1.05 0.96 0.95 0.92 0.71 0.65 0.68 0.63 0.69 0.65 0.62 0.64

8.5 10.8 13.3 16.0 13.0 16.2 20.3 24.0 13.0 16.8 20.3

1.02 1.03 1.02 1.00 0.28 0.31 0.29 0.30 0.28 0.27 0.29

-0.16 -0.14 -0.12 -0.09 -0.07 -0.06 -0.04 -0.04 -0.06 -0.05 -0.04

2.18 2.16 2.13 2.10 0.94 0.99 0.93 0.95 0.94 0.90 0.97

16.5 20.4 25.2 31.2 16.5 20.4 25.2

0.14 0.15 0.15 0.14 0.14 0.15 0.15

-0.04 -0.03 -0.02 -0.02 -0.03 -0.03 -0.02

0.64 0.65 0.64 0.61 0.64 0.67 0.65

Equations developed from information provided by Majhi and Atherton (1999).

 (2T + τ )2  m m  x 4 − 2(Tm + τ m )  T  Tm  x5 = m  . 3 2 (2Tm + τm ) x 3   −1− 2   2Tm τ m

175

b720_Chapter-03

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176 Rule

Kc

Td

Kc

Ti

Td

1.2τ m K m Tm

0.5τ m

0.15τ m K m Tm

18

Chidambaram (2000a). Model: Method 1

19

Comment

Ti

2

β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = α ;

φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = K c ; bf 3 = 1 ; 0.8(Tm + 0.4 τ m ) Huang et al. Kmτm (2000). ( 0 . 6 T m + 0.4 τ m ) Model: Method 1 Kmτm

bf 4 = 0 ; a f 5 = 1 .

Regulator tuning. Tm + 0.4 τ m

0 Servo tuning.

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = af 3 =

18

Kc =

1 Km

0.4Tm τm + af 3 , Tm + 0.4τm

0.4Tm τm ; af 4 = 0 ; K0 = 0 .   (Tm + 0.4τm )min 20, 0.4Tm τm   Tm + 0.4τm 

2     4.114 − 6.575 τm + 3.342 τm   , T   Tm   m  

2  τ   K cTi τ Ti = Tm 0.311 + 1.372 m − 0.545 m   , Td = ; [ Tm T 16 . 016 − 11.7(τm Tm )]   m    

α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm ) + 8.431(τm Tm ) , τm Tm ≤ 0.5 , 2

3

α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 ; K 0 = K c .

19

α = 0.6762 − 0.4332

τ τm τ m 1. 2 τ m , . ≤ 0.8 ; b = 0.1493 , m = 0.9 ; K 0 = Tm Tm Tm K m Tm

b720_Chapter-03

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FA


Rule

Kc 20


Ti

Kc

177

Comment

Td

Direct synthesis: frequency domain criteria Ti Td M s = 1.25

β = 1 ; χ = 0 ; δ = 0 ; N not specified; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0 .

Åström and Hägglund (2006), pp. 252, 265. Model: Method 56

21

Kc

Ti

Td

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; 2 ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = x1 , a f 4 = 0.5x1 ; K 0 = 0 .

Padhy and Majhi (2006a). Model: Method 39

22

Kc

x1 1 + x1x 2 τm

0

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ; b f 3 = 1 ; b f 4 = 0.5τm ; a f 5 = 0 .

20

Kc =

2 x 2 2 (τm Tm ) − 4 + 8x1x 2 + 0.5(τm Tm )x 2 2 (4 + 2 τm Tm )

[

]

K m x 2 2 (τm Tm )2 + 4 + 4 x1x 2 (τm Tm )

,

  2 τ  2 x 2 m − 4 + 8x1x 2    Tm τ Tm , Ti = Tm  0.5 m +  , Td = T   τ (4 + 2 τ m Tm )x 2 2 Tm m   4 + 2 m x 22  + 2    Tm  τ m 2 x 2 2 [τ m Tm ] − 4 + 8x1x 2    x1 = 0.5 +

0.25 x2

2

,

0.416 + 0.956 Tm τm τm τ ; ≤ 2.51 ; x1 = 1 , m > 2.51 ; x 2 = 1 + 0.0498 τm Tm Tm Tm

α = 1 − 0.517(τm Tm )

0.279

, τm Tm ≤ 10.64 ; α = 0 , τm Tm > 10.64 .

0.4τm + 0.8Tm + 0.6 x1 1  Tm + 0.5x1  21 Kc = (τm + 0.5x1 ) , 0.2 + 0.45  , Ti = Km  τm + 0.1Tm + 0.55x1 τm + 0.5x1  0.5(τm + 0.5x1 )(Tm + 0.5x1 ) T T Td = , d ≤ x1 ≤ d . 0.3(τm + 0.5x1 ) + (Tm + 0.5x1 ) 30 3 22

Kc =

(A − 1) [T

2φm + π(A m − 1) 2K m τm

2

m

m

]

− 0.7071 Tm τm , x1 =

Tm − 0.7071 Tm τm 1.4142 Tm τm + 1

and

 2φ + π(A m − 1)   2A m x 2 = 0.7854A m  m  1 − 2 π  2 A m − 1  

(

)

 2φm + π(A m − 1)    .  2 A m 2 − 1 

(

)

b720_Chapter-03

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178 Rule

Kc

Ti

Td

Comment

0.1 ≤ τm ≤ 10 , 23 Eriksson and Ti Td Kc 0.1 ≤ Tm ≤ 10 . Johansson α = 1 , τ T ≤ 1 , α = 0 , τ T > 1 ; β = 1 ; χ = 0 ; δ =0 ; N =∞; (2007). m m m m Model: Method 2 bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ,

a f 3 = 0.1 ωg , a f 4 = 0.0025 ωg 2 , τm Tm ≤ 0.25 ; a f 3 = 0.2τm , 2

a f 4 = 0.01τm , τm Tm > 0.25 , ωg = gain crossover frequency; K0 = 0 .

Robust

Tm K m (0.5τ m + λ )

Harris and Tyreus (1987). Model: Method 1

0

Tm

λ ≥ maximum [0.8τ m , Tm ]

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ; K 0 = 0. 24

Ho et al. (2001). Model: Method 1

Kc

0

Tm

1.551Tm KmAmτm

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ; K 0 = 0. 25

Lee and Edgar (2002). Model: Method 1

0

Ti

Kc

α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu , b f 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .

23

Kc = Ti =

 0.01Tm (0.4Tm + 11) 1  0.4Tm − 0.04  , + 0.16  , Td = ( ( 0 . 4 Tm − 0.04 ) τm ) + 0.16 K m  τm 

(0.35T

m

2

100((0.4Tm − 0.04) τ m ) + 0.16

) (

3

2

+ 4Tm + 50 τm − 0.11Tm − 1.5Tm + 1.5

)

τm 2 .

Tm 24 Kc = , TCL = −0.5τ m + 0.25τ m 2 + ωg −2 , K m (0.5τm + TCL ) ωg = frequency at which the Nyquist curve has a magnitude of unity. 25

Kc =

(1 + 0.25K u K m )Ti K m (λ + τm )

2

, Ti =

Tm − 0.25K u K m τm τm + . 1 + 0.25K u K m 2(λ + τm )

b720_Chapter-03

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FA


Rule

Kc 26


Kc

Ti

Td

Ti

Td

Comment

α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu , bf 3 = 1 , bf 4 = 0 , a f 5 = 0 . 27

Visioli (2002). Model: Method 1

Tm + 0.5τ m

Kc

Tm τ m 2Tm + τ m

α = 0 , β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 =

0.05 ≤

τm ≤1 Tm

Tm , 1 + K0

 λτ m  λτ m Tm ; ε = 0;φ = 0; ϕ = 0; af1 =  + Tm  , a f 2 = 2(λ + τm )  2(λ + τm )  b f 2 = b f 1 , a f 3 = a f 1 , a f 4 = a f 2 ; for minimum IAE (regulator), K 0 = 0.2171(Tm τm )

1.481

; b f 3 = 1 ; b f 4 = 0 ; a f 5 = Tm .

Tm Cooper (2006a), K (λ + 0.5τ ) Tm 0.5τm m m (2006b). α = −0.5τm Tm , χ = 0 , a f 1 = 0 (Cooper, 2006a); Model: Method 1 Tm λ (Cooper, 2006b). α = 0 , χ = −0.5τm Tm , a f 1 = 2(λ + τm ) Tm + 0.5τm K m (λ + 0.5τm )

Tm + 0.5τm

α = 0 , χ = 0 , af1 =

Tm τm 2Tm + τm

Cooper (2006b)

Tm λ (Cooper, 2006b). 2(λ + τm )

β = 1 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 2 = 0 ; a f 3 = 0 ; ε = α ; φ = χ ; ϕ = 0 ; bf 2 = 0 , a f 3 = a f 1 , a f 4 = 0 ; K 0 = 0 .

λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning); λ > [Tm ,8τm ] (‘moderate’

tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning).

26

2

K c = x1Ti , Ti =

1 + 0.25K u K m Tm − 0.25K u K m τm τm + , x1 = 1 + 0.25K u K m 2(λ + τm ) K m (λ + τm )

3 4 2  0.25K u K m τm 2 τm τm τ (T − 0.25K u K m τm )  Td = x1  − + + m m . 2  2(1 + 0.25K K ) 6(λ + τ ) 4(λ + τ ) 2(1 + 0.25K u K m )(λ + τm )  u m m m  2Tm + τm 1 27 Kc = 2K m (λ + τm ) (1 + K 0 )

179

b720_Chapter-03

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180

3.4 FOLPD Model with a Zero

G m (s ) =

K (1 − sTm 4 )e −sτ m K m (1 + sTm 3 )e − sτm or G m (s) = m 1 + sTm1 1 + sTm1


R(s)

−

+

 1   K c 1 +  T is  

Y(s)

U(s)

Model

Table 18: PI controller tuning rules – FOLPD model with a zero K (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s ) = m or G m (s) = m 1 + sTm1 1 + sTm1 Rule

Slätteke (2006). Model: Method 3

Kc

Comment

Ti

Minimum performance index: regulator tuning Minimum IE, subject x1 (Tm1 + 0.333τm ) x T + x 4 τm x 2 τm 3 m1 to M s . K m Tm 3τm x 5Tm1 + x 6 τm x1

x2

x3

x4

x5

x6

Ms

0.09 0.16 0.23 0.28

4 5 1 1

6 23 100 88

1 4 17 15

1 9 11 12

21 105 94 85

1.1 1.2 1.3 1.4

b720_Chapter-03

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FA


Kc 1

3

3

Kc =

Ti

Model: Method 1

Tm1

K m (τm + Tm3 )

2

Direct synthesis: frequency domain criteria Model: Method 1; Ti 4 Kc Tm1 > Tm 4 .

Tm1 + 0.5τm (λ + τm )K m

Marchetti and Scali (2000). Model: Method 1

2

x1Tm1 K mTm 4

4Tm1

Desbiens (2007), p. 90.

Kc =

Comment

Direct synthesis: time domain criteria ‘Smaller’ τm Tm 4 . Kc Tm1 2 ‘Larger’ τm Tm 4 . Kc

Chidambaram (2002), p. 157. Model: Method 1 Sree and Chidambaram (2003a). Slätteke (2006). Model: Method 3

1

Ti

Tm1 + 0.5τm (λ + τm + 2Tm 4 )K m

1.571

(

)

.

(

)

.

K m τm 1 + 3.142 Tm 4 τm 2 2.358

K m τm 1 + 4.712 Tm 4 τm 2

Robust Tm1 + 0.5τm

Negative zero.

Tm1 + 0.5τm

Positive zero.

x1 = value of the ‘inverse jump’ of the closed loop system; x1 < K m Tm 4 Tm1 .

Ti =

(x1Tm1

Tm 4 )[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )] ; (x1Tm1 Tm 4 )(1 − x 2 ) − x 2

T 0.98x1Tm1 0.95x1Tm1 ≤ x2 ≤ , 0.1 ≤ m 4 ≤ 0.3 ; x1Tm1 + Tm 4 x1Tm1 + Tm 4 Tm1 0.3x1Tm1 T 0.1x1Tm1 ≤ x2 ≤ , m4 > 1 . x1Tm1 + Tm 4 Tm1 x1Tm1 + Tm 4 4

Kc =

Ti ω Km

2

Tm1 ω2 + 1

, with ω given by solving (for

(TiTm 4 ) ω4 + (Ti 2 + Tm 42 )ω2 + 1 φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm1 ω) − τ m ω ; 2

Ti = Tm1 + 0.175τ m , τm Tm1 < 2 ; Ti = 0.65Tm1 + 0.35τ m , τ m T m1 > 2 .

181

b720_Chapter-03

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FA


182


  1 1 + Td s  G c (s) = K c 1 + Ti s   1 + Tf s E(s)

R(s)

−

+

  1 1 K c 1 + + Td s   Ti s  1 + Tf s

U(s)

Model

Y(s)

Table 19: PID controller tuning rules – FOLPD model with a zero K (1 + sTm 3 )e −sτm K (1 − sTm 4 )e −sτ m G m (s) = m or G m (s) = m 1 + sTm1 1 + sTm1 Rule

Kc

Ti

Comment

Td

Direct synthesis: time domain criteria Sree and Chidambaram (2003a). Jyothi et al. (2001). Model: Method 1

1

Kc

2

Kc

3

Kc

0

Tm1

Model: Method 1

Direct synthesis: frequency domain criteria τm Tm3 < 1 0 Tm1 τm Tm3 > 1 Robust

Chang et al. Tm1 (1997). K m (TCL + τ m ) Model: Method 2

1

Kc =

2

Kc =

Tm1 T (λ − τ m ) . , Tf = m 4 K m (2Tm 4 + λ + τm ) 2Tm 4 + λ + τm 1.57Tm1 K m τm

3

Tm1

Kc = K m τm

T  1 + 9.870 m3   τm  0.79Tm1 T  1 + 2.467 m3   τm 

2

2

.

.

0

Tf = Tm 3

b720_Chapter-03

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FA


K m e − sτ m

3.5 SOSPD model

2

Tm1 s 2 + 2ξ m Tm1s + 1

or

183

K m e −sτ m (1 + Tm1s )(1 + Tm 2 s )


R(s)

+

−

 1   K c 1 + Ti s  

U(s)

Y(s)

SOSPD model

Table 20: PI controller tuning rules – SOSPD model K m e − sτ m K m e − sτ m or (1 + Tm1s)(1 + Tm 2 s) Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule

Kc

Comment

Ti

Minimum performance index: regulator tuning Model: Method 1 x1 K m x 2Tm1

Minimum IAE – Lopez (1968), pp. 63, 69–74.

Coefficients of K c and Ti deduced from graphs. These are representative results. 0.5 0.6 0.8 1.0 1.5 2.0 4.0

ξm

x1

x2

τm Tm1 = 0.1

4.8 4.2 5.7 3.4 7.8 2.8 9.7 2.3 15 1.7 21 1.4 35

x1

x2

x1

x2

x1

x2

x1

x2

x1

x2

x1

x2

-

τm Tm1 = 0.2

2.2 3.3 2.7 3.0 3.9 2.7 5.1 2.4 8.3 1.9 11.5 1.7 27 1.2

τm Tm1 = 0.5

0.76 2.1 1.0 2.3 1.6 2.4 2.1 2.4 3.7 2.4 5.4 2.4 12.5 2.4

τm Tm1 = 1.0

0.33 1.2 0.50 1.6 0.82 2.2 1.2 2.5 2.1 2.8 3.0 3.3 6.6 3.7

τm Tm1 = 2.0

0.23 1.2 0.34 1.6 0.52 2.2 0.70 2.8 1.15 3.8 1.65 4.4 3.4 5.9

τm Tm1 = 5.0

0.32 2.6 0.34 2.9 0.38 3.3 0.46 3.8 0.62 5.0 0.80 6.3 1.5 9.6

τm Tm1 = 10.0

0.34 5.0 0.35 5.3 0.38 5.6 0.39 5.9 0.46 7.1 0.52 8.3 0.90 -

Minimum IAE – Murata and Sagara (1977). Model: Method 4; K c , Ti deduced from graphs.

x1 K m

Tm1 = Tm 2

x 2Tm1

x1

x2

τm Tm1

x1

x2

τm Tm1

2.6 1.3 1.0

2.5 2.5 2.6

0.2 0.4 0.6

0.8 0.7

2.7 2.8

0.8 1.0

b720_Chapter-03

FA


184 Rule

Minimum IAE – Harriott (1988). Model: Method 1; coefficients of Ti deduced from graphs.

Minimum IAE – Harriott (1988). Model: Method 1; coefficients of Ti deduced from graphs.

Minimum IAE – Harriott (1988). Model: Method 1 Minimum IAE – Shinskey (1994), p. 158. Minimum IAE – Shinskey (1996), p. 48. Model: Method 1; τm Tm1 = 0.2 .

1

17-Mar-2009

Kc

Ti

Comment

0.5K u

0.4Tu

0.1 ≤ τm Tm1 ≤ 2 .

0.5K u

x 2Tu

Alternative.

Tm1 = Tm 2 ;

x2

τm Tm1

x2

τm Tm1

1.10 0.85 1.45 1.10 1.75 1.25

0.1 0.2 0.1 0.2 0.1 0.2

0.65 0.52 0.75 0.55 0.90

0.5 1.0 0.5 1.0 0.5

0.5K u τm Tm1

x2

Tm 2 Tm1 = 0.2 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 1.0

x 2Tu τm Tm1

x2

1.35 0.1 0.65 0.5 Tm 2 Tm1 = 0.2 0.85 0.2 2.00 0.1 0.80 0.5 Tm 2 Tm1 = 0.5 1.10 0.2 0.60 1.0 2.25 0.1 1.00 0.5 Tm 2 Tm1 = 1.0 1.25 0.2 Disturbance input subsequent to the smaller time constant and delay terms. Tm1 = Tm 2 ; 0.45K u 1.1Tu τ T = 0.2 . m

m1

Random load disturbance through a filter with time constant = 2Tm1 . Kc

Ti

Model: Method 1

0.77Tm1 K m τm

2.83(τ m + Tm 2 )

Tm 2 Tm1 = 0.1

1

0.70Tm1 K m τm 0.80Tm1 K m τm 0.80Tm1 K m τm

2.65(τ m + Tm 2 )

2.29(τ m + Tm 2 )

1.67(τ m + Tm 2 )

Tm 2 Tm1 = 0.2 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 1.0

3Tm1    −  (τ m + Tm 2 )    Kc = , Ti = τ m  0.5 + 3.5 1 − e . Tm1         −     τ + T   K m (τm + Tm 2 ) 50 + 551 − e m m 2        

100Tm1

b720_Chapter-03

17-Mar-2009

FA

Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Huang et al. (1996). Model: Method 1

2

2

Kc

Ti

Kc

Ti

Comment 0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τm Tm1 ≤ 1 .

Note: equations continued into the footnote on p. 186. τm τ T  1  T − 3.491 m 2 − 25.3143 m m2 2  Kc = 6.4884 + 4.6198 K m  Tm1 Tm1 Tm1  +

−0.9077 −0.063 0.5961   τm   τm  τ  1   0.8196 m      5 . 2132 7 . 2712 − − T  T  T  Km    m1   m1   m1   

+

T 1   − 18.0448 m 2 T Km   m1 

+

T T 1  0.4937 m 2 + 19.1783 m 2 Km  Tm1 τm 

+

τ 1  8.4355 m Km  Tm1 

+

τ m Tm 2  Tm 2 τm  2 1  − 0.7241e Tm1 − 2.2525e Tm1 + 5.4959e Tm1  ;  Km   

  

 Tm 2   Tm1

0.7204

  

T + 5.3263 m 2  Tm1  τm     Tm1 

0.557

− 17.6781

  

1.0049

T + 13.9108 m 2  Tm1

0.8529

+ 12.2494

τm Tm1

 Tm 2   Tm1

  

Tm 2 Tm1

  

1.005 

 τm     Tm1 

  

0.5613 

  

1.1818 

  

2  τ   T τ Ti = Tm1 0.0064 + 3.9574 m + 4.4087 m 2 − 6.4789 m    Tm1 Tm1  Tm1    2 3  T  τ   τ T + Tm1  − 12.8702 m m2 2 − 1.5083 m 2  + 9.4348 m    Tm1  Tm1   Tm1   

 T + Tm1 17.0736 m 2 Tm1  

2

 τm  τ  + 15.9816 m  T T m1  m1 

 Tm 2   Tm1

2

T   − 3.909 m 2  Tm1 

  

3

  

4 3 2   τ  T τ  T τ  + Tm1 − 10.7619 m  − 10.864 m 2  m  − 22.3194 m   m 2 Tm1  Tm1    Tm1   Tm1  Tm1   3 4 5  T  τ   τ T  + Tm1 − 6.6602 m  m 2  + 6.8122 m 2  + 7.5146 m   Tm1  Tm1    Tm1   Tm1   

 T + Tm1 2.8724 m 2 Tm1  

4

T  τm    + 11.4666 m 2 T  Tm1  m1 

  

2

3

τ   τm    + 11.1207 m  T  Tm1   m1 

  

2

2

  

 Tm 2   Tm1

  

3

  

185

b720_Chapter-03

FA


186 Rule Minimum IAE – Huang et al. (1996). Model: Method 1

3

  τ  T + Tm1  − 1.2174 m  m 2   Tm1  Tm1   T + Tm1  − 0.112 m 2 Tm1  

Kc

Ti

Comment

Kc

Ti

0.05 ≤ τm Tm1 ≤ 1 .

4

T   − 4.3675 m 2  Tm1 

5

T  τm   + 1.0308 m 2  T  Tm1  m1 

3   τ  T + Tm1 − 3.4994 m   m 2   Tm1   Tm1  3

17-Mar-2009

3

0.4 ≤ ξ m ≤ 1 ;

5

τ    − 2.2236 m   Tm1  

6

τ    − 1.9136 m   Tm1  

τ    − 1.5777 m   Tm1  

2

 Tm 2   Tm1

4

6

  

 Tm 2   Tm1

  

2

  

4

 τ  + 1.1408 m T m1 

 Tm 2   Tm1

Note: equations continued into the footnote on p. 187. τm τ  1  − 5.5175ξ m − 26.5149ξ m m  Kc =  − 10.4183 − 20.9497 Km  Tm1 Tm1  1 . 4439 0 . 1456 0.3157  τ   τm   τm  1  42.7745 m       + + + 10 . 5069 15 . 4103 T  T  T  Km    m1   m1   m1    1 3.7057 4.5359 1.9593 + 34.3236ξ m − 17.8860ξ m − 54.0584ξ m Km

[

1 + Km

]

−0.0541 4.7426    τm  τ   22.4263ξ m  m    + 2.7497ξ m  T  Tm1    m1     

+

1 Km

 T  τ 1.8288  τ m    − 17.1968 m ξ m 2.7227 + 1.0293ξ m m1  50.2197ξ m Tm1 τ m    Tm1 

+

1 Km

τ τm  ξm m   − 16.7667e Tm1 + 14.5737e ξm − 7.3025e Tm1  ,    

2  τ  τ  τ Ti = Tm1 11.447 + 4.5128 m − 75.2486ξm − 110.807 m  − 12.282ξ m m  Tm1 Tm1    Tm1    3 2  τ  τ   + Tm1 345.3228ξ m 2 + 191.9539 m  + 359.3345ξ m  m     Tm1   Tm1   

4   τm   τm 2 3     + Tm1 − 158.7611 ξ m − 770.2897ξ m − 153.633 Tm1 Tm1      

  

5

.  

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum IAE – Shinskey (1996), 0.48K u p. 121. Model: Method 1 Minimum ISE – x1 K m McAvoy and Johnson (1967). Model: Method 1; ξ m Tm1 x 1 τm coefficients of K c 1 0.5 0.8 and Ti deduced from 1 4.0 5.7 graphs. 1 10.0 13.6

Comment

Ti

τm Tm1 = 0.2 ,

0.83Tu

Tm 2 Tm1 = 0.2 .

x 2τm x2

1.82 12.5 25.0

Coefficient values: ξ m Tm1 x 1 x 2 τm 4 4

0.5 4.3 3.45 4.0 27.1 6.67

ξ m Tm1 τm

7 7

x2

0.5 7.8 3.85 4.0 51.2 5.88

3 2   τ  τ  τ + Tm1  − 412.5409ξ m  m  − 414.7786ξ m 2  m  + 485.0976 m ξ m 3  Tm1    Tm1   Tm1    5 4  τ   τ  + Tm1 864.5195ξ m 4 + 55.4366 m  + 222.2685ξ m  m     Tm1    Tm1   3 2   τ  τ  τ  + Tm1 275.116ξ m 2  m  + 205.2493 m  ξ m 3 − 479.5627 m ξ m 4     Tm1   Tm1   Tm1    6 5   τ  τ  + Tm1  − 473.1346ξ m 5 − 6.547 m  − 43.2822ξ m  m  + 99.8717ξ m 6     Tm1   Tm1    4 3 2   τ  τ  τ  + Tm1  − 73.5666 m  ξ m 2 − 56.4418 m  ξ m 3 − 37.497 m  ξ m 4     Tm1   Tm1   Tm1   

  τ + Tm1 160.7714 m ξ m 5  . T m1  

x1

187

b720_Chapter-03

17-Mar-2009

FA


188 Rule

Kc

Ti

Comment

Minimum ISE – Lopez (1968), pp. 63, 69–74.

x1 K m

x 2Tm1

Model: Method 1


ξm

x2

x1

x2

τm Tm1 = 0.1

6.5 4.8 7.7 4.2 10.5 3.4 13 2.9 20.5 2.1 28 1.6

x1

x2

x1

x2

x1

x2

x1

-

-

τm Tm1 = 0.2

3.1 4.2 3.9 3.8 5.4 3.3

7

x2

x1

x2

x1

2.9 11 2.4 15 2.0 33 1.4

τm Tm1 = 0.5

1.25 2.9 1.55 2.9 2.2 2.9 2.9 2.9 4.8 2.9 6.7 2.7 14.5 2.5

τm Tm1 = 1.0

0.63 2.2 0.83 2.5 1.2 2.9 1.65 3.1 2.7 3.7 3.8 3.8

τm Tm1 = 2.0

0.42 1.9 0.54 2.4 0.75 3.0 1.0 3.6 1.6 4.5 2.3 5.9 4.5 6.9

8

4.2

τm Tm1 = 5.0

0.45 3.5 0.48 3.8 0.56 4.4 0.65 5.3 0.85 6.7 1.05 7.7 2.0 11.6

τm Tm1 = 10.0

0.46 6.3 0.48 6.7 0.53 7.4 0.56 8.0 0.63 9.1 0.75 10.6 1.15 16.1

Minimum ITAE – Lopez (1968), pp. 63, 69–74.

x1 K m

Model: Method 1

x 2Tm1


ξm

x2

x1

x2

τm Tm1 = 0.1

2.9 3.0 3.8 2.7 5.5 2.3 7.4 2.0 11.5 1.5 16.5 1.2

x1

x2

x1

x2

x1

x2

x1

x2

x1

x2

x1

-

0.8

τm Tm1 = 0.2

1.35 2.4 1.9 2.4 2.8 2.2 3.8 2.0 6.5 1.7 9.5 1.5 23 1.1

τm Tm1 = 0.5

0.42 1.3 0.65 1.6 1.15 1.9 1.7 2.0 3.0 2.1 4.5 2.1 10.5 2.0

τm Tm1 = 1.0

0.21 0.8 0.34 1.2 0.64 1.8 0.93 2.1 1.7 2.7 2.5 2.9 5.6 3.4

τm Tm1 = 2.0

0.18 0.9 0.26 1.3 0.42 1.9 0.58 2.4 0.98 3.3 1.4 4.0 3.2 5.4

τm Tm1 = 5.0

0.25 2.3 0.29 2.5 0.34 2.9 0.38 3.4 0.52 4.1 0.66 5.6 1.4 8.7

τm Tm1 = 10.0

0.28 4.3 0.30 4.5 0.32 4.9 0.33 5.6 0.38 6.3 0.46 7.4 0.77 -

Minimum ITAE – Lopez et al. (1969). Model: Method 8; coefficients of K c

x1 K m ξm

τm Tm1

x1

x 2Tm1 x2

and Ti deduced from 0.5 0.1 3.0 2.86 0.5 1.0 0.2 0.83 graphs. 0.5 10.0 0.3 4.0

Coefficient values: ξm τ m x1 x 2 Tm1 1 1 1

0.1 7.0 2.00 1.0 0.95 2.22 10.0 0.35 5.00

ξm

4 4 4

τm Tm1

x1

x2

0.1 40.0 0.83 1.0 6.0 3.33 10.0 0.75 10.0

b720_Chapter-03

17-Mar-2009

FA


Rule Minimum ITAE – Chao et al. (1989). Model: Method 1 Minimum ITAE – Hassan (1993). Model: Method 8

4

Kc =

4

Kc

Ti

Kc

Ti

Comment

0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 5

0.5 ≤ ξ m ≤ 2 ,

Ti

Kc

0.09578 + 0.6206ξ m − 0.1069ξ m 2  τm     2ξ T  Km  m m1 

0.1 ≤ τm Tm1 ≤ 4 .

−1.108 + 0.2558ξ m − 0.0579 ξ m 2

,

(

Ti = 2ξ m Tm1 1.31 − 0.0914ξ m + 0.07464ξ m 5

189

2

)

 τm     2ξ T   m m1 

−0.6441+ 0.7626 ξ m − 0.1193ξ m 2

.

Formulae correspond to graphs of Lopez et al. (1969). K c is obtained as follows: log[K m K c ] = −0.0099458 + 1.9743700ξm − 3.8984310

τm − 1.1225270ξ m 2 Tm1

2

τ   τ 2 τ + 2.3493280 m  + 1.9126490ξ m m + 0.0754568ξ m  m  T T T m1  m1   m1 

2

2

τ  2 τ 3 − 0.5594610ξ m  m  − 0.7930846ξ m m + 0.2412770ξ m T T m1  m1  3

3

τ  τ   2 τ − 0.3567954 m  − 0.0024730ξ m  m  + 0.0478593ξ m  m  T T T  m1   m1   m1 

2

2

− 0.1354322ξ m

3

3

  τm 3 τ 3 τ − 1.1756470ξ m  m  − 0.0096974ξ m  m  ; Tm1 T T  m1   m1 

Ti is obtained as follows:

 T  τ log  i  = 0.9321658 − 0.7200651ξ m − 3.4227010 m + 0.0432084ξ m 2 T T m1  m1  2

τ   τ 2 τ + 1.4965440 m  + 5.3636520ξ m m − 0.0848431ξ m  m  T T T m1  m1   m1 

2

2

τ  2 τ 3 − 1.6011510ξ m  m  − 2.1777800ξ m m + 0.0404586ξ m T T m1  m1  3

3

τ  τ   2 τ − 0.1769621 m  + 0.0912008ξ m  m  + 0.1501474ξ m  m  T T T  m1   m1   m1  2

+ 0.3033341ξ m

3

2

3

  τm 3 τ 3 τ + 0.1753407ξ m  m  − 0.0622614ξ m  m  . Tm1 T T  m1   m1 

b720_Chapter-03

Rule Minimum ITSE – Chao et al. (1989). Model: Method 1 Nearly minimum IAE, ISE, ITAE – Hwang (1995).

Kc =

6

Kc

Ti

Kc

Ti

Comment

0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 7

Decay ratio = 0.2; Model: Method 14

Ti

Kc

0.3366 + 0.6355ξ m − 0.1066ξ m 2  τm     2ξ T  Km  m m1 

−1.0167 + 0.1743ξ m − 0.03946 ξ m 2

,

(

Ti = 2ξ m Tm1 1.9255 − 0.3607ξ m + 0.1299ξ m 7

FA


190

6

17-Mar-2009

0.2 ≤ τm Tm1 ≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 .

[

 0.622 1 − 0.435ωH τ m + 0.052(ωH τm )2 K c = 1 −  K H K m (1 + K H K m ) 

Ti =

[

]K  

[

)

 τm     2ξ T   m m1 

−0.5848 + 0.7294 ξ m − 0.1136 ξ m 2

.

,

(

K c (1 + K H K m )

0.0697ωH K m 1 + 0.752ωH τm − 0.145ωH 2 τm 2

 0.724 1 − 0.469ωH τm + 0.0609(ωH τm )2 K c = 1 −  K H K m (1 + K H K m ) 

Ti =

H

2

]K  

H

, K c (1 + K H K m )

(

0.0405ωH K m 1 + 1.93ωH τm − 0.363ωH 2 τm 2

]

) , ε < 2.4 ;

) , 2.4 ≤ ε < 3 ;

 1.26(0.506)ωH τ m 1 − 1.07 ε + 0.616 ε 2  K H , K c = 1 −   K H K m (1 + K H K m )   K c (1 + K H K m ) Ti = , 3 ≤ ε < 20 ; 0.0661ωH K m (1 + 0.824 ln[ωH τm ]) 1 + 1.71 ε − 1.17 ε 2

[

 1.09 1 − 0.497ωH τm + 0.0724(ωH τm )2 K c = 1 −  K H K m (1 + K H K m ) 

Ti =

]K  

(

H

)

,

(

K c (1 + K H K m )

0.054ωH K m − 1 + 2.54ωH τm − 0.457ωH 2 τm 2

) , ε > 20 .

b720_Chapter-03

17-Mar-2009

FA


Rule

Minimum IAE – Gallier and Otto (1968). Model: Method 25. Minimum IAE – Murata and Sagara (1977). Model: Method 4; K c , Ti deduced from graphs. Minimum IAE – Huang et al. (1996). Model: Method 1

8

Kc

Comment

Ti

Minimum performance index: servo tuning x1 K m x 2 (Tm1 + Tm 2 + τ m ) Tm1 = Tm 2

Representative coefficient values (deduced from graphs): τm 2Tm1 x1 x2 τm 2Tm1 x1 x2 0.053 3.6 0.11 2.3 0.25 1.35 x1 K m

1.35 1.17 0.93

0.67 1.50 4.0

0.76 0.72 0.53 0.60 0.43 0.51 Tm1 = Tm 2

x 2Tm1

x1

x2

τm Tm1

x1

x2

τm Tm1

1.4 1.0 0.8

2.3 2.3 2.4

0.2 0.4 0.6

0.6 0.5

2.4 2.5

0.8 1.0

8

0 < Tm 2 Tm1 ≤ 1 ;

Ti

Kc

0.1 ≤ τm Tm1 ≤ 1 .

Note: equations continued into the footnote on p. 192. 1  T τm τ T  Kc = + 2.6647 m 2 + 9.162 m m2 2  − 13.0454 − 9.0916 K m  Tm1 Tm1 Tm1  +

−1.0169 3.5959 3.6843   τm  τ   τm  1  0.3053 m       + − 1 . 1075 2 . 2927 T  T  T  Km    m1   m1   m1   

+

T 1  − 31.0306 m 2 T Km   m1 

+

T T 1   − 0.6418 m 2 + 18.9643 m 2 τm Km  Tm1 

+

τ 1  28.155 m Km  Tm1 

+

τ m Tm 2  τm Tm 2  2 1  4.8259e Tm1 + 2.1137e Tm1 + 8.4511e Tm1  ,  Km   

  

 Tm 2   Tm1

0.8476

  

T − 13.0155 m 2  Tm1  τm     Tm1 

0.801

− 2.0067

  

2.6083

T + 9.6899 m 2  Tm1

−0.2016

τm Tm1

− 39.7340  Tm 2   Tm1

  

Tm 2 Tm1

  

2.9049 

 τm     Tm1 

  

1.3293 

3.956 

  

2  τ  τ T  T τ Ti = Tm1 0.9771 − 0.2492 m + 0.8753 m 2 + 3.4651 m  − 3.8516 m m2 2   Tm1 Tm1 Tm1   Tm1   

  

191

b720_Chapter-03

FA


192 Rule Minimum IAE – Huang et al. (1996). Model: Method 1

 T + Tm1 7.5106 m 2   Tm1 

9

Kc

Ti

Kc

Ti

2

Comment 0.4 ≤ ξ m ≤ 1 ; 0.05 ≤ τm Tm1 ≤ 1 .

3

 τ  T  − 7.4538 m  + 11.6768 m 2 Tm1 T m 1   

 τ + Tm1  − 10.9909 m Tm1  

 Tm 2   Tm1

 T + Tm1  − 18.1011 m 2 Tm1  

 τm  τ    + 6.2208 m  T m 1    Tm1 

2

 τm     Tm1 

2

 Tm 2   Tm1

 T + Tm1  − 2.2691 m 2   Tm1 

  

3

 T + Tm1 − 4.728 m 2 Tm1  

4

 T  − 7.1990 m 2   Tm1

5

 τm  T   + 1.1395 m 2 T m 1    Tm1

3   τ  T + Tm1 1.0885 m   m 2   Tm1   Tm1 

3

2

 Tm 2   Tm1

  

 τm     Tm1 

+

1 Km

6

4

6

  

 Tm 2   Tm1

  

4

 τ  + 4.5398 m T m1 

2

  

 Tm 2   Tm1

3     − 10.95 − 1.8845 τm − 3.4123ξ m + 4.5954ξm τm − 1.7002 τm   T   Tm1 Tm1   m1   

2  τ  τ  1   − 2.1324ξ m  m  − 14.4149ξ m 2  m  − 0.7683ξ m 3  T  T  Km    m1   m1   

0.421 0.1984 1.8033   τm  τ   τm  1  7.5142 m       3 . 7291 5 . 3444 + + T  T  T  Km    m1   m1   m1    1 19.5419 1.0749 1.1006 + − 0.0819ξ m − 3.603ξ m + 7.1163ξ m Km

+

[

  

  

Note: equations continued into the footnote on p. 193. Kc =

3

  

5

 Tm 2   Tm1

  

4

τ    + 1.1496 m   Tm1  

2

 Tm 2   Tm1

3

τ    + 0.6385 m   Tm1  

 τ   + 3.1615 m    Tm1 

  

2

5

τ   τm    − 8.387 m  T m 1  Tm1   

4

 τ  + 21.9893 m T m1 

 τ  T  − 4.7536 m  + 14.5405 m 2 T Tm1 m 1    2

  

3

 T + Tm1 15.8538 m 2   Tm1 

4

2

τ    + 8.2567 m   Tm1  

 T  − 16.1461 m 2   Tm1 3

  τ  T + Tm1 − 16.651 m  m 2   Tm1  Tm1 

9

17-Mar-2009

]

  

5

.  

b720_Chapter-03

17-Mar-2009


+

−0.6753 −0.1642  τ   τm  1  1.9948  τ m  3.206ξ m  m       7 . 8480 11 . 3222 − ξ + ξ m m  T  T  T  Km   m1   m1   m1  

+

τm ξm τm T  1  2.4239e Tm1 + 3.4137e ξm + 1.0251e Tm1 − 0.5593ξ m m1  , Km  τm   

2  τ   τ Ti = Tm1 2.4866 − 23.3234 m + 5.3662ξm + 65.6053 m   Tm1   Tm1    3  τ   τ + Tm1 29.0062ξ m m − 24.1648ξ m 2 − 83.6796 m   Tm1   Tm1   

2    τm  τ    + 43.1477 m ξ m 2 + 51.9749ξ m 3  + Tm1 − 135.9699ξ m   Tm1    Tm1    4 3 2   τm   τm  2  τm          + Tm1 86.0228 + 70.4553ξ m  + 153.4877ξ m  Tm1  Tm1  Tm1         5   τm   τm 3 4     + Tm1 − 125.0112 ξ m − 68.5893ξ m − 62.7517 Tm1 Tm1       4 3 2    τm   τm  2  τm         ξm3  + Tm1 27.6178ξ m  − 152.7422ξ m  + 20.8705       Tm1   Tm1   Tm1    6   τm    τm  4 5       + Tm1 54.0012 ξ m + 58.7376ξ m + 13.1193 Tm1  Tm1        5 4    τm   τm  6      ξm 2  + Tm1 20.2645ξ m  − 23.2064ξ m − 61.6742      Tm1   Tm1    3 2    τm   τm  τ 3      ξ m 4 + 20.4168 m ξ m 5  . + Tm1 136.2439 ξ m − 95.4092   Tm1    Tm1   Tm1   

FA

193

b720_Chapter-03

17-Mar-2009

FA


194 Rule

Kc

Comment

Ti

x1 x 2τm Minimum ISE – Coefficient values: Keviczky and Csáki 0.1 0.2 0.5 1.0 ξ (1973). τ m Tm1 x1 x 2 x1 x 2 x1 x 2 x1 x 2 Model: Method 1; representative K c , Ti 0.1 0.26 1.5 0.55 2 2.2 3.5 6 6 0.2 0.21 1.3 0.4 1.5 1.5 2.8 3 4.5 coefficients estimated 0.5 0.15 1.2 0.25 1.3 0.8 2 2 3.5 from graph; K m = 1 . 1.0 0.12 1.1 0.2 1.2 0.65 1.8 1.2 3.3 2.0 0.1 1.0 0.16 1.2 0.4 1.9 0.9 3.5 5.0 0.1 1.2 0.16 1.4 0.4 3.0 0.65 5.0 10.0 0.15 1.6 0.2 2.0 0.45 5.5 0.6 7.5 Other coefficient values: τm τm x1 x2 ξ x1 x2 T T m1

m1

0.2 0.5 0.5 1.0 1.0 Minimum ITAE – Chao et al. (1989). Model: Method 1 Minimum ITSE – Chao et al. (1989). Model: Method 1

10

Kc =

30 13 30 7 14 10

15 14 30 14 30

5.0 5.0 10.0 5.0 10.0

2.0 2.0 5.0 5.0 10.0 10.0

15 30 17 32 19 34

x2

14 9 9 8 5 7 3 6.5 1.8 7 1.0 8 0.7 10

ξ 5.0 10.0 5.0 10.0 5.0 10.0

Ti

Kc

0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 11

Ti

Kc

0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 .

0.2763 + 0.3532ξ m − 0.06955ξ m Km

2

 τm     2ξ T   m m1 

(

−0.3772 − 0.2414 ξ m + 0.03342 ξ m 2

,

Ti = 2ξ m Tm1 0.9953 + 0.07857ξ m + 0.005317ξ m 11

4 8 1.9 3.4 1.1 2.0

2.0 x1

0.3970 + 0.4376ξ m − 0.08168ξ m Kc = Km

(

2

 τm     2ξ T   m m1 

2

)

 τm     2ξ T   m m1 

−0.3276 + 0.3226 ξ m − 0.05929 ξ m 2

.

−0.5629 − 0.1187 ξ m + 0.01328ξ m 2

,

)

 τm   Ti = 2ξ m Tm1 1.3414 − 0.1487ξ m + 0.07685ξ m 2    2ξ m Tm1 

−0.4287 + 0.1981ξ m − 0.01978 ξ m

2

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc 12

Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 14

12

195

Comment

Ti Ti

Kc

2

For ξ m ≤ 0.776 + 0.0568

τ  τm + 0.18 m  : Tm1  Tm1 

Coefficient values: x3 x4 x5

x1

x2

x6

x7

0.822 x8

-0.549 x9

0.122 x 10

0.0142 x 11

6.96 x 12

-1.77 x 13

0.786 x 14

-0.441 x 15

0.0569 x 16

0.0172 x 17

4.62 x 18

-0.823 x 19

1.28 x 20

0.542 x 21

-0.986 x 22

0.558 x 23

0.0476 x 24

0.996 x 25

2.13 x 26

-1.13

1.14

-0.466

0.0647

0.0609

1.97

-0.323

Decay ratio = 0.1; 0.2 ≤ τm Tm1 ≤ 2.0 ; 0.6 ≤ ξ m ≤ 4.2 .

[

 x 1 + x 2ωH τm + x 3 (ωH τm )2 K c = 1 − 1  K H K m (1 + K H K m ) 

]K  

H

, Ti =

[


]K  

H

(

x 4ωH K m 1 + x 5ωH τm + x 6ωH 2 τm 2

) , ε < 2.4 ;

, Ti =

[

K c (1 + K H K m )

]

K c (1 + K H K m )

(

x10ωH K m 1 + x11ωH τm + x12ωH 2 τm 2

) , 2.4 ≤ ε < 3 ;

 x x ωH τ m 1 + x15 s + x16 ε 2  K H , K c = 1 − 13 14   K H K m (1 + K H K m )  

Ti =

[

K c (1 + K H K m ) , 3 ≤ ε < 20 ; x17 ωH K m (1 + x18 ln[ωH τm ]) 1 + x19 s + x 20 s 2


]K  

(

H

)

,

Ti =

(

K c (1 + K H K m )

x 24ωH K m − 1 + x 25ωH τm + x 26ωH 2 τm 2

) , ε > 20 .

b720_Chapter-03

17-Mar-2009

FA


196 Rule

Kc

Comment

Ti

2

Nearly minimum IAE, ISE, ITAE – Hwang (1995) – continued.

For ξ m > 0.889 + 0.496

τ  τm + 0.26 m  : Tm1  Tm1 


x1

x2

x6

x7

0.794 x8

-0.541 x9

0.126 x 10

0.0078 x 11

8.38 x 12

-1.97 x 13

0.738 x 14

-0.415 x 15

0.0575 x 16

0.0124 x 17

4.05 x 18

-0.63 x 19

1.15 x 20

0.564 x 21

-0.959 x 22

0.773 x 23

0.0335 x 24

0.947 x 25

1.9 x 26

-1.07

1.07

-0.466

0.0667

0.0328

2.21

-0.338 2

τ  τ For ξ m > 0.776 + 0.0568 m + 0.18 m  and Tm1  Tm1  2

τ  τ ξ m ≤ 0.889 + 0.496 m + 0.26 m  : Tm1  Tm1 


x1

x2

x6

x7

0.789 x8

-0.527 x9

0.11 x 10

0.009 x 11

9.7 x 12

-2.4 x 13

0.76 x 14

-0.426 x 15

0.0551 x 16

0.0153 x 17

4.37 x 18

-0.743 x 19

1.22 x 20

0.55 x 21

-0.978 x 22

0.659 x 23

0.0421 x 24

0.969 x 25

2.02 x 26

-1.11

1.11

-0.467

0.0657

0.0477

2.07

-0.333

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Minimum ISTSE – Fukura and Tamura (1983). Model: Method 1

13

τm Tm1 = 2.11

0.462(τm + Tm1 )

undefined

14

τm Tm1 = 3.92 G c = 1 (Tis ) ;

x 2 K m Tm1

Tm1 > Tm 2 .

     1 1 Tm 2  + 0. 2   + , 1.5  Tm1  Tm1    1 + 2 Tm 2   2 + 0.2    τm  τm   

     1 1 Tm 2  + Ti = [Tm1 + 0.36τm ]  1.5  Tm1  Tm1   1 + 2 Tm 2   2 + 0.2    τm  τm    14

τm Tm1 = 1.27

0.420(τm + Tm1 )

1.14 K m

Bryant et al. (1973). Model: Method 1

τm Tm1 = 0.60

0.368(τm + Tm1 )

1.69 K m 0.82 K m

1  Tm1 0.56 τm Km 

0.05 ≤ τm Tm1 ≤ 1 .

0.268(τm + Tm1 )

3.61 K m

Decay ratio ≈ 33% .

Kc =

Tm 2 ≤ Tm1 ;

Ti

Direct synthesis: time domain criteria 13.9 K m 0.140(τm + Tm1 ) τm Tm1 = 0.22

Ream (1954). Model: Method 1; Tm1 = Tm 2 .

13

Comment

Ti

Kc

197

   3.6 Tm 2  1 +  τm T   m 1 + 0.05   T m1  

T  0.3 m1   τm 

0.25

.

Representative x 2 values, summarised as follows, are deduced from graphs: τ m Tm1

x 2 value ξm

0.5

1.0

2.0

3.0

4.0

5.0

6.0

0.4 0.6 0.8 0.9 1.0 1.1 1.2 1.5 1.9

7.94 8.93 9.80 12.2 15.4

5.13 5.99 5.81 9.01 9.90 10.8 13.3 16.7

5.81 6.94 7.30 7.30 11.5 12.3 13.2 15.4 18.9

7.46 8.40 9.17 10.0 13.9 14.5 15.4 18.2 21.3

9.17 10.0 11.9 14.1 16.4 16.9 17.9 20.0 23.8

10.9 11.8 15.4 19.6 18.9 19.6 20.4 22.7 25.6

12.7 14.1 20.0 28.6 -

b720_Chapter-03

17-Mar-2009


198 Rule Bryant et al. (1973) – continued.

Kc

Ti

Comment

x1 K m τ m

Tm1

Tm1 > Tm 2

τ m Tm1

Coefficient values (deduced from graph): x1 τ m Tm1 x1

0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67 15

Kuwata (1987). Model: Method 1

15

FA

0.08 0.09 0.09 0.11 0.14 0.16 0.18 0.23

1.0 2.0 10.0

0.15 0.24 0.33

Critically damped dominant pole.

1.0 2.0 10.0

0.31 0.47 0.64

Damping ratio (dominant pole) = 0.6.

x1 K m

x 2 K m Tm1

Tm1 > Tm 2 Tm1 = Tm 2

16

Kc

Ti

17

Kc

Ti

Representative x 1 values, summarised as follows, are deduced from graphs: τ m Tm1

x 1 value ξm

1.0

2.0

3.0

4.0

5.0

0.4 0.006 0.076 0.103 0.115 0.6 0.039 0.093 0.114 0.122 0.8 0.055 0.110 0.127 0.114 0.093 0.9 0.139 0.147 0.122 0.093 0.070 Representative x 2 values, summarised as follows, are deduced from graphs: τ m Tm1

x 2 value ξm

1.0

0.4 0.6 5.13 0.8 5.99 0.9 5.81 0 . 4 2 T + τ 0. 5 ( ) 16 m1 m Kc = − , Ti K m (2Tm1 + τm − x1 ) K m

2.0

3.0

4.0

5.0

6.0

5.81 6.94 7.30 7.30

7.46 8.40 9.17 10.0

9.17 10.0 11.9 14.1

10.9 11.8 15.4 19.6

12.7 14.1 20.0 28.6

= 1.25x1 - 0.25(2Tm1 + τm ) , with x1 =

17

Kc =

6.0 0.122 0.120 0.075 0.052

(2Tm1 + τm )2 − 0.267τm [6Tm12 + 6Tm1τm + τm 2 ] .

0.4(2ξ m Tm1 + τm ) 0. 5 − , Ti = 1.25x1 - 0.25(2ξm Tm1 + τm ) , with K m (2ξ m Tm1 + τm − x1 ) K m x1 =

(2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Rotach (1995). Model: Method 9

3.35 K m

3.32τm

Pomerleau and Poulin (2004). Model: Method 6 Vítečková (2006). Model: Method 5

Damping factor for oscillations to a disturbance input = 0.75. Tm1 = Tm 2 ; Tm1 1.5Tm1 K m (Tm1 + τ m ) OS < 10%. 18

19

Kc ≥

20

Five criteria are fulfilled.

Ti

Kc

1.571Tm1 . K m (2.718τm + 1.5Tm1 )

2   x1  1  0.7Tm 2 (Tm1 + Tm 2 ) + Tm12   τ  , Ti = (Tm1 + 0.7Tm 2 )1 + 0.1 m  ; Kc = 1+     Km Tm1Tm 2  Tm1 + 0.7Tm 2 Tm1      x 1 values are obtained as follows:

τ m Tm1

x 1 value

20

Tm1 = Tm 2

1.571Tm1

Kc

Direct synthesis: frequency domain criteria Maximise crossover 19 frequency. Ti Kc

Hougen (1979), pp. 345–346. Model: Method 1 Hougen (1988). Model: Method 1

18

Comment

Tm 2 Tm1

0.1

0.2

0.3

0.4

0.5

0.1 0.3 0.5 1.0

0.31 0.6 0.65 0.7

0.19 0.42 0.5 0.6

0.15 0.35 0.45 0.5 τ m Tm1

0.11 0.28 0.46 0.45

0.09 0.23 0.32 0.38

Tm 2 Tm1

0.6

0.7

0.8

0.9

1.0

0.1 0.3 0.5 1.0

0.085 0.2 0.29 0.34

0.075 0.18 0.26 0.32

0.07 0.16 0.24 0.31

0.065 0.155 0.23 0.29

0.06 0.15 0.22 0.28

Equations for K c deduced from graph; 

 1 Kc = 10  Km 

 1 Kc = 10  Km

T 0.73 log10  m 2  τm

   + 0.65    

T 0.48 log10  m 2  τm

   + 0.05    

Ti = Tm1 + 0.7Tm 2 + 0.12τm .



 Tm 2  τm

   + 0.29    

 Tm 2  τm

   − 0.06    

 0.61 log10  T 1  , m 2 = 0.1 ; K c = 10  Tm1 Km 

 0.41 log10  T 1  , m 2 = 0.5 ; K c = 10  Tm1 Km

,

Tm 2 = 0.2 ; Tm1

,

Tm 2 =1. Tm1

199

b720_Chapter-03

17-Mar-2009


200 Rule

Kc 21

Somani et al. (1992). Model: Method 1; suggested A m : 1.75 ≤ A m ≤ 3.0

Ti

Kc

τm Tm1

Alternative.

x 2τm

x1 K m A m

τm Tm 2

Comment

Ti

x1

Coefficient values: τm τm x2 Tm 2 Tm1

0.1 0.147 11.707 12.5 0.1 0.440 10.091 8.33 0.1 0.954 10.315 6.25 0.1 1.953 11.070 5.00 0.1 4.587 12.207 4.17 0.1 13.89 13.337 3.70 0.125 0.121 11.318 12.5 0.125 0.404 8.613 8.33 0.125 0.897 8.510 6.25 0.125 1.842 8.997 5.00 0.125 4.219 9.840 4.17 0.125 11.628 10.707 3.70 0.25 0.064 14.494 11.11

21

FA

1.00 1.00 1.00 1.00 1.67 1.67 1.67 1.67 1.67 2.5 2.5 2.5 2.5

x1

0.082 12.389 0.437 3.462 1.016 2.370 2.075 2.087 0.169 6.871 0.581 2.773 1.242 1.930 2.445 1.624 4.651 1.957 0.156 7.607 0.338 4.015 0.559 2.797 1.182 1.848

x2

6.25 5.00 4.17 3.57 5.00 4.17 3.57 3.13 3.13 4.55 4.17 3.85 3.33

For 25Tm1Tm 2 Ti > 1 , K c ≈ 1 +  m1  x 1 2 KmAm  τm 

2

  

2

 T −1  Tm1 + m2 2.944 − tan  τ τm  m 

T 1 +  m 2  τm

T 1 +  m 2  τm

2

  or  2

2

  T T   2.944 − m1 − m 2  ; τ τm  m  

2

 2  x 1 , with 

 τ τ x 1 = 0.981 + 0.5 m + m  T T m2   m1

 τ τ  + 0.945 + 0.25 m + m T T m2   m1

 τ τ  + 0.945 m + m T T m2   m1

 τ τ + 0.5 m + m  Tm1 Tm 2  

 τ τ  + 0.945 − 0.25 m + m   Tm1 Tm 2

 τ τ  + 0.945 m + m   Tm1 Tm 2

2

2

           

0.333

0.333

.

b720_Chapter-03

17-Mar-2009

FA


Rule Somani et al. (1992) – continued.

Ti

x1 K m A m

x 2τm

τm Tm1

0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 Tan et al. (1996). Model: Method 2

x1

Coefficient values: τm τm x2 Tm 2 Tm1

0.256 6.504 0.667 5.136 1.406 4.961 3.021 5.174 6.452 5.502 0.062 14.918 0.376 4.353 0.909 3.274 1.880 3.025 3.401 3.038

8.33 6.25 5.00 4.17 3.70 8.33 6.25 5.00 4.17 3.70 Ti

Tm1 K m (Tm1 + τ m )

Tm1

−ωTm1 τm

Schaedel (1997). Model: Method 1

22

Kc = Ti =

23

Ti =

(

x1Ti ωφ 1 + x1Tm1ωφ

(

A m 1 + x1Ti ωφ x 1ωφ

tan [− 2 tan

)

)2

2

23

x1

x2

0.075 16.187 4.17 0.238 5.623 3.85 0.433 3.448 3.57 0.667 2.52 3.33 0.951 2.014 3.13 0.073 17.650 3.85 0.235 5.996 3.57 0.424 4.524 3.33 0.649 2.641 3.13 0.917 2.093 2.94 Tm1 = Tm 2 Tm1 > 5Tm 2 ;

minimum φ m = 58 0 . Minimum φ m = 250

Ti

24

Kc

Ti

‘Normal’ design.

25

Kc

Ti

‘Sharp’ design.

, x1 = 0.8,

τm τ < 0.5 ; x1 = 0.5, m > 0.5 ; ωφ < ωu ; Tm1 Tm1

1 −1

5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 10.0

Kc

22

Poulin et al. (1996). Model: Method 1

Comment

Kc

τm Tm 2

201

x1Tm ωφ − x1τ m ωφ − φ m

].

4Tm1 (τm + Tm 2 ) , 0.16Tm1 ≤ Tm 2 + τ m ≤ 0.3Tm1 ; ω = frequency 1.3Tm1 − 4(τm + Tm 2 )

where phase of G m ( jω)G c ( jω) is maximum; G m =

− K m e − sτ m . (1 + Tm1s )(1 + Tm 2s )

(2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) .

24

Kc =

0.5Ti , Ti = K m (2ξ m Tm1 + τm − Ti )

25

Kc =

0.375 4ξm 2Tm12 + 2ξ mTm1τm + 0.5τm 2 − Tm12 , Km Tm12 + 2ξ m Tm1τm + 0.5τm 2 2

Ti =

2

2

2

4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 . 2ξm Tm1 + τm

b720_Chapter-03

Rule Leva et al. (2003). Model: Method 1

27

28

29

30

Kc =

Kc =

Kc =

Kc =

Comment

Kc

Ti

Tm1 2 K m (Tm 2 + τ m )

4(Tm 2 + τ m )

Hägglund and Åström (2004). Model: Method 1; Tm1 = Tm 2 .

Kc =

FA


202

26

17-Mar-2009

26

Kc

Ti

27

Kc

Ti

28

Kc

Ti

29

Kc

Ti

30

Kc

Ti

2.01 − 0.80φ

( ) 

 G jω 1 + (15.45 − 6.30φ) p φ Km  

Tm1 >> Tm 2 + τ m ; symmetrical optimum principle. 1.920 < φ < 2.356 ; M s = 1.2 .

φ = 2.09 radians (‘good’ choice). 2.094 < φ < 2.443 ; M s = 1.4 . φ = 2.269 radians (‘reasonable’ choice). 2.356 < φ < 2.705 ; M s = 2. 0 .

6.283(1.72φ − 2.55)

, Ti =

( ) 

 G jω G p jωφ 1 + (3.44φ − 5.20 ) p φ  Km    0.33 6.61 , Ti = . 2   G p jωφ  G p jωφ  1 + 2.26  G p jωφ  ωφ 1 + 2.01 Km   Km       2.50 − 0.92φ 6.283(1.72φ − 3.05) , Ti =  G p jωφ   G jω 1 + (10.75 − 4.01φ)  G p jωφ 1 + (3.44φ − 6.10 ) p φ Km   Km     0.41 5.36 , Ti = . 2   G p jωφ  G p jωφ  1 + 1.65  G p jωφ  ωφ 1 + 1.71 Km   Km       3.43 − 1.15φ 6.283(0.86φ − 1.50) , Ti =  G p jωφ   G jω 1 + (11.30 − 4.01φ)  G p jωφ 1 + (1.72φ − 2.90) p φ Km   Km    

( )

( )

ωφ

( ) 

2

( ) 

2

. ωφ

 

( )

( )

( )

 

( )

( )

( )

( )

( )

.

2

( )

 

. ωφ

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Hägglund and Åström (2004) – continued. Desbiens (2007), p. 90.

31

Kc

32

Kc

Ti

Comment

Ti

φ = 2.53 radians (‘good’ choice). Tm1 > Tm 2 ; Model: Method 1

Ti

Robust

Tm1 + Tm 2 + 0.5τ m K m τ m (2λ + 1)

Brambilla et al. (1990). Model: Method 1; λ values obtained from graph.

31

32

Kc =

0.1 0.2 0.5 1.0 2ξ mTm1 + 0.5τm K m (2λ + τm )

Kc =

0.52

( ) 

 G jω 1 + 1.15 p φ Km  

Ti Km

 

( )

G p jωφ

, Ti =

3.0 1.8 1.0 0.6

2.0 5.0 10.0

λ 0.4 0.2 0.2

2ξ m Tm1 + 0.5τm

4.25

( ) 

 G jω 1 + 1.45 p φ Km  

(Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2 Ti 2ω2 + 1

τm ≤ 10 Tm1 + Tm 2

Coefficient values: τm λ Tm1 + Tm 2

τm Tm1 + Tm 2


0.1 ≤

Tm1 + Tm 2 + 0.5τ m

 

.

2

ωφ

, with ω given by solving (for

φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω . 2  T  τ T  τ Ti = Tm1 1 + 0.175 m + 0.3 m 2  + 0.2 m 2  , m < 2 ; Tm1 Tm1 Tm1  Tm1    2   Tm 2  τm Tm 2  τ m    + 0.3 + >2. 0 . 2 Ti = Tm1  0.65 + 0.35 ,   Tm1  Tm1 Tm1  Tm1   

203

b720_Chapter-03

17-Mar-2009

FA


204 Rule

Kc

Kristiansson (2003). Model: Method 1

33

Kc

34

Kc

Panda et al. (2004). Model: Method 1

35

Kc

Ti

37

Kc

Ti

Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0.5 .



Comment

Ti 2ξ m Tm1

M max = 1.7 36

Ultimate cycle x 2 Tu

x 1K u x1

x2

τm Tu

x1

x2

τm Tu

0.08 0.09 0.03

0.32 0.14 0.03 x 1K u

0.05 0.10 0.15

0.07 0.13 0.18

0.04 0.08 0.10 x 2 Tu

0.25 0.30 0.35

x1

x2

τm Tu

x1

x2

τm Tu

0.01 0.04 0.08 0.12

0.29 0.19 0.15 0.14 x 1K u

0.05 0.10 0.15 0.20

0.16 0.18 0.21 0.22

0.14 0.15 0.15 0.15 x 2 Tu

0.25 0.30 0.35 0.40

x1

x2

τm Tu

x1

x2

τm Tu

0.21 0.25 0.26

1.13 0.70 0.43

0.15 0.20 0.25

0.25 0.29 0.23 0.21 0.225 0.17

x1

x2

x1

x2

0.225 0.155 0.225 0.155

Kc =

2ξ m Tm1  1  x1 − x12 − 1 + K m τm  M max 2

34

Kc =

2ξ m Tm1  T T T 2 1 + 0.1913 m 2 − 0.346 + 0.3826 m 2 + 0.0366 m 22 K m τm  τm τm τm 

35

Kc =

2 4ξ 2 T + ξ m τ m + 0.5Tm1 8ξm Tm1 + 2ξ m τm + Tm1 , Ti = m m1 . 2λK m 2ξ m

37

x2

0.225 0.155 0.225 0.155

33

36

 T  , x1 = 1 + m 2 τm 

0.30 0.35 0.40

x1

0.21 0.12 0.225 0.14 0.225 0.155

 1 1 − 1 − M max 2 

τm Tu

0.40 0.45 0.50 τm Tu

0.45 0.50

τm Tu

0.45 0.50

 . 

 .  

  T λ ≥ maximum 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) ,1.7 τm  . T m1   Tm 2 0 . 558 T T m 2 m1 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) + 0.828 + 0.812 0.5τ m Tm1 Tm1 + 1.208Tm 2 + , Kc = λK m λK m

Ti = 0.828 + 0.812

Tm 2 0.558Tm 2Tm1 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) + + 0.5τ m . Tm1 Tm1 + 1.208Tm 2

b720_Chapter-03

17-Mar-2009

FA


Rule Thyagarajan and Yu (2003).

38

39


38

39

Ti

Comment

Kc

2Tm1

Tm1 = Tm 2 ; Model: Method 20

Kc

Other tuning 1.95 tan (0.5φ) ωφ

Kc

0.413 G p ( jω2.27 )

3  T   T  2 0.0200 m1  + 0.2714 m1    Tu   Tu  . Kc =  2  Tm1   +1 K m 0.2947   Tu  2.31 Kc = . 2 1 + tan (0.5φ) G p ( jωφ )

[

]

4.182 ω 2.27

Tm1 = Tm 2 ; Tm1 >> τ m .

φ = 2.27 radians (‘reasonable’ choice).

205

b720_Chapter-03

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FA


206

  1 + Td s  3.5.2 Ideal PID controller G c (s) = K c 1 + Ti s   E(s)

R(s)

+

−

  U(s) 1 K c 1 + + Td s  Ti s  

Table 21: PID tuning rules – SOSPD model G m (s) = G m (s ) = Rule Auslander et al. (1975).

1

Kc 1

Tm1 2 s 2 + 2ξ m Tm1s + 1

  Tm 2  T  m1

(Tm1 − Tm 2 )2

− 1+

Tm 2 Tm1 − Tm 2

 2  Tm 2 (Tm1 − Tm 2 )  Tm1  1+

Tm 2 Tm1 − Tm 2

Tm 2

  Tm 2  T  m1

 Tm 2    Tm1 

Tm 1   Tm1 −Tm 2     

Tm 2

 Tm1 −Tm 2  Tm 2  −    Tm1

Tm1  Tm1 −Tm 2

−

Tm1 Tm1 − Tm 2

 Tm1 −Tm 2  Tm 2  −   Tm1 

Tm1   Tm1 −Tm 2     

Tm1  Tm1 −Tm 2

 Tm 2    Tm1 

Model: Method 3

 Tm1 −Tm 2  Tm 2  −    Tm1

1

2Tm1Tm 2  Tm1   − ln Tm1 − Tm 2  Tm 2 

Comment

Td

Process reaction x1 +2τm 0.25Ti

 1.2 T T T  Kc = {τm + m1 m 2 ln m1  K m (Tm1 − Tm 2 )  Tm1 − Tm 2 Tm 2 

x1 =

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e −sτ m Ti

Kc

Y(s)

SOSPD model

Tm 2

−

Tm1 Tm1 − Tm 2

Tm1   Tm1 −Tm 2     

 Tm 2    Tm1 

Tm 2  Tm1 −Tm 2

 Tm 2    Tm1 

Tm 2  Tm1 −Tm 2

2

.

}−1 ,

b720_Chapter-03

17-Mar-2009

FA

Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Alfaro Ruiz (2005a).

2

ξm τm Tm1 = 0.1

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

Tm1 = Tm 2 ; 1.0 ≤ x1 ≤ 1.67 .

Minimum performance index: regulator tuning Representative 5.8 tuning values; 0.36Tu 0.21Tu Km Tm 2 = τ m 5 = 0.1Tm1 . 0.29Tu 0.29Tu Km

Minimum IAE – Wills (1962b) – deduced from graph. Model: Method 1 Minimum IAE – Lopez (1968), pp. 79–90.

207

x1 K m

x 2Tm1

Model: Method 1

x 3Tm1

Coefficients of K c , Ti and Td deduced from graphs. These are representative results. 0.5 0.6 0.8 x1

x2

x3

x1

x2

x3

x1

x2

x3

27

-

0.28

28

-

0.26

30

-

0.26

τm Tm1 = 0.2

9.5

0.61

0.48

10.5

0.59

0.45

11.5

0.57

0.41

τm Tm1 = 0.5

2.25

0.97

0.92

2.6

1.00

0.82

3.2

1.05

0.69

τm Tm1 = 1.0

0.78

1.11

1.40

1.0

1.25

1.20

1.4

1.47

0.94

τm Tm1 = 2.0

0.39

1.27

1.40

0.52

1.56

1.30

0.77

1.92

1.15

τm Tm1 = 5.0

0.42

2.7

1.40

0.45

2.9

1.50

0.53

3.0

1.70

τm Tm1 = 10.0

0.38

4.9

1.15

0.41

5.3

1.40

0.47

5.7

1.90

1.0

ξm τm Tm1 = 0.1

x1

x2

31

-

1.5 x1

x2

0.24 35

x3

-

2.0 x3

4.0

x1

x2

x3

x1

x2

x3

0.22 40

-

-

68

-

-

τm Tm1 = 0.2

13.0 0.59 0.43 16.0 0.56 0.30 19.5 0.56 0.25 47 0.53

τm Tm1 = 0.5

4.0 1.06 0.59 6.0 1.11 0.46 7.8 1.14 0.39 16.5 1.14 0.29

-

τm Tm1 = 1.0

1.85 1.56 0.82 3.0 1.75 0.68 4.3 1.85 0.60 9.0 1.92 0.52

τm Tm1 = 2.0

1.05 2.3 1.10 1.65 2.6 1.05 2.3

2.9 1.00 4.8

τm Tm1 = 5.0

0.62 3.6 1.90 0.85 4.3 2.10 1.1

4.8 2.15 2.15 6.5

τm Tm1 = 10.0

0.52 6.1 2.35 0.60 6.9

3.3 0.98 2.3

2.9 0.70 7.4 3.45 1.2 10.0 4.0

τ   16.57 + 20.83 m   x 2 . 0 T T 2  m1 m1 K c = 1 0.70 +  , Ti =  τm Km  τm  1 11 . 36 +  Tm1 

τ     2.65 + 3.33 m T τ , T =  m1  m d  τm 1 11 . 36 +   Tm1  

  τ .  m  

b720_Chapter-03

17-Mar-2009

FA


208 Rule

Kc

Ti

Td

Comment

Minimum IAE – Shinskey (1988), p. 151. Model: Method 2

0.62 K u

0.38Tu

0.15Tu

Tm 2 = 0.25 Tm 2 + τ m

0.68K u

0.33Tu

0.19Tu

Tm 2 = 0.5 Tm 2 + τ m

0.79 K u

0.26Tu

0.21Tu

Tm 2 = 0.75 Tm 2 + τ m

Ti

Td

Model: Method 1

Minimum IAE – Kang (1989). T 0.05 ≤ L ≤ 7.0. Tm1 Minimum ISE – Lopez (1968), pp. 79–90. ξm

3

Kc

x1 , x 2 …. x14 , x15 values are specified on pp. 127–165 of Kang

(1989), for τm Tm1 = 0.1, 0.2 ….. 0.9, 1.0 with Tm 2 Tm1 = 0.1 , 0.2…..0.9, 1.0. x1 K m

Model: Method 1

x 3Tm1


3

x 2Tm1

x2

x3

x1

x2

x3

x1

x2

x3

τm Tm1 = 0.1

29

0.21

0.34

30

0.21

0.33

31

0.21

0.31

τm Tm1 = 0.2

10.0

0.38

0.56

10.5

0.38

0.54

12.0

0.38

0.48

τm Tm1 = 0.5

2.4

0.67

1.10

2.8

0.69

1.10

3.2

0.74

0.83

τm Tm1 = 1.0

0.86

0.87

1.70

1.1

0.95

1.45

1.55

1.11

1.20

τm Tm1 = 2.0

0.45

1.1

2.00

0.57

1.3

1.80

0.85

1.6

1.60

τm Tm1 = 5.0

0.44

2.4

2.00

0.49

2.6

2.20

0.57

2.9

2.40

τm Tm1 = 10.0

0.45

5.3

1.50

0.48

5.6

1.85

0.53

5.9

2.50

Kc =

Ti =

T   −x2 L    1  T  T   x1 − e Tm1 x 3 cos x 4 L  − x 5 sin  x 4 L  , Km    Tm1   Tm1  

Tm1

T T  x8 L  x 10 m 2   T  Tm1  + x 9e TL sin  x11 L + x12  x 6 + x 7 1 − e  T   m1     1 Load disturbance model = . 1 + sTL

, Td =

T  x 15 L  1  x13 + x14e Tm1  ,  Tm1   

b720_Chapter-03

17-Mar-2009

FA


209

Rule

Kc

Ti

Td

Comment

Minimum ISE – Lopez (1968), pp. 79–90 – continued. ξm

x1 K m

x 2Tm1

x 3Tm1

Model: Method 1

Coefficients of K c , Ti and Td deduced from graphs. These are representative results.

1.0 x1

τm Tm1 = 0.1

1.5

x2

x3

x1

2.0

x2

x3

x1

x2

4.0 x3

x1

x2

x3

32 0.21 0.30 38 0.21 0.25 42 0.21 0.23 70 0.21

-

τm Tm1 = 0.2

13 0.38 0.44 17 0.38 0.36 21 0.38 0.31 39 0.38 0.20

τm Tm1 = 0.5

4.2 0.77 0.74 6.2 0.80 0.57 8.5 0.82 0.50 17.5 0.83 0.37

τm Tm1 = 1.0

2.0 1.18 1.05 3.2 1.33 0.85 4.4 1.39 0.77 9.5 1.49 0.66

τm Tm1 = 2.0

1.1

2.6

1.2

τm Tm1 = 5.0

0.66 3.1

2.6 0.90 3.7

2.8 1.20 4.2 2.85 2.25 5.3

3.0

τm Tm1 = 10.0

0.58 5.9

3.0 0.67 6.3

4.0 0.80 6.7

Minimum ITAE – Lopez (1968), pp. 79–90.

x1 K m

x 2Tm1

x 3Tm1

ξm

1.8

1.5

1.8

2.1 1.35 2.9

2.3 1.30 5.2 -

1.25 8.3

-

Model: Method 1


x2

x3

x1

x2

x3

x1

x2

x3

τm Tm1 = 0.1

24

-

0.25

24

-

0.25

24

0.50

0.225

τm Tm1 = 0.2

8.8

0.71

0.44

9.2

0.74

0.41

10.0

0.74

0.35

τm Tm1 = 0.5

2.1

1.15

0.84

2.5

1.15

0.74

3.2

1.22

0.60

τm Tm1 = 1.0

0.72

1.22

1.20

0.93

1.39

1.00

1.35

1.61

0.82

τm Tm1 = 2.0

0.36

1.30

1.15

0.48

1.59

1.10

0.75

2.04

1.00

τm Tm1 = 5.0

0.40

2.7

1.20

0.44

2.9

1.30

0.52

3.2

1.45

τm Tm1 = 10.0

0.34

4.3

0.98

0.38

5.0

1.30

0.43

-

1.75

1.0

ξm x1 τm Tm1 = 0.1

x2

1.5 x1

x2

26 0.53 0.20 31

x3

-

2.0 x1

x2

0.175 36

x3

-

4.0 x1

x2

x3

0.15 68

x3

-

0.1

τm Tm1 = 0.2

11.3 0.77 0.32 14.5 0.71 0.25 18.2 0.68 0.205 36 0.57 0.14

τm Tm1 = 0.5

3.8 1.25 0.52 5.8 1.28 0.40 7.7 1.21 0.34 16.2 1.21 0.26

τm Tm1 = 1.0

1.8 1.75 0.70 2.9 1.92 0.58 4.0 2.00 0.54 8.7 2.13 0.46

τm Tm1 = 2.0

0.36 2.4 0.96 0.48 2.9 0.92 0.73 3.2 0.88 0.97 3.6 0.84

τm Tm1 = 5.0

0.40 3.6

τm Tm1 = 10.0

0.34 5.9

1.6 0.44 4.5 -

0.38 6.9

1.7 0.53 5.3 1.85 0.58 6.9 -

0.43 7.7

-

0.48

-

2.0 -

b720_Chapter-03

Rule Minimum ITAE – Lopez et al. (1969). Model: Method 23

Bohl and McAvoy (1976b). Minimum ITAE – Hassan (1993).

5

FA


210

4

17-Mar-2009

Comment

Kc

Ti

Td

x1 K m

x 2 Tm1

x 3 Tm1

Representative coefficient values (from graphs): x1 x2 x3 ξm τ m Tm1 25 0.7 0.35 25 1.8 9.0

0.5 1.3 5 0.5 1.7 2

0.25 1.2 1.0 0.2 0.7 0.45

0.5 0.5 0.5 1.0 1.0 4.0

0.1 1 10 0.1 1 1

4

Kc

Ti

Td

Minimum ITAE; Model: Method 1

5

Kc

Ti

Td

Model: Method 8

Tm 2 Tm1 = 0.12,0.3,0.5,0.7,0.9 ; τm Tm1 = 0.1,0.2,0.3,0.4,0.5 .  10 τ m     −1.2096 + 0.1760 ln    Tm1   10.9507  10τm    Kc =    K m  Tm1    

 T   10 τ m     0.1044 + 0.1806 ln  10 m 2  − 0.2071 ln    10Tm 2   Tm1   Tm1      ,   Tm1    

 10 τ m     0.7750 − 0.1026 ln    10τm   Tm1    Ti = 0.2979Tm1     Tm1    

 10 Tm 2   10 τ m     + 0.0081 ln   0.1701+ 0.0092 ln    T   10Tm 2   Tm1   m1     ,   Tm1    

 10 τ m     0.6025 − 0.0624 ln    10τm   Tm1    Td = 0.1075Tm1     Tm1    

 10 Tm 2   10 τ m     + 0.0128 ln   0.4531− 0.0479 ln    T   10Tm 2   Tm1   m1     .   Tm1    

Note: equations continued into the footnote on p. 211. 0.5 ≤ ξ m ≤ 2 ; 0.1 ≤ τm Tm1 ≤ 4 . K c is obtained as follows: log[K m K c ] = 1.9763370 − 0.6436825ξm − 5.1887660

τm + 0.4375558ξ m 2 Tm1

2

τ   τ 2 τ + 2.9005550 m  + 3.1468010ξ m m − 0.1697221ξ m  m  T T T m1  m1   m1 

2

2

τ  τ 3 − 0.8161808ξ m  m  − 1.2048220ξ m 2 m − 0.0810373ξ m T T m1  m1  3

3

τ  τ   2 τ − 0.4444091 m  + 0.0319431ξ m  m  + 0.1054399ξ m  m  T T T  m1   m1   m1 

2

b720_Chapter-03

17-Mar-2009

FA


2

+ 0.1652807ξ m

3

3

  τm 3 τ 3 τ + 0.1175991ξ m  m  − 0.0375245ξ m  m  ; Tm1 T T  m1   m1 

Ti is obtained as follows:

 T  τ log  i  = −0.7865873 + 0.6796885ξm + 2.1891540 m − 0.3471095ξ m 2 T T m1  m1  2

τ   τ 2 τ − 1.9003610 m  − 0.7007801ξ m m + 0.3077857ξ m  m  T T T m1  m1   m1 

2

2

τ  τ 3 + 0.8566974ξ m  m  − 0.2535062ξ m 2 m + 0.0412943ξ m T T m1  m1  3

3

τ  τ   2 τ + 0.3484161 m  − 0.1626562ξ m  m  − 0.0661899ξ m  m  T T T  m1   m1   m1 

2

2

+ 0.2247806ξ m

3

3

  τm 3 τ 3 τ − 0.2470783ξ m  m  + 0.0493011ξ m  m  , Tm1 T T  m1   m1 

Td is obtained as follows:

T  τ log  d  = −0.6726798 − 0.2072477ξ m + 2.6826330 m + 0.0807474ξm 2 T T m1  m1  2

τ   τ 2 τ − 1.7707830 m  − 1.6685140ξ m m + 0.0845958ξ m  m  T T T m1  m1   m1 

2

2

τ  τ 3 + 0.7159307ξ m  m  + 0.5631447ξ m 2 m − 0.0225269ξ m T T m1  m1  3

3

τ  τ   2 τ + 0.2821874 m  − 0.0616288ξ m  m  − 0.0626626ξ m  m  T T T  m1   m1   m1  2

− 0.0372784ξ m 3

2

3

  τm 3 τ 3 τ − 0.0948097ξ m  m  + 0.0272541ξ m  m  . Tm1 T T  m1   m1 

211

b720_Chapter-03

FA


212 Rule Minimum ITAE – Sung et al. (1996).

6

17-Mar-2009

Kc =

1 Km

6

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 11

2.001 0.766   τ  Tm1    − 0.67 + 0.297 Tm1   , m < 0.9 or   ξ 2 . 189 + m τ  τ    Tm1 m  m     

Kc =

2 0.766  τ T   τ 1  ξ m  , m ≥ 0. 9 ; − 0.365 + 0.260 m − 1.4  + 2.189 m1  Km   Tm1  τm    Tm1  

0.520   τ τ  − 0.3 , m < 0.4 or Ti = Tm1 2.212 m    Tm1  Tm1    2

 τ τ Ti = Tm1{−0.975 + 0.910 m − 1.845  + x1} , m ≥ 0.4 with T T m1   m1 ξm  − τ  0.15 + 0.33 m Tm1 x1 = 1 − e   

Td =

ξm  − 1 − e − 0.15 + 0.939(Tm1  

 2      5.25 − 0.88 τm − 2.8   ; T     m1    

Tm1 ; 1.171  0.530   1.121     T T τm )  1.45 + 0.969 m1   − 1.9 + 1.576 m1  τ  τ    m    m     0.05
0.9 . − 0.544 + 0.308 m + 1.408 m  Km  Tm1   Tm1   

Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or Ti = Tm1[1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 .

Td =

(τ  − m 1 − e  

0.05
0.649 + 0.58

τ  τm − 0.005 m  . Tm1  Tm1 

[

2 2  0.738 1 − 0.415ωH 2 τm + 0.0575ωH 2 τm Kc =  KH2 −  K m (1 + K H 2 K m ) 

Ti = 21

[

 

[

K c (1 + K H 2 K m )

(

ωH 2 K m 0.0124 1 + 4.05ωH 2 τm − 0.63ωH 2 2 τm 2

 1.15(0.564 )ωH 2 τ m 1 − 0.959 ε 2 + 0.773 ε 2 2 Kc =  KH2 −  K m (1 + K H 2 K m ) 

Ti = 22

] ,

] ,  

K c (1 + K H 2 K m )

(

ωH 2 K m 0.0335(1 + 0.947 ln[ωH 2 τm ]) 1 + 1.9 ε 2 − 1.07 ε 2 2


Ti =

] ,  

(

K c (1 + K H 2 K m )

ωH 2 K m 0.0328 − 1 + 2.21ωH 2 τm − 0.338ωH 2 2 τm 2

). ). ).

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc Kc

Ti

25

Kc

Ti

26

Kc

Ti

27

Kc

Ti

23

Hwang (1995) – continued.

23

Ti

Comment

Td 24

Decay ratio = 0.1; τ 0.2 ≤ m ≤ 2.0 ; Tm1

0.471K u K m ωu

0. 6 ≤ ξ m ≤ 4. 2 .

[


] ,  

K c (1 + K H 2 K m )

Ti =

(

ωH 2 K m 0.009 1 + 9.7ωH 2 τm − 2.4ωH 2 2 τm 2

2

24

25

ξ ≤ 0.649 + 0.58

2

[


[

] ,  

[

(

K c (1 + K H 2 K m )

ωH 2 K m 0.0153 1 + 4.37ωH 2 τ m − 0.743ωH 2 2 τ m 2

 1.22(0.55)ωH 2 τ m 1 − 0.978 ε 2 + 0.659 ε 2 2 Kc =  KH2 −  K m (1 + K H 2 K m ) 

Ti = 27

).

τ  τ  τm τ − 0.005 m  , ξ > 0.613 + 0.613 m + 0.117 m  . Tm1 T T m 1 m 1    Tm1 

Ti = 26

217

] ,  

K c (1 + K H 2 K m )

(

ωH 2 K m 0.0421(1 + 0.969 ln[ωH 2 τm ]) 1 + 2.02 ε 2 − 1.11 ε 2 2

2 2  1.111 − 0.467ωH 2 τm + 0.0657ωH 2 τm Kc =  KH2 −  K m (1 + K H 2 K m ) 

Ti =

] ,  

(

K c (1 + K H 2 K m )

ωH 2 K m 0.0477 − 1 + 2.07ωH 2 τm − 0.333ωH 2 2 τm 2

). ). ).

b720_Chapter-03

17-Mar-2009

FA


218 Rule

Kc

Ti

Comment

Td

Minimum performance index: other tuning

Minimum

Wilton (1999). Model: Method 1

∫ [e (t) + x ∞

2

0

x1 Km

2 2

]

K m 2 (y( t ) − y ∞ )2 dt ; Tm1 > Tm 2 .

Tm1 + Tm 2

Tm 2 1 + Tm 2 Tm1

Coefficient values (obtained from graphs): Tm 2 Tm1 = 0.25 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 0.75 x1

x2

x1

x2

x1

x2

1250.1 0.0001 22.946 0.04 20.396 0.04 22.946 0.04 4.8990 0.2 4.8990 0.2 4.8990 0.2 2.1213 1 2.1213 1 2.1213 1 180.01 0.0001 230.01 0.0001 300.01 0.0001 8.1584 0.04 9.1782 0.04 10.708 0.04 2.9394 0.2 3.1843 0.2 3.4293 0.2 1.2728 1 1.3435 1 1.4142 1 90.004 0.0001 100.00 0.0001 112.01 0.0001 4.5891 0.04 4.6911 0.04 5.354 0.04 2.0331 0.2 2.0821 0.2 2.1311 0.2 0.8768 1 1.0324 1 1.0748 1 T T 28 i d K

τ m Tm1 = 0.1

τ m Tm1 = 0.5

τ m Tm1 = 1

Keane et al. c (2000). Model: Method 1 Minimum ITAE servo plus ITAE regulator, with minimum M max and with good noise attenuation; Tm1 = Tm 2 . Huang and Jeng (2003).

28

29

Kc =

29

2ξ m Tm1

Kc

(ln K u )[Tu + Tm1 (1 + ln K u )] , Tu K m

Td =

ξ m ≤ 1.1 Model: Method 17

Td

Tu Tm1 , Tu + Tm1 (1 + ln K u )

Ti =

(ln K u )[Tu + Tm1 (1 + ln K u )] . (ln K u )(1 + ln K u ) + [ln K u + ln(1 + eT − K )][ln(K u + 1.33419τm ) − 1]

Kc =

1.2858Tm1ξ m  τm    T  K m τm  m1 

m

u

0.0544

, Td =

[(− 0.0349ξm + 1.0064)Tm1 + (0.4196ξm − 0.1100)τm ]2 2Tm1ξm

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Huang and Jeng (2003) – continued.

30

Kc

Ti

Td

Comment

Ti

Td

ξ m > 1.1

Servo or regulator tuning – minimum IAE. x1 K m

Direct synthesis: time domain criteria x 2 Tu x 3 Tu

Wills (1962a). Representative coefficient values (deduced from graphs): Model: Method 1 x x2 x3 x1 x2 x3 x1 x2 x3 1 Servo response; 2.5 2.9 0.035 1.6 1.4 0.143 0.2 0.48 0.143 ξ =1. 2.75 2.9 0.073 16 1.4 0.28 8 0.48 0.36 Tm1 = Tm 2 ; 3.2 2.9 0.143 10 1.4 0.72 0.09 0.28 0.073 τ m = 0.1Tm1 . 20 2.9 0.285 0.42 0.70 0.073 0.1 0.28 0.143 12 2.9 0.57 0.4 0.70 0.143 0.11 0.28 0.28 1.4 1.4 0.035 10 0.70 0.36 1.5 1.4 0.073 2 0.70 0.70 Tm1 + Tm 2 + 31 Step disturbance. Van der Grinten Td Kc 0.5τm (1963). Stochastic Model: Method 1 32 disturbance. Ti Td Kc

Pemberton (1972b).

30

2(Tm1 + Tm 2 ) 3K m τ m

0.5890  τm    Kc = K m τm  Tm 2 

0.0030

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

Model: Method 1

2    0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm + Tm1  ,   τ m  

2

Ti = 0.0052

Tm 2 + 0.8980Tm 2 + 0.4877 τm + Tm1 , τm

2    0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm    τm Td = Tm1  . 2 T  0.0052 m 2 + 0.8980T + 0.4877 τ + T  m2 m m1   τm   31

Kc =

(T + T )τ + 2Tm1Tm 2 . 1  T +T   0.5 + m1 m 2  , Td = m1 m 2 m   τm + 2(Tm1 + Tm 2 ) Km  τm 

32

Kc =

T + Tm 2 − ωd τ m τ e − ωd τm ; Ti = − ωmτ x1 , x 1 , x1 = 1 − 0.5e− ωd τ m + m1 e Km τm e d m

− ωd τ m   1 − 0.5e − ωd τ m + Tm1Tm 2e (T + T )  (Tm1 + Tm 2 )τm  m1 m 2  . Td = x1

219

b720_Chapter-03

17-Mar-2009

FA


220 Rule

Kc

Ti

Td

Pemberton (1972a), (1972b). Model: Method 1

(Tm1 + Tm 2 ) K mτm

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

Pemberton (1972b). Model: Method 1

2(Tm1 + Tm 2 ) 3K m τ m

Tm1 + Tm 2

λTm1 + Tm 2 K m (1 + λτ m )

Tm1 + Tm 2

Chiu et al. (1973). Model: Method 1

Comment

Tm1 + Tm 2 4

Tm1Tm 2 Tm1 + Tm 2

0.1 ≤

Tm1 ≤ 1.0 , Tm 2

0.2 ≤

τm ≤ 1.0 . Tm 2

0.1 ≤

Tm1 ≤ 1.0 . Tm 2

0.2 ≤

τm ≤ 1.0 . Tm 2

λ variable; suggested values are 0.2, 0.4, 0.6 and 1.0. Appropriate for “small τ m ”.

Mollenkamp et 2ξ m Tm1 2ξ m Tm1 Tm1 al. (1973). K m (TCL + τ m ) 2ξ m Model: Method 1 Smith et al. Tm1 > Tm 2 λTm1 (1975). T T m 1 m 2 ( ) K m 1 + λτ m Model: Method 1 λ = pole of specified FOLPD closed loop response. Suyama (1992). Tm1 + Tm 2 Tm1Tm 2 Model: Method 1 OS = 10% T + T m1 m2 2K m τ m Tm1 + Tm 2 Rotach (1995). Model: Method 9 Wang and Clements (1995).

33

Kc =

1.06τm

Damping factor for oscillations to a disturbance input = 0.75. 2ξ m Tm1 2λξ m Tm1 Tm1 K m (1 + λτ m ) 2ξ m

Gorez and Klàn (2000). Model: Method 1 Vítečková (1999), Vítečková et al. (2000a). Model: Method 1

1.84τm

8.47 K m

33

Kc

Model: Method 22; underdamped response. Non-dominant 2ξ m Tm1 Tm1 time delay. 2ξ m

x 1 (Tm1 + Tm 2 ) Kmτm

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

x1

OS (%)

x1

OS (%)

x1

OS (%)

0.368 0.514 0.581 0.641

0 5 10 15

0.696 0.748 0.801 0.853

20 25 30 35

0.906 0.957 1.008

40 45 50

2ξ m Tm1 . K m (2ξ mTm1 + τm )

Closed loop overshoot (OS) defined. Overdamped process; Tm1 > Tm 2 .

b720_Chapter-03

17-Mar-2009

FA


Rule Vítečková (1999), Vítečková et al. (2000a) – continued.

Skogestad (2004b). Model: Method 1

35

Kc =

x 1 ξ m Tm1 K mτm

2ξ m Tm1

Tm1 2ξ m

OS (%)

0.74Tm1 K mτm 34

Kc

1.0128ξ m Tm1 K mτm 35

2ξdes m Tm1

( 2ξ

Comment

x1

OS (%)

x1

OS (%)

1.392 1.496 1.602 1.706

20 25 30 35

1.812 1.914 2.016

40 45 50

2ξ m Tm1

Underdamped process; 0. 5 < ξ m ≤ 1 .

Recommended TCL = τ m .

Tm1 2ξ m

A m = 3.14 , φ m = 61.4 0 , M max = 1.59

Chen and Seborg (2002).

K m τm

Td

x1

Bi et al. (2000).

Kc =

Ti

0.736 0 1.028 5 1.162 10 1.282 15 2ξ m Tm1 K m (TCL + τ m )

Vítečková et al. (2000b). Model: Method 5 Arvanitis et al. (2000a).

34

Kc

des m

Kc

2Tm1

0.5Tm1

Tm1 = Tm 2

2ξ des m Tm1

0.5Tm1 ξdes m

1.9747 K m τ m

0.5064Tm1 K mτm


Ti

Td

2

Model: Method 1

).

+1

1 [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2 τm , Km (TCL + τm )3

[(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm , Tm1Tm 2 + (Tm1 + Tm 2 + τm )τm 2 3T T T + Tm1Tm 2 τm (3TCL + τm ) − (Tm1 + Tm 2 + τm )TCL 3 Td = CL m1 m 2 [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm Ti =

Tise −sτ m  y = .   3  d desired K c (TCLs + 1)

,

221

b720_Chapter-03

17-Mar-2009

FA


222 Rule

Kc

Chen and Seborg (2002) – continued. Chidambaram (2002), p. 156. Skogestad (2004c), Manum (2005). Model: Method 1

36

Ti

Td

Ti

Td

2ξ m Tm1

0.5Tm1 ξm

4(λ + τm )

Tm 2 Td

Kc 2

1.333ξ m Tm1 K m τm

Tm1 K m (λ + τm ) 37

Kc

4(λ + τm ) + Tm 2

38

Kc

Ti

39

Kc

Ti

Comment

Model: Method 1

0.5Tm1 ξ m

τm x1 (Tm1 + Tm 2 ) Tm1Tm 2 ≤2 Trybus (2005). T + T m 1 m 2 Tm K m τm Tm1 + Tm 2 Model: Method 1 x1 OS x1 OS x1 OS x1 OS x1 OS x1 OS Tm1 > Tm 2 0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25%

36

Kc = Ti = Td =

37

38

39

Kc =

(

)

1 2ξ mTm1τm + Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm , Km (TCL + τm )3

(2ξ

m Tm1τ m

)

+ Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm Tm12 + (2ξ m Tm1 + τm )τm

Tise −sτ m  y ,   = , 3  d desired K c (TCLs + 1)

3TCL 2Tm12 + Tm12 τm (3TCL + τm ) − (2Tm1ξm + τm )TCL 3

(2ξ

m Tm1τ m

)

+ Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm

.

 Tm1 Tm 2 + 4(λ + τm ) Tm 2  , Td = Tm 2 1 + . 2 4K m (λ + τm )  4(λ + τm ) 

 2Tm1ξ m Tm1 [ξ m (Tm1 + 4(λ + τm )ξ m ) + 2Tm1 (1 − ξ m )] K c = max  ,  , ξm < 1 ; 2 ( ) K λ + τ  m  4K m (λ + τ m ) m Ti = max{2Tm1ξ m , ξ m Tm1 + 4(λ + τ m )(2 − ξ m )} .

   Tm1 Tm1 + 4(λ + τm ) ξ m + ξm 2 − 1    2T ξ      m1 m K c = max  ,  , ξm > 1 ; 4K m (λ + τm )2   K m (λ + τm )   Ti = max{2Tm1ξm , Tm1 + 4(λ + τm )} .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

 x1  Tm1  + 0.5  Trybus (2005) – Tm1 + 0.5τm K m  τm  continued. x1 OS x1 OS x1 OS

Boudreau and McMillan (2006).

Td

Comment

0.5Tm1τm Tm1 + 0.5τm

τm >2 Tm

Ti

x1

OS

x1

OS

x1

223

OS

0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25% Tm1 40 Model: Method 1 Tm 2 Ti K m (λ + τm ) λ = τm (maximum load rejection capability); otherwise, λ is a

multiple of Tm . Vítečková (2006). Model: Method 5

≥

Sample results: A m = 2.0 ,

2Tm1

0.5Tm1

1.0472Tm1 K mτm

2Tm1

0.5Tm1

Ti

Td

τ m / Tm < 2ξ m

Tm 2

Tm1 > Tm 2

41

Kc

ω p Tm1

42

Ti

φ m = 45 o . A m = 3. 0 ,

φ m = 60 o .

AmKm

2

Ti = Tm1 ; for an overdamped response, Ti >

Kc =

Tm1 = Tm 2

0.5Tm1

1.5708Tm1 K mτm

Ho et al. (1994). Model: Method 1 Ho et al. (1995a). Model: Method 1

41

2Tm1

Direct synthesis: frequency domain criteria πTm1 2Tm1 0.5Tm1 Tm1 = Tm 2 AmK mτm

Hang et al. (1993a). Model: Method 10; τm > 0.3 Tm1 (Yang and Clarke, 1996).

40

0.736Tm1 K m τm

4Tm1 λ + τm

1  T  1 +  m1  λ + τ m  

2

2  πξm   2ωp Tm1  πξm 2  + π − 2ωp τm  , Ti = Tm12  + π − 2ωp τm  ,  Tm1ωp   πA m  ωpTm1 π   

Td =

42

.

1

Ti = 2ωp −

4ωp 2 τ m π

1 + Tm1

.

πTm1

  πξ 2 m + πTm1 − 2Tm1ωp τm    ωp  

.

b720_Chapter-03

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FA


224 Rule

Kc

Ho et al. (1995a) – continued.

0.7854Tm1 K mτm

Tm1

Tm 2

0.5236Tm1 K mτm

Tm1

Tm 2

Ti

Comment

Td

Sample results: A m = 2.0 ,

φ m = 45 o . A m = 3. 0 ,

φ m = 60 o .

Ho et al. (1997).

43

Kc

Ti

Td

Model: Method 1

Leva et al. (1994).

44

Kc

Ti

0.25Ti

Model: Method 21

43

Kc =

2 πA m K m

 τ  2  πξm + π − 2 m  , Ti = Tm1 (πξm + π − 2τm ) ,  Tm1  π  Td =

2 τ τ πTm1 ; m ≤ 1 , 2ξ m ≥ m ; Tm 2(πξ mTm1 + πTm1 − 2τm ) Tm

  π + π 2 + 8πτ m ξ m  Tm1 φm <  44

Kc =

ωcn Ti Km

(1 + ω T ) + 4ξ ω (1 − T T ω ) + T ω 2

cn

2 2 m1

2

m

i d

cn

2

cn

2 2

2

i

Tm12 2

(

)

,

cn

ωcn =

Ti =

  A m 2 − 1 − 2 πA m (A m − 1)   4A m

 2ξ m τ m Tm1 (0.5π − φ m )  1  4.07 − φ m + tan −1   ; τm   (0.5π − φ m )2 Tm1 2 − τ m 2  

   2ξ ω T  2 tan 0.5ωcn τm + φm − 0.5π − tan −1  m2 cn2 m1  ,  ω T − 1   ωcn    cn m1    3τ m  ξ m + ξ m 2 − 1  ); φ m = 70 0 (at least). ξ m ≤ 1 or ξ m > 1 (with 0.5π − φ m > Tm1  

b720_Chapter-03

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FA


Rule

Kc

Leva et al. (1994) – continued. Wang et al. (1999).

45

ξ m Tm1 K mτm

Wang and Shao (1999a). Model: Method 16

x1

ξ m Tm1 K mτm

2

ω T K c = cn m1 K m Td

Comment

Ti

Td

Ti

Td

2ξ m Tm1

0.5Tm1 ξm 0. 5

2ξ m Tm1

Model: Method 16 46

Tm1 ξm

Some coefficient values: Comment x1

x1

Comment

1.571

A m = 2 , φ m = 45

0

0.785

A m = 4 , φ m = 67.50

1.047

A m = 3 , φ m = 60 0

0.628

A m = 5 , φ m = 72 0

2

45

Kc

ωcn +

1  2ξ m ξ m 2 − 1 − 1 2   Tm1  2

ωcn + z 2 ωcn =

,

 2ξ m τ m Tm1 (0.5π − φ m )  1  4.07 − φ m + tan −1   , τm   (0.5π − φ m )2 Tm1 2 − τ m 2   ωcn z= ,       ωcn Tm1 tan φ m − 0.5π + ωcn τ m + tan −1      ξ m + ξ m 2 − 1       

2 Tm1z +  ξm + ξ m − 1  Tm1  , T = Ti = , ξm > 1 d z ξm + ξ m 2 − 1  Tm1z +  ξ m + ξm 2 − 1      3τ m  3τ m 2  ξ m − ξ m − 1 < 0.5π − φ m ≤ with  Tm1  Tm1 46

ξ m > 0.7071 or 0.05
1, ξ m ≥ 1

ξ + ξ 2 − 1  . m  m 

b720_Chapter-03

Rule

Kc

Wang (2000). Model: Method 1.

ξ m Tm1 K mτm

Wang and Shao (2000b). Chen et al. (1999b). Model: Method 1

0.05
1 , for ξm > 1 ;

ξm τm ξ τ < 0.15 or m m > 1 , for 0.7071 < ξ m < 1 . Tm1 Tm1

 e − τ m x1 0.368  2Tm1ξm , min   , x1 = 2Tm1 (K m ξm − Tm1 ) , ξm > 1 or τm  Km  x1 x1 =

49

Kc =

50

Kc =

51

Kc =

1.508  2ξ m Tm1   , suggested M max = 1.6. 1.451 − K m τm  M max  Tm1Tm 2

8K m (Tm 2 + τm )2

, Tm1 ≥ 4(Tm 2 + τ m ) .

Tm1 (5Tm 2 + 4 τ m ) 8K m (Tm 2 + τ m )2

, Td =

4Tm 2 (Tm 2 + τ m ) , Tm1 ≥ 8(Tm 2 + τ m ) . 5Tm 2 + 4τ m

Tm1 , ξm < 1 . ξm

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Åström and Hägglund (2006), pp. 245–246. Lee et al. (2006). Model: Method 1

52

Kc

Ti K m (λ + τm )

Ti

Td

Comment

Ti

Td

Model: Method 1; M s = M t = 1.4 .

53

227

Td

Ti

Desired closed loop response =

e − sτ m

(λs + 1)2

; recommended λ = 0.5τm .

Robust 54

Brambilla et al. (1990). Model: Method 1

Kc

Td

Representative coefficient values (deduced from graph): τm τm τm λ λ λ Tm1 + Tm 2 Tm1 + Tm 2 Tm1 + Tm 2 0.1 0.2 0.5

52

Ti

0.50 0.47 0.39

1.0 2.0 5.0

0.29 0.22 0.16

10.0

0.14

1  Tm1 T T T  + 0.18 m 2 + 0.02 m1 m 2  , 0.19 + 0.37 Km  τm τm τm  Tm1 Tm 2 Tm1Tm 2 + 0.18 + 0.02 0.19 + 0.37 τm τm τm , Ti = 0.48 Tm1 Tm 2 T T + 0.03 2 − 0.0007 2 + 0.0012 m1 3m 2 τm τm τm τm

Kc =

 T T  T T  0.29τm + 0.16Tm1 + 0.20Tm 2 + 0.28 m1 m 2  m1 m 2  τm  Tm1 + Tm 2 Td =  . T T T T 0.19 + 0.37 m1 + 0.18 m 2 + 0.02 m1 m 2 τm τm τm 2

2

53

Ti = 2ξm Tm1 +

54

Kc =

2

τm τm , Td = + 2(λ + τm ) 2(λ + τm )

Tm1 −

Tm1 + Tm 2 + 0.5τm , Ti = Tm1 + Tm 2 + 0.5τm , K m τm (2λ + 1) Td =

3

τm 6(λ + τm ) . Ti

τm Tm1Tm 2 + 0.5(Tm1 + Tm 2 )τm , 0.1 ≤ ≤ 10 . Tm1 + Tm 2 Tm1 + Tm 2 + 0.5τm

b720_Chapter-03

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FA


228 Rule

Kc

Lee et al. (1998). Model: Method 1 desired closed loop response =

Ti K m (2λ + τm )

e − τ ms

(λs + 1)2

Comment

Td

55

Ti

Td

56

Ti

Td

57

Ti

Td

58

Ti

Td

λ = maximum [0.25τm ,0.2Tm1 ] (Panda et al., 2004). Ti K m (λ + τm )

.

Ti

Huang et al. 2ξ m Tm1 Tm1 (1998), Rivera 2ξ m Tm1 K m (τ m + λ) 2ξ m and Jun (2000). Model: Method 1 λ specified graphically for different OS and TR values (Huang et al., 1998). 59 Marchetti and Model: T Td Kc i Method 1 Scali (2000). Smith (2003). Tm1 Tm1 Tm 2 λ ∈ [ τ m , Tm 2 ] , Model: Method 1 K (λ + τ ) Tm1 ≥ Tm 2 . m m Kristiansson (2003).

60

Ti = 2ξ m Tm1 −

56

Ti = Tm1 + Tm 2 −

58

59

60

2ξ m Tm1

Tm1 2ξm

Model: Method 1

2λ2 − τm 2 T 2 τm 3 , Td = Ti − 2ξ mTm1 + m1 − . 2(2λ + τm ) Ti 6Ti (2λ + τm )

55

57

Kc

2λ2 − τm 2 τm 3 T T , Td = Ti − Tm1 − Tm 2 + m1 m 2 − . 2(2λ + τm ) Ti 6Ti (2λ + τm )

 τm 3  2  − T m 1  τm 2 6(λ + τm )  Ti = 2ξm Tm1 + , Td = Ti − 2ξ m Tm1  . 2(λ + τm ) Ti       3     Tm1Tm 2 − τm  τm 6(λ + τm )  Ti = Tm1 + Tm 2 + , Td = Ti − (Tm1 + Tm 2 ) . 2(λ + τm ) Ti       2ξ T + 0.5τm T (T + ξ m τm ) K c = m m1 , Ti = 2ξ m Tm1 + 0.5τm , Td = m1 m1 . K m (2λ + τm ) 2ξ m Tm1 + 0.5τm 2

Kc =

2ξmTm1  1  1 −  . Recommended values of M max : 1.4,1.7,2.0. K m τm  M max 

b720_Chapter-03

17-Mar-2009

FA


Rule Shamsuzzoha et al. (2006c). Model: Method 1 Shamsuzzoha and Lee (2007a). Model: Method 1

61

Kc =

Ti

Td

Comment

61

Kc

Ti

Td

λ , ξ not specified.

62

Kc

Ti

Td

λ = x1Tm

Ti , K m (4λξ + τm − x1 )

Ti = Tm1 + Tm 2 + x1 −

Td =

Kc

229

4λ2ξ 2 + 2λ2 − x 2 + x1τm − 0.5τm 2 , 4λξ + τm − x1

x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −

3

2

0.167 τm − 0.5x1τm + x 2 τm + 4λ3ξ 4λξ + τm − x1 Ti −

2

τ

4λ2ξ 2 + 2λ2 + x1τm − 0.5τm 2 − x 2 , 4λξ + τm − x1

τ

2

m 2 2  − m 2λξ  − Tm1 2 λ 2  λ − 2λξ + 1 e Tm 2 + Tm 2 2 − Tm12 − + − 1 e T Tm1  m 2 T 2 T  T 2 T  m1 m2  m1   m2  , x1 = Tm 2 − Tm1



x 2 = Tm 2 62

Kc =

Td =

2  

λ2

 Tm12 

2 τm  2λξ  − Tm1 − +1 e − 1 + x1Tm1 .   Tm1  

Ti 6λ2 − x 2 + x1τm − 0.5τm 2 , , Ti = Tm1 + Tm 2 + x1 − K m (4λ + τm − x1 ) 4 λ + τ m − x1 x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −

3

2

0.167 τm − 0.5x1τm + x 2 τm + 4λ3 4λ + τm − x1 Ti 2

6λ2 + x1τm − 0.5τm − x 2 , 4λ + τ m − x 1 τm τm 4 4     λ  − Tm1  λ  − Tm 2 2   e   Tm12 1 − 1 T 1 e − − 1 − − m2       Tm1   Tm 2   ,   x1 = Tm 2 − Tm1 τm 4   λ  − Tm 2  + x1Tm 2 .  e x 2 = Tm 2 2 1 − 1 −   Tm 2   

b720_Chapter-03

FA


230 Rule Shamsuzzoha and Lee (2007a) – continued.

Wills (1962a). Model: Method 1 Quarter decay ratio tuning. Tm1 = Tm 2 ; τ m = 0.1Tm1 .

Bohl and McAvoy (1976b).

63

17-Mar-2009

Kc

Ti

Comment

Td

Coefficients of x1 (deduced from graph): τm Tm

Ms

1.8 0.27 0.29 0.31 0.33

1.9 0.18 0.20 0.22 0.24 0.25 0.27

1.8 1.9 0.2 0.34 0.28 0.3 0.35 0.29 0.4 0.38 0.32 0.5 0.41 0.34 0.6 0.46 0.38 0.7 0.51 0.41 0.8 0.55 0.43 Ultimate cycle x 2 Tu x 3Tu

x1 K m x1

x2

10 8 14 33 18

2.9 1.4 2.9 2.9 1.4 63

τm Tm

Ms

2.0 0.12 0.15 0.17 0.19 0.21 0.22 0.23

2.0 0.25 0.26 0.28 0.30 0.33 0.36 0.38

0.9 1.0 1.25 1.5 2.0 2.5 3.0

Coefficient values (deduced from graph): x3 x1 x2 x3 x1 x2 0.036 0.036 0.068 0.143 0.068

28 6 2 0.6 30 Ti

Kc

τ   2.0302 + 0.9553 ln  m    Tu   2.2689  τm  Kc =    K m  Tu    

1.4 0.67 0.48 0.28 0.67

0.143 0.068 0.068 0.068 0.143 Td

18 0.9 18 18 13

0.48 0.28 0.67 0.48 0.28

x3

0.143 0.143 0.24 0.24 0.28

Model: Method 1

τ   1.3135+ 0.1809 ln (K u K m )+ 0.5219 ln  m   (K u K m )  Tu   ,  

τ   − 0.0517 − 0.0643 ln  m    τm   Tu   Ti = 0.2693Tu      Tu   

τ   0.5914 −0.0832 ln (K u K m )+ 0.0687 ln  m   (K u K m )  Tu   ,  

τ   − 4.2613−1.5855 ln  m    τm   Tu    Td = 0.0068Tu      Tu   

τ   −1.1436 −0.2658 ln (K u K m )−1.1100 ln  m   (K u K m )  Tu   .  

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

x 1K u x 2 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 1 , x 2 deduced Tu from graphs. 0.47 0.23 0.22 0.15 0.29 0.17 ξ m = 0. 5 . 0.45 0.21 0.19 0.18 0.225 0.17 0.35 0.18 0.10 0.30 0.27 0.19 x 1K u x 2 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 1 , x 2 deduced Tu from graphs. 0.45 0.30 0.14 0.20 0.34 0.20 ξm = 0.7 . 0.42 0.25 0.11 0.25 0.31 0.19 0.38 0.22 0.09 0.30 0.28 0.18 x 1K u x 2 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 1 , x 2 deduced Tu from graphs. 0.45 0.47 0.12 0.20 0.37 0.24 ξm = 1 . 0.43 0.37 0.10 0.25 0.33 0.21 0.40 0.30 0.09 0.30 0.30 0.19 x 1K u x 2 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 1 , x 2 deduced Tu from graphs. 0.46 0.80 0.11 0.20 0.38 0.29 ξ m = 1. 5 . 0.44 0.52 0.09 0.25 0.35 0.22 0.42 0.38 0.08 0.30 0.30 0.19 4 + x1 1 64 Landau and Voda Kc x1 ω1350 (1992). 3K u 0.51Tu Model: Method 1 1.9 K u

64

Kc =

4 + x1 x1 . 4 2 2 G p ( jω1350 )

0.64Tu

Comment

Td

x 3 Tu x3

τm Tu

x1

x2

x3

τm Tu

0.08 0.35 0.28 0.19 0.03 0.50 0.01 0.41 0.02 0.47

x 3 Tu x3

τm Tu

x1

x2

x3

τm Tu

0.07 0.35 0.28 0.18 0.03 0.50 0.06 0.40 0.04 0.45

x 3 Tu x3

τm Tu

x1

x2

x3

τm Tu

0.07 0.35 0.28 0.18 0.03 0.50 0.06 0.40 0.04 0.45

x 3 Tu x3

τm Tu

x1

x2

x3

τm Tu

0.07 0.35 0.28 0.18 0.03 0.50 0.06 0.40 0.04 0.45 4 1 1 ≤ x1 ≤ 2 4 + x1 ω1350 0.13Tu 0.16Tu

ωu τ m ≤ 0.16 0.16 < ωu τ m

≤ 0.2

231

b720_Chapter-03

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FA


232


 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s R(s) E(s)

−

+

U(s) 1 Tf s + 1

  1 K c 1 + + Td s  Ti s  

SOSPD model

Table 22: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule

Kc

Lim et al. (1985).

1

Arvanitis et al. (2000a). Model: Method 1 Panda et al. (2004). Model: Method 1

Huang et al. (2005a). Model: Method 18 Huang and Jeng (2005).

1

Kc =

2

Kc =

Tm1 2 s 2 + 2ξ m Tm1s + 1 Ti

(

Td

Comment

Direct synthesis: time domain criteria Model: Method 1 Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm1

Kc

Kc

ξ m Tm1 K mτm

2ξ m Tm1

Tm1 2ξ des m

TCL 2 >

τm 2

Tm1 , 2 Nξ m N not defined. Direct synthesis: frequency domain criteria ξ m Tm1 Tm1 0.025Tm1 Tf = 2ξ m Tm1 K mτm 2ξ m ξm Tm1 2ξ m

Tf =

A m ≈ 3 , φ m ≈ 60 0

1.1ξ m Tm1 K mτm

2ξ des m Tm1 + τm

)

, Tf =

0.5Tm1 ξm

2ξ m Tm1

2 TCL 2 Tm1 + Tm 2 . , Tf = K m (2ξTCL 2 + τ m ) 2ξTCL 2 + τ m

K m 2ξ des m TCL 2

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e − sτ m

2ξ des m Tm1

2

Y(s)

(

2 2TCL 2 2 − τ m 2 2ξ des m TCL 2 + τ m

).

Model: Method 1; Tf = 0.01τ m .

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Rule

Kc

Hang et al. (1993a). Model: Method 10 Rivera and Jun (2000). Model: Method 1; λ not specified. Lee and Edgar (2002).

2Tm1 + τ m 2(λ + τ m )K m

Zhang et al. (2002b). Model: Method 1; 5% overshoot: λ = 0.5τ m .

3

Ti

Td

Robust Tm1 + 0.5τ m Tm1 τ m 2Tm1 + τ m

Model has a repeated pole (Tm1 ) . 2ξ m Tm1 λ

2ξ m Tm1

0.5Tm1 ξ m

2ξ m Tm1

Tm1 2ξ m

Kc

Ti

Td

Tm1 + Tm 2 K m (τ m + 2λ )

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

2ξ m Tm1 K m (2 τ m + λ ) 3

233

Comment λ > 0.25τ m .

Tf =

λτ m . 2(λ + τ m )

Tf = τ m

Tf =

τmλ 2τ m + λ

Model: Method 1 Tf =

λ2 ; 2λ + τ m

H ∞ method. 4

Tm1Tm 2 Tm1 + Tm 2

Kc

Ti K m (2λ + τm )

5

Tm1 + Tm 2 Td

Ti

Tf =

λτ m λ + 2τ m

Maclaurin method.

2

Kc =

1 0.5τm − λ2 + (2ξ m Tm1 + K m (2λ + τm ) 2λ + τ m

Tm 2 −

 τm3 0.5τ m 2 − λ2 +  2ξ m Tm1 + 6(2λ + τ m )  2λ + τ m

 0.5τ m 2 − λ2  ),  2λ + τ m 

2

Ti = 2ξm Tm1 +

0.5τm − λ2 + 2λ + τ m


Tm 2 −

Td = 4

5

Kc =

Tm 2 −


 0.5τ m 2 − λ2  .  2λ + τ m 

Tm1Tm 2 , H 2 method. K m (Tm1 + Tm 2 )(λ + 2τm )

Ti = Tm1 + Tm 2 −

2

2

2λ − τ m , Td = 2(2λ + τm )

τm 3 2λ2 − τm 2 12λ + 6τm − , Tf = 0 . Ti 2(2λ + τm )

Tm1Tm 2 −

 0.5τ m 2 − λ2  ,  2λ + τ m 

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234 Rule

Kc

Ti

Td

Comment

2ξ m Tm1

Tm1 2ξ m

Model: Method 1

Kristiansson (2003).

6 7

Kc

Panda et al. (2004). Model: Method 1

8

Kc

Ti

Td

9

Kc

Ti

Td

6

Kc =

7

Kc =

8

Kc =

Kc

2ξ m Tm1  1  x1 − x12 − 1 + K m τm  M max 2

 T  1  , x1 = 1 + f 1 − 1 − τm  M max 2 

2ξ m Tm1  T T T 2 1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12  ; K m τm  τm τm τm    for suggested M max = 1.7 , x 2 = 10 . 2

Td =

 T  , Tf = m1 . x2 

2

2ξ m Tm1 + 0.25Tm1 + 0.5ξ m τm 4ξ T + 0.5Tm1 + ξ m τm , Ti = m m1 , K m (λ + 2ξ m Tm1 ) 2ξ m

(Tm1 + 2ξm τm )Tm1ξm 2

4ξm Tm1 + ξ m τm + 0.5Tm1

, Tf =

λ (Tm1 + 2ξ m τ m ) , 4ξ m λ + 4ξ m τ m + 2Tm1

  T  λ = maximum 0.25 m1 + τm ,0.4ξ mTm1  .  2ξ m    9

1 Kc = Km

0.558Tm 2Tm1 Tm 2 + 0.172Tm1e − (6.9 Tm 2 / Tm1 ) + + 0.5τm Tm1 + 1.208Tm 2 Tm1 , T λ + 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) Tm1

0.828 + 0.812

Ti = 0.828 + 0.812

0.558Tm 2Tm1 Tm 2 + 0.172Tm1e− (6.9Tm 2 / Tm1 ) + + 0.5τm , Tm1 + 1.208Tm 2 Tm1

    1.116Tm 2Tm1 Tm 2 + τm  + 0.172Tm1e − (6.9Tm 2 / Tm1 )   0.828 + 0.812 T 1 . 208 T + T m1 m2 ,   m1 Td =  Tm 2 1.116Tm 2Tm1 − (6.9 Tm 2 / Tm1 ) + 0.344Tm1e + + τm 1.656 + 1.624 Tm1 Tm1 + 1.208Tm 2 λ[1.116Tm1Tm 2 + τ m (Tm1 + 1.208Tm 2 )] , Tf = 2(λ + τ m )(Tm1 + 1.208Tm 2 ) + 2.232Tm 2 Tm1

  0.279Tm 2Tm1 T λ = maximum  + 0.25τm ,0.1656 + 0.1624 m 2 + 0.0344Tm1e− (6.9Tm 2 / Tm1 )  . T + 1 . 208 T T m2 m1   m1

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Rule Naşcu et al. (2006). Model: Method 19

Kc

Ti

Td

2ξ m Tm1 K m (2TCL 2 + τm )

2ξ m Tm1

0.5Tm1 ξm

Desired servo closed loop transfer function =

235

Comment 2

Tf =

TCL 2 2TCL 2 + τm

e − sτ m

(TCL 2s + 1)2

;

suggested TCL 2 values: 0.4ξ m Tm1 , ξm Tm1 . Ultimate cycle

Huang et al. (2005a).

10

K c = 0.080

10

Kc

Ti

Td

Tf = 0.05Td

A m ≈ 3 , φ m ≈ 60 0 ; Model: Method 18.

^ ^     ^ ^ K u Tu  6.283τm   6.283τm  = sin  , T 0 . 159 K T sin i u u   , ^ ^ K m τm  T   T  u u    

    1 + K^ cos 6.283τm   u   ^ ^   T  Tu  u   . Td = 0.159 ^      Ku   6.283τm   sin    ^  T    u    

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236

  Td s 1 + 3.5.4 Controller with filtered derivative G c (s) = K c 1 + sT  Ti s 1+ d  N  E(s)

R(s)

−

+

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

  U(s)   s 

SOSPD model

Table 23: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule

Kc

Hang et al. (1994). Model: Method 1

AmKm

Y(s)

K m e −sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e − sτ m Tm1 2 s 2 + 2ξ m Tm1s + 1 Ti

ω p Tm1

     

Td

Comment

Direct synthesis: frequency domain criteria 1 N = 20; Tm 2 Ti Tm1 > Tm 2 .

0.7854Tm1 K mτm

Sample result: Tm1 Tm 2

A m = 2.0 ,

φ m = 45 o . Robust

Hang et al. Tm1 (1994). K m (TCL + τ m ) Model: Method 1 2 Zhong and Li Kc (2002).

1

1

Ti = 2ωp −

2

2 4ωp τm

π

1 + Tm1

Tm1

Tm 2

N = 20; Tm1 > Tm 2 .

Ti

Td

Model: Method 1

.

2

Kc =

2 x1 + τm 2ξ m Tm1 (2 x1 + τm ) − x1 , Ti = , K m ( 2 x1 + τ m ) 2 2ξm Tm1 (2 x1 + τm ) − x12  2 2ξ T x 2 ( 2 x 1 + τ m ) − x 1 4  2x + τ Td = Ti Tm1 − m m1 1  , N = 1 2 m Td . (2x1 + τ m ) 2 x1  

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Rule Wang (2003), pp. 50–54. Lavanya et al. (2006a). Model: Method 13; N = 10.

3

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

2ξm Tm1 K m (λ + τm )

2ξ m Tm1

0.5ξm Tm1

3

Ti K m (2λ + τm )

4

λ = max [0.25τm ,0.2Tm1 ]

2

2 ξ T ( 2 x + τ ) − x1 2ξ m Tm1 (2 x1 + τm ) − x1 , Ti = m m1 1 m , 2 x1 + τ m K m ( 2 x1 + τ m ) 2

Td =

4

τm < 1; Tm1

λ not specified. τm >1 Tm1

Td

Ti

2

Kc =

0.1 ≤

237

Ti = 2ξm Tm1 −

2

2

Tm12 (2 x1 + τm )

2ξ m Tm1 (2x1 + τm ) − x12

2λ − τ m , Td = Ti − 2ξ mTm1 + 2(2λ + τm )

Tm12 −

τm 3 6(2λ + τm ) . Ti

−

2x + τ x12 , N = 1 2 m Td . 2x1 + τm x1

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238

   1  1 + Td s    3.5.5 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   E(s)

R(s)

+

−

     1  1 + Td s  U(s)  K c 1 + Td  Ti s   s 1 + N  

SOSPD model

Table 24: PID controller tuning rules – SOSPD model G m (s) = G m (s ) = Rule

Kc

Y(s)

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e −sτ m 2 2

Tm1 s + 2ξ m Tm1s + 1 Ti

Td

Comment

Minimum performance index: regulator tuning 1 0.954 0.780 0.982 Approximately Tm1 > Tm 2 ;  x2   x2    minimum IAE – 0.817  x1      0 . 602 x 0 . 903 x 1 1 x  x  x  N = 8. K  1 Huang and Chao  1 m  2  (1982). Also given by Chao et al. (1989); Model: Method 1. Minimum IAE – Shinskey (1988), p. 149. Model: Method 1; N not specified.

1

0.80Tm1 K mτm

1.5(Tm 2 + τ m )

Tm 2 0.60(Tm 2 + τ m ) T + τ = 0.25 m2 m

0.77Tm1 K mτm

1.2(Tm 2 + τ m )

0.70(Tm 2 + τ m )

0.83Tm1 K mτm

0.75(Tm 2 + τ m )

Tm 2 0.60(Tm 2 + τ m ) T + τ = 0.75 m2 m

 τ 1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − . T 1 . 349 ( T T ) + m1 m2   m2

Tm 2 = 0.5 Tm 2 + τ m

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Kc

Ti

239

Comment

Td

2 Minimum IAE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 3 Minimum IAE – Ti Td Tm 2 τm ≤ 3 Kc Shinskey (1994), 2.5Tm1 K m τm τ m + 0.2Tm 2 τ m + 0.2Tm 2 Tm 2 τm > 3 p. 159. Model: Method 1; N not specified. Minimum IAE – τm Tm1 = 0.2 , Shinskey (1996), 0.59 K u 0.36Tu 0.26Tu Tm 2 Tm1 = 0.2 . p. 121. Model: Method 1; N not specified. 0.85Tm1 K m τm 1.98τ m 0.86τ m Tm 2 Tm1 = 0.1 Minimum IAE – 0.87Tm1 K m τm 2.30τ m 1.65τ m Tm 2 Tm1 = 0.2 Shinskey (1996), 1.00Tm1 K m τm 2.50τ m 2.00τ m Tm 2 Tm1 = 0.5 p. 119. Model: Method 1 1.25T K τ 2.75τ 2.75τ T T = 1. 0 m1

m m

m

m

m2

m1

τm Tm1 = 0.2 ; N not specified.

Minimum ISE – 0.667Tm1 K m τm Model: Method 1 Tm1 Tm 2 Haalman (1965). 0 Tm1 > Tm 2 . M s = 1.9, A m = 2.36, φ m = 50 ; N = ∞ .

2

− 0.01133 + 0.5017ξm − 0.07813ξ m Kc = Km

2

(

 τm     2ξ T   m m1 

Ti = 2ξ m Tm1 1.235 − 0.4126ξ m + 0.09873ξ m

(

Td = 2ξ m Tm1 1.214 − 0.6250ξ m + 0.1358ξm 3

Kc =

2

2

)

−1.890 + 0.7902 ξ m − 0.1554 ξ m 2

)

,

 τm     2ξ T   m m1 

 τm     2ξ T   m m1 

−0.03793 + 0.3975ξ m − 0.05354 ξ m 2

0.5696 − 0.8484 ξ m + 0.0505ξ m 2

100      48 + 57 1 − e   

1.2 Tm1 − τm

   K m τm   Tm1 

2  1 + 0.34 Tm 2 − 0.2  Tm 2       τm  τm   

T  − m1  Ti = τ m 1.5 − e 1.5τ m  

    

,

. ,

1.2 Tm1  T    − − m2    1 − e τm   , Td = 0.56τm 1 − e τ m  + 0.6Tm 2 . 1 0 . 9 +            

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240 Rule Minimum ISE – McAvoy and Johnson (1967).

ξm

Kc

Ti

Td

Comment

x1 K m

x 2τm

x 3τ m

Model: Method 1

Representative coefficient values (deduced from graph): x1 x2 x3 ξm x1 x2 Tm1 Tm1 τm τm

N = 20

1 1 1

0.5 4.0 10.0

0.7 7.6 34.3

0.97 3.33 5.00

0.75 2.03 2.7

N = 10

1 1 1

0.5 4.0 10.0

0.9 8.0 33.5

1.10 4.00 6.25

0.64 1.83 2.43

4 4 7 7 4 4 7 7

0.5 4.0 0.5 4.0 0.5 4.0 0.5 4.0

3.0 22.7 5.4 40.0 3.2 23.9 5.9 42.9

1.16 1.89 1.19 1.64 1.33 2.17 1.39 1.89

x3

0.85 1.28 0.85 1.14 0.78 1.17 0.78 1.04

0.881 4 0.883 0.881 Approximately Tm1 > Tm 2 ;  x2  1.101  x1   x2    minimum ISE –     1.134x1   0.563x1     N = 8. Huang and Chao K m  x 2   x1   x1  (1982). Also given by Chao et al. (1989); Model: Method 1. 5 Minimum ISE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 1.036 6 0.595 1.006 Approximately Tm1 > Tm 2 ;  x2   x2  0.745  x1    minimum ITAE     0 . 771 x 0 . 597 x 1 1     N = 8. Km  x2  – Huang and  x1   x1  Chao (1982). Also given by Chao et al. (1989); Model: Method 1.

4

5

τ  1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − . T 1 . 349 ( T T ) + m1 m2   m2 0.2538 + 0.3875ξ m − 0.04283ξ m 2  τm    Kc =  2ξ T  Km  m m1 

(

Ti = 2ξ m Tm1 1.8283 − 0.8083ξm + 0.1852ξ m

(

Td = 2ξ m Tm1 1.077 − 0.4952ξ m + 0.1062ξ m 6

2

2

)

−1.652 + 0.5416 ξ m − 0.09544 ξ m 2

)

,

 τm     2ξ T   m m1 

 τm     2ξ T   m m1 

0.1363+ 0.2342 ξ m − 0.01445ξ m 2

,

0.4772 + 0.0442 ξ m + 0.01694 ξ m 2

τ  1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − . T 1 . 349 ( T T ) + m1 m2   m2

.

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Rule Bohl and McAvoy (1976b). Minimum ITAE – Chao et al. (1989).

7

Kc

Ti

Td

Comment

7

Kc

Ti

Td = Ti

Minimum ITAE; Model: Method 1

8

Kc

Ti

Td

0.8 ≤ ξ m ≤ 3

0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.

τm Tm1 = 0.1,0.2, 0.3,0.4,0.5; Tm 2 Tm1 = 0.12,0.3 ,0.5,0.7,0.9. N = ∞ .

 τ   −1.1445 + 0.1100 ln  10 m    Tm1   5.5030  τm   Kc =  10  K m  Tm1      τ   0.6212 − 0.0760 ln  10 m    τm   Tm1    Ti = 1.8681Tm1 10     Tm1   

8

241

− 0.1033 + 0.5347ξm − 0.07995ξ m Kc = Km

2

  Tm 2 10  Tm1 

  

  Tm 2 10  Tm1 

  

 τm     2ξ T   m m1 

(

Ti = 2ξ m Tm1 0.9096 − 0.1299ξ m + 0.04179ξm

(

)

2

 τ   T  0.1648+ 0.1709 ln  10 m 2  − 0.2142 ln  10 m    Tm 1    Tm1 

  

 τ   T  0.3144 − 0.0062 ln  10 m 2  + 0.0085 ln  10 m    Tm 1    Tm1 

  

−1.798 + 0.7191ξ m − 0.1416 ξ m 2

)

,

 τm     2ξ T   m m1 

τm  2   Td = 2ξ m Tm1 1.2226 − 0.6456ξ m + 0.1373ξ m   ξ 2  m Tm1 

−0.1277 + 0.5220 ξ m − 0.08629 ξ m 2

,

0.4780 − 0.03653ξ m + 0.04523ξ m 2

; N = 8.

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242 Rule

Kc

Ti

Comment

Td

9 Chao et al. Ti Td 0.3 < ξ m < 0.8 Kc (1989) – N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 . continued. 10 0.884 0.907 0.869 Approximately Tm1 > Tm 2 ; x  x    minimum ITSE – 0.994  x1  0.570x1  2  1.032x1  2  N = 8.    x1  Huang and Chao Km  x2   x1  (1982). Also given by Chao et al. (1989); Model: Method 1. 11 Minimum ITSE Ti Td 0.8 ≤ ξ m ≤ 3 Kc – Chao et al. N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989).

9

1.3292 − 2.8527ξm + 2.0365ξ m 2  τm    Kc =  2ξ T  Km  m m1 

−1.6159 − 0.06616 ξ m + 0.5351ξ m 2

,

(

)

 τm   Ti = 2ξm Tm1 exp(x1 ) + 1.4094 − 4.4581ξ m + 3.4857ξ m 2 ln  2ξ m Tm1 

(

+ 0.1160 − 0.8142ξ m + 0.7402ξ m

2

2

  τ m   , ln   2ξ m Tm1 

)

(

x 1 = 3.3368 − 7.7919ξ m + 4.3556ξ m

(

)

2

);

2

).

 τm   Td = 2ξ m Tm1 exp(x 2 ) + 1.8243 − 5.6458ξ m + 4.6238ξ m 2 ln  2ξ m Tm1  2

τ m  2    , + 0.2866 − 1.4727ξ m + 1.2893ξ m ln   2ξ m Tm1 

(

)

(

x 2 = 2.3054 − 6.4328ξ m + 3.9743ξ m 10

11

τ  1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − . 1.349 + (Tm1 Tm 2 )   Tm 2 0.1034 + 0.4857ξ m − 0.0709ξ m 2  τm    Kc =  2ξ T  Km  m m1 

(

2

(

)

Ti = 2ξ m Tm1 1.5144 − 0.6060ξ m + 0.1414ξm

−1.713 + 0.6238ξ m − 0.1178ξ m 2

)

,

 τm     2ξ T   m m1 

τm  2   Td = 2ξ m Tm1 1.0783 − 0.4886ξ m + 0.1038ξm   ξ 2  m Tm1 

0.1008 + 0.2625ξ m − 0.02203ξ m 2

,

0.4316 + 0.08163ξ m + 0.009173ξ m 2

.

b720_Chapter-03

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FA


Rule Chao et al. (1989) – continued. Approximately minimum IAE – Huang and Chao (1982).

12

Kc 12

Kc

Ti

Td

Comment

Ti

Td

0. 3 < ξ m < 0. 8

243

N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 . Minimum performance index: servo tuning 1.00 1.032 Tm1 > Tm 2 ; 0.673  x1   x2  13     0 . 502 x T 1 N = 8.   x  i Km  x2   1

Also given by Chao et al. (1989); Model: Method 1.

1.8342 − 3.8984ξ m + 2.7315ξm 2  τm    Kc =  2ξ T  Km  m m1 

(

−1.5641+ 0.1628ξ m + 0.09337 ξ m 2

,

)

 τm   Ti = 2ξm Tm1 exp(x1 ) + 0.9475 − 3.6318ξ m + 2.9969ξ m 2 ln  2ξ m Tm1 

(

+ − 0.06927 − 0.3875ξ m + 0.4220ξ m

2

2

  τ m   , ln   2ξ m Tm1 

)

(

2

)

2

).

x 1 = 3.2026 − 7.4812ξ m + 4.3686ξ m ;

(

)

 τm   Td = 2ξm Tm1 exp(x 2 ) + 1.0983 − 1.5794ξ m + 1.0744ξ m 2 ln  2ξ m Tm1 

(

+ 0.01553 − 0.01414ξ m − 0.009083ξ m

(

2

2

  τ m   , ln   2ξ m Tm1 

)

x 2 = 1.7393 − 3.7971ξ m + 1.7608ξ m 13

 τ 1.398 x1 Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − T x   Tm 2 1.349 + m1 0.148 2  + 0.979  T m2   x1 

  .   

b720_Chapter-03

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244 Rule

Kc

Ti

Comment

Td

14 Minimum IAE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 0.964 15 0.971 1.028 Approximately Tm1 > Tm 2 ;  x2  0.787  x1   x2    minimum ISE –     0 . 594 x 1 . 678 x 1 1 x     N = 8. K m  2  Huang and Chao  x1   x1  (1982). Also given by Chao et al. (1989); Model: Method 1. 16 Minimum ISE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989).

14

− 0.1779 + 0.6763ξ m − 0.1264ξ m Kc = Km

2

 τm     2ξ T   m m1 

(

−0.9941+ 0.1669 ξ m − 0.04381ξ m 2

, x

)

 τm  1  , Ti = 2ξ m Tm1 0.2857 + 0.4671ξm − 0.08353ξ m 2    2ξ m Tm1  2

3

x 1 = 0.1665 − 0.3324ξ m + 0.1776ξ m − 0.02897ξ m ;

(

)

 τm   Td = 2ξm Tm1 exp(x 2 ) + − 2.0552 + 2.786ξ m − 0.5687ξ m 2 ln  2ξ m Tm1  2

  τ m   , + − 0.5008 + 0.6232ξ m − 0.1430ξ m 2 ln   2ξ m Tm1 

(

)

(

x 2 = − 0.8776 + 0.4353ξ m − 0.1237ξ m 15

16

2

).

2

);

τ  1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − . T 1 . 349 ( T T ) + m1 m2   m2 0.1446 + 0.4302ξ m − 0.07501ξm 2  τm    Kc =  2ξ T  Km  m m1 

−1.214 + 0.3123ξ m − 0.06889 ξ m 2

;

(

)

 τm   Ti = 2ξm Tm1 exp(x1 ) + − 2.2502 + 3.7015ξ m − 1.7699ξ m 2 + 0.2680ξ m 3 ln  2ξ m Tm1 

(

)

+ − 0.4823 + 0.9479ξm − 0.4913ξ m + 0.07792ξ m [ln (0.5τm ξTm1 )] ,

(

2

)

3

(

x 1 = − 0.1256 + 0.1434ξ m − 0.03277ξ m

Td = 2ξm Tm1 0.9105 − 0.3874ξ m + 0.08662ξ m (0.5τm ξm Tm1 ) , 2

2

x2

2

x 2 = 0.0779 + 0.2855ξm − 0.01985ξm .

b720_Chapter-03

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FA


Rule

Kc

Ti

245

Comment

Td

0.995 1.049 Approximately Tm1 > Tm 2 ;    x2  17 minimum ITAE 0.684  x1    0 . 491 x T N = 8. 1 i  K m  x 2  – Huang and  x1  Chao (1982). Also given by Chao et al. (1989); Model: Method 1.

Minimum ITAE – Chao et al. (1989).

18

Kc

Ti

Td

0.8 ≤ ξ m ≤ 3

19

Kc

Ti

Td

0.3 < ξ m < 0.8

N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1.

17

18

 τ 1.398 x1 Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − T x   Tm 2 1.349 + m1 0.114 2  + 0.986  T m2   x1 

− 0.2641 + 0.7736ξ m − 0.1486ξ m Kc = Km

2

 τm     2ξ T   m m1 

(

  .   

−0.7225 − 0.04344 ξ m − 0.001247 ξ m 2

; x

)

 τm  1  , Ti = 2ξ m Tm1 0.22257 + 0.7452ξ m − 0.1451ξm 2    2ξ m Tm1  2

x 1 = 0.03138 − 0.04430ξ m + 0.01120ξ m ;

(

)

 τm   Td = 2ξm Tm1 exp(x 2 ) + − 1.2256 + 1.8544ξ m − 0.3366ξ m 2 ln  2ξ m Tm1  2

  τ m   , + − 0.2315 + 0.3402ξ m − 0.06757ξ m 2 ln   2ξ m Tm1 

(

)

(

x 2 = 0.05515 − 0.7031ξ m + 0.1433ξ m 19

Kc =

1.1295 − 2.8584ξ m + 2.1176ξ m 2  τm     2ξ T  Km  m m1 

(

Ti = 2ξ m Tm1 8.6286 − 20.70ξm + 13.203ξm

2

).

2

−0.9904 − 2.5311ξ m + 2.8008ξ m 2

)(0.5τ

; ξ m Tm1 ) 1 , x

m

2

(

)

x 1 = 0.7962 − 2.4809ξ m + 1.7611ξ m ;

Td = 2ξm Tm1 exp(x 2 ) + 3.2228 − 12.034ξ m + 9.2547ξ m 2 ln (0.5τm ξm Tm1 )

(

)

+ 0.6366 − 3.0208ξ m + 2.4603ξ m [ln (0.5τm ξ m Tm1 )] , 2

(

2

2

)

x 2 = 3.9285 − 12.874ξ m + 8.6434ξ m .

b720_Chapter-03

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FA


246 Rule

Kc

Ti

Td

Comment

1.00 1.028 Approximately Tm1 > Tm 2 ;  x2  0.718  x1  20 minimum ITSE –     0 . 596 x T 1 N = 8.   x  i Huang and Chao K m  x 2   1 (1982). Also given by Chao et al. (1989); Model: Method 1. 21 Minimum ITSE Ti Td 0.8 ≤ ξ m ≤ 3 Kc – Chao et al. N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989).

20

21

 τ 1.398 x1 Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1  m + 1.039 − T  x2   Tm 2 1.349 + m1 0.063  + 1.061  T m2   x1 

0.0820 + 0.4471ξm − 0.07797ξ m 2  τm    Kc =  2ξ T  Km  m m1 

(

)

  .   

−0.9246 + 0.07955ξ m − 0.02436 ξ m 2

; x

 τm  1  , Ti = 2ξ m Tm1 0.5930 + 0.1457ξ m − 0.02062ξ m 2    2ξ mTm1  2

x 1 = −0.2201 + 0.1576ξ m − 0.03202ξ m ;

(

)

 τm   Td = 2ξm Tm1 exp(x 2 ) + − 1.0018 + 1.5914ξ m − 0.2831ξ m 2 ln  2ξ m Tm1 

(

+ − 0.2291 + 0.3064ξ m − 0.0634ξ m

(

2

2

  τ m   , ln   2ξ m Tm1 

)

x 2 = − 0.6534 − 0.1770ξ m − 0.0240ξ m

2

).

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Chao et al. (1989) – continued.

22

Kc

Td

Comment

Ti

Td

0.3 < ξ m < 0.8

N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 .

Smith et al. (1975). Model: Method 1

Direct synthesis: time domain criteria Tm1 Tm 2 λ = 1 TCL ; λTm1 K m (λτ m + 1) N not specified.

x 1Tm1 K mτm

Smith et al. (1975). Model: Method 1; N = 10.

ξm

ρ

6 1.75 1.75 1.5 1.0

≥ 0.02 0.27 0.07 0.11 0.25

Tm1

ρ=

Tm 2

τm Tm1 + Tm 2

Coefficient values (deduced from graph): ρ ρ x1 ξm x1 ξm

0.7 0.3 0.8Tm1 Tm 2 Hougen (1979), τm pp. 348–349. Model: Method 1 0.84Tm1 0.8 Tm 2 0.2 τm

22

Ti

247

0.51 0.46 0.36 0.33 0.46

3 1.75 1.5 1.5 1.0

0.5Tm1 + Tm 2 23

0.50 0.42 0.44 0.28 0.48

≥ 0.04 0.13 0.33 0.08 0.17 3

(

Ti = 2ξ m Tm1 6.3891 − 15.773ξ m + 10.916ξm

2

≥ 0.06 0.09 0.17 0.50 0.13

x1

0.45 0.39 0.38 0.40 0.49

τ m Tm1Tm 2

N = 10

Td

N = 30

Ti

0.9666 − 2.3853ξm + 2.4096ξm 2  τm    Kc =  2ξ T  Km  m m1 

2 1.75 1.5 1.0 1.0

−1.533 − 0.05442 ξ m +10.916 ξ m 2

; x

)

 τm  1    2ξ T  ,  m m1  2

x 1 = 0.04175 − 1.3860ξ m + 1.4053ξ m ;

(

)

 τm   Td = 2ξm Tm1 exp(x 2 ) + 2.277 − 8.9402ξ m + 7.7176ξ m 2 ln  2ξ m Tm1 

(

+ 0.3465 − 1.9274ξ m + 1.8030ξ m

(

2

2

  τ m   , ln   2ξ m Tm1 

)

x 2 = 3.2620 − 10.789ξ m + 7.6420ξ m 23

Ti = 0.53Tm1 + 1.3Tm 2 , Td = 2.4(τm Tm1Tm 2 )

0.28

.

2

).

b720_Chapter-03

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FA


248 Rule

Kc

Hougen (1988). Model: Method 1 Sklaroff (1992).

Ti

Td

Comment N = 10; τ m < Tm1 .

24

Kc

0.5Tm1 + Tm 2

Td

25

Kc

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

Also given by Åström and Hägglund (1995), pp. 250–251. Model: Method 8. N = 8 – Honeywell UDC6000 controller. 26 N not defined. Ti Td K

Schaedel (1997). c Model: Method 1 Skogestad (2003). 0.5Tm1 K m τm N =∞.



24

 1 Kc = 10  Km

T x 1 − x 2 log10  m 2  Tm1

    

min(Tm1 ,8τ m )

Model: Method 24

Tm 2

. Coefficients of K c deduced from graphs:

τ m Tm 2

0.1

0.3

0.5

1

2

x1

0.9

0.42

0.24

-0.10

-0.39

x2

0.7

0.7

0.68

0.70

0.69

Td = (Tm1 + Tm 2 )10 Td = (Tm1 + Tm 2 )10 25

Kc =

26

Kc =

   τ   0.33 log10  m  − 0.40     Tm1 

   τ   0.31 log10  m  − 0.63     Tm1 

,

Tm 2 = 0.1 ; Tm1

, 0.2 ≤

Tm 2 ≤1. Tm1

3 .   3τ m   K m 1 + Tm1 + Tm 2   0.375 Km

2

2

2

2

4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 , (2ξmTm1 + τm )2 +  1 − 1(2ξmTm1 + τm )Tm1 2(2ξm 2 − 1) − x N  2

2

2

2

x = 4ξ m Tm1 + 2ξ m Tm1 τ m + 0.5τ m − Tm1 ; 2

Ti = Td =

2

2

2

4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 ; 2ξm Tm1 + τm

(2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc 27

Panda et al. (2004).

Kc

Tm1 2K m τ m

Ti

Td

Comment

2ξ m Tm1

Tm1 2ξm

Tm1

Tm 2

N = 8; Model: Method 8 N not defined; Model: Method 1

Tm 2

N =∞; Tm1 > Tm 2 .

Skoczowski (2004). Tm1 28 Kc Model: Method 1 Huang and Jeng T 0.495 m1 + 0.2 (2005). 0.4τ m + 0.9Tm1 τm Model: Method 1 Km Skogestad (2006b). Model: Method 1

27

Kc =

28

Kc ≤

29

Tm 2

N =∞;  4T  min Tm1 , m1  Tm 2 T m1 > Tm 2 . Km   Slow control; ‘acceptable’ disturbance rejection performance. Direct synthesis: frequency domain criteria Tm1 Tm1 Tm 2 K m (Tm1 + τ m ) N = 5; Tm1 ≤ 5Tm 2 ; minimum φ m = 550 .

ωp Tm1 KmAm x 1Tm1 K mτm

Ti

Tm1

Tm1

Tm 2

29

N =∞.

A m = 1.571 x1 ; φ m = 0.5π − x 1 .

Sample result ( A m = 2.0; φ m = 45 ): 0.785Tm1 K m τm

Tm1

2Tm1  1 + 2 x12 − 2 x1 1 + x12  , with x1 ≈ τm   1

2ωp −

Tm1 = Tm 2 ;

0

Tm 2

3 .  1.5τm   K m 1 +   ξm Tm1 

Ti =

Tm 2 τm

1


Yang and Clarke (1996). Model: Method 10 O’Dwyer (2001a). Model: Method 1; N =∞.

N = 100

4ωp 2 τm π

+

1 Tm1

.

ln (1 OS)

π + [ln (1 OS)]2 2

.

249

b720_Chapter-03

17-Mar-2009

FA


250 Rule

Kc

Majhi and Litz (2003).

30

Kc

Ti

Td

Comment

Ti

Tm1

N=∞

Model: Method 10; Tm1 = Tm 2 . Ultimate cycle

Bohl and McAvoy (1976b).

30

Kc =

31

2φm + π(A m − 1) Tm1 , 2 K m τm 2 Am − 1

(

)

Ti = 1+

31


Td = Ti

Ti

Kc

Tm1 τm

Tm1 .   Am Am [ ] ( ) ( ( ) ) 2 φ + π A − 1 1 − 2 φ + π A − 1   m m m m 2 Am2 − 1   π A m − 1

(

τ   1.2264 + 0.4568 ln  m    Tu   0.5458  τm  Kc =    K m  Tu    

)

(

)

τ   1.2190 − 0.0958 ln (K u K m )−0.1007 ln  m   (K u K m )  Tu   ,  

τ   −1.9314 − 0.6221 ln  m    τm   Tu   Ti = 0.0393Tu      Tu   

τ   −0.0703−0.1125 ln (K u K m )−0.2944 ln  m   (K u K m )  Tu   .  

b720_Chapter-03

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FA


251



E(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

  2  b f 0 + b f 1s + b f 2 s  1 + a f 1s + a f 2 s 2 s 


U(s) SOSPD model

Table 25: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Huang et al. (2003). Model: Method 12

Kc

0.8ξm Tm1 K m τm

Y(s)

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e − sτ m 2

Tm1 s 2 + 2ξ m Tm1s + 1 Ti

Comment

Td

Direct synthesis: time domain criteria 2ξ m Tm1 0.5Tm1 ξm

b f 0 = 1 ; b f 1 not defined; b f 2 = 0 ; N = ∞ ; a f 1 = a f 2 = 0 .

Robust Jahanmiri and Fallahi (1997). Model: Method 7

2ξ m Tm1 K m (τ m + λ)

2ξ m Tm1

b f 1 = 0.5τ m ,

Tm1 2ξ m

a f1 =

λτ m . 2( τ m + λ )

N = ∞ , b f 0 = 1 , b f 2 = 0 , a f 2 = 0 ; λ = 0.25τ m + 0.1ξ m Tm1 . Kristiansson (2003). Model: Method 1

1

Kc =

1

Kc

2ξ m Tm1

M max = 1.7

Tm1 2ξ m

2

b f 0 = 1 , b f 1 = b f 2 = 0 , a f 1 = 0.08Tm1 , a f 2 = 0.01Tm1 , N = ∞ .

2ξ m Tm1  T T T 2 1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12 K m τm  τm τm τm 

 .  

   

b720_Chapter-03

17-Mar-2009

FA


252 Rule Shi and Lee (2004).

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

0.5τm K m (λ + τm )

0.5τm

0.167 τm

N=∞

2

Shamsuzzoha and Lee (2006a). Model: Method 1

af1 =

2 τmλ τm λ , af 2 = , b f 0 = 1 , b f 1 = 2ξ m Tm1 , 2(τm + λ ) 12(τm + λ )

2

b f 2 = Tm1 . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 .

2

Kc =

 1 + ω−6dB τ m ω− 6dB 0.5   Tm1 + Tm 2 −  , bf 0 = 1 , b f 1 = ,   2ω−6dB ω− 6 dB  K m (2 + ω− 6dBτm ) 

bf 2 = Ti = Tm1 + Tm 2 −

τm τm 1 , a f1 = , af 2 = ; 4ω −6dB 2ω−6dB 2ω−6dB (1 + ω −6dB τ m )

2 0.5 T T ω − 0.5[(Tm1 + Tm 2 )ω− 6dB − 0.5] , Td = m1 m 2 − 6dB , ω− 6dB ω− 6 dB [(Tm1 + Tm 2 )ω− 6dB − 0.5]

N=

0.5[(Tm1 + Tm 2 )ω−6dB − 0.5]

Tm1Tm 2 ω−6dB 2 − 0.5[(Tm1 + Tm 2 )ω −6dB − 0.5]

.

b720_Chapter-03

17-Mar-2009

FA




    T s β d α + K Td  c  1 s +   N   E(s)

R(s)

−

+

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

+

SOSPD model

Table 26: PID controller tuning rules – SOSPD model G m (s) =

Y(s)

K m e − sτ m or (1 + sTm ) 2

K m e − sτ K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) Tm1 2 s 2 + 2ξ m Tm1s + 1 m

Rule

Kc

Ti

Td

Comment

Minimum performance index: regulator tuning 1.18Tm1 K m τm 2.20τ m 0.72τ m Tm 2 Tm1 = 0.1

Minimum IAE – 1.25Tm1 K m τm Shinskey (1996), 1.67Tm1 K m τm p. 119. Model: Method 1 2.5T K τ m1 m m

253

2.20τ m

1.10τ m

Tm 2 Tm1 = 0.2

2.40τ m

1.65τ m

Tm 2 Tm1 = 0.5

2.15τ m

2.15τ m

Tm 2 Tm1 = 1.0

τm Tm1 = 0.2 ; α = 0 ; β = 1 ; N = ∞ .

b720_Chapter-03

17-Mar-2009


254 Rule

Minimum IAE – Shinskey (1996), p. 121. Model: Method 1 Minimum IE – Shen (1999). Model: Method 1

Td

Comment

Ti

Td

Tm 2 τm ≤ 3

τ m + 0.2Tm 2

τ m + 0.2Tm 2

Tm 2 τm > 3

α = 0 ;β =1; N = ∞ .

0.85K u

0.35Tu

2

Ti

Kc

− 1.5 Tm1 τ m

m m

Km = 1

Td

m1

− Tm1 1.5 τ m

m

m

Tm 2 Tm1 = 0.2 .

α = 0 ;β =1; N = ∞ .

)] (38 + 40[1 − e ( )(K τ T )(1 + 0.34[T τ ( ) ]), T = τ (0.5 + 1.4[1 − e ]) (1 + 0.48[1 − e ( ) ) + 0.6T . T = 0.42τ (1 − e d

τm Tm1 = 0.2 ,

0.17Tu

100

Kc = i

2

Ti

Kc

1 Minimum IAE – Kc Shinskey (1994), 3.33Tm1 K m τm p. 159. Model: Method 1

1

FA

m2

m

] − 0.2[Tm 2

τm ]2

),

− Tm 2 τ m

− 1.2 Tm1 τ m

m2

x1 , x 2 , x 3 and x 4 are given by the following equation: 2

3

τ  τ  τ  ξ τ exp[a + b m  + c m  + d m  + e m m + fξ m + gξ m 2 Tm1 T T T  m1   m1   m1  3

2

3

2 τ  τ  τ  ξm τm 2 2 + i m  ξ m + j m  ξ m + k  m  ξ m ] ; T T T Tm1  m1   m1   m1  K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4

+h

are given below. 1.4 ≤ M s ≤ 2 ; β = 0 ; N = ∞ . a b c d e f g h i j k

x1

x2

x3

x4

1.40 -11.60 14.17 -6.44 8.28 -0.07 0.13 -4.66 -0.99 -3.03 2.89

2.43 -2.50 -4.16 2.85 -0.30 0.1 -1.72 15.51 -7.98 -8.45 4.66

1.32 -0.51 0.70 -1.06 0.02 -0.09 -7.43 1.77 2.64 4.77 -3.74

-1.44 1.27 1.34 -0.37 1 -5.97 -0.78 -10.63 18.85 -10.97 6.44

b720_Chapter-03

17-Mar-2009

FA


Rule Minimum IE – Shen (2000). Model: Method 1

3

3

Kc

Ti

Td

Comment

Kc

Ti

Td

Km = 1

255


3

 τm   τ m   τm   + c    + eξ m + f ξ m 2 exp[a + b   τ + T  + d τ + T  τ + T m1  m1  m1   m  m  m 2

3

 τm   τm   τm  ξ m + h   ξ m + i  ξ m + g τ + T τ + T τ + T m1  m1  m1   m  m  m 2

3

 τm  2  τm   τm  ξ m + k   ξ m 2 + l  ξ m 2 ] ; + j τ + T τ + T τ + T m1  m1  m1   m  m  m K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4

are given below. M s = 2 ; β = 0 ; N = ∞ ; 0 < ξ m < 1 ; 0 ≤ τm Tm1 ≤ 1 . a b c d e f g h i j k l

x1

x2

x3

x4

1.73 -17.51 30.73 -24.69 0.15 -0.12 3.30 33.07 -37.64 2.63 -34.57 36.88

2.28 0.87 -24.00 15.09 -0.01 -0.02 -8.03 45.21 -22.48 3.89 -22.39 8.32

1.43 -1.94 14.82 -21.44 -0.25 0.16 -1.06 -43.39 51.16 -3.25 36.04 -34.95

-1.60 4.30 -12.16 26.54 1.33 -1.08 -17.56 38.13 -56.45 14.80 -39.33 51.18

b720_Chapter-03

FA


256 Rule Minimum IE – Shen (2000). Model: Method 1 Madhuranthakam et al. (2008). Model: Method 1

4

17-Mar-2009

4

5

Kc

Ti

Td

Comment

Kc

Ti

Td

Km = 1

Kc

Ti

Td

Tm1 > Tm 2 ; τm < Tm1 + Tm 2 .


3

 τm   τ m   τm   + c  + d  + eξ m + f ξ m 2 exp[a + b    τ +T  m1   τm + Tm1   τm + Tm1   m 2

3

 τm   τm   τm  ξ m + h   ξ m + i  ξ m + g τ + T τ + T τ + T m1  m1  m1   m  m  m 2

3

 τm  2  τm   τm  ξ m + k   ξ m 2 + l  ξ m 2 ] ; + j τ + T τ + T τ + T m1  m1  m1   m  m  m K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4

are given below. M s = 2 ; β = 0 ; N = ∞ ; 0.4 ≤ ξ m ≤ 1 ; 1 < τm Tm1 ≤ 15 . a b c d e f g h i j k l 5

Kc =

x1

x2

x3

x4

128.9 -560.3 769.2 -338.9 -324.8 194.3 1421 -1981.6 895.0 -848.1 1186.1 -538.1

92.8 -391 521.6 -225.5 -224.5 137.5 963.9 -1310.6 574.4 -590.8 808.0 -356.7

-3.1 46.3 -107.4 63.1 5.2 -2.5 -79.6 176.0 -103.6 36.2 -78.4 45.8

-119.9 539.5 -785.1 371.8 233.9 -115.7 -1068 1560.5 -741.5 529.2 -774.6 369.2

0.6202  Tm1 + Tm 2  1 +   K m  τm 

0.9931

,

2       τm τm τ   ,  + 13.81 Ti = τm 1 + m  4.566 − 14.906 τ +T +T    m1 m2   m  τm + Tm1 + Tm 2   Tm1   

Td = 0.0921(τ m + Tm 2 )

1.4849

τm  T + Tm 2  1 + m1   Tm1  τm 

, N = 10, α = 0 , β = 1 .

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Madhuranthakam et al. (2008). Model: Method 1

6

Ti

Kc

Comment

Td

Minimum performance index: servo tuning Tm1 > Tm 2 ; Ti Td τm < Tm1 + Tm 2 .

Minimum performance index: servo/regulator tuning x1 K m x 2Tm1 x 3Tm1 N=∞

Taguchi et al. (1987). x Model: Method 1 1 11.47 3.39 1.70 1.06 0.75 0.58 0.48 Minimum ITAE – Pecharromán (2000a), Pecharromán and Pagola (2000). Model: Method 15; Km = 1; Tm1 = 1 ; ξm = 1 .

x2

x3

α

β

0.74 1.17 1.35 1.39 1.37 1.34 1.32 x1K u

0.40 0.67 0.89 1.06 1.18 1.25 1.28

0.64 0.57 0.47 0.35 0.21 0.08 -0.05

0.84 0.80 0.75 0.69 0.62 0.55 0.46 x 2 Tu

x1

x2

α

0.147

1.150

0.170

1.013

τm Tm1

x1

x2

x3

α

β

τm Tm1

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.41 0.38 0.35 0.33 0.33 0.35 0.37

1.32 1.34 1.38 1.53 1.75 2.26 2.82 0

1.28 1.25 1.21 1.12 1.06 1.08 1.19

-0.15 -0.24 -0.30 -0.37 -0.37 -0.32 -0.24

0.37 0.27 0.18 -0.04 -0.19 -0.30 -0.18

1.6 1.8 2.0 2.5 3.0 4.0 5.0

Coefficient values: φc x1

x2

α

φc

0.280

0.512

0.281

− 730

− 140 0

0.291

0.503

0.270

− 630

0.1713 1.0059 0.4002 − 139.7 0 0.195 0.880 0.386 − 1330

0.297

0.483

0.260

− 52 0

0.303

0.462

0.246

− 410

0.210

0.720

0.342

− 1250

0.307

0.431

0.229

− 30 0

0.234

0.672

0.345

− 115 0

0.317

0.386

0.171

− 19 0

0.249

0.610

0.323

− 105

0

0.324

0.302

0.004

− 10 0

0.262

0.568

0.308

− 94 0

0.320

0.223

-0.204

− 60

0.274

0.545

0.291

− 84 0

0.411

− 146

0

0.401

1.0409

6

Kc =

0.5723  Tm1 + Tm 2   1 +  τm K m  

, 1.6501

Ti = 0.2476

  τm (τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2  τm Tm1  

Td = 0.0943(τ m + Tm 2 )

,

1.4636

T + Tm 2  τm  1 + m1   Tm1  τm 

, N = 10, α = 0 , β = 1 .

257

b720_Chapter-03

17-Mar-2009

FA


258 Rule

Kc

Ti

Td

Comment

Minimum ITAE – Pecharromán and Pagola (2000).

0.7236K u

0.5247Tu

0.1650Tu

N = 10; Model: Method 15

Minimum ITAE – Pecharromán (2000a). Model: Method 15; Km = 1; Tm1 = 1 ;

β = 1 ; N = 10; ξm = 1 ; 0.1 < τ m < 10 .

Minimum ITAE – Pecharromán (2000b), p. 145. Model: Method 15; Km = 1; Tm1 = 1 ;

β = 1 ; N = 10; ξm = 1 .

α = 0.5840 , β = 1 , φc = −139.650 , K m = 1 ; Tm1 = 1 ; ξ m = 1 . x1K u x1

x 2 Tu

x 3Tu

Coefficient values and other data: α x2 x3

φc

0.803

0.509

0.167

0.585

− 146 0

0.727

0.524

0.165

0.584

− 140 0

0.672

0.532

0.161

0.577

− 134 0

0.669

0.486

0.170

0.550

− 1250

0.600

0.498

0.157

0.543

− 1150

0.578

0.481

0.154

0.528

− 1050

0.557

0.467

0.149

0.504

− 930

0.544

0.466

0.141

0.495

− 84 0

0.537

0.444

0.144

0.484

− 730

0.527

0.450

0.131

0.477

− 630

0.521

0.440

0.129

0.454

− 52 0

0.515

0.429

0.126

0.445

0.509

0.399

0.132

0.433

− 410

0.496

0.374

0.123

0.385

− 19 0

0.480

0.315

0.112

0.286

− 10 0

0.430

0.242

0.084

0.158

− 60

x1K u x1

x 2 Tu

− 30 0

x 3Tu


φc

0.7965

0.5139

0.1638

0.5854

− 146.10

0.7139

0.5275

0.1625

0.5844

− 140.00

0.6675

0.5109

0.1600

0.5771

− 134.00

0.6540

0.4240

0.1704

0.5498

− 125.00

0.5996

0.4736

0.1629

0.5428

− 115.00

0.5801

0.4660

0.1573

0.5276

− 105.00

0.5604

0.4545

0.1512

0.5042

− 93.00

0.5500

0.4520

0.1470

0.4951

− 84.00

b720_Chapter-03

17-Mar-2009

FA


Rule Minimum ITAE – Pecharromán (2000b), p. 145 – continued.


7

Kc =

Ti

Td

x1K u

x 2 Tu

x 3Tu


x1


0.4321

0.1491

0.4844

− 73.00

0.5312

0.4431

0.1361

0.4769

− 63.00

0.5247

0.4356

0.1325

0.4544

− 52.0 0

0.5189

0.4260

0.1281

0.4452

− 41.00

0.5153

0.3971

0.1324

0.4327

− 30.00

0.4987

0.3680

0.1227

0.3855

− 19.00

0.4846

0.3131

0.1146

0.2860

− 10.00

0.4610

0.2471

0.0966

0.1577

− 6.00

Direct synthesis: time domain criteria Ti 0 Ti

7

Kc

8

Kc

9

Kc

Ti

Td

10

Kc

Ti

Td

0.8(2Tm + τm ) 1 − , x1 = K m (2Tm + τm − x1 ) K m

2

+ 4Tm τm + τm 2Tm + τm

m

 

α = 1; β = 1; N =∞.

2

].

(2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] ;

[

 0.833 2Tm12 + 4ξm Tm1τm + τm 2 Ti = 1 −  (2ξmTm1 + τm )2  9

α =1

(2Tm + τm )2 − 0.267τm [6Tm 2 + 6Tm τm + τm 2 ] ;

] 1.667[2T

 0.833 2Tm 2 + 4Tm τm + τm 2 Ti = 1 −  (2Tm + τm )2  0.8(2ξ m Tm1 + τm ) 1 Kc = − , K m (2ξ m Tm1 + τm − x1 ) K m x1 =

φc

0.5451

[

8

Comment

Kc

] 1.667[2T

2 m1

 

+ 4ξ mTm1τm + τm 2ξm Tm1 + τm

2

].

2 2 1.20 x1 1 x x x − (2Tm + τm )x 2 − ; Ti = 1.667 2 − 4.167 23 ; Td = 0.833x 2 1 3 , 2 Km x2 Km x1 x1 x1 − 0.833x 2 2 3

Kc =

2

2

2

2

3

x1 = 2Tm + 4Tm τm + τm , x 2 = 6Tm τm + 6Tm τm + τm . 10

(

3

Kc =

)

2 2 1.20 x1 1 2.778 x1 − 2.5x 2 x 2 = − ; ; x1 = 2Tm12 + 4ξm Tm1τm + τm 2 ; T i K m x 22 K m x14

x1 − (2ξm Tm1 + τm )x 2 2

Td = 0.833x 2

x13 − 0.833x 2 2

, x 2 = 6Tm12 τm + 6ξ m Tm1τm 2 + τm 3 .

259

b720_Chapter-03

17-Mar-2009

FA


260 Rule Taguchi and Araki (2000). Model: Method 1

Kc

Ti

Td

Comment

11

Kc

Ti

0

ξm = 1

12

Kc

Ti

0

ξ m = 0.5

τ m Tm ≤ 1.0 ; OS ≤ 20% .

Taguchi and Araki (2000). Model: Method 1

14

Kc

Ti

Td

ξm = 1

15

Kc

Ti

Td

ξ m = 0.5

τ m Tm1 ≤ 1.0 ; OS ≤ 20% ; N not specified.

11

Kc =

 1  0.5316 ,  0.3717 +  [τm Tm ] + 0.0003414  Km 

(

)

Ti = Tm 2.069 − 0.3692[τm Tm ] + 1.081[τm Tm ] − 0.5524[τm Tm ] , 2

3

α = 0.6438 − 0.5056(τm Tm ) + 0.3087(τm Tm ) − 0.1201(τm Tm ) . 2

12

Kc =

3

 0.05627 1   0.1000 + , 2   Km  [(τm Tm ) + 0.06041] 

(

)

Ti = Tm 4.340 − 16.39[τm Tm ] + 30.04[τm Tm ] − 25.85[τm Tm ] + 8.567[τm Tm ] , 2

3

4

α = 0.6178 − 0.4439(τm Tm ) − 7.575(τm Tm ) + 9.317(τm Tm ) − 3.182(τm Tm ) . 2

14

Kc =

3

4

 0.6978 1  1.389 + , K m  [(τm Tm1 ) + 0.02295]2 

(

)

Ti = Tm1 0.02453 + 4.104[τm Tm1 ] − 3.434[τm Tm1 ] + 1.231[τm Tm1 ] , 2

3

4  τ  τ  τ   τ Td = Tm1  0.03459 + 1.852 m − 2.741 m  + 2.359  m  − 0.7962  m   , Tm1  Tm1   Tm1   Tm1    2

3

α = 0.6726 − 0.1285(τm Tm1 ) − 0.1371(τm Tm1 ) + 0.07345(τm Tm1 ) , 2

3

β = 0.8665 − 0.2679(τm Tm1 ) + 0.02724(τm Tm1 ) . 2

15

Kc =

 0.5013 1   0.3363 + , K m  [(τm Tm1 ) + 0.01147]2 

( (0.03392 + 2.023[τ

) ] ),

Ti = Tm1 − 0.02337 + 4.858[τm Tm1 ] − 5.522[τm Tm1 ] + 2.054[τm Tm1 ] , Td = Tm1

2

Tm1 ] − 1.161[τm Tm1 ] + 0.2826[τm Tm1 2

m

3

3

α = 0.6678 − 0.05413(τm Tm1 ) − 0.5680(τm Tm1 ) + 0.1699(τm Tm1 ) , 2

β = 0.8646 − 0.1205(τm Tm1 ) − 0.1212(τm Tm1 ) . 2

3

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Arvanitis et al. (2000a). Model: Method 1 Huang et al. (2000).

16

ξ τm

Kc = λ α=

17

Kc

Comment

Direct synthesis: frequency domain criteria 0 < x1 < 1 ; Tm1 Ti suggested   T x1  2ξm + m1  x1 = 0.5 . τm   α = 1, β = 1, N = ∞ . Model: Method 1 Ti Td

(

where 17

Kc

Td

x1Tm1  T  (x1Tm1 τm )(2ξm + (Tm1 τm ))  2ξm + m1  , Ti = ,  K m τm  τm  1  Tm1 (2ξ m τm + Tm1 ) 4ξ 2 (1 − x1 ) + x1   − x2 1 + τm  τm 2 

Kc =

x2 =

16

Ti

8(1 − x1 )

2

Tm1 Km

)

  Tm1 (2ξm τm + Tm1 ) 2ξ2 (1 − x1 ) + x1 Tm1 (2ξm τm + Tm1 ) + τm  , τm τm  

(2ξ m τ m + Tm1 ) + 2

(

)

2  1  2  Tm1  2   (2ξ m τ m + Tm1 )2  ≥ 0 . x 2 1 x + ξ ( − ) 1 1 2 K   τm   m  

2 2Tm1ξ m + 0.4τm T + 0.8Tm1ξ m τm , Ti = 2Tm1ξ m + 0.4τm , Td = m1 , K m τm 0.8Tm1ξm τm

2Tm1ξ m 0.5Tm1 2 −1, β = − 1 , N = min[20,20Td ] ; 2 2Tm1ξ m + 0.4τ m Tm1 + 0.8Tm1ξ m τ m

λ = 0.6 , τmξ m Tm1 ≤ 1 , A m = 2.83 ; λ = 0.7 , 1 < τmξ m Tm1 ≤ 2 , A m = 2.43 ; λ = 0.8 , τmξ m Tm1 > 2 , A m = 2.13 .

261

b720_Chapter-03

FA


262 Rule

Kc 18


Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 5 . Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 7 .

Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 1. 0 .

Kc =

Td =

Td

Comment

Ti

Td

λ = x1Tm

τm Tm

Ms

1.9 0.18 0.20 0.22 0.24 0.25 0.27

2.0 0.12 0.15 0.17 0.19 0.21 0.22 0.23

1.8 0.2 0.34 0.3 0.35 0.4 0.38 0.5 0.41 0.6 0.46 0.7 0.51 0.8 0.55 Ultimate cycle 0 x 2 Tu

x 1K u x1

0.05 0.11

x2

0.01 0.03 0.07 0.11

x2

1.9 0.28 0.29 0.32 0.34 0.38 0.41 0.43

2.0 0.25 0.26 0.28 0.30 0.33 0.36 0.38

0.9 1.0 1.25 1.5 2.0 2.5 3.0

α =1

τm Tu

x1

x2

τm Tu

x1

0.25 0.30

0.16 0.18 x 2 Tu

0.23 0.18

0.35 0.40 0

0.20 0.20

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.05 0.10 0.15 0.20

0.14 0.17 0.19 0.20 x 2 Tu

0.14 0.14 0.14 0.14

0.25 0.30 0.35 0.40 0

0.20 0.20

0.14 0.14

0.45 0.50

0.83 0.39 x 1K u

x1

τm Tm

Ms

0.27 0.17 0.14 0.14 x 1K u

x2

τm Tu

0.15 0.45 0.14 0.50 α =1

α =1

x1

x2

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.04 0.11 0.17 0.20

0.95 0.53 0.38 0.30

0.05 0.10 0.15 0.20

0.21 0.21 0.21 0.20

0.23 0.20 0.17 0.15

0.25 0.30 0.35 0.40

0.20 0.20

0.14 0.14

0.45 0.50

2 Ti 6λ2 − x 3 + x1τm − 0.5τm , Ti = Tm1 + Tm 2 + x1 − x 2 , x 2 = , K m (4λ + τm − x1 ) 4 λ + τ m − x1

x 3 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 −

[

Tm1 (1 − (λ Tm1 )) e − τ m 2

x1 =

Kc

Ti

N = ∞ , α = 0.6 , β = 0 . Coefficients of x1 (deduced from graph): 1.8 0.27 0.29 0.31 0.33

18

17-Mar-2009

[

4

Tm1

x 3 = Tm 2 (1 − (λ Tm 2 )) e − (τ m 2

4

]

3

2

0.167 τm − 0.5x1τm + x 3τm + 4λ3 4λ + τm − x1 − x2 , Ti

[

− 1 − Tm 2 (1 − (λ Tm 2 )) e − τ m Tm 2 − Tm1

Tm 2 )

2

]

− 1 + x1Tm 2 .

4

Tm 2

],

−1

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Td

Comment

Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 1. 5 .

x 1K u

x 2 Tu

0

α =1

Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 5 ;

x1

0.15 0.23 0.27 x1

x2

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.10 0.15 0.20

0.27 0.24 0.22 x 2 Tu

0.38 0.27 0.20

0.25 0.30 0.35

0.20 0.20 0.20

0.17 0.15 0.14

0.40 0.45 0.50

0.80 0.60 0.47 x 1K u x2

x3

0.42 0.24 0.19 α = 1; β = 1; 0.27 0.15 0.17 N =∞. Kuwata (1987); x 1K u x 1 , x 2 deduced x1 x 2 x 3 from graphs. ξm = 0.7 ; 0.54 0.32 0.14 α = 1; β = 1; 0.37 0.25 0.12 N =∞. 0.29 0.19 0.08 Kuwata (1987); x 1K u x 1 , x 2 deduced x1 x 2 x 3 from graphs. ξm = 1 ; α = 1 ; 0.61 0.42 0.12 β =1; N = ∞ . 0.47 0.34 0.10 0.38 0.27 0.08 Kuwata (1987); x 1K u x 1 , x 2 deduced x1 x 2 x 3 from graphs. ξ m = 1. 5 ; α = 1 ; 0.67 0.48 0.11 β =1; N = ∞ . 0.54 0.43 0.09 0.43 0.34 0.08 Oubrahim and 19 Leonard (1998). 0. 6 K u Model: Method 1

19

α = 1 − 1.3 α = 1−

τm Tu

x1

x 3 Tu x2

x3

τm Tu

x1

x2

x3

τm Tu

x1

x2

x3

τm Tu

0

0.50

x3

τm Tu

0

0.50

x3

τm Tu

0.18 0.19 0.11 0.07 0.30 0.23 0.17 0.10 0 0.36 x 2 Tu

τm Tu

x1

x 3 Tu x2

x3

τm Tu

0.18 0.22 0.16 0.03 0.36 0.20 0.14 0.23 0.20 0.14 0.01 0.42 0.30 0.20 0.14 0 0.47 x 2 Tu x 3 Tu τm Tu

x1

x2

x3

τm Tu

x1

x2

0.18 0.30 0.20 0.06 0.36 0.20 0.14 0.23 0.21 0.17 0.025 0.42 0.30 0.20 0.14 0.01 0.47 x 2 Tu x 3 Tu τm Tu

x1

x2

x3

τm Tu

x1

x2

0.18 0.33 0.25 0.07 0.36 0.20 0.14 0 0.50 0.23 0.25 0.18 0.03 0.42 0.30 0.20 0.15 0.02 0.47 Tm1 = Tm 2 ; 0.5Tu 0.125Tu 0.1 < τm Tm1 < 3

 16 − K m K u 16 − K m K u , β = 1 − 1.69 17 + K m K u  17 + K m K u

2

  , 10% overshoot.  

38 1.717 , 20% overshoot. , β =1− 29 + 3.5K m K u (1 + 0.121K m K u )2

263

b720_Chapter-03

17-Mar-2009

FA


264


R(s)

+

F1 (s)

U(s)

Y(s)

Model

−

F2 (s) Table 27: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Huang et al. (1996).

1

Kc 1

Kc

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )

K m e − sτ m 2 2

Tm1 s + 2ξ m Tm1s + 1 Ti

Td

Comment

Minimum performance index: regulator tuning 0 Model: Method 1 Ti

Minimum IAE. Equations continued into the footnote on pp. 265 and 266; 0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τ m Tm1 ≤ 1 . Kc =

1 Km

 τm τ T  T + 76.6760 m 2 − 3.3397 m m2 2  0.1098 − 8.6290 Tm1 Tm1 Tm1  

1 + Km

−0.8058 0.6642 2.1482    τm   τm  τ   1.1863 m      + 20.3519 + 23.1098 T  Tm1  Tm1    m1      

1 + Km

 T  − 52.0778 m 2 T   m1 

  

0.8405

T − 12.1033 m 2  Tm1

  

2.1123 

  

b720_Chapter-03

17-Mar-2009

FA


+

τ 1  9.4709 m Km  Tm1 

+

T 1   − 19.4944 m 2 Km  Tm1 

 Tm 2   Tm1

  

0.5306

Tm 2 Tm1

+ 13.6581

 τm     Tm1 

−0.4500

− 28.2766

 τm     Tm1  τm Tm1

−1.0781 

  

 Tm 2   Tm1

  

1.1427 

  

τ m Tm 2 τm Tm 2   2 T 1  Tm1 Tm 1 + − 19.1463e + 8.8420e + 7.4298e Tm1 − 11.4753 m 2  ; Km  τm    2  τ  T τ T  τ Ti = Tm1  − 0.0145 + 2.0555 m + 0.7435 m 2 − 4.4805 m  + 1.2069 m m2 2   Tm1 Tm1 Tm1   Tm1   

 T + Tm1 0.2584 m 2   Tm1 

 τ  T  + 7.7916 m  − 6.0330 m 2 Tm1   Tm1 

 τ + Tm1 3.9585 m T  m1 

 Tm 2   Tm1

 T + Tm1 7.0004 m 2 Tm1  

T  τm   − 2.7755 m 2  T  Tm1  m1 

 T + Tm1 3.1663 m 2   Tm1 

T τ    + 2.4311 m  − 0.9439 m 2 T  Tm1  m1  

2

3

2

T   − 3.0626 m 2  Tm1  3

 T + Tm1  − 2.4506 m 2 Tm1  

4

 τm     Tm1 

3

τ    − 7.0263 m   Tm1     

2

4

T  τm   − 0.2227 m 2  T  Tm1  m1  3

 τ  − 0.5494 m T m1 

  

4

  

2

 τ  T  τm   − 1.5769 m  m 2  T  Tm1  Tm1  m1 

5

2   τ  T + Tm1 1.9228 m   m 2   Tm1   Tm1 

2

  

2

 τm     Tm1 

 Tm 2   Tm1

  

  

5

  

3

  

4

;  

2  τ  τ T  τ T x1 = Tm1 − 0.0206 + 0.9385 m + 0.7759 m 2 − 2.3820 m  + 2.9230 m m2 2   Tm1 Tm1 Tm1   Tm1   

 T + Tm1 − 3.2336 m 2   Tm1 

 τ + Tm1 2.7095 m T  m1 

2

3

 τ  T  + 7.2774 m  − 9.9017 m 2 Tm1 T   m1 

 Tm 2   Tm1

2

T   + 6.1539 m 2  Tm1 

3

 τm     Tm1 

τ    − 11.1018 m   Tm1  

2

  

4

  

  

3

  

265

b720_Chapter-03

17-Mar-2009

FA


266

 T + Tm1 10.6303 m 2 Tm1  

3

τ   τm   + 5.7105 m   T  Tm1   m1 

2

 τ + Tm1  − 7.9490 m Tm1  

 Tm 2   Tm1

 T + Tm1  − 4.4897 m 2 Tm1  

T  τm   − 7.6469 m 2  T  Tm1  m1 

4

2   τ m   Tm 2     + Tm1 2.2155 Tm1   Tm1   

 T + Tm1 0.519 m 2 Tm1  

3

T   − 6.6597 m 2  Tm1 

4

τ    + 8.0849 m   Tm1     

2

 τm     Tm1 

3

5

2

2

τ    − 2.274 m   Tm1  

 τ  T   + 5.0694 m  m 2  Tm1  Tm1 

T  τm   − 1.1295 m 2  T  Tm1  m1 

4   τ m   Tm 2     + Tm1 2.2875 Tm1   Tm1   

 Tm 2   Tm1

5

  

  

4

T   + 4.1225 m 2  Tm1 

6

3

  

3

τ    − 1.6307 m   Tm1  

τ    + 0.9524 m   Tm1  

6

 Tm 2   Tm1

3

2

 Tm 2   Tm1

  

  

5

  

4

  

 τ  − 0.9321 m T m1 

 Tm 2   Tm1

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .

  

5

;   = 0;

b720_Chapter-03

17-Mar-2009

FA


Rule Huang et al. (1996).

2

2

Kc

Ti

Td

Comment

Kc

Ti

0

Model: Method 1

267

Minimum IAE. Equations continued into the footnote on p. 268. 0.4 ≤ ξ m ≤ 1 , 0.05 ≤ τ m Tm1 ≤ 1 . Kc =

 τ  1  τ − 35.7307 + 14.19 m + 1.4023ξ m + 6.8618ξ m  m   K m  Tm1  Tm1  

+

3 τ  τ 1   − 0.9773 m  + 55.5898ξ m  m T  T Km   m1   m 

+

−1.6624  τ  τ 1   53.8651 m ξ m 2 + 11.4911ξ m 3 + 0.8778 m  T  Km  Tm1   m1   

+

−0.6951 −0.4762  τ   τm  1  − 29.8822 m    + 53 . 535 − 16.9807ξ m 1.1197      Km    Tm1   Tm1   

+

−2.1208  τ  1    − 25.4293ξ m 1.4622 − 0.1671ξ m 58981 + 0.0034ξ m  m  T  Km    m1   

1 + Km +

1 Km

  

0.086

τ  − 3.3093ξ m  m   Tm1 

2

  

τm   τ τ  − 25.0355 m ξ m 3.0836 − 54.9617 m ξ m 1.2103 − 0.1398e Tm1  Tm1 Tm1     τ m ξm  ξ T   − 8.2721e ξm + 6.3542e Tm1 + 1.0479 m m1  ; τm    

2  τ  τ  τ Ti = Tm1 0.2563 + 11.8737 m − 1.6547ξ m − 16.1913 m  − 9.7061ξ m m   Tm1 Tm1   Tm1    3 2  τ  τ   τ  + Tm1 3.5927ξ m 2 + 19.5201 m  − 14.5581ξ m  m  + 2.939ξ m 2  m    Tm1   Tm1   Tm1  

4 3   τm    τm  3       + 50.5163ξ m  + Tm1 − 0.4592ξ m − 34.6273 Tm1  Tm1        2   τm 2  τm  3 4     + ξ − ξ 8 . 6966 6 . 9436 + Tm1 8.9259ξ m  m m Tm1  Tm1     

b720_Chapter-03

268

17-Mar-2009


5 4 3  τ   τ  τ  + Tm1 27.2386 m  − 20.0697ξ m  m  − 42.2833ξ m 2  m     Tm1    Tm1   Tm1   2   τ  τ  + Tm1 8.5019 m  ξ m 3 − 12.2957 m ξ m 4 + 8.0694ξ m 5 − 2.7691ξ m 6     Tm1   Tm1    6 5   τ  τ  τ + Tm1  − 7.7887 m  + 2.3012ξ m  m  + 4.6594 m ξ m 5  Tm1    Tm1   Tm1    4 3 2   τ  τ  τ  + Tm1 8.8984 m  ξ m 2 + 10.2494 m  ξ m 3 − 5.4906 m  ξ m 4  ;    Tm1   Tm1   Tm1    2  τ  τ  τ x1 = Tm1 − 0.021 + 3.3385 m + 0.185ξ m − 0.5164 m  − 0.9643ξ m m   Tm1 Tm1   Tm1    3 2  τ  τ   τ  + Tm1  − 0.8815ξ m 2 + 0.584 m  − 12.513ξ m  m  + 1.3468ξ m 2  m    Tm1   Tm1   Tm1  

4 3    τm   τm  3 4       − ξ 15 . 3014 3 . 1988 + ξ + Tm1 2.3181ξ m − 5.2368 m m T  Tm1    m1      2 5   τm   τm 2  τm  3       − 2.0411 ξ m + 3.4675 + Tm1 11.9607ξ m  Tm1  Tm1 Tm1        4 3 2    τm   τm  2  τm  3         + Tm1 − 0.8219ξ m  − ξ − ξ 15 . 0718 1 . 8859 m m T  T  Tm1    m1  m1       6 5   τm    τm   τm  4 5         + 0.529ξ m  ξ m + 2.2821ξ m − 0.9315 + Tm1 0.4841 Tm1  Tm1  Tm1         4 3    τm   τm  6 2 3       ξ 7 . 1176 ξ + + Tm1 − 0.6772ξ m − 1.4212 m m T  Tm1    m1      2    τm  τm 4 5  ;   ξ + ξ 0 . 5497 + Tm1 − 2.3636 m m Tm1  Tm1     

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .

FA

b720_Chapter-03

17-Mar-2009

FA


Rule Madhuranthakam et al. (2008). Model: Method 1

3

Kc

Ti

Td

Kc

Ti

0

269

Comment Tm1 > Tm 2 ; τm < Tm1 + Tm 2 .

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 .


4

Minimum performance index: servo tuning Tm1 > Tm 2 ; 0 Ti τm < Tm1 + Tm 2 .

Kc

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 .

3

0.6202  Tm1 + Tm 2   1 + Kc =  K m  τm 

0.9931

1.4849

τ  T + Tm 2   , x1 = 0.0921(τ m + Tm 2 ) m 1 + m1  Tm1  τm 

,

2       τm τm τ   .  + 13.81 Ti = τm 1 + m  4.566 − 14.906 τ +T +T    m1 m2   m  τm + Tm1 + Tm 2   Tm1    1.0409

4

Kc =

0.5723  Tm1 + Tm 2   1 +  K m  τm 

, x1 = 0.0943(τm + Tm 2 ) 1.6501

Ti = 0.2476

  τm (τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2  Tm1 τ m  

.

1.4636

τm  Tm1 + Tm 2   1 +  Tm1  τm 

,

b720_Chapter-03

FA


270 Rule Huang et al. (1996).

5

17-Mar-2009

Kc

Ti

Td

Comment

Kc

Ti

0

Model: Method 1

5

Minimum IAE. Equations continued into the footnote on p. 271; 0 < Tm 2 Tm1 ≤ 1 , 0.1 ≤ τ m Tm1 ≤ 1 .

Kc =

1  T τm τ T  − 137.8937 m 2 − 122.7832 m m2 2  7.0636 + 66.6512 K m  Tm1 Tm1 Tm1 

+

0.0865 2.6405 T  τm  τ  1  26.1928 m    33 . 6578 + 3.0098 m 2 + T  T  Km  m 1 m 1  Tm1     

+

T 1   − 10.9347 m 2 T Km   m1 

+

T 1  34.3156 m 2 Km  Tm1 

+

τ 1  152.6392 m Km  Tm1 

+

τ m Tm 2 τm Tm 2  2 τ 1  − 57.9370e Tm1 + 10.4002e Tm1 + 6.7646e Tm1 + 7.3453 m Km   Tm 

  

2.345

 τm     Tm1   Tm 2   Tm1

T + 141.511 m 2  Tm1

−0.9450

  

  

1.0570

+ 29.4068

− 70.1035

Tm 2 Tm1

 τm     Tm1 

− 47.9791

τm Tm1

 Tm 2   Tm1

0.8866

  

  

1.0309 

  

Tm 2   τm  

−0.9282 

  

0.8148 

     

−0.4062 

;  

2  τ  τ T  τ T Ti = Tm1 0.9923 + 0.2819 m − 0.2679 m 2 − 1.4510 m  − 0.6712 m m2 2   Tm1 Tm1 Tm1   Tm1   

 T + Tm1 0.6424 m 2   Tm1 

2

3

 τ  T  + 2.504 m  + 2.5324 m 2 Tm1 T   m1 

 τ + Tm1 2.3641 m Tm1  

 Tm 2   Tm1

T   + 2.0500 m 2  Tm1 

 τ + Tm1 0.8519 m Tm1  

 Tm 2   Tm1

 T  − 1.3496 m 2 Tm1 

 τ + Tm1  − 2.4216 m Tm1  

2

 Tm 2   Tm1

4

3

3

 τm     Tm1 

2

  

τ    − 1.8759 m   Tm1  

4

  

3

τ   τm    − 3.4972 m  T  Tm1   m1 

T   − 3.1142 m 2  Tm1 

4

τ    + 0.5862 m   Tm1  

5

  

2

 Tm 2   Tm1

  

2

  

b720_Chapter-03

17-Mar-2009

FA


 T + Tm1 0.0797 m 2 Tm1  

4

T  τm    + 0.985 m 2 T  Tm1  m1 

2   τ  T + Tm1 1.2892 m   m 2   Tm1   Tm1 

  

2

 τm     Tm1 

3

T   + 1.2108 m 2  Tm1 

  

271

3

  

5

;  

2  τ  τ T  τ T x1 = Tm1 0.0075 + 0.3449 m + 0.3924 m 2 − 0.0793 m  + 2.7495 m m2 2   Tm1 Tm1 Tm1   Tm1   

 T + Tm1 0.6485 m 2   Tm1 

 τ  T  + 0.8089 m  − 9.7483 m 2 Tm1 T   m1 

 τ + Tm1 3.4679 m Tm1  

 Tm 2   Tm1

2

3

 T + Tm1 12.0049 m 2 Tm1    T + Tm1 10.0045 m 2   Tm1 

2

T   − 5.8194 m 2  Tm1  3

2

 Tm 2   Tm1

5

  

2

3

τ   τm    + 7.0404 m  T  Tm1   m1 

  τ  T + Tm1 1.4294 m  m 2   Tm1  Tm1   T + Tm1 1.1839 m 2 Tm1  

3

τ   T  + 0.3520 m  − 6.3603 m 2 T Tm1  m1  

 T + Tm1  − 3.2980 m 2   Tm1 

4

 T  − 6.9064 m 2   Tm1 5

 τm  T   + 1.7087 m 2 T  m1   Tm1

2

τ    − 1.0884 m   Tm1  

τ   τm    − 1.4056 m  T  Tm1   m1 

4

 τm     Tm1 

2

 Tm 2   Tm1

  

4

  

2

 τ  − 3.7055 m T m1 

 τm     Tm1    

  

3

  

3

  

5

 τ   + 1.7444 m    Tm1 

4

6

  

 Tm 2   Tm1

  

2

  

3 3 2 4   τ m   Tm 2   τ m   Tm 2  τm          + Tm1 − 1.2817   T  − 2.1281 T   T  + 1.5121 T T  m1  m1   m1   m1   m1   α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ;

a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .

  

4

 τ   + 0.0471 m    Tm1 

6

 Tm 2   Tm1

 Tm 2   Tm1

  

5

;  

b720_Chapter-03

FA


272 Rule Huang et al. (1996).

6

17-Mar-2009

6

Kc

Ti

Td

Comment

Kc

Ti

0

Model: Method 1

Minimum IAE. Equations continued into the footnote on p. 273. 0.4 ≤ ξ m ≤ 1 ; 0.05 ≤ τ m Tm1 ≤ 1 . Kc =

1 Km

  τ  τ − 8.1727 − 32.9042 m + 31.9179ξm + 38.3405ξm  m  Tm1   Tm1 

−1.9009 0.1571 1.2234   τm   τm  τ  1   + 0.2079 m       29 . 3215 35 . 9456 + + T  T  T  Km   m1  m1  m1       1 0.1311 1.9814 1.737 + − 21.4045ξ m + 5.1159ξ m − 21.9381ξ m Km

+

[

]

+

−0.1303 1.2008   τm  τ  1   − 17.7448ξ m  m     26 . 8655 + ξ m  T  Km    Tm1   m1   

+

τm  τ τ 1  − 52.9156 m ξ m 1.1207 − 22.4297 m ξ m 0.3626 − 3.3331e Tm1  Km  Tm1 Tm1   

1 + Km

τ m ξm  ξ T  8.5175e ξm − 1.5312e Tm1 + 0.8906 m m1  [Huang (2002)]; τm    

2  τ  τ  τ Ti = Tm1 1.1731 + 6.3082 m − 0.6937ξ m + 8.5271 m  − 24.7291ξ m m   Tm1 Tm1   Tm1    3 2   τ  τ  + Tm1  − 6.7123ξ m  m  + 7.9559ξ m 2 − 32.3937 m  + 5.5268ξ m 6     Tm1   Tm1   

4 3    τm   τm  2  τm          166 . 9272 36 . 3954 + ξ + ξ + Tm1 − 27.1372 m m  T  T  Tm1   m1  m1       2   τm 2  τm  3 3 4     − ξ − ξ + ξ 22 . 6065 1 . 6084 29 . 9159 + Tm1 − 94.8879ξ m  m m m Tm1  Tm1      5 4 3   τm   τm  2  τm          − 93.8912ξ m  − 84.3776ξ m  + Tm1 49.6314 Tm1  Tm1  Tm1         2    τm  4  τm  3 5       ξ − ξ 25 . 1896 19 . 7569 ξ − + Tm1 110.1706 m m m T  Tm1    m1     

b720_Chapter-03

17-Mar-2009


6 5 4   τ  τ  τ  + Tm1  − 12.4348 m  − 11.7589ξ m  m  + 68.3097 m  ξ m 2     Tm1   Tm1   Tm1    3 2   τ  τ  τ + Tm1  − 17.8663 m  ξ m 3 − 22.5926 m  ξ m 4 + 9.5061 m ξ m 5  ; Tm1    Tm1   Tm1    2  τ  τ  τ x1 = Tm1 0.0904 + 0.8637 m − 0.1301ξ m + 4.9601 m  + 14.3899ξ m m   Tm1 Tm1   Tm1    3 2   τ  τ  + Tm1 0.7170ξ m 2 − 12.5311 m  − 42.5012ξ m  m  + 17.0952ξ m 4     Tm1   Tm1   

4   τm   2  τm  3       − 6.9555ξ m − 12.3016 + Tm1 − 21.4907ξ m  Tm1  Tm1        3 2    τm  τm 2  τm  3       + ξ 7 . 5855 19 . 1257 + ξ + Tm1 102.9447ξ m  m m T  Tm1  Tm1   m1      5 4 3   τm   τm  2  τm          − 110.0342ξ m  − 17.2130ξ m  + Tm1 10.8688 Tm1  Tm1  Tm1         2    τm  4  τm  3 5 6       ξ − ξ + ξ 16 . 7073 16 . 2013 5 . 4409 ξ − + Tm1 50.6455 m m m m T  Tm1    m1      6 5 4    τm   τm   τm  2         ξ 10 . 9260 29 . 4445 + − ξ + Tm1 − 0.0979 m m T  T  Tm1    m1  m1       3 2    τm   τm  τm 3 4 5  ;     ξ + ξ 24 . 1917 6 . 2798 ξ − + Tm1 21.6061 m m m T  Tm1  Tm1   m1     

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 .

FA

273

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274 Rule

Majhi and Atherton (1999), Atherton and Boz (1998). Model: Method 1

Kc

7

Ti

Comment

Td

Direct synthesis: time domain criteria Coefficient x3 0 values deduced 1 1 1 + + from graphs. τm Tm 2 Tm1

Kc

α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = x1 ; b f 3 = 1 , bf 4 = x 2 , a f 5 = 0 . x3

Minimum ISE – servo – Atherton and Boz (1998).

1.2 2.6 4.5 6.4 1.8 4.2 6.9 10.0 Minimum ITSE – 1.8 servo – Majhi 4.2 and Atherton 6.9 (1999). 10.0 Minimum ISTSE 2.3 – servo – 5.2 Atherton and Boz 8.4 (1998). 12.4 Minimum ISTSE 2.3 – servo – Majhi 5.2 and Atherton 8.7 (1999). 12.4

7

x4

x5

x6

x3

x4

x5

x6

0.58 0.91 0.89 0.98 0.17 0.22 0.25 0.26 0.17 0.22 0.25 0.26 0.08 0.11 0.14 0.13 0.08 0.11 0.12 0.13

-0.81 -0.55 -0.30 -0.22 -0.31 -0.16 -0.11 -0.08 -0.29 -0.16 -0.11 -0.08 -0.16 -0.10 -0.07 -0.04 -0.16 -0.10 -0.06 -0.04

2.36 2.49 2.18 2.23 1.08 0.96 0.95 0.92 1.05 0.96 0.95 0.92 0.71 0.65 0.68 0.63 0.69 0.65 0.62 0.64

8.5 10.8 13.3 16.0 13.0 16.2 20.3 24.0 13.0 16.8 20.3

1.02 1.03 1.02 1.00 0.28 0.31 0.29 0.30 0.28 0.27 0.29

-0.16 -0.14 -0.12 -0.09 -0.07 -0.06 -0.04 -0.04 -0.06 -0.05 -0.04

2.18 2.16 2.13 2.10 0.94 0.99 0.93 0.95 0.94 0.90 0.97

16.5 20.4 25.2 31.2 16.5 20.4 25.2

0.14 0.15 0.15 0.14 0.14 0.15 0.15

-0.04 -0.03 -0.02 -0.02 -0.03 -0.03 -0.02

0.64 0.65 0.64 0.61 0.64 0.67 0.65

Equations developed from information provided by Majhi and Atherton (1999). (T T + τmTm1 + τmTm 2 )3 , x = 1 1 − (Tm1Tm 2 + τmTm1 + τmTm 2 )3 x  , K c = x 4 m1 m 2  1 5 2 2 2 K m  τm Tm1 Tm 2 K m τm 2Tm12Tm 2 2   (T T + τ T + τ T )2  m m1 m m2  m1 m 2 x 6 − (Tm1 + Tm 2 − τ m )  τ m Tm1Tm 2   τm + Tm 2 ≤ 0. 8 . x2 =  , 3 Tm1 (T T + τm Tm1 + τ m Tm 2 ) x 5   1 − m1 m 2   τ m 2 Tm12 Tm 2 2  

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Rule 8

Hansen (1998). Model: Method 1

Kc

Ti

Td

Kc

Ti

0

275

Comment

χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = K c − (1 K m ) , b f 4 = x 2 , a f 5 = 0 .

9 Ti Td Kc Arvanitis et al. 10 Ti Td Kc (2000a). Model: Method 1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b = 0 , a = T , a = T T ; f1 f1 i f2 i d

ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 .

8

Kc =

[

[

2 Tm1Tm 2 + (Tm1 + Tm 2 )τm + 0.5τm 2

]

3

9K m Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm 2 + 0.167 τm 3

]

2

,

Ti = 1.54 x1 , α = 0.802 , nearly zero overshoot; Ti = 1.27 x1 , α = 0.711 , nearly

minimum IAE; Ti = 1.75x1 , α = 0.857 , conservative tuning; x1 = x2 = 9

10

[

3 Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm + 0.167 τm

[T

2

m1Tm 2 + (Tm1 + Tm 2 )τ m + 0.5τ m

[

[

2

]

2 Tm1Tm 2 + (Tm1 + Tm 2 )τm + 0.5τm 2

3

]

],

2

3K m Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm 2 + 0.167 τm 3

]−

Tm1 + Tm 2 + τm . Km

Kc =

x1Tm1  T  Tm1τm  2ξm + m1  , 0 < x1 < 1 ; suggested x1 = 0.4 ; Td = ; τm  K m τm  x1 ( 2ξ m τm + Tm1 )

Ti =

  1  τm  x + 2ξ 2  − 1 + 2ξ 2  2 x x2  1  

Kc =

1 (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2 , K m τm TCL 2 2 + τm (2ξTCL 2 + τm )

  1   1   − 1 x 2 + ξ2  − 1   , with x  x    1   1   2 2   1 τm τm 1+ + ξ2  − 1 ≥ 0 , x 2 = 1 + . x1Tm1 (2ξ m τm + Tm1 ) x1Tm1 (2ξm τm + Tm1 )  x1 

(2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2 , τm 2 + τm 2Tm1 (2ξ m τm + Tm1 ) 2 2 Tm1 [TCL 2 + τm (2ξTCL 2 + τm )] Td = . (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 22 Ti =

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276 Rule

Kc

Ti

Td

Kc

Ti

0

11

Huang et al. (2000). Model: Method 1

Comment

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 x 2 , x 2 not specified.

Åström and Hägglund (2006), pp. 252, 265. Model: Method 24

11

Kc = λ

12

Kc

Direct synthesis: frequency domain criteria Ti Td M s = M t = 1.4

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 ; a f 3 = x1 , Td 30 ≤ x1 ≤ Td 3 ; 2

a f 4 = 0 . 5 x1 ; K 0 = 0 .

2 2Tm1ξ m + 0.4τm T + 0.8Tm1ξ m τm , Ti = 2Tm1ξ m + 0.4τm , x1 = m1 ; K m τm 0.8Tm1ξm τm

λ = 0.5, τm ξ m Tm1 ≤ 1 , λ = 0.6,1 < τm ξm Tm1 ≤ 2 , λ = 0.7, τmξ m Tm1 > 2 , servo; λ = 0.6, τm ξ m Tm1 ≤ 1 , λ = 0.7,1 < τm ξ m Tm1 ≤ 2 , λ = 0.8, τm ξm Tm1 > 2 , regulator. 12

1  Tm1 T + 0.5x1 T (T + 0.5x1 )  + 0.18 m 2 + 0.02 m1 m 2 0.19 + 0.37 , Km  τm + 0.5x1 τm + 0.5x1 τm + 0.5x1  Tm1 T + 0.5x1 T (T + 0.5x1 ) 0.19 + 0.37 + 0.18 m 2 + 0.02 m1 m 2 τm + 0.5x1 τm + 0.5x1 τm + 0.5x1 Ti = , Tm1 (Tm 2 + 0.5x1 ) Tm 2 + 0.5x1 0.48 Tm1 0 . 0012 0 . 0007 + + 0.03 − τm + 0.5x1 (τm + 0.5x1 )3 (τm + 0.5x1 )2 (τm + 0.5x1 )2

Kc =

Td =

Tm1 (Tm 2 + 0.5x1 ) τm + 0.5x1 Tm1 Tm 2 + 0.5x1 Tm1 (Tm 2 + 0.5x1 ) 0.19 + 0.37 + 0.18 + 0.02 τ m + 0. 5 x 1 τm + 0.5x1 τ m + 0. 5 x 1

0.29(τm + 0.5x1 ) + 0.16Tm1 + 0.2(Tm 2 + 0.5x1 ) + 0.28

 Tm1 + Tm 2 + 0.5x1    T +T +τ +x . m2 m 1  m1

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277

3.6 SOSPD Model with a Zero K (1 + sTm 3 )e −sτ m K m (1 + Tm 3 s )e − sτ m G m (s) = m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1

or G m (s) =

K m (1 − Tm 4 s )e − sτ m K m (1 − Tm 4 s )e −sτ m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1


+

 1   K c 1 + Ti s  

E(s)

−

U(s) Model

Y(s)

Table 28: PI controller tuning rules – SOSPD model with a zero K m (1 − Tm 4s )e −sτ m K m (1 + Tm3s )e −sτ m K m (1 − Tm 4s )e −sτ m G m (s ) = or or 2 2 (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 Tm1 s + 2ξ m Tm1s + 1 Rule Chien et al. (2003). Model: Method 1

Kc 1

Kc

2

Kc

Ti

Direct synthesis: time domain criteria Tm1 Tm1 > Tm 2 Ti

Pomerleau and Poulin 1 Tm1 (2004). K m Tm1 + Tm 4 + τm Model: Method 3

1

Kc =

Comment

1.5Tm1

OS < 10%; Tm1 = Tm 2 .

Tm1 , K m (1.414x1 + Tm 4 + τm ) x1 = 0.707Tm 2 + 0.5 4(Tm 4 τm + Tm 2Tm 4 + Tm 2 τm ) + 2Tm 2 . 2

2

Kc =

Ti , K m (2.414x1 + Tm 4 + τm − Ti ) 4.828ξ m Tm1x1 + 2ξ mTm1 (Tm 4 + τm ) + Tm 4 τm − 2.414 x1 , 2ξ m Tm1 + Tm 4 + τm 2

Ti =

x1 obtained numerically from the solution of a cubic equation.

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278 Rule

Kc

Direct synthesis: frequency domain criteria Ti Tm1 > Tm 2 3 Kc

Khodabakhshian and Golbon (2004).


Comment

Ti

Also Desbiens (2007), p. 90. Model: Method 1 Robust Negative zero. 2ξ m Tm1 + 0.5τm 2ξ mTm1 + 0.5τm K m (2λ + τm ) 2ξm Tm1 + 0.5τm K m (2λ + τm + 2Tm 4 )

Positive zero.

2ξ m Tm1 + 0.5τm

Ultimate cycle Luyben (2000). Tm 4 Tm1 values.

x 1 values.

3

Kc =

Ti Km

K u x1 K m

Model: Method 1

Tu x 2

0.2

0.4

0.8

1.6

3.9

3.3

2.8

3.0

τm Tm 4 = 0.2

3.8

3.4

3.0

2.9

τm Tm 4 = 0.4

4.1

3.6

3.2

3.1

τm Tm 4 = 0.6

4.3

3.8

3.4

3.2

τm Tm 4 = 0.8

4.8

4.3

3.7

3.3

τm Tm 4 = 1.0

5.2

4.6

3.9

3.4

τm Tm 4 = 1.2

5.5

5.0

4.4

3.7

τm Tm 4 = 1.4

6.0

5.4

4.7

3.8

τm Tm 4 = 1.6

(Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2 (TiTm 4 )2 ω4 + (Ti 2 + Tm 4 2 )ω2 + 1

 1 − 10−0.1M max cos −1   2 

4

, with ω given by solving

  = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω)  

− tan −1 (Tm1 ω) − τ m ω with recommended M max = 0dB (Khodabakhshian and Golbon, 2004); ω given by solving, for φm = 58.13 (Desbiens, 2007): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω ; 2  T  T  τ τ Ti = Tm1 1 + 0.175 m + 0.3 m 2  + 0.2 m 2  , m < 2 ; Tm1 Tm1 Tm1  Tm1    2   Tm 2  T  τ τm    Ti = Tm1  0.65 + 0.35 + 0.2 m 2  , m > 2 . + 0.3  Tm1 Tm1 Tm1  Tm1    4

x 2 = (0.5 + 1.56[Tm 4 Tm1 ]) + (3.44 − 1.56[Tm 4 Tm1 ])(τm Tm 4 ) ; Tm1 = Tm 2 .

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E(s)

−

+

  1 K c 1 + + Td s  T s i  

U(s)

Y(s) Model

Table 29: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm3 )e −sτ m K m (1 − sTm 4 )e −sτ m K m (1 − sTm 4 )e −sτ m G m (s ) = or or (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule

Kc

Wang et al. (1995a).

1

Ti

Td

Comment

Minimum performance index: servo tuning p 0 q 2 − p 2 p1 Model: Method 1 p 1 p 0 q 1 − p1 q1 − 1 − 1 r p0 p 0 q 1 − p1 p 0 Km p0 2

p 0 = Tm1 + Tm 2 + Tm 4 + τm , p1 = Tm1Tm 2 + 0.5τm (Tm1 + Tm 2 ) − 0.5τm Tm 4 ,

p 2 = 0.5Tm1Tm 2 τ m , q 1 = Tm1 + Tm 2 + 0.5τ m , q 2 = Tm1Tm 2 + 0.5τ m (Tm1 + Tm 2 ) ,

r = δ0 + δ1

T  T    τ  T   T  τ τm + δ 2  m 2  + δ3  m 4  + δ4  m  + δ7  m 2  + δ9  m 4  m Tm1  Tm1   Tm1    Tm1   Tm1   Tm1  Tm1

 τ   τ  T  T   T  T  T  T + δ7  m  + δ5  m 2  + δ8  m 4  m 2 + δ9  m  + δ8  m 2  + δ6  m 4  m 4 ;   Tm1   Tm1   Tm1   Tm1  Tm1   Tm1   Tm1  Tm1 Minimum IAE: δ 0 = 3.5550 , δ1 = −3.6167 , δ 2 = 2.1781 , δ 3 = −5.5203 , δ 4 = 1.4704 , δ 5 = −0.4918 , δ 6 = 2.5356 , δ 7 = −0.4685 , δ 8 = −0.3318 , δ 9 = 1.4746 .

Minimum ISE: δ 0 = 3.9395 , δ1 = −3.2164 , δ 2 = 1.6185 , δ 3 = −5.8240 , δ 4 = 1.0933 , δ 5 = −0.6679 , δ 6 = 2.5648 , δ 7 = −0.2383 , δ 8 = −0.0564 , δ 9 = 1.3508 .

Minimum ITAE: δ 0 = 3.2950 , δ1 = −3.4779 , δ 2 = 2.5336 , δ 3 = −5.5929 , δ 4 = 1.4407 , δ 5 = −0.3268 , δ 6 = 2.7129 , δ 7 = −0.5712 , δ 8 = −0.5790 , δ 9 = 1.5340 .

279

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280 Rule

Kc

Chidambaram (2002), p. 157. Model: Method 1

2

Kc

3

Kc

4

Kc

5

Ti

Comment

Td

Direct synthesis: time domain criteria ‘Smaller’ Tm1Tm 2 Tm1 + Tm 2 τm Tm 4 . Tm1 + Tm 2 ‘Larger’ τm Tm 4 .

0.5

2ξ m Tm1

Tm1 ξm

Kc

‘Smaller’ τm Tm 4 . ‘Larger’ τm Tm 4 .

Direct synthesis: frequency domain criteria

Wang et al. (2000b), (2001a). Model: Method 2

6

2ξ m Tm1

Kc

Tm1

2

M s = 1.5

Robust

Marchetti and 2ξ mTm1 + 0.5τm Scali (2000). K m (2λ + τm ) Model: Method 1 7 Kc

2

3

4

5

6

Kc = Kc =

1.571(Tm1 + Tm 2 )

2ξ m Tm1 + 0.5τm Tm1 (Tm1 + ξm τm ) 2ξm Tm1 + 0.5τm

(

)

.

(

)

.

)

.

)

.

K m τm 1 + 3.142 Tm 4 τm 2 2.358(Tm1 + Tm 2 )

K m τm 1 + 4.712 Tm 4 τm 2

Negative zero. Positive zero.

2

Kc =

3.142ξ mTm1

(

K m τm 1 + 3.142 Tm 4 τm 2 2

Kc = Kc =

4.716ξ m Tm1

(

K m τm 1 + 4.712 Tm 4 τm 2

1 cos θ 2ξ m Tm1ω ; ω is determined by solving the M s cos ωτ m − tan −1 (x1ω) K x 2ω2 + 1 m 1

[

]

 cos θ   = tan −1 (x1ω) − ωτ m − 0.5π , with equation: tan −1    M s − sin θ 

( (

) )

 x τ ω2 − 1 cos(ωτ m ) − ωτ m sin (ωτ m )  θ = tan −1  1 m 2  ; x1 = Tm3 or −Tm 4 , as appropriate.  x1τm ω − 1 sin (ωτ m ) + ωτ m sin (ωτ m )  7

Kc =

2ξm Tm1 + 0.5τm . K m (2λ + τm + 2Tm 4 )

b720_Chapter-03

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Rule

Kc

Ti Lee et al. (2006). K (T + τ ) m CL m Model: Method 1 9 Kc

8

8

Kc =

Ti

Td

Recommended τm ≤ TCL ≤ 2τm .

Ti

Td

Recommended τ m ≤ λ ≤ 2τm .

2

Desired closed loop transfer function = 9

Comment

0 . 5τ m T 0. 5 τ m τ (0.1667 τm − 0.5Tm 3 ) , Td = + m1 + m − Tm3 ; TCL + τm Ti Ti (TCL + τm ) TCL + τm 2

Ti = 2ξm Tm1 − Tm3 +

Td

Ti

281

2

2

e − sτ m . TCLs + 1

2 T (τ − λ ) + 0.5τm Ti , Ti = 2ξm Tm1 + m 4 m , K m (λ + τm + 2Tm 4 ) λ + τm + 2Tm 4

Tm 4 (τm − λ ) + 0.5τm T τ (0.1667 τm + 0.5Tm 4 ) + m1 − m ; λ + τm + 2Tm 4 Ti Ti (λ + τm + 2Tm 4 ) 2

Td =

Desired closed loop transfer function =

(1 − sTm 4 )e −sτ . (1 + sTm 4 )(1 + sλ ) m

2

2

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282


  1 1 + Td s  G c (s) = K c 1 + Ti s   Tf s + 1 R(s)

+

E(s)

U(s)   1 1 K c 1 + + Td s   Ti s  Tf s + 1

−

Y(s)

Model

Table 30: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm3 )e −sτ m K m (1 + Tm3s )e −sτ m K m (1 − sTm 4 )e −sτ m G m (s ) = or or (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule Chien et al. (2003). Model: Method 1; Tf = Tm3 .

Kc

Ti

0.829ξm Tm1 K m τm 1

Kc

2

Kc

Pomerleau and Tm1 1 Poulin (2004). K m Tm1 + τ m Model: Method 3

Tm1

Tm1

0

Ti

0

1.5Tm1

0

(

Tm1 > Tm 2

Tm1 = Tm 2 , Tf = Tm 3 ; OS < 10%.

)

2 2 , x 1 = 0.707T m 2 + 0.5 2 T m 2 + τ m + 4T m 2 τ m .

1

Kc =

2

Kc =

Ti , K m (2.414 x1 + τm − Ti )

Ti =

4.828ξm Tm1x1 + 2ξ m Tm1τm + 0.5τm − 2.414 x1 , 2ξm Tm1 + τm

K m (1.414 x1 + Tm3 + τm )

Comment

Td

Direct synthesis: time domain criteria 2ξ m Tm1 Tm1 2ξ m

2

2

x1 is obtained numerically from the solution of a cubic equation.

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Li et al. (2004). Model: Method 1

Kc

Ti

Td

Kc

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

2ξm Tm1 K m (λ + τm )

2ξ m Tm1

3

283

Comment

Robust Bialkowski (1996). Model: Method 1 Chen et al. (2006). Model: Method 1 Chen et al. (2006).

3

Kc =

4

Kc

0.6K u

Tm1 + Tm 2

Tm1 2ξ m Tm1Tm 2 Tm1 + Tm 2


Tf = Tm3 or Tm 4 ; λ not defined. τm ≤ λ ≤ 10τm Tf = 0.0125Tu

2 T − Tm 4 τm Tm1 + Tm 2 ; , Tf = CL 2 K m (2TCL 2 + Tm 4 + τm ) 2TCL 2 + Tm 4 + τm

desired closed loop transfer function = − 4

Kc =

(λ − τ m )Tm 4 Tm1 + Tm 2 . , Tf = K m (λ + 2Tm 4 + τ m ) λ + 2Tm 4 + τ m

(1 − sTm 4 )e−sτ (TCL 2s + 1)2

m

.

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284

   Td s  1  + 3.6.4 Controller with filtered derivative G c (s) = K c 1 +  Ti s 1 + Td s    N   R(s)

+

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

E(s)

−

  U(s)   s 

Y(s) Model

Table 31: PID controller tuning rules – SOSPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m K m (1 + sTm3 )e −sτ m or or m or G m (s) = m 2 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s + 2ξmTm1s + 1 (1 + sTm1 )(1 + sTm 2 ) K m (1 − sTm 4 )e −sτ m Tm12s 2 + 2ξ mTm1s + 1 Rule

Kc

Chien et al. (2003). Model: Method 1

0.829ξm Tm1 K m τm 2

Kc

Ti

Td

Comment

Direct synthesis: time domain criteria 1 2ξ m Tm1 − Tm 3 Td 2ξ m Tm1 − 0.1Td

Td

N = 10

2

1

Td =

Tm1 Tm1 − Tm3 , N = −1 . 2ξ m Tm1 − Tm3 Tm 3 (2ξ mTm1 − Tm3 )

2

Kc =

Tm1 2 , x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td ; K m (2 x1 + Tm 4 + τm )

2 Td = Tm1 10ξ m − 3.02 11ξm − 1  , ξm > 0.302 .  

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Kc

Ti

Td

Comment

Td

Tm1 > Tm 2 > Tm 3

285

Robust 3

Kc Chien (1988). ξ 2 T m m1 − Tm 3 Model: Method 1 K m (λ + τ m )

Ti 2ξ m Tm1 − Tm 3

4

Td

5

Kc

Ti

Td

6

Kc

Ti

Td

Tm1 > Tm 2 > Tm 4

N = 10; λ = [Tm1 , τ m ] (Chien and Fruehauf, 1990).

3

Kc =

4

Td =

5

Tm1 + Tm 2 − Tm3 T T − (Tm1 + Tm 2 − Tm3 )Tm3 . , Ti = Tm1 + Tm 2 − Tm3 , Td = m1 m 2 K m (λ + τm ) Tm1 + Tm 2 − Tm 3

Tm2 1 − (2ξ m Tm1 − Tm 3 )Tm3 . 2ξ m Tm1 − Tm3

Tm 4 τm Tm 4 τm λ + Tm 4 + τm , Ti = Tm1 + Tm 2 + , K m (λ + Tm 4 + τm ) λ + Tm 4 + τm

Tm1 + Tm 2 + Kc =

Td =

6

Tm 4 τm Tm1Tm 2 + . λ + Tm 4 + τm Tm1 + Tm 2 + (Tm 4 τm (λ + Tm 4 + τm ))

Tm 4 τm Tm 4 τm λ + Tm 4 + τm , Ti = 2ξm Tm1 + , K m (λ + Tm 4 + τm ) λ + Tm 4 + τm

2ξ m Tm1 + Kc =

Td =

2 Tm 4 τm Tm1 + . λ + Tm 4 + τm 2ξ m Tm1 + (Tm 4 τm (λ + Tm 4 + τm ))

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286

   1  1 + Td s    3.6.5 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   R(s)

+

     1  1 + Td s  U(s)  K c 1 + Td  Ti s   s 1 + N  

E(s)

−

Y(s) Model

Table 32: PID controller tuning rules – SOSPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m or m G m (s) = m (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Rule

Kc

Zhang (1994). Model: Method 1

Tm1 K m τm

Chien et al. (2003). Model: Method 1; Tm1 > Tm 2 .

0.414Tm1 K m τm


Kc =

Comment

Td

Direct synthesis: time domain criteria Tm1 Tm 2

‘Small’ τ m ; ‘good’ servo response. N = Tm 2 Tm 3 or N = ∞ .

1

Tm1

Tm 2

N = Tm 2 Tm3

N = 10

Kc

Direct synthesis: frequency domain criteria Tm1 Tm1 Tm 2 N = Tm 2 Tm 3 K m (Tm1 + τ m ) Minimum φ m = 90 0 ; φ m = 60 0 , τ m = Tm1 . 2

1

Ti

Kc

Tm1

Tm 2

Minimum φ m = 650 .

Tm1 , K m (2 x1 + Tm 4 + τm ) 2

x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2Tm 4 + 0.4Tm 2 τm + 0.04Tm 2 . 2

Kc =

T (T + 2Tm 4 ) Tm1 , φ m = 56 0 when Tm 4 = Tm1 = τ m . , N = m 2 m1 K m (Tm1 + 2Tm 4 + τ m ) Tm1Tm 4

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Rule O’Dwyer (2001a). Model: Method 1

Kc

Ti

Td

x1Tm1 K m τm

Tm1

Tm 2

Comment

A m = 0.5π x 1 ; φ m = 0.5π − x 1 ; N = Tm 2 Tm 4 ; sample result:

0.785Tm1 K mτm

Tm1

Tm 2

A m = 2.0; φ m = 450 .

Robust



Chien (1988). Model: Method 1; Tm1 > Tm 2 > Tm 4 .


Bialkowski (1996). Model: Method 1

Tm1 > Tm 2 ; N = 10; [Tm1 , τ m ] λ ∈ Tm1 (Chien and Tm1 Tm 2 K m (λ + τ m ) Fruehauf, 1990). Tm 2 1.1(Tm1 − Tm 3 ) Tm1 > Tm 2 > Tm 3 Tm 2 K m (λ + τ m ) N = 11; [Tm1 , τ m ] λ ∈ Tm1 1.1(Tm 2 − Tm 3 ) Tm1 (Chien and K m (λ + τ m ) Fruehauf, 1990). Tuning rules converted to this equivalent controller architecture. N = 10; Tm 2 Tm1 Tm 2 [Tm1 , τ m ] λ ∈ K m (λ + τ m ) (Chien and Tm1 Tm 2 Tm1 Fruehauf, 1990). K m (λ + τ m )

Tm 2 K m (λ + τ m )

Tm 2

Tm1

Tm 2 K m (λ + τ m )

Tm 2

3

Td

Tm1 K m (λ + τ m )

Tm1

4

Td

N = 11; λ ∈ [T, τ m ] , T = dominant time constant (Chien and Fruehauf, 1990). Tuning rules converted to this equivalent controller architecture. Tm1 Tm 2 Tm1 > Tm 2 Tm1 K m (λ + τm )

3

Td = 1.1Tm1 +

1.1Tm 4 τm . λ + Tm 4 + τm

4

Td = 1.1Tm 2 +

1.1Tm 4 τ m . λ + Tm 4 + τ m

N = Tm 2 Tm3 or N = Tm 2 Tm 4 .

287

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288


  Td s 1 + G c (s) = K c 1 + T  Ti s 1+ d  N  R(s) +

E(s)

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

  2  b f 0 + b f 1s + b f 2 s  1 + a f 1s + a f 2 s 2 s 


U(s) Model

Y(s)

Table 33: PID controller tuning rules – SOSPD model with a zero

G m (s) = Rule

K m (1 + sTm 3 )e −sτ m 2

Tm1 s 2 + 2ξ m Tm1s + 1

Kc

or

K m (1 − sTm 4 )e−sτ m

Tm12s 2 + 2ξ mTm1s + 1

Ti

Td

Comment

0.167 τm

Negative zero.

Robust


0.5τm K m (λ + τm ) 1

0.5τm

Positive zero.

Kc

0.5(λ + Tm3 )τm + Tm3λ − 0.083τm + 0.5Tm3τm or λ + τm 2

af1 =

0.5(λ + Tm 4 )τm + Tm 4λ − 0.083τm − 0.5Tm 4 τm ; λ + τm + 2Tm 4 2

af1 =

af 2 =

0.5τm Tm3 (λ − 0.167 τm ) 0.5τm Tm 4 (λ + 0.167 τm ) or a f 2 = ; λ + τm λ + τm + 2Tm 4 2

b f 0 = 1 , b f 1 = 2ξ m Tm1 , b f 2 = Tm1 , N = ∞ . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 .

1

Kc =

0.5τm . K m (λ + τm + 2Tm 4 )

   

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289


    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

Y(s)

Model

+

Table 34: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm 3 )e −sτ m K m (1 − sTm 4 )e −sτ m K m (1 + sTm3 )e−sτ m G m (s) = or or or (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 K m (1 − sTm 4 )e −sτ m Tm12s 2 + 2ξ m Tm1s + 1 Rule

Kc


1

Ti

Kc =

α = 0 , β = 1 , N = 10.

0.4938  Tm1 + Tm 2 − Tm 3  1 +   K m  τm 

Ti = 12.585

Comment

Minimum performance index: regulator tuning Ti Td Tm1 > Tm3 Kc

1.3594

1

Td

, 0.1 ≤

τm ≤ 0.66 ; τm + Tm1 + Tm 2 − Tm3

Tm 2 (Tm1 + Tm 2 − Tm3 )  Tm1 + Tm 2 − Tm3  1 +    τm τm  

2.4858

,

Td = 6.465(Tm1 + Tm 2 − Tm 3 )

Tm 2  Tm1 + Tm 2 − Tm3  1 +   τm  τm 

2.9185

.

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290 Rule Madhuranthakam et al. (2008). Model: Method 1 Chien et al. (2003). Model: Method 1 Huang et al. (2000). Model: Method 1

Kc 2

Kc

Ti

Kc =

Comment

Td

Minimum performance index: servo tuning Ti Td Tm1 > Tm3

α = 0 , β = 1 , N = 10. Direct synthesis: time domain criteria 3

4

Ti

Kc

Kc

α = 0 ; β =1; N = 10.

Td

Direct synthesis: frequency domain criteria N = min[20, 2Tm1 ξ m + 0.4τ m Td 20Td ]

1.2206

2

FA

0.5254  Tm1 + Tm 2 − Tm 3  1 +   K m  τm 

, 0.1 ≤

τm ≤ 0.66 ; τm + Tm1 + Tm 2 − Tm3

1.4569

 T + Tm 2 − Tm3   Ti = 0.5657 τm 1 + m1  τm  

,

Td = 6.7572(Tm1 + Tm 2 − Tm3 ) 3

Kc =

Tm 2 τm

 T + Tm 2 − Tm 3  1 + m1    τm  

Tm1 , x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td 2 ; K m (2 x1 + Tm 4 + τm )

2 Ti = 2ξ m Tm1 − 0.1Td ; Td = Tm1 10ξ m − 3.02 11ξm − 1  , ξm > 0.302 .  

4

Kc = λ α=

2 T + 0.8Tm1ξ m τm 2Tm1ξ m + 0.4τm 2T ξ + 0.4τm ; or K c = λ m1 m , Td = m1 K m (τm + 2Tm 3 ) K m (τm + 2Tm 4 ) 0.8Tm1ξm τm

2Tm1ξ m 0.5Tm1 2 −1 ; β = −1 ; 2Tm1ξ m + 0.4 τ m Tm1 2 + 0.8Tm1ξ m τ m

λ = 0.6,

τm τ ξ m ≤ 1 , A m = 2.83 ; λ = 0.7,1 < m ξ m ≤ 2 , A m = 2.43 ; Tm1 Tm1

λ = 0.8,

τm ξ m > 2 , A m = 2.13 . Tm1

2.9057

.

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Rule Jyothi et al. (2001). Model: Method 1; α = 0 , β =1, N =∞.

Kc

Ti

Td

Comment

5

Kc

Tm1 + Tm 2

τm Tm3 < 1

6

Kc

Tm1Tm 2 Tm1 + Tm 2

7

Kc

τm Tm3 < 1

8

Kc

Tm1 2ξ m

0.5(Tm1 + Tm 2 )

0.5Tm1Tm 2 Tm1 + Tm 2

τm Tm 4 < 1

Tm1 + Tm 2

Tm1Tm 2 Tm1 + Tm 2

2ξ m Tm1

Tm1 2ξ m

2ξ m Tm1

Tm1 + Tm 2 K m Tm 4 9

0.5

Kc

Tm1 + Tm 2 K m Tm 4 10

Kc

ξm Tm1 K m Tm 4 11

5

6

7

8

9

10

11

Kc = Kc = Kc = Kc = Kc = Kc = Kc =

Kc

1.57(Tm1 + Tm 2 ) K m τm 1 + 9.870(Tm3 τm )2 0.79(Tm1 + Tm 2 )

K m τm 1 + 2.467(Tm3 τm )2 3.14ξ m Tm1 K m τm 1 + 9.870(Tm3 τm )2 1.57ξ m Tm1 K m τm 1 + 2.467(Tm3 τm )2 1.57(Tm1 + Tm 2 )

K m τm 1 + 2.467(Tm 4 τm )2 0.79(Tm1 + Tm 2 )

K m τm 1 + 2.467(Tm 4 τm )2 1.57ξ m Tm1 K m τm 1 + 2.467(Tm 4 τm )2

. . . . . . .

τm Tm3 > 1

τm Tm3 > 1

τm Tm 4 > 1 τm Tm 4 < 1 τm Tm 4 > 1 τm Tm 4 < 1 τm Tm 4 > 1

291

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292


R(s)

+

F1 (s)

U(s)

−

Y(s)

Model F2 (s)

Table 35: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm 3 )e −sτ m K m (1 − sTm 4 )e −sτ m G ( s ) = or G m (s) = m Tm12s 2 + 2ξ mTm1s + 1 Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule

Kc 1

Huang et al. (2000). Model: Method 1

Kc

Ti

Td

Comment

Direct synthesis: time domain criteria 2Tm1ξ m + 0.4τ m 0

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 x 2 , x 2 not specified.

Chien et al. (2003). Model: Method 1

2

Kc

Tm1

0

Tm1 > Tm 2

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = Tm 2 , a f 4 = 0.1Tm 2 .

1

Kc = λ

2 2Tm1ξ m + 0.4τm 2T ξ + 0.4τm T + 0.8Tm1ξ m τm ; or K c = λ m1 m , x1 = m1 K m ( τm + 2Tm3 ) K m (τm + 2Tm 4 ) 0.8Tm1ξ m τm

λ = 0.5, τ m ξ m Tm1 ≤ 1 ; λ = 0.6,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.7, τmξ m Tm1 > 2 ; servo. λ = 0.6, τmξ m Tm1 ≤ 1 ; λ = 0.7,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.8, τm ξm Tm1 > 2 ; regulator. 2

Kc =

Tm1 , x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2 (Tm 4 + τm ) + 0.04Tm 2 2 . K m (2 x1 + Tm 4 + τm )

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3.7 TOSPD Model G m (s) = K m

G m (s) = G m (s) =

1 + b 1s + b 2 s 2 + b 3 s 3 1 + a 1s + a 2 s 3 + a 3 s 3

e −sτ m or

K m e −sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 )

(T

K m (Tm 3 s + 1)e −sτ m

m1

)

s + 2ξ m Tm1s + 1 (Tm 2 s + 1)

2 2


R(s)

+

−

 1   K c 1 +  T is  

U(s)

TOSPD model

Y(s)

Table 36: PI controller tuning rules – third order system plus time delay model K m e − sτ m 1 + b1s + b 2s 2 + b3s3 −sτ m G m (s ) = K m e or or 3 3 (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) 1 + a1s + a 2s + a 3s K m (Tm3s + 1)e −sτ m

(T

2 2 m1 s

Rule Minimum IAE – Murata and Sagara (1977). Model: Method 2; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). Model: Method 2; K c , Ti deduced from graphs.

)

+ 2ξ m Tm1s + 1 (Tm 2s + 1)

Kc

Comment

Ti

Minimum performance index: regulator tuning x1 K m x 2Tm1 Tm1 = Tm 2 = Tm3 x1

1.3 0.9 0.7

x2

τm Tm1

x1

x2

τm Tm1

3.4 0.2 0.6 3.6 0.8 3.4 0.4 0.6 3.8 1.0 3.5 0.6 Minimum performance index: servo tuning x1 K m x 2Tm1 Tm1 = Tm 2 = Tm3

x1

x2

τm Tm1

x1

x2

τm Tm1

0.9 0.7 0.6

3.0 3.0 3.1

0.2 0.4 0.6

0.5 0.5

3.3 3.5

0.8 1.0

293

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294 Rule

Kc


1

Comment

Ti

Direct synthesis: time domain criteria Ti Tm1 = Tm 2 = Tm3

Kc

Direct synthesis: frequency domain criteria Hougen (1979), pp. 350–351. Model: Method 1; Tm1 ≥ Tm 2 ≥ Tm 3 .

0.7 Km

2

Vrančić et al. (1996), Vrančić (1996), p. 121. Model: Method 1

Vrančić et al. (2004a). Model: Method 1 Vrančić et al. (2004b).

1

Kc =

 Tm1     τm 

3

0.333

τm Tm1 ≤ 0.04

Kc

0.5A 3 ( A1A 2 − K m A 3 )

A3 A2

0.5A 3 ( A1A 2 − K m A 3 )

4

undefined 5

τ m Tm1 > 0.04

Ti

A m ≥ 2, φ m ≥ 60 0 (Vrančić et al., 1996).

Ti

A1 A 3 A1A 2 − K m A 3

G c = 1 (Tis )

Ti

Model: Method 1

Kc

0.4(3Tm1 + τm ) 0.5 − , Ti = 1.25x1 − 0.25(3Tm1 + τm ) , with K m (3Tm1 + τm − x1 ) K m x1 =

(3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12 τm + 9Tm1τm 2 + τm3 ] .

0.333 1   Tm1  T + Tm 2 + Tm 3   + 0.8 m1 0.7 , Ti = 1.5τm 0.08 Tm1 (Tm 2 + Tm3 ) .  2K m   τm  ( Tm1Tm 2Tm 3 )0.333   

2

Kc =

3

A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m

(

(

2

),

2

3

)

A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m .

A1A3 (A1A 2 − K m A 3 ) ) (A1A 2 − K m A3 )2 + K m A3 (A1A 2 − K m A3 ) + 0.25K m 2A32

4

Ti =

5

Kc =

A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3

.

2

A 3K m 2 − 2A1A 2 K m + A13

,

A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3 2

Ti = 2A1

2

(A3K m − 2A1A 2 K m +

A13 )(1 +

K c K m )2

.

b720_Chapter-03

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FA


Rule

Kc

Ti

Comment

Robust


a1 + 0.5τm (3λ + τm )K m a1 + 0.5τm (3λ + τm + 2Tm3 )K m

b1 = b 2 = b3 = 0 or a 1 + 0. 5 τ m

b 2 = b3 = 0; b1 = Tm 3 . b 2 = b3 = 0 ; b1 = −Tm3 .

Ultimate cycle x 2 Tu

Non-oscillatory x 1K u TOSPD model – x1 x 2 τm Tu x 1 x 2 τm Tu x 1 x 2 τm Tu Kuwata (1987). x 1 , x 2 deduced from 0.13 0.40 0.05 0.22 0.21 0.25 0.225 0.155 0.45 0.17 0.33 0.10 0.225 0.19 0.30 0.225 0.155 0.50 graphs. 0.19 0.30 0.15 0.225 0.17 0.35 0.21 0.24 0.20 0.225 0.16 0.40 Yu (2006), Ziegler–Nichols. 0.45K u 0.83Tu pp. 107–112. 0.33K u 2Tu Model: Method 3 Oscillatory TOSPD model.

295

b720_Chapter-03

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296


+

E(s)

−

  U(s) 1 K c 1 + + Td s  Ti s  

Table 37: PID controller tuning rules – TOSPD model G m (s) = G m (s ) =

Rule

Kc 1

Polonyi (1989). Model: Method 1

Jones and Tham (2006). Marchetti and Scali (2000).

Kuwata (1987); x 1 , x 2 deduced from graphs.

1

2

3

Kc

Y(s)

TOSPD model

K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )

K m e − sτ m 1 + a1s + a 2s3 + a 3s3 Ti

Comment

Td

Minimum performance index: other tuning Ti τm Tm1 > 10(Tm 2 + τ m )

Standard form optimisation – binomial. 2 Ti τm Kc Standard form optimisation – minimum ITAE. Robust 3 Tm1 + Tm 2 + Tm3 Td Kc a1 + 0.5τm

(3λ + τm )K m

a 1 + 0. 5 τ m

a 2 + 0.5a1τm a1 + 0.5τm


Ultimate cycle x 2 Tu x 3 Tu

x 1K u x1

x2

x3

τm Tu

x1

x2

x3

0.45 0.46 0.45 0.43

0.57 0.47 0.40 0.33

0.16 0.13 0.11 0.10

0.05 0.10 0.15 0.20

0.41 0.38 0.35 0.32

0.29 0.25 0.22 0.20

0.08 0.08 0.07 0.06

τm Tu

x1

x2

x3

τm Tu

0.25 0.29 0.19 0.04 0.45 0.30 0.28 0.18 0.03 0.50 0.35 0.40

 T T T K c = 1 − 4 m 2 + m 2  m1 , Ti = 4 6Tm 2 τm + τm .  6τm Tm3  τm   Tm 2 T T + m 2  m1 , Ti = 2.7 3.4Tm 2 τm + τm . K c = 1 − 2.1  3.4τm Tm3  τm  T + Tm 2 + Tm3 T T + Tm 2Tm 3 + Tm1Tm3 K c = m1 , Td = m1 m 2 , λ not specified. K m (λ + τm ) Tm1 + Tm 2 + Tm3

b720_Chapter-03

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297


  1 1 + Td s  G c (s) = K c 1 +  Ti s  Tf s + 1 R(s)

E(s)

−

+

U(s) Y(s)   1 1 K m e − sτ K c 1 + + Td s  2 3 1 + a 1s + a 2 s + a 3 s  Ti s  Tf s + 1 m

Table 38: PID controller tuning rules – TOSPD model G m (s) = Rule Schaedel (1997). Model: Method 1

1

Kc =

Kc 1

Td

1 + a 1s + a 2 s 2 + a 3 s 3 Comment

Direct synthesis: frequency domain criteria Ti Td Tf = Td N ; 5 ≤ N ≤ 20 .

2 2 0.375Ti a − a 2 + a1τm + 0.5τm , Ti = 1 , T   a1 + τm − Td K m  a1 + τm + d − Ti  N  

2

Td =

Kc

Ti

K m e − sτ m

2

3

a 2 + a1τm + 0.5τm a + a 2τm + 0.5a1τm + 0.167 τm . − 3 a1 + τ m a 2 + a1τm + 0.5τm 2

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298

  Td s 1 + 3.7.4 Controller with filtered derivative G c (s) = K c 1 +  Ti s 1 + Td  N  R(s)

E(s)

−

+

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

U(s)

K m e − sτ 2

Rule Schaedel (1997).

1

Kc =

Td = N=

Kc 1

Kc

Y(s)

m

1 + a 1s + a 2 s + a 3 s

Table 39: PID controller tuning rules – TOSPD model G m (s) = Ti

Td

   s  

3

K m e − sτ m 1 + a 1s + a 2 s 2 + a 3 s 3 Comment

Direct synthesis: frequency domain criteria Model: Method 1 Ti Td

2 2 0.375Ti a − a 2 + a1τm + 0.5τm , Ti = 1 , T   a1 + τm − Td K m  a1 + τm + d − Ti  N  

a 2 + a1τm + 0.5τm 2 a 3 + a 2 τm + 0.5a1τm 2 + 0.167 τm 3 , − a1 + τ m a 2 + a1τm + 0.5τm 2 Td  x + τ − Ti  − 1 m + 2  

(x1 + τm − Ti )2 + 0.375 TiTd 4

,

Kv

0.05 ≤ x1 ≤ 0.2 , K v = prescribed overshoot in the manipulated variable for a step response in the command variable.

b720_Chapter-03

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FA




    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

+

TOSPD model

Y(s)

Table 40: PID controller tuning rules – third order system plus time delay model 1 + b 1s + b 2 s 2 + b 3 s 3 −sτ m K m e − sτ m G m (s) = K m e or G m (s) = 3 3 (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) 1 + a 1s + a 2 s + a 3 s Rule Taguchi and Araki (2000).

1

Kc =

Kc 1

Ti

Td

Comment

Minimum performance index: servo/regulator tuning 0 Model: Method 1 Ti K c

OS ≤ 20% ; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 .

 1  0.7399  0.2713 + ,  Km  (τm Tm1 ) + 0.5009 

(

Ti = Tm1 2.759 − 0.003899[τm Tm1 ] + 0.1354[τm Tm1 ]

2

α = 0.4908 − 0.2648(τm Tm1 ) + 0.05159(τm Tm1 ) . 2

),

299

b720_Chapter-03

Rule Taguchi and Araki (2000).

2

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

OS ≤ 20% ; N not specified; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 .

Kuwata (1987). Tm1 = Tm 2 = Tm3 . Kuwata (1987). Model: Method 1

Kc =

FA


300

2

17-Mar-2009

3

Kc

4

Kc

Direct synthesis: time domain criteria 0 Ti α = 1; Model: Method 1 Ti

Tm1 = Tm 2 = Tm3

Td

α = 1, β = 1, N = ∞ .

 1  1.275  0.4020 + ,  [τm Tm1 ] + 0.003273  Km 

( (0.8335 + 0.2910[τ

)

Ti = Tm1 0.3572 + 7.467[τm Tm1 ] − 12.86[τm Tm1 ] + 11.77[τm Tm1 ] − 4.146[τm Tm1 ] , Td = Tm1

2

3

m

)

Tm1 ] − 0.04000[τm Tm1 ] , 2

α = 0.6661 − 0.2509[τm Tm1 ] + 0.04773[τm Tm1 ] ,

4

2

β = 0.8131 − 0.2303[τm Tm1 ] + 0.03621[τm Tm1 ] . 2

3

Kc = x1 =

0.8(3Tm1 + τm ) 1 − , with K m (3Tm1 + τm − x1 ) K m

(3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12τm + 9Tm1τm 2 + τm3 ] ;

[

 0.833 6Tm12 + 6Tm1τm + τm 2 Ti = 1 −  (3Tm1 + τm )2  4

Kc =

] 1.667[6T

2 m1

 

+ 6Tm1τm + τm 3Tm1 + τm

2

].

x 1.20 x13 1 x 2 x 2 − (3T + τ )x − , Ti = 1.667 2 − 4.167 23 , Td = 0.833x 2 1 3 m1 m 2 2 ; 2 x1 Km x2 Km x1 x1 − 0.833x 2 2

2

3

2

2

3

x1 = 6Tm1 + 6Tm1τm + τm , x 2 = 6Tm1 + 18Tm1 τm + 9Tm1τm + τm .

b720_Chapter-03

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FA


Rule

Kc

Vrančić (1996), pp. 136–137.

5

Ti

Kc

x1

Kc =

x1

x2

τm Tu

x1

x2

τm Tu

0.05 0.10 0.15

0.20 0.21 0.21 x 2 Tu

0.22 0.20 0.18

0.20 0.25 0.30

0.21 0.20 0.20

0.16 0.15 0.14 Tm1 = Tm 2

0.35 0.40 0.45 = Tm3

(1 − [1 − α] )(K

2 m

τm Tu

x1

x2

x3

0.13 0.12 0.11 0.10

0.05 0.10 0.15 0.20

0.38 0.32 0.28 0.23

0.26 0.22 0.18 0.16

0.08 0.07 0.04 0.02

A 3 + A13 − 2K m A1A 2 x1 =

Kc =

), K

(1 − [1 − α] )(K

2 m

A 3 + A13 − 2K m A1A 2

A1

Ti =

2

Km +

mA3

x1

τm Tu

x2

x3

τm Tu

0.25 0.21 0.15 0.01 0.45 0.30 0.20 0.14 0 0.50 0.35 0.40

− A1A 2 < 0 ;

(K m A3 − A1A 2 )2 − (1 − [1 − α]2 )A3 (K m 2A3 + A13 − 2K m A1A 2 ) ;

A1A 2 − K m A 3 + x1

2

x 3 Tu

x3

A1A 2 − K m A 3 − x1

2

Tm1 = Tm 2 = Tm3

τm Tu

0.33 0.29 0.26 x 1K u

Kuwata (1987); x1 x 2 x 1 , x 2 deduced from graphs. α = 1 , β = 1 , 0.69 0.41 0.60 0.38 N =∞. 0.51 0.36 0.43 0.31

5


x2

0.14 0.17 0.19

Comment

Td

Direct synthesis: frequency domain criteria Model: Method 1 0 Ti

x 1K u

Kuwata (1987); x 1 , x 2 deduced from graphs. α = 1.

301

(

1 KK 2 + c m 1 − [1 − α ] 2K c 2

(

)

), K

mA3

− A1A 2 > 0 .

with

)

A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a1 − b1τm + 0.5τm , 2

(

2

3

)

A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ;

0.2 ≤ α ≤ 0.5 – good servo and regulator response (p. 139).

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302


R(s)

+

F1 (s)

U(s)

−

Y(s)

Model F2 (s)

Table 41: PID controller tuning rules – third order system plus time delay model K m e − sτ m G m (s ) = (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) Rule

Kc

Ti

Kc

Ti

Td

Comment

Td

0.5τm ≤ λ ≤ 3τm

Robust 1

Jones and Tham (2006). Model: Method 1

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;

ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K0 =

1

Tm1 + Tm 2 + Tm3 − Kc =

0.4 ; bf 3 = 1 , bf 4 = 0 , a f 5 = 0 . K m (τm + 0.75)

0.4τm τm + 0.75

K m (λ + τm )

Tm1 + Tm 2 + Tm3 − , Ti =

1+

0.4τm τm + 0.75

0.4τm τm + 0.75

Td =

,

Tm1Tm 2 + Tm1Tm 3 + Tm 2Tm3 . 0.4τm Tm1 + Tm 2 + Tm 3 − τm + 0.75

b720_Chapter-03

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303

3.8 Fifth Order System Plus Delay Model K (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm G m (s ) = m 1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5

(

)

  1 + Td s  3.8.1 Ideal PID controller G c (s) = K c 1 +  Ti s  U(s) E(s)

R(s)

−

+

(

Y(s)

) )

K m 1 + b1s + b 2s 2 + b 3s 3 + b 4s 4 + b 5s 5 e − sτ 1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5

  1 K c 1 + + Td s   Ti s 

(

m

Table 42: PID controller tuning rules – fifth order model with delay K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m G m (s) = m 1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5

(

Rule

Kc

Vrančić et al. (1999).

1

1

Kc

)

Ti

Comment

Td

Direct synthesis: frequency domain criteria Model: Method 1 Ti Td

Note: equations continued into the footnote on p. 304. Kc =

a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1

[

2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + y 2

(

)

] , with

y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and 2

2

3

y 2 = (a1 − b1 ) τ m + (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td ; 2

Ti =

[a

2

3

2

a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1

2 1

− a1b1 − a 2 + b 2 + (a1 − b1 )τm + 0.5τm 2 − (a1 − b1 + τm )Td

Td is found by solving the following equation: 2

3

];

4

Td x1 + 360 x 2 − 360τm x 3 + 180τm x 4 − 60τm x 5 − 15τm x 6 − 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0 , with 5

4

6

3

x 1 = 360[a 1 b 2 − a 1 (a 2 b1 + a 3 + b1b 2 + b 3 ) 2

2

2

2

+ a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 ) 2

− a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 ) 2

2

2

− a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 ] 4

3

2

2

− 360τ m [a 1 b1 − a 1 (a 2 + b1 + 2 b 2 ) + 3a 1 (a 3 + b1 b 2 ) 2

+ a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2 b 4 )

7

b720_Chapter-03

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FA


304

2

− a 2 ( b 1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ] 2

4

3

2

2

+ 180τ m [a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1 b 2 + b 3 ) 2

2

− a 2 ( b1 + 2 b 2 ) + a 3 b1 + 2a 4 + b1 b 3 + 2( b 2 − b 4 )] 3

3

2

2

+ 60τ m [4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4b 2 ) +4a 2 b1 − a 3 − 5b1 b 2 + b 3 ] 4

2

5

2

6

+ 15τ m [ 9a 1 −14a 1 b1 − 4a 2 + 5b1 + 4 b 2 ] − 42 τ m ( b1 − a 1 ) + 7τ m , 4

3

x 2 = a 1 b 3 − a 1 ( a 2 b 2 + a 1 b 3 + a 4 + b1 b 3 ) 2

2

2

+ a 1 (a 2 b1 + a 2 ( 2a 3 + b1 b 2 − 3b 3 ) + a 3 ( b1 + b 2 ) + 2a 4 b1 + a 5 + b 2 b 3 − b 5 ) 3

2

2

2

− a 1 ( a 2 + a 2 ( b 1 − 2 b 2 ) + a 2 ( 2a 3 b 1 − 2 b 1 b 3 + b 2 − b 4 ) 2

2

2

+ 2a 3 + a 3 ( 2 b1 b 2 − 3b 3 ) + a 4 ( b1 + b 2 ) + a 5 b1 − b1 b 5 + b 3 ) 2

2

2

+ a 2 a 3 + a 2 [a 3 ( b1 − 2 b 2 ) − a 5 − b1 b 4 + b 5 ] + a 3 b1 2

+ a 3 ( a 4 − b1 b 3 + b 2 − b 4 ) + a 4 ( b1 b 2 − b 3 ) + a 5 b 2 − b 2 b 5 + b 3 b 4 , 4

3

x 3 = a 1 b 2 − a 1 ( a 2 b1 + a 3 + b1 b 2 + b 3 ) 2

2

2

2

+ a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 ) 2

−a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 ) 2

2

2

− a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 , 4

3

2

x 4 = a 1 b1 − a 1 ( a 2 + b1 + 2 b 2 ) 2

2

+ 3a 1 (a 3 + b1 b 2 ) + a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2b 4 ) 2

− a 2 ( b1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ) , 4

3

2

2

x 5 = a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1b 2 + b 3 ) 2

2

− a 2 ( b 1 + 2 b 2 ) + a 3 b 1 + 2a 4 + b 1 b 3 + 2( b 2 − b 4 ) , 3

2

2

x 6 = 4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4 b 2 ) +4a 2 b1 −a 3 − 5b1 b 2 + b 3 , 2

2

x 7 = 9a 1 − 14a 1 b1 − 4a 2 + 5b1 + 4b 2 .

b720_Chapter-03

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305

3.8.2 Controller with filtered derivative

  Td s 1  G c (s) = K c 1 + +  Ti s 1 + (Td N )s 

E(s) R(s)

+

−

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

Y(s)  U(s)  2 3 4 5 − sτ m  K m 1 + b1s + b 2s + b 3s + b 4s + b 5s e  1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5 s 

(

) )

(


(

Rule

)

Kc

Ti

Comment

Td

Direct synthesis: frequency domain criteria Vrančić (1996), pp. 118–119. Model: Method 1

1

A3

Ti =

2

A 2 − Td A1 −

Td =

Ti 2(A1 − Ti )

Td N

1

N < 10

Td

Ti

,

− (A 32 − A 5A1 ) +

(A

2 3

)

4 (A3A 2 − A5 )(A5A 2 − A 4A3 ) N , 2 (A3A 2 − A5 ) N 2

− A 5A1 −

(


2

),

( ) (b − a + A a − A a + A a − b τ + 0.5b τ + 0.167b τ + 0.042τ ) (a − b + A a − A a + A a − A a + b τ − 0.5b τ ) + K (0.167 b τ − 0.042b τ + 0.008τ ) . 2

3

A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m , A4 = Km A5 = K m

2

4

4

3 1

2 2

1 3

3 m

3

2 m

4

1 m

m

2

5

5

4 1

3 2

2 3

1 4

4 m

3 m

3

m

2 m

4

1 m

5

m

b720_Chapter-03

FA


306 Rule Vrančić et al. (1999).

2

17-Mar-2009

Kc =

[

2

Kc

Ti

Td

Comment

Kc

Ti

Td

8 ≤ N ≤ 20 ; Model: Method 1

a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1

2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + (a1 − b1 ) 2 τ m + y 2

(

)

] , with

y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and 2

2

3

2 3 2 y 2 = (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td − 3

Td 2 (a1 − b1 + τm ) ; N

2

a1 − a1 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1

Ti =

2  2 T  2 a1 − a1b1 − a 2 + b 2 + (a1 − b1 )τ m + 0.5τ m − (a1 − b1 + τ m )Td − d  N   Td is found by solving the following equation:

4

3

2

Td N −3z1 + Td N −2 z 2 + Td N −1z3 + Td x1 + 360 x 2 − 360τm x 3 + 180τm x 4 − 60τm x 5 − 15τm x 6 − 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0 2

2

5

4

6

7

with x 1 , x 2 , x 3 , x 4 , x 5 , x 6 and x 7 as defined in Table 42; z 1 , z 2 and z 3 are defined as follows: 3

2

z 1 = 360[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ] 2

2

3

+ 360τ m [a 1 − a 1 b1 − a 2 + b 2 ] + 180τ m (a 1 − b1 ) + 60τ m , 3

2

z 2 = 360(a 1 − b1 )[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ] 3

2

2

+ 360τ m [2a 1 − 3a 1 b1 − a 1 (3a 2 − b1 − 2b 2 ) + 2a 2 b1 + a 3 − b1 b 2 − b 3 ] 2

2

2

3

4

+ 180τ m (3a 1 − 4a 1 b1 − 2a 2 + b1 + 2 b 2 ) + 240τ m (a 1 − b1 ) + 60τ m , 2

3

4

5

z 3 = z 4 + z 5 τ m + z 6 τ m + z 7 τ m + 135(a 1 − b1 ) τ m + 27τ m , with 4

3

2

2

z 4 = −360[a 1 b1 − a 1 (a 2 + b1 + b 2 ) + a 1 ( 2(a 3 + b1 b 2 ) − a 2 b1 ) 2

2

2

+ a 1 ( a 2 + a 2 ( b 1 + b 2 ) − a 3 b 1 − 2a 4 − b 1 b 3 − b 2 + b 4 ) − a 2 ( a 3 + b1 b 2 ) + a 4 b1 + a 5 + b 2 b 3 − b 5 ] , 4

3

2

2

z 5 = 360[a 1 − 3a 1 b1 + a 1 ( 2( b1 + b 2 ) − a 2 ) + a 1 (3a 2 b1 − a 3 − 3b1 b 2 ) 2

2

− a 2 ( b1 + b 2 ) + a 4 + b1 b 3 + b 2 − b 4 ] , 3

2

2

z 6 = 540[a 1 − 2a 1 b1 − a 1 (a 2 − b1 − b 2 ) + b1 (a 2 − b 2 )] and 2

2

z 7 = 180[ 2a 1 − 3a 1 b1 − a 2 + b1 + b 2 ] .

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Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Vrančić et al. (2004a), Vrančić and Lumbar (2004).

3

Kc

Ti

Td

Comment

Kc

Ti

Td

8 ≤ N ≤ 20 ; Model: Method 1

3

307

K c , Ti and Td are determined by solving the following equations:

0.5 K c . Td =

A2 (A 4 A1 − A 5 K m ) − 0.5 A 4 (A1A 2 − A 3 K m ) A1 Km 2

2

K c 1 + 2K m K c + K m K c = Ti 2 A1 + K m 2 K cTd

(

)

and 2

A 3K m − A1 K m K cTd − A 2 A1 + A 2 K m K c Td + x 2A1A 2 K m −

[

,

2

2

Kc =

2

A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m

A13

− A 3K m

2

(

, with

][

)

]

x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd , and with 2

2

2

(


( (b (a

2

3

)

2

),

A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m , A4 = K m

A5 = K m

2

4

5

− a 4 + A3a1 − A 2a 2 + A1a 3 − b3τm + 0.5b 2 τm + 0.167 b1τm3 + 0.042τm 4

− b5 + A 4a1 − A3a 2 + A 2a 3 − A1a 4 + b 4 τm − 0.5b3τm

(

2

)

3

) 4

5

)

+ K m 0.167 b 2 τm − 0.042b1τm + 0.008τm .

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308



    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

−

+

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

Y(s) Model

+


(

Rule

)

Kc

Ti

Comment

Td

Direct synthesis: frequency domain criteria Vrančić et al. (2004a), Vrančić and Lumbar (2004).

1

1

Ti

Kc

Model: Method 1; 8 ≤ N ≤ 20 .

Td

Note: equations continued into the footnote on the next page. K c , Ti and Td are determined by solving the following equations: A A 0.5 2 (A 4 A1 − A 5 K m ) − 0.5 4 (A1A 2 − A 3 K m ) A1 Km K c . Td = , 2 2 A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m K c 1 + 2K m K c + K m 2 K c 2 = and Ti 2 A1 + K m 2 K cTd

(

Kc =

A 3K m −

)

A12 K m K cTd

− A 2 A1 + A 2 K m 2 K c Td + x

2A1A 2 K m − A13 − A 3K m 2

[

(

)

, with

][

]

x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd , and with 2

2

2

(


(

2

3

)

A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ,

2

),

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( (a

A 4 = K m b 4 − a 4 + A 3 a 1 − A 2 a 2 + A 1a 3 − b 3 τ m + 0.5b 2 τ m 2 + 0.167b1 τ m 3 + 0.042 τ m 4

A5 = K m

5

− b 5 + A 4 a 1 − A 3a 2 + A 2 a 3 − A1a 4 + b 4 τ m − 0.5b 3 τ m

(

2

)

3

4

309

) 5

)

+ K m 0.167 b 2 τ m − 0.042b1 τ m + 0.008τ m ;

α = 1, β = 1.

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310

3.9 General Model G m (s) =

K m e − sτ m (1 + sTm1 ) n


R(s)

−

+

 1   K c 1 + Ti s  

U(s)

K m e − sτ

Y(s)

m

(1 + sTm1 ) n

Table 45: PI controller tuning rules – general model G m (s) = Rule Minimum IAE – Murata and Sagara (1977). n = 4. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 5. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 4. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 5. Model: Method 3; K c , Ti deduced from graphs.

Kc

K m e − sτ m (1 + sTm1 ) n

Comment

Ti

Minimum performance index: regulator tuning x1 K m x 2Tm1 x1

x2

τm Tm1

x1

x2

τm Tm1

0.9 0.7 0.6

4.0 4.0 4.2

0.2 0.4 0.6

0.5 0.5

4.4 4.7

0.8 1.0

x1 K m

x 2Tm1

x1

x2

τm Tm1

x1

x2

τm Tm1

0.7 0.6 0.5

4.5 4.7 4.9

0.2 0.4 0.6

0.5 0.4

5.2 5.5

0.8 1.0

Minimum performance index: servo tuning x1 K m x 2Tm1 x1

x2

τm Tm1

x1

x2

τm Tm1

0.7 0.6 0.5

4.5 4.7 4.9

0.2 0.4 0.6

0.5 0.4

5.2 5.5

0.8 1.0

x1 K m

x 2Tm1

x1

x2

τm Tm1

x1

x2

τm Tm1

0.6 0.5 0.5

4.1 4.3 4.6

0.2 0.4 0.6

0.5 0.4

5.0 5.3

0.8 1.0

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Kc

Kuwata (1987).

1

Davydov et al. (1995).

Kc = x1 =

2

Kc =

Comment

Direct synthesis: time domain criteria Model: Method 1 Ti Kc

Model: Method 1; ξ = 0.9. Direct synthesis: frequency domain criteria τ m + Tar 0.5 ‘Normal’ design. 0.5 Km n −1 n ( n + 1)( τ m + Tar ) 0.375  n + 1  ‘Sharp’ design.   2n K m  n − 1 2

[

Schaedel (1997). Model: Method 2

1

Ti

311

Ti

Kc

]

0.4(nTm1 + τm ) 0.5 − , Ti = 1.25x1 - 0.25(nTm1 + τm ) , with K m (nTm1 + τm − x1 ) K m

(nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] .

  τ*m , Ti = (nTm1 + τ m )0.624 − 0.467      nTm1 + τ m K m  4.345 − 0.151 nT + τ m1 m   1

τ*m

   

n   (n − 1)! 1 ( ) n 1 ! with τ*m = τ m + Tm1 (n − 1) − . + − n −1 − (n −1) n −i (n − 1) e ( ) (i − 1)!  i =1 n − 1

∑

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312


R(s)

+

−

U(s)

Y(s) Model

Table 46: PID controller tuning rules – general model Comment Kc Ti Td

Rule

Gorez and Klàn (2000), Klán (2000). Model: Method 4

  1 K c 1 + + Td s  T s i  

Direct synthesis: time domain criteria Td 1

Ti K m (Ti + τm )

 τ  1 − m   T  ar   0.25Ti

Ti τm Tar

Ti

< 0.25Ti

2

Alternative tuning. Alternative tuning. Klán (2000).

τ  τ 1 + 1 + 2 m  − 2 m Tar  Tar  1 General delayed model. Ti = Tar , 2 2 T  T K  2τ    τ  Td = ar (Ti + Tcr ) cr + Ti − Taa − m 1 − c 1 + m   ,  Ti  Tar 2Tar  K m  3Ti     Tar = average residence time of the process (which equals Tm + τ m for a FOLPD

process, for example); Taa = additional apparent time constant;

 K  τ  Tcr = 1 − c 1 + m  τm .  K m  2Ti 

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Rule Gorez and Klàn (2000). Lee and Teng (2003).

313

Kc

Ti

Td

Comment

2

Kc

Ti

Td

Model: Method 4

3

Kc

Ti

Td

Model: Method 1

2 x1 K m 2Tm1 0.5Tm1 0.2 ≤ τm Tm1 ≤ 2 Trybus (2005). Model: Method 1 τm n Tm1 0.20 0.22 0.25 0.29 0.33 0.40 0.50 0.67 1.0 2.0

2

x1

0.23 0.23 0.22 0.22 0.21 0.21 0.20 0.18 0.16 0.12

3

x1

0.14 0.14 0.14 0.14 0.14 0.13 0.13 0.12 0.11 0.09

4

x1

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.07

5

General delayed model has a positive zero. K c =

Ti = Tar

τ 1 + 1 + 2 m  Tar

Ti , K m [Ti + (1 + x1 )τm ]

2

 τ  τ   − 2x1 m 1 − (3 + 2.5x1 ) m   Tar  Tar  τm (1 + 1.5x1 )  − , 2 Tar

 Tar  Tcr 1 + Ti − Taa − Tca  , (Ti + Tcr ) Ti  Tar  1 + x1 K c τ m K m Tar with Tar = average residence time of the process, Taa = additional apparent time

Td =

constant and x1 is a normalised area parameter characterising the inverse response of the process.   K  K  2τ   τ 2 τ  Tcr = 1 − (1 + x1 ) c 1 + m  τm ; Tca = 1 − (1 + x1 ) c 1 + m   m . K m  2Ti  Km  3Ti   2Tar   3

General delayed model. The three dominant desired poles of the closed loop response are − ξωCL ± jωCL 1 − ξ 2 and − x1ωCL . G p ( jω) evaluated at − ξωCL + jωCL 1 − ξ 2

equals x 2∠φ2 , with G p ( jω) evaluated at − x1ωCL equalling − x 3 . Then,

[

2

Kc = − Ti =

[

x 2 x 3 x12

[

]

−1

2

2

2

1

3

−1

− 2 x1ξ + 1 sin cos ξ

−1

−1

3

−1

3

−1

1 2

ωCL

] [

−1

[

]

];

[cos ξ + φ ]+ x sin[cos ξ − φ ]+ x x sin[2 cos ξ] ; x ω {x sin [cos ξ] + x [sin (cos ξ − φ ) + x sin φ ]} x x sin [cos ξ] + x [x sin (cos ξ + φ ) − sin φ ] [x x sin(cos ξ + φ ) + x sin(cos ξ − φ ) + x x sin(2 cos ξ)] .

2 x1 x 3 sin

1 CL

Td =

]

x1 x 3 sin cos −1 ξ + φ2 + x 3 sin cos −1 ξ − φ2 + x1x 2 sin 2 cos −1 ξ

3

2

3

2

2

1

−1

1

−1

−1

1 2

2

2

2

2

1 2

−1

b720_Chapter-03

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FA


314 Rule Trybus (2005) – continued.

Kc

Ti

Td

Comment

(1 + τm ) x1 Km

Tm1 + 0.5τm

0.5Tm1τm Tm1 + 0.5τm

τm >2 Tm1

τm Tm1

2.2

2.5

2.9

3.3

4.0

5.0

6.7

10

20

n

x1

0.12 0.13 0.15 0.16 0.19 0.21 0.26 0.32 0.43

3

x1

0.09 0.10 0.11 0.13 0.15 0.17 0.21 0.26 0.37

4

x1

0.07 0.08 0.09 0.10 0.12 0.14 0.17 0.22 0.33

5

G m (s) = K m e −sτ m (1 + sTm1 ) . n

Direct synthesis: frequency domain criteria

Skoczowski and Tarasiejski (1996).

4

Td

Model: Method 1

0.5(n − 1)Tm1

λ ≥ nTm1

Tm1

Kc

Robust

Larionescu (2002). Model: Method 1

4

nTm1 K m (λ + τm )

ωg Tm1 K m e − sτ m G m (s ) = . Kc ≤ n Km (1 + sTm1 )

nTm1

G m (s) = K m e −sτ m (1 + sTm1 ) . n

(1 + ω

g

2

Tm12 2

1 + ωg Td

)

n −1

, Td =

2

n −1 Tm1 , n ≥ 2 with n+2

 2n + 4 τm 4n + 2  − Tm1  n 2 − 2n − 2 + + φm  ± b π Tm1 π   ωg = , and with   + τ − 4 n 2 2 n 2 2 2 m + φm  2Tm1 n − 4n + 3 + π Tm1 π   2

b = Tm1

 2 2n + 4 τ m 4 n + 2  + φm  + c , with  n − 2n − 2 + π Tm1 π  

2  4n + 2  τ m  2n − 2    + φm  , c = 4(n + 2)1 + φm  n 2 − 4n + 3 + π π  Tm1  π     φ m = specified phase margin.

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315


 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s E(s)

R(s)

U(s)   1 1 K m e − sτ K c 1 + + Td s  (1 + sTm1 ) n  Ti s  Tf s + 1 m

+

−

Y(s)

Table 47: PID controller tuning rules – general model G m (s) = Rule Schaedel (1997). Model: Method 2

1

Ti =

Kc

Ti

Td

K m e − sτ m (1 + sTm1 ) n

Comment

Direct synthesis: frequency domain criteria 1 Td Tf = Td N , 0.563  n + 1  Ti   5 ≤ N ≤ 20 . Km  n − 2

3(n + 1) (τm + Tar ) , Td = n + 1 (τm + Tar ) . 5n − 1 6n

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316


  Td s 1 + K c 1 + T  Ti s 1+ s d  N 

E(s)

−

+

     

U(s)

K m e − sτ (1 + sTm1 ) n m

Table 48: PID controller tuning rules – general model G m (s) = Rule

Kc

Ti

     

Y(s)

K m e − sτ m (1 + sTm1 ) n

Comment

Td

Direct synthesis: time domain criteria Davydov et al. (1995). ξ = 0.9; N = Km ; Model: Method 1

1

2

Kc

Ti

0.25Ti

τ*m > 0.5 nTm1 + τm

Kc

Ti

Td

τ*m < 0. 5 nTm1 + τm

0.5(n − 1)Tm1

λ ≥ nTm1 ; N = 5.

Robust Larionescu (2002). Model: Method 1

1

Kc =

nTm1 K m (λ + τm )

nTm1

   τ*m  , , Ti = (nTm1 + τm )0.911 − 0.716 nTm1 + τm         K m  3.540 − 0.718  nTm1 + τm   1

τ*m

  (n − 1)! + (n − 1)! n 1 with τ*m = τm + Tm1 (n − 1) − . n −1 − (n −1) n −i (n − 1) e (n − 1) (i − 1)!  i =1

∑

2

Kc =

1  τ*m K m  2.766  nTm1 + τ m 

  0.150τ*m , Ti = (nTm1 + τ m )0.559 −   nT + τ    m1 m − 0.521  

  .  

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3.9.5 Two degree of freedom structure 1   Td s 1 + U (s ) = K c  1 +  Ti s 1 + Td  N 


    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

−

+

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

+

Y(s) K m e − sτ (1 + sTm1 )n

Table 49: PID controller tuning rules – general model G m (s) = Rule

Kc


1

Kc = x1 =

1

Kc

2

Kc

Ti

Kc =

m

K m e − sτ m (1 + sTm1 ) n

Comment

Direct synthesis: time domain criteria 0 Ti Td

0.8(nTm1 + τm ) 1 − , α = 1, K m (nTm1 + τm − x1 ) K m

(nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] ,

[

 0.833 n ( n − 1)Tm12 + 2nTm1τm + τm 2 Ti = 1 −  (nTm1 + τm )2  2

Td

Ti

] 1.667[n(n − 1)T

2 m1

2

]

+ 2nTm1τm + τm . nTm1 + τm

 

 x 2 − (nT + τ m )x 2 x x 1.20 x13 1 − , Ti = 1.6671 − 2.5 22  2 , Td = 0.833x 2 1 3 m1 , 2  Km x2 Km x1  x1 x 1 − 0.833x 2 2 

α = 1 , β = 1 , N = ∞ ; x1 = n (n − 1)Tm12 + 2nTm1τm + τm 2 , x 2 = n (n − 1)(n − 2 )Tm1 + 3n (n − 1)Tm1 τm + 3nTm1τm + τm . 3

2

2

3

317

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318

3.10 Non-Model Specific


E(s)

+

−

 1   K c 1 + Ti s  

U(s)

Non-model specific

Y(s)

Table 50: PI controller tuning rules – non-model specific Rule

Kc

Ti

Comment

Ziegler and Nichols (1942). Idzerda et al. (1955).

0.45K u

Ultimate cycle 0.83Tu

Quarter decay ratio.

Johnson (1956).

0.5K u

Hwang and Chang (1987).

0.83K u

1

3TuK c Tu

 5.22 5.22   Tu − T1   Tu p1 , T1 = decay rate, period measured under proportional control 0.45K u

1 p1

when K c = 0.5K u .

1

Parr (1989).

0. 5K u

0.43Tu

p. 191. p. 192.

Parr (1989).

0.33K u

2Tu

Chen and Yang (1993). Hang et al. (1993b), p. 59. Pessen (1994).

0.36K u

5.25Tu

0.25K u

0.25Tu

0.25K u

0.167Tu

McMillan (1994).

0.3571K u

Tu

Dominant time delay process. Dominant time delay process. p. 90.

TuK c = period of the continuous oscillation signal (of the measured variable) generated when K c = 0.83K u , with Ti being decreased until continuous oscillations exist.

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Åström and Hägglund (1995), pp. 141–142.

Kc

Ti

Comment

0.4698K u

0.4373Tu

A m = 2, φ m = 20 0 .

0.1988K u

0.0882Tu

A m = 2.44, φ m = 610 .

0.2015K u

0.1537Tu

A m = 3.45, φ m = 46 0 .

Calcev and Gorez (1995).

Edgar et al. (1997), p. 8-15. Klán and Gorez (2000). ABB (2001).

0.3536 K u

0.1592Tu

0.59 K u

0.81Tu

0.5 K m

0.5Tu

0. 5K u

0.8Tu

Robbins (2002).

Prokop et al. (2005).

0.3K u

2

Ti

0.3K u

3

Ti

0.5 K m

Wang et al. (1995b). Vrančić et al. (1996), Vrančić (1996), p. 121. Vrančić (1996), p. 140. Friman and Waller (1997).

4

φm = 150 , large τ m . Minimum IAE – regulator. K m assumed known.

Minimum IAE – servo. Good response – regulator. K m assumed known.

Direct synthesis: frequency domain criteria Ti

Kc

0.5A 3 ( A1A 2 − K m A 3 )

A3 A2

A m ≥ 2, φ m ≥ 60 0 (Vrančić et al., 1996).

0.45K u

A3 A2

Modified Ziegler– Nichols method.

0.4830

3.7321 ω1500

A m > 2,

G p ( jω1500 )

2

Ti = (0.24 + 0.111K u K m )Tu .

3

Ti = (0.27 + 0.054K u K m )Tu .

4

0.4Tu

φm = 450 , small τ m

  1 jωCL G CL ( jωCL ) 1 Kc = Im   , Ti = ωCL ωCL  G p ( jωCL ){1 − G CL ( jωCL )}

φ m > 150 .

  jωCLG CL ( jωCL ) Im  ( ) { ( ) } ω − ω G j 1 G j  CL CL   p CL .   jωCLG CL ( jωCL ) Rl    G p ( jωCL ){1 − G CL ( jωCL )}

319

b720_Chapter-03

17-Mar-2009


320 Rule

Kc


KuKm

K 1350 K m K 1350 K m

6

Ti =

0.50K u K m − 0.36 . (0.33K u K m − 0.17)ωu

7

Ti =

9

10

Ti =

K c = K135 0

K u K m > 0. 5

7

Ti

K u K m < 0.1 ;

K 1350 K m ≤ 0.1 . K u K m < 0.1 ;

Ti

K 1350 K m > 0.1 .

KuKm ω u (0.18K u K m + 1)

K u K m ≤ 10

Kc

Ti

K u K m > 10

10

Kc

Ti

1.6 ≤ M s ≤ 1.9

(0.315K1350 K m − 0.175)ωu 5.4K1350 K m − 13.6

Ti

9

20K135 0 K m − 160

(1.32K1350 K m − 3.2)ωu

6

8

2

Kristiansson and 0.33K u K m − 0.15 Lennartson (2002a). K (0.18K K + 1) m u m

1.18K u K m − 1.72 . (0.33K u K m − 0.17)ωu

0.1 ≤ K u K m ≤ 0.5

2

5.4 K 1350 K m − 13.6

Ti =

Ti

2

20 K 1350 K m − 160

5

5

2

0.50K u K m − 0.36


Comment

Ti

1.18K u K m − 1.72

KuKm

8

FA

.

.

0.09K1350 K m + 0.25 0.09K1350 K m + 1.2

, Ti =

(

K1350 K m

ω1350 0.09K1350 K m + 1.2

).

     K 0 ω150 0  K 0 0.075  Kc = 0 . 2 + , Ti = ω150 0 0.06 + 1.6 150 − 0.06 150   K150 0 Km  Km  Km  + 0.05    K m  

   

2

.  

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc


11


12


13

Kc

1.6(Tar − τm )

0.15 Ku

0.8Tu 1 + 3.7(K u K m )

0.16 Ku

Tu 1 + 4.5(K u K m )

0.81K 37%

Other tuning rules 1.29T37%

0.05 ≤

K 1500 Km

≤ 0. 9 ;

K 1500 Km

≤ 0. 9 ;

K u > 0.2K m (close to optimal tuning); M s = M t = 1.4 .

37% decay ratio.

[0.63K 33% ,0.97 K 33% ] [1.1T33% ,1.3T33% ]

(

33% decay ratio.

0.90K 37%

T37%

37% decay ratio.

1.19 K 37%

0.92T37%

37% decay ratio.

1.02 K 37%

0.92T37%

37% decay ratio.

  0.075 2 0.2 +  K150 0 K m + 0.05   Kc = K m 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m

(

K 0  K c = 2Tar  0.7 − 0.45 150 Km  K c = 1.6(Tar − τm )

0.05 ≤

Ti

Kc

)

)

(

Ti =

13

≤ 0. 9 ;

K u ≥ 0.2 K m (robust); K u ≥ 0.5K m (close to optimal tuning).

MacLellan (1950).

12

Km

1.60 ≤ M max ≤ 1.85 .

Rutherford (1950).

[

K 1500

1.60 ≤ M max ≤ 1.85 .

Åström and Hägglund (2006), p. 239.

11

0.05 ≤

Ti

1.65 ≤ M max ≤ 1.85 .


Aikman (1950).

Comment

Ti

Kc

321

)2 ]

,

[

(

2

)

(

ω150 0 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m

   K  ω150 0  0.075  , T = 2T  0.7 − 0.45 150 0  + 0 . 2 i ar   K  K150 0  Km  m  + 0.05   K m  

 ω150 0  0.075 0.2 + . K m  K150 0 K m + 0.05 

(

)

 .  

)2 ]

.

b720_Chapter-03

17-Mar-2009

FA


322 Rule

Kc

Ti

Comment

Young (1955).

K 37%

< 1.1Tu

37% decay ratio.

K x% 0.5 + 0.361 ln x

Tx % 1.2 1 + 0.025 ln 2 x

0.999 K 25%

0.814T25%

Parr (1989), p. 191.

0.667 K 25%

T25%

Example: x = 25 i.e. quarter decay ratio. Quarter decay ratio.

McMillan (1994), pp. 42–43.

0.42 K 25%

T25%

‘Fast’ tuning.

0.33K 25%

T25%

‘Slow’ tuning.

Wade (1994), p. 114.

0.7515K 25%

0.7470T25%

ECOSSE team (1996b). Hay (1998), p. 189.

0.65K 50%

0.7917T50%

K 25%

T25%

Bateson (2002), p. 616.

K 25%

Tu

McMillan (2005), p. 62.

0.25K 25%

Chesmond (1982), p. 400.

sin φ m

Åström (1982).

G p ( jω900 )

Leva (1993).


 τ 0.1 + 65.31 m  T25% tan φ m ω900 Ti

0.5

4 ω1350

G p ( jω1350 ) 0.25

1. 6 ω1350

G p ( jω1350 ) Hägglund and Åström (1991).

14

Kc =

^

^

0.25 K u

tan (φm − φω − 0.5π )

G p ( jω) 1 + tan (φm − φω − 0.5π ) 2

‘Modified’ ultimate cycle method.

T25%

Kc

14

x = 100(decay ratio).

0.2546 Tu

, Ti =

  

2

tan( φ m − φ ω − 0.5π) >0 Alfa-Laval Automation ECA400 controller. Alfa-Laval Automation ECA400 controller – large delay. Model: Method 2

tan (φm − φω − 0.5π ) . ω

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

NI Labview (2001). Model: Method 2

0.4 K u ^

0.18 K u

Jones et al. (1997). Model: Method 2

^

^

‘Little’ overshoot.

∧

Model: Method 5

∧

0.333 A p

2 Tu

0.2 hTu sin φ m Ap

0.16Tu tan φ m ^

h Ap

x1 Tu

0.4614  K m h  K m  A p 

Majhi (1999), pp. 132–137.

0.5380  K m h  K m  A p  2.5465h sin φ m

Lee et al. (2005).

‘Some’ overshoot.

1.6 Tu

Majhi (1999), pp. 132–137.

Tang et al. (2002). Model: Method 4

^

0.8 Tu

0.3 K u

0.5787


0.8 Tu

^

Cox et al. (1994).

^

0.8 Tu

0.13 K u

Pagola and Pecharromán (2002). Parr (1989).

Comment

Ti

^

323

0.5554

0.9395

^

^

Tu  K m h  3.7736  A p  ^

0.7958 ≤ x1 ≤ 1.5915.

Tu  K m h  3.2541  A p 

0.5752

tan φ m ^

A p x 1 ω900 A m

ω900

0.45 1 + Tm 2 x12 Km

5.215 x1

Minimum ISTE – servo.

0.5468

Minimum ISTE – regulator. 0.81 ≤ x 1 ≤ 1 ; x 1 = 1 , ‘small’ τ m ; x 1 = 0.91 , ‘large’ τ m .

Equivalent to Ziegler and Nichols (1942). 15

15

Representative result for FOLPD process model. Other formulae may be deduced, equivalent to other ultimate cycle tuning rules, for this process model. In addition, tuning rules may also be deduced for process models in SOSPD form, FOLIPD form, FOLIPD form (with a negative zero) and SOSIPD form, as well as processes of 4th, 5th and 6th order without an explicit time delay term. x1 =

2 τm

16(τm Tm ) + 66 − 181(τm Tm )2 + 762(τm Tm ) + 3081

(τm

Tm ) + 17

, 0.1 ≤

τm ≤ 2.0 . Tm

b720_Chapter-03

17-Mar-2009

FA


324

  1 3.10.2 Ideal PID controller G c (s) = K c 1 + + Td s    Ti s E(s)

R(s)

+

−

  U(s) 1 K c 1 + + Td s   Ti s 

Non-model specific

Y(s)

Table 51: PID controller tuning rules – non-model specific Rule

Kc

Ziegler and Nichols (1942).

[ 0. 6 K u , K u ]

Ti

Td


Farrington Tu [ 0.1Tu , 0.25Tu ] [ 0.33K u ,0.5K u ] (1950). McAvoy and 0.54 K u Tu 0.2Tu Johnson (1967). Atkinson and 0.25K u 0.75Tu 0.25Tu Davey (1968). 20% overshoot – servo response. Ku 0.5Tu 0.125Tu Carr (1986); 0.6667 K u Tu 0.167Tu Pettit and Carr (1987). 0. 5K u 1.5Tu 0.167Tu Tu

0.2Tu

0. 5K u

Tu

0.25Tu

0. 5K u

0.34Tu

0.08Tu

Tinham (1989).

0.4444 K u

0.6Tu

0.19Tu

Blickley (1990).

0. 5K u

Tu

Corripio (1990), p. 27.

0.75K u

0.63Tu

Parr (1989), pp. 190,191,193.

0. 5K u

[0.125Tu , 0.167Tu ] 0.1Tu

Comment Quarter decay ratio.

Underdamped. Critically damped. Overdamped.

ξ ≈ 0.45

Less than quarter decay ratio response. Quarter decay ratio. Quarter decay ratio.

b720_Chapter-03

17-Mar-2009

FA


Åström (1982).

Kc

Td

Ti

K u cos φ m

1

Ti

Comment

x1Ti

0.87 K u

Representative results: 0.55Tu 0.14Tu

φ m = 30 0

0.71K u

0.77Tu

0.19Tu

φ m = 450

0.50K u

1.19Tu

0.30Tu

φ m = 60 0

Ku Am

Arbitrary

4 π 2 Ti

K u cos φ m

x1Td

Hang and Åström (1988b).

K u sin φ m

Tu (1 − cos φ m ) π sin φ m

Tu (1 − cos φ m ) 4π sin φ m


0.35K u

0.77Tu

0.19Tu

0.4698K u

0.4546Tu

0.1136Tu

0.1988K u

1.2308Tu

0.3077Tu

0.2015K u

0.7878Tu

0.1970Tu

0.27 K u

2.40Tu

1.32Tu

0.906K u

0.5Tu

0.125Tu

φ m = 250

0.866K u

0.5Tu

0.125Tu

φ m = 30 0

0. 5K u

0.5Tu

0.125Tu

p. 90.

0.5Tu

0.125Tu

p. 21.

0.1592Tu

0.0398Tu

Åström and Hägglund (1984).


Chen and Yang (1993). De Paor (1993). McMillan (1994)

McMillan (2005) [0.3K u ,0.5K u ]

1

2

Calcev and Gorez (1995).

0.3536K u

ABB (1996).

0.625K u

325

Tu 2

Td

Specify A m . Specify φ m .

Am ≥ 2 , φ m ≥ 450 . A m = 2, φ m = 20 0 . A m = 2.44, φ m = 610 . A m = 3.45, φ m = 46 0 .

0

φ m = 45 , ‘small’ τ m ; φ m = 150 , ‘large’ τ m . 0.5Tu

0.083Tu

‘P+D’ tuning rule.

T Ti =  tan φm + 4 x1 + tan 2 φm  u . x1 = 0.25 (Åström, 1982);   4απ x1 = 0.5 (Seki et al., 2000); 0.1 ≤ x1 ≤ 0.3 (Seki et al., 2000). T Td =  tan φ m + 1 + tan 2 φ m  u , x1 not specified.   π

b720_Chapter-03

17-Mar-2009

FA


326 Rule

Kc

Ti

Td

Luo et al. (1996).

0.48K u

0.5Tu

0.125Tu

[0.213Tu ,

[0.133Tu ,

1.406Tu ]

0.469Tu ]

0.46K u

2.20Tu

0.16Tu

0. 4 K u

0.5Tu

0.125Tu

Based on work by Tyreus and Luyben (1992). p. 26.

0. 5K u

Tu

0.125Tu

p. 27.

[0.32 K u ,0.6K u ]

Karaboga and Kalinli (1996). Luyben and Luyben (1997), p. 97. Tan et al. (1999a). Tan et al. (1999a).

Tighter damping than quarter decay ratio tuning. 0. 4 K u 0.333Tu 0.083Tu

Wojsznis et al. (1999). Yu (1999), p. 11. Tan et al. (2001).

Robbins (2002).

Smith (2003).

0.125Tu

Some overshoot.

0. 2 K u

0.5Tu

0.125Tu

No overshoot.

Kc

Ti

0.25Ti

0.33K u

0.5Tu

0.333Tu

0. 2 K u

0.5Tu

0.333Tu

0.45K u

4

Ti

0.25Tu

0.45K u

5

Ti

0.25Tu

0.625Tu

0.1Tu

0.625Tu

0.1Tu

0.6Tu 1 + 2(K u K m )

Td

0.75K u


Ti ωu G 'c ( jωu ) 2

0.5Tu

[K u ,1.67K u ]

Alfaro Ruiz (2005a).

Kc =

0.33K u 3

Chau (2002), p. 115.

3

Comment

2

0.25Ti ωu + 1

6

Kc

, Ti = 0.3183Tu tan

‘Just a bit of’ overshoot. No overshoot. Minimum IAE – servo. ‘Good’ response – regulator. Quarter decay ratio. K u > 0.2K m (close to optimal tuning); M s = M t = 1. 4 .

0.5π + ∠G 'c ( jωu ) , 2

G 'c ( jωu ) = desired frequency response of controller G c ( jω) at ωu . 4

Ti = (0.24 + 0.111K m K u )Tu .

5

Ti = (0.27 + 0.054K m K u )Tu .

6

1 Kc = Ku

4     0.3 − 0.1 K u   , Td = 0.15Tu 1 − (K u K m ) . K   1 − 0.95(K u K m )   m  

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

Rutherford (1950).

1.25K 37%

T37%

Other rules 0.125T37%

1.12 K 37%

T37%

0.25T37%

1.77 K 37%

1.29T37%

0.16T37%

1.67 K 37%

1.29T37%

0.16T37%

MacLellan (1950). Harriott (1964), pp. 179–180. Lipták (1970), pp. 846–847.

1.49 K 37%

0.98T37%

0.24T37%

K 25%

0.167T25%

0.667T25%

K 25%

0.667T25%

0.167T25%

Kc

Ti

0.25Ti

0.999 K 25%

0.488T25%

0.122T25%

K 25%

0.67T25%

0.17T25%

37% decay ratio.

0.83K 25%

0.5T25%

0.1T25%

0.67 K 25%

0.5T25%

0.1T25%

‘Slow’ tuning.

Wade (1994), [ 1.002 K 25% , p. 114. 1.67 K

0.45T25%

0.1125T25%

0.8497 K 50%

0.475T50%

0.1188T50%

K 25%

0.667T25%

0.167T25%

K 25%

0.6667Tu

0.1667Tu

7

Parr (1989), p. 191. McMillan (1994), p. 43.

ECOSSE team (1996b). Hay (1998), p. 189. Bateson (2002), p. 616. Rotach (1994).

8

37% decay ratio.

Quarter decay ratio. Quarter decay ratio. x = 100(decay ratio). Example: x = 25 i.e. quarter decay ratio. Quarter decay ratio. ‘Fast’ tuning.

Chesmond (1982), p. 400.

7

Comment

Td

8

Kc

25%

]

Ti

− 0.5

Tx % K x% , Ti = . 0.5 + 0.361ln x 2 1 + 0.025 ln 2 x M max 2 Kc = G p ( jωM max ) , Ti = − 2  dφωM M max − 1 max ωM max 2   dωM max

Kc =

  

.

dφ ωM

max

dω M max

‘Modified’ ultimate cycle.

327

b720_Chapter-03

17-Mar-2009


328 Rule Rotach (1994) – continued. García and Castelo (2000).

Ti

Td

9

Kc

Ti

Td

10

Kc

Ti

0.25Ti

^

0. 4 K u

Parr (1989).

0.5 A p

^

M max 2 − 1

Td = −

11

Kc = Kc =

^

Td =

^

0.159 Tu

∧

∧

Tu

0.25 Tu

Ti

Td

G p ( jωr ) ,

dφωr      0.5 1 1 −1 2 [ ωr + φ ωr  ]. + φωr   + tan  π − sin −1 1 + tan  π − sin M dωr  M ωr  max max    

(

cos 1800 + φm − ∠G p ( jω1 )

) , T = 2[sin(180 + φ − ∠G ( jω )) + 1] . ω cos(180 + φ − ∠G ( jω ) ) 0

i

G p ( jω1 ) 1 + x12 x 2 2 + sin 2 φm

, with x1 =

m

0

1

1

x3 = Ti =

Self-regulating processes. Non selfregulating processes.

^

0.083 Tu

Tu

Kc

ω1 < ωu

2 ,  dφω     1 1 −1 −1 2 r + φω r  ωr ωr + φω r  − tan π − sin 1 + tan  π − sin M max M max  dωr     

Ti = −

10

11

M max

Kc =

^

0.3333 Tu

0. 4 K u

Zhang et al. (1996a).

Comment

Kc

Lloyd (1994). Model: Method 3

9

FA

x3 1 + x 32

p

m

p

1

π 2 2 , x2 = Ap − ε4 , 4h

x 2 − cos φm 2

1

2

x 2 + sin φm

sin φm or x 3 =

cos φm − x 2 x 2 2 + sin 2 φm

A x −x Am2 − 1 Am2 − 1 or Ti = ; Td = m 4 2 1 or A mωg (x 4 − A m x1 ) A mωg (x 4 + A m x1 ) ωg A m − 1

(

A m x 4 + x1

(

2

) , with x

ωg A m − 1

4

)

∈ [A m x1 + 1.0, A m x1 + 1.2] or

∧  ∧  ∧  ∧     ∧  x 4 ∈   ωpc − ωu  ωu  − 0.4,   ωpc − ωu  ωu  + 0.4 , ωu obtained from an ideal        

∧  ∧   relay experiment. x 4 = A m x1 + 1.2 or x 4 =   ωpc − ωu  ωu  + 0.4 (minimum IAE,    regulator).

sin φm ;

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Jones et al. (1997). Shin et al. (1997). Majhi (1999), pp. 132–137. Majhi (1999), pp. 132–137. NI Labview (2001). Model: Method 2

12

Kc =

0.7490h A p

^

Model: Method 2

^

x1 Tu

0.125 Tu

0.2526 ≤ x1 ≤ 0.5053 . 12

Kc

Ti

x 3Ti

Model: Method 2

13

Kc

Ti

Td

14

Kc

Ti

Td

Minimum ISTE – servo. Minimum ISTE – regulator. Quarter decay ratio. ‘Some’ overshoot.

∧

∧

0.6 K u

∧

0.5 Tu

∧

0.12 Tu

∧

0.25 K u

∧

0.5 Tu

0.12 Tu

ρ

,

 ∧   1  1    − x 2  x 3 ωu Ti − ∧ x1  x 3ωu Ti −  − y 2 − ρx 2  ωu Ti     T ω u i   

(x 2 − x1 ) + (x1 − x 2 )2 + 4 x 3ρx1ωu + x 3 y2 ωu  ρx1 + ∧

 ωu



Ti =

Comment

Td

Ti

329

y 2  ∧   ωu 

,

∧   2 x 3ρx1ωu + x 3 y 2 ωu   

∧    ωu − ωu  2 1 1    1− ξ , x1 = − , x 2 = ∧ cos ∠G p ( j ωu )  , ρ =  Ku ωu ξ   Ku ∧ 1   y 2 = ∧ sin ∠G p ( j ωu )  . Typical x 3 : 0.1; typically 0.3 ≤ ξ ≤ 0.7 .   Ku

∧

13

Kc =

 ∧ K h K h 1  0.5585 m  + 0.03835 , Ti = 0.259 Tu  m   Ap   Ap  Km       

0.516

, ∧

(

)

.

∧

(

)

.

Td = 0.124 Tu K m h A p 14

Kc =

1 Km

  ∧ K h K h 0.7837 m  − 0.0526 , Ti = 0.257 Tu  m   Ap   Ap       

0.403

0.197

, Td = 0.131Tu K m h A p

1.109

b720_Chapter-03

17-Mar-2009

FA


330 Rule

Kc

NI Labview (2001) – continued. Chen and Yang (1993).

Ti ∧

15

0.5 Tu

0.12 Tu

‘Little’ overshoot.

4Td

Td

Model: Method 6

∧

0.15 K u

0.25 + 0.87 x1 0.25 + x12

Comment

Td ∧

2.5465h sin φ m 16 ^ Tang et al. x 2Td 0.81 ≤ x 1 ≤ 1 Td A p x 1 ω900 A m (2002). Model: Method 4 x 1 = 1 , ‘small’ τ m ; x 1 = 0.91 , ‘large’ τ m ; 1.5 ≤ x 2 ≤ 4 .

Dutton et al. (1997).

17

Tue

0.5K eu

0.25Tue

pp. 576–578.

Direct synthesis: time domain criteria Ramasamy and Ti 18 Sundaramoorthy K (τ + T ) Td Ti m CL CL (2007).

2

15

x1 =

π Ap − εr

, 2ε r = relay deadband, d r = relay output amplitude;

4d r

− x 2 ± x 22 + 1

Td =

16

2

^

2 ωu

, x2 =

0.43 − 0.5x1 . 0.25 + 0.87 x1

− cot φ m + cot 2 φ m + 4 x 2

Td =

^

.

2 ω90 0 17

K *c

K eu =

(8 − r )2 1−

, Tue =

6.283  (8 − r )3.5  1 −  ωd  1110 

0.286

with r = ratio of the height of the first peak

55

to the height of the first trough of the closed loop response, when controller gain K *c is applied; ωd = damped natural frequency. 18

Desired closed loop transfer function = e −sτ CL 1 + sTCL . 2

2

−

Ti = t − τCL + −

 τCL , Td =  Ti + τCL 2(τCL + TCL ) 

−   τCL − t  − σ 2 _ τCL 3  , − − t  +  2Ti 6Ti (τCL + TCL ) 

with t = mean and σ 2 = variance of the process impulse response.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ramasamy and Sundaramoorthy (2007) – continued.

19

Kc

21

Td

Ti

Td

Kc

max

22

Åström and Hägglund (1995), p. 217. Streeter et al. (2003), Keane et al. (2005).

19

23

24

Comment

Direct synthesis: frequency domain criteria 0 < KmKu < ∞ ; Ti Td M = 1.4.

20


Ti

Kc

Ti

Td

Kc

Ti

Td

(

0 < KmKu < ∞ ; M max = 2.0. 0 < KmKu < ∞ ; M max = 2.0.

)

Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 . Kc =

2

2 − 2TCL1

τCL Ti , Ti = t − τCL + , K m (τCL + 2ξCL TCL1 ) 2(τCL + 2ξCL TCL1 ) −

2

 Td =  Ti + τCL 

−   τCL − t  − σ 2 _ − τCL 3  − t  +  , with t = mean − 2Ti 6Ti (τCL + 2ξCL TCL1 ) 

and σ 2 = variance of the process impulse response. 20

21

22

23 24

  1 Tuning rules converted from formulae developed for G c (s) = K c  b + + Td s  . Ti s  

(

2

)

(

)

(

2

)

(

2

)

K c = 0.19e −1.61κ + 2.5 κ K u , Ti = 0.44e −2.9 κ + 3.14 κ Tu , Td = 0.29e0.84 κ −5.6 κ Tu

  1 Tuning rules converted from formulae developed for G c (s) = K c  b + + Td s  . T s i  

(

2

2

)

(

2

)

K c = 0.18e −1.04 κ +1.08 κ K u , Ti = 0.15e −0.74 κ + 0.26 κ Tu , Td = 0.60e −1.96 κ + 0.68 κ Tu .

(

K c x1 = 0.72e

(

Ti 0.59e = x1

−1.6 κ +1.2 κ 2

−1.3 κ + 0.38 κ 2

)K

)([ 0.72e

[0.72e

− 0.0012340Tu − 6.1173.10 , −1.6 κ +1.2 κ 2

−1.6 κ +1.2 κ

2

Td x1 =

u

−6

2

)K

u

− 0.040430e

( )

0.108e− 3κ +1.76 κ K u Tu − 0.0026640 eTu 2

]

− 0.0012340Tu − 6.1173.10−6 Tu −1.3 κ + 0.38 κ 2

(

log 1.6342 log K u

0.72K u e −1.6 κ +1.2 κ − 0.0012340Tu − 6.1173.10− 6

)

]K

,

u 2

, x1 = 0.25e0.56 κ − 0.12 κ +

Ku . eK u

331

b720_Chapter-03

17-Mar-2009

FA


332


 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s R(s) E(s)

−

+

  1 K c 1 + + Td s  Ti s  

Rule

Kc =

K 0 K m  150  Km

U(s)

Non-model specific

Y(s)

Table 52: PID controller tuning rules – non-model specific Comment Kc Ti Td


1

1 Tf s + 1

1

Kc

Minimum performance index: other tuning Ti Td

‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 .

0.60  K 0 K 0  + 0.07  0.32 + 1.6 150 − 0.8 150 K  m   Km

   

2

1.5 , 2  K150 0  K150 0     ω150 0 0.32 + 1.6 − 0.8 Km K m       0.6667 , Td = 2  K150 0  K150 0     ω150 0 0.32 + 1.6 − 0.8 Km K m       0.4 Tf = K 0 K 0 K 0  ω150 0  150 + 0.07  0.32 + 1.6 150 − 0.8 150 Km  Km  Km  

,

  

Ti =

Tf =

 K 0 ω150 0 20 150 Km 

1  K 0  K 0 + 0.5 0.32 + 1.6 150 − 0.8 150  K m   Km

2

2

   K   min 6 + m ,25   K150 0        

2

  

.

or

b720_Chapter-03

17-Mar-2009

FA


Kc

Ramasamy and Sundaramoorthy (2007).

2

2

Kc

3

Kc

Ti

333

Comment

Td

Direct synthesis: time domain criteria Applies to x1 + Tf Td processes with x 3 + Tf dominant Tf numerator x 2 + Tf 1+ dynamics. x2

Desired closed loop transfer function = e −sτ CL 1 + sTCL .

Kc =

2 −  Tf  τCL , Tf not specified; 1 +  , x1 = t − τCL + K m (τCL + TCL )  x1  2(τCL + TCL )

Ti

2

−   τ − t  − σ 2 3 −  CL τCL  x1 + τCL − t +  + Tf − 2 x1 6 x1 (τCL + TCL ) Td = , Tf 1+ 2 _ τCL t − τCL + 2(τCL + TCL ) −

with t = mean and σ 2 = variance of the process impulse response (treated as a statistical distribution). 3

(

)

Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 . Kc =

 Tf  τCL x2 , 1 +  , x 2 = t − τCL + K m (τCL + 2ξCLTCL1 )  x 2  2(τCL + 2ξCL TCL1 ) −

2

2 − 2TCL1

2

 x 3 =  x 2 + τCL 

−   τCL − t  − σ 2 _ − τCL3  , with t = mean − − t  +  2x 2 6 x 2 (τCL + 2ξCLTCL1 ) 

and σ 2 = variance of the process impulse response (treated as a statistical distribution).

b720_Chapter-03

FA


334 Rule

Kc

Wang et al (1995b). Kristiansson and Lennartson (2002b).

4

17-Mar-2009

Ti

4

Kc

5

Kc

Td

Comment

Direct synthesis: frequency domain criteria Ti Td Ti

0.4444Ti

Plant does not contain an integrator; 1.6 ≤ M max ≤ 1.9 .

For a stable process, K m is assumed known; for an integrating process, K m and τ m are assumed known. K c =

  1 jωCL G CL ( jωCL ) Im  , ωCL  G p ( jωCL ){1 − G CL ( jωCL )}

  jωCL G CL ( jωCL ) Im  G j 1 G j ω − ω ( ) { ( ) }  1 CL CL   p CL Ti = ωCL   jωCL G CL ( jωCL ) Rl    G p ( jωCL ){1 − G CL ( jωCL )} +

   jωCL G CL ( jωCL ) j2ωCL G CL ( j2ωCL ) 1    Rl   − Rl   , 3   G p ( jωCL )[1 − G CL ( jωCL )]   G p ( j2ωCL )[1 − G CL ( j2ωCL )]  

      j2ωCLG CL ( j2ωCL ) jωCLG CL ( jωCL )  Rl  − Rl   G p ( j2ωCL )[1 − G CL ( j2ωCL )]  1   G p ( jωCL )[1 − G CL ( jωCL )] Td =  . 3   jωCL G CL ( jωCL )  Im     G p ( jωCL )[1 − G CL ( jωCL )]  

5

Kc = Ti =

Tf =

  0.48 1.5 − 0.15  K150 0 K m + 0.07 

(

[

)

(

)

(

K m 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m

[

1.5

(

)

(

)2 ]

ω150 0 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m

[

(

,

)2 ]

,

  0.48 − 0.15    K150 0 K m + 0.07

(

)

(

)

ω150 0 0.34 + 1.5 K 150 0 K m − 0.7 K 150 0 K m

)2 ]2 min{3 + 2(K m

) }

K 150 0 ,25

.

b720_Chapter-03

17-Mar-2009

FA

Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule 6


Kc

Ti

Td

7

Kc

Ti

Td

8

Kc

Ti

Td

Comment

M max ≈ 1.7

Plant contains an integrator. Ultimate cycle 0.5Tu 0.125Tu

[0.3K u ,0.5K u ]

( [ (

Tf =

Kc =

Ti =

8

Td

  0.45 1. 5  − 0.1 1.5  K150 0 K m + 0.07  , T = , Kc = i K m 0.86 K150 0 K m + 0.44 ω150 0 0.86 K150 0 K m + 0.44

Td =

7

Ti

Plant does not contain an integrator; 1.65 ≤ M max ≤ 1.85 .

McMillan (2005).

6

Kc

Kc =

Td =

ω1500 Km

)

[ (

]

)

]

)

0.6667 1 , Tf = or K150 0 K 1500 K 1500      ω150 0 0.86 + 0.44 ω1500 0.86 + 0.44 2 + 14  Km Km Km        0.45 − 0.1  K K 0 . 07 +  150 0 m 

(

[ (

)

)

]

{

(

) }

ω150 0 0.86 K150 0 K m + 0.44 2 min 3 + 2 K m K150 0 ,25 3.333ω150 0

e

Km 3.333 ω150 0

3.5 −

3.5 −

2.94ω150 0 Km 2.94 ω150 0

(

6.4 −

9ω

e

4.4 −

3.5 −

G p jω

3ω150 0

150 0

(

(

G p jω150 0

6.5ω

150 0

Km

3ω150 0 Km

)

G p jω150 0

Km

1 3ω150 0 Km

(

150 0

Km

(

)

, Tf =

, Td =

)

G p jω

150 0

(

)

)

Km

≤ 0.91 .

0.083 ω1500 , G p jω1500 ≥ 0.7 ; ω1500 Km

(

)

, Ti =

(

2.94 ω150 0

3.5 −

3ω150 0   0.11 4.4− G p jω150 0  , Tf = e 3.5 − ω1500 Km  

(

K 1500

3ω150 0  0.834  G p jω150 0  . 3.5 − ω150 0  Km 

)

G p jω150 0

; 0.04 ≤

)

G p jω1500 < 0.7 , 1.7 ≤ M max ≤ 1.8 .

1 3ω150 0 Km

6.5ω

150 0

Km

(

)

(

G p jω150 0

G p jω

1500

)

,

)

,

335

b720_Chapter-03

17-Mar-2009

FA


336

   Td s  1  + 3.10.4 Controller with filtered derivative G c (s) = K c 1 +  Ti s 1 + Td s    N     Td s 1 + K c 1 + T  Ti s 1+ d  N 

E(s)

R(s)

−

+

 U(s)  Non-model   specific s 

Y(s)

Table 53: PID controller tuning rules – non-model specific Comment Td Kc Ti

Rule

Direct synthesis: frequency domain criteria Ti 6.283 x1ω φω > φm − π

Leva (1993). N not specified.

1

Kc

2

Kc

x1Td

Td

6 ≤ x1 ≤ 10

Yi and De Moor (1994). Vrančić (1996), p. 130.

3

Kc

4Td

Td

N not defined.

4

Td

N < 10

1

2

Ti 2(A1 − Ti )

ωTi

Kc =

2

 2π   + ω2Ti 2 G p ( jω) 1 − ωTi x1   ωTi Kc = , 2 G p ( jω) 1 − ω2Ti Td + ω2Ti 2

(

4

Kc = Ti =

0.5 cos θ G p ( jωφ )

, Td =

A3

, Ti =

tan (φm − φω − 0.5π) 2π ω 1+ tan (φm − φω − 0.5π ) x1

, x1 = 10 .

)

Td = 3

Ti

(

A 2 − Td A1 − Td

2

− x1ω + x12ω2 + 4x1ω2 tan 2 (φm − φω − 0.5π ) 2 x1ω2 tan (φm − φω − 0.5π )

, φω < φm − π .

tan θ + 4 + tan 2 θ , θ = 2250 − ∠G p ( jωφ ) . 2ωφ N

), 2

Td =

− (A 3 − A 5A1 ) +

(A

2 3

)

2

− A 5A1 − (4 N )(A 3A 2 − A 5 )(A 5A 2 − A 4 A 3 )

(2 N )(A3A 2 − A5 )

.

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Ti

5

Kc

Ti

Vrančić (1996), p. 134.

6

Kc

Vrančić et al. (1999). Lennartson and Kristiansson (1997). Kristiansson and Lennartson (1998a).

7

Vrančić (1996), pp. 120–121.

5

Kc =

(A

2 3

)

Kc =

7

Kc =

Comment

Td

A3A4 − A 2 A5 A 3 2 − A1 A 5

N ≥ 10

Ti

x1Ti

N = 10 ; 0.2 ≤ x1 ≤ 0.5 .

Kc

Ti

Td

8 ≤ N ≤ 20

8

Kc

Ti

0.4Ti

K u K m ≥ 1.67

9

Kc

Ti

0.4Ti

(

0.5A3 A32 − A1A5

)

− A1A5 (A1A 2 − A3K m ) − (A3A 4 − A 2 A5 )A12

,

Ti = 6

337

(A

(

A3 A 32 − A1A 5

2 3

)

)

− A1A5 A 2 − (A3A 4 − A 2 A 5 )A1

A − A 2 2 − 4x1A1A 3 0.5Ti . , Ti = 2 A1 − K m Ti 2 x1A1 A3

2   T 2 2 A1A 2 − A 0 A 3 − Td A1 − d A 0 A1    N   Td is obtained by solving:

A3

, Ti =

2

A 2 − Td A1 −

(

Td Km N

,

)

A 0 A 3 4 A1A 3 3 A 0 A 5 − A 2 A 3 2 2 Td + Td − Td + A 3 − A1A 5 Td + (A 2 A 5 − A 3A 4 ) = 0 . N N3 N2 8

Kc = KuKm Ti =

9

N=

[

K u K m + 2.5 12K u 2 K m 2 − 35K u K m + 30 Kc 3

2

− 0.053ωu + 0.47ωu − 0.14ωu + 0.11

Kc = KuKm Ti =

12K u 2 K m 2 − 35K u K m + 30

,N=

K u K m + x1 12K u 2 K m 2 − 35K u K m + 30 Kc

− 0.525ωu 3 + 0.473ωu 2 − 0.143ωu + 0.113 x1 KmKu

2.5  35 30  + 12 − . K m K u  K m K u K m 2 K u 2 

12K u 2 K m 2 − 35K u K m + 30

[

,

],

];

ωu ≤ 0.4 ; KuKm

 35 30  + 12 −  ; x 1 = 3 , K u K m > 10 ; x 1 = 2.5 , K u K m < 10 . K m K u K m 2 K u 2  

.

b720_Chapter-03

Rule

Kc

Kristiansson and Lennartson (1998a) – continued. Kristiansson and Lennartson (1998b). Kristiansson and Lennartson (2000).

Kc

11

Kc =

2

Km Ku

2

0.4Ti

Ti

11

Kc

Ti

0.4Ti

12

Kc

Ti

0.333Ti

13

Kc

Ti

Td

K c as in footnote 9; Ti =

7.71

10

Kc =

0. 1 ≤ K u K m

≤ 0.5

Kc 3

− 0.185ωu + 1.052ωu

12K u 3K m 2 − 35K u 2 K m + 30K u

[

3

K m K u 3 + 2.5 12K u 2 K m 2 − 35K u K m + 30 3

2

Kc =

N = 2.5

2

ωu ≥ 0.4 . − 0.854ωu + 0.309 K u K m ,

ωu 9.14 > 0.45; − + 3.14 , K u K m > 1.67 , KuKm K mK u

Ti = 13

Comment

Td

Ti

Ti = 12

FA


338

10

17-Mar-2009

[

KcKm Ku 2

2

2

(1.1K u K m − 2.3K u K m + 1.6)

− 0.63ωu + 0.39ωu 2 + 0.15ωu + 0.0082

.

],

ωu 0.95K m K u − 2K m K u + 1.4 2

Kc 3

]

,N =

2.5 KmKu

 35 30  + 12 − . K m K u K m 2 K u 2  

2

2

K m K u (−20 + 13K m K u )(1 + 0.37 K m K u ) 2  1.6( −20 + 13K m K u )(1 + 0.37 K m K u )  − 1 ,  2 2   1.1K m K u − 2.3K m K u + 1.6

Ti =

2 2 K m K u (1.1K u K m − 2.3K u K m + 1.6)  1.6( −20 + 13K m K u )(1 + 0.37 K m K u )  − 1 ,  ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2  1.1K m 2 K u 2 − 2.3K m K u + 1.6 

Td =

K m K u (1.1K u K m − 2.3K u K m + 1.6) ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2

2

2

  ( −20 + 13K m K u ) 2 (1 + 0.37 K m K u ) 2   2 2 2 (1.1K m K u − 2.3K m K u + 1.6)  − 1 ,   1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) −1   2 2 1.1K m K u − 2.3K m K u + 1.6  

N=

2 2 K K (1.1K u K m − 2.3K u K m + 1.6) Td . , Tf = m u Tf ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Kristiansson and Lennartson (2000) – continued.

14

Kc = Ti =

Ti

Td

Comment

14

Kc

Ti

Td

K u K m > 0.5

15

Kc

Ti

Td

K u K m < 0.1

2 2  (1.1K u K m − 2.3K u K m + 1.6) 2  4.8K m K u (1 + 0.37 K m K u ) − 1 , 2 2 2 2 2  3ωu K m K u (1 + 0.37 K m K u )  1.1K m K u − 2.3K m K u + 1.6  2 2  (1.1K u K m − 2.3K u K m + 1.6)  4.8K m K u (1 + 0.37 K m K u ) − 1 ,  2 2 2 3ωu (1 + 0.37 K m K u )  1.1K m K u − 2.3K m K u + 1.6 

  3K m 2 K u 2 (1 + 0.37 K m K u ) 2   2 2 2 2 2 (1.1K u K m − 2.3K u K m + 1.6)  (1.1K m K u − 2.3K m K u + 1.6) , Td = 1 −   4.8K m K u (1 + 0.37 K m K u ) 3ωu (1 + 0.37 K m K u ) 2 −1   2 2 1.1K m K u − 2.3K m K u + 1.6  

N= 15

2 2 Td 1.1K u K m − 2.3K u K m + 1.6 . , Tf = Tf 3ωu (1 + 0.37 K m K u ) 2

Kc =

 20.8(1.8 + 0.3K m K135 0 )  − 1 ,  13K m (1.8 + 0.3K m K1350 )  ( −6 + 3.7 K m K1350 ) 

Ti =

(−6 + 3.7 K1350 K m )K1350 K m  20.8(1.8 + 0.3K m K135 0 )  − 1 ,  13ω1350 (1.8 + 0.3K m K135 0 ) 2  (−6 + 3.7 K m K1350 ) 

(−6 + 3.7 K1350 K m ) 2

2

  169(1.8 + 0.3K m K 0 ) 2 135   2 (−6 + 3.7 K m K135 0 )K m K1350  (−6 + 3.7 K m K1350 )  Td = − 1 ,  13ω1350 (1.8 + 0.3K m K135 0 ) 2  20.8(1.8 + 0.3K m K1350 ) −1    (−6 + 3.7 K m K1350 )   (−6 + 3.7 K m K1350 ) K m K1350 T . N = d , Tf = Tf 13ω1350 (1.8 + 0.3K m K1350 ) 2

339

b720_Chapter-03

Rule

Kc

Kristiansson and Lennartson (2002a).

Ti

Td

Comment

16

Kc

Ti

Td

K u K m ≤ 10

17

Kc

Ti

Td

K u K m > 10

K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ]x1 , Km (0.4K m K u + 0.75) 2 ( 4 + K m K u )

Kc = Ti =

K m K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ] , ωu (0.4K m K u + 0.75) 2 (4 + K m K u )

Td =

K mK u (4 + K m K u ) , ωu [1.5(4 + K m K u )(0.4K m K u + 0.75) − K m K u x1 ] 1 + N −1

(

2

N=

K K x1 Td , , Tf = m u Tf ωu (0.4K m K u + 0.75) 2 (4 + K m K u ) 2

2

[

Km

(0.44K m K1350 + 1.4) 2 x 2

[

K m K135 0 1.5(0.44K m K135 0 + 1.4) x 2 − K m K u x 3

Td =

(0.44K m K135 0 + 1.4) 2 x 2

ω135 0 K m K1350 ω1350

[1.5x (0.44K 2

2

N=

]

K1350 1.5(0.44K m K135 0 + 1.4) x 2 − K m K1350 x 3 x 3

Kc =

Ti =

)

2

x 1 = 0.13 + 0.16K m K u − 0.007 K m K u . 17

FA


340

16

17-Mar-2009

m K135 0

,

],

x2 , + 1.4) − K m K1350 x 3 1 + N −1

](

)

2

K m K1350 x3 Td , , Tf = ω135 0 Tf (0.44K m K1350 + 1.4) 2 x 2

x 2 = min(14 + 0.5K m K 1350 ,25) , x 3 = 1.35 + 0.35K m K 1350 − 0.006K m 2 K 1350 2 .

b720_Chapter-03

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 1  1 + sTd  3.10.5 Classical controller G c (s) = K c 1 +  Ti s  1 + s Td N R(s)

+

E(s)

−

Rule

   1  1 + sTd  K c 1 + Td Ti s   1 + s N 

 U(s)   Non-model  specific  

Y(s)

Table 54: PID controller tuning rules – non-model specific Comment Td Kc Ti Minimum performance index: regulator tuning

Minimum IAE – Edgar et al. (1997), p. 8-15.

Pessen (1953). N =∞. Pessen (1954). N =∞. Atkinson (1968). N =∞.

0.56K u

0.39Tu

0.25K u


0. 2 K u

0.25Tu

N = 10

0.14Tu

‘Optimum’ servo response.

0.5Tu

‘Optimum’ regulator response – step changes. 0.33K u 0.5Tu 0.33Tu 0.750K u

0.75Tu


0.25Tu

0.375K u

‘Rather underdamped response’ – p. 396. 0.75Tu 0.25Tu

Kinney (1983).

0.25K u

‘Good results’ – p. 396. 0.5Tu 0.12Tu

Corripio (1990), p. 27. Hang et al. (1993b), p. 58.

0. 6 K u

0.5Tu

0.125Tu

10 ≤ N ≤ 20

0.35K u

1.13Tu

0.20Tu

N not specified.

0. 5K u

1.2Tu

0.125Tu

0.25K u

1.2Tu

0.125Tu

‘Aggressive’ tuning. ‘Conservative’ tuning.

St. Clair (1997), p. 17. N ≥ 1 .

N not specified.

341

b720_Chapter-03

FA


342 Rule

Kc

Ti

Td

Comment

Harrold (1999).

0.67 K u

0.5Tu

0.125Tu

N not specified.

Tan et al. (1999a), p. 26.

0.3K u

0.15Tu

0.25Tu

N=∞

O’Dwyer (2001b) – representative results – Åström and Hägglund (1995) equivalent, p. 142. N =∞. Tan et al. (2001).

1

17-Mar-2009

Kc =

Ti =

Ziegler–Nichols ultimate cycle equivalent. A m = 2,

0.2349 K u

0.2273Tu

0.2273Tu

0.0944 K u

0.6154Tu

0.6154Tu

0.1008K u

0.3939Tu

0.3939Tu

φ m = 46 0 .

Ti

0.25Ti

N=∞

1

Kc

Ti ωu G 'c ( jωu ) Ti 2ωu 2 + 1 0.0625Ti 2ωu 2 + 1

,

 2∠G 'c ( jωu ) + π  2 2 6.25ωu + 4ωu tan 2   − 2.5ωu 2    2∠G 'c ( jωu ) + π  ωu 2 tan   2  

,

G 'c ( jωu ) = desired controller frequency response at ω = ωu .

φ m = 20 0 . A m = 2.44, φ m = 610 . A m = 3.45,

b720_Chapter-03

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343



+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

U(s)   2 b + b s + b s Non-model f1 f2  f0  1 + a f 1s + a f 2 s 2 specific s 

Kristiansson et al. (2000).

1

Kc

‘Almost’ optimal tuning rule.

0.625  G p ( jωu ) ωu 0.37 + Km  

Tf = x1 ,

Y(s)

Minimum performance index: other tuning Ti Td M s ≈ 1.7

2   G p ( jωu ) G p ( j ωu )   − + 1.6 1.6 2 . 3 1 . 1 2   Km Km 1.6   Kc = , Ti =    G p ( jωu ) G p ( jωu )  ωu 0.37 + K m 0.37 + Km  Km     

Td =

   

Table 55: PID controller tuning rules – non-model specific Comment Kc Ti Td

Rule

1


G p ( jω u ) Km

   

, x1 =

≤ 0.5 ; Tf =

   

,

2   G (j ) 1.6 G p ( jωu ) − 2.3 p ωu + 1.1 2   Km Km  

 G p ( jωu ) ωu 0.37 +  Km 

   

2

 G ( j ωu ) 13 − 20 p  Km 

   

,

x1 G p ( j ω u ) , > 0. 5 ; Km 3 2

b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = Tf , a f 2 = Tf , N = ∞ .

b720_Chapter-03

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FA


344 Rule Kristiansson (2003).

2

Kc

Ti

Td

Kc

Ti

Td

Comment

‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 . Direct synthesis: time domain criteria

Ramasamy and Sundaramoorthy (2007).

2

Kc =

3

x1 + Tf

Kc

Td

0.60 K 0 K 0  K 0 K m  150 + 0.07  0.32 + 1.6 150 − 0.8 150 Km    Km  Km

   

,

2

  

1.5

Ti =

, 2  K150 0  K150 0     ω150 0 0.32 + 1.6 − 0.8  Km   K m    0.6667 , bf 0 = 1 , bf 1 = 0 , bf 2 = 0 , N = ∞ , Td = 2  K150 0  K150 0     ω150 0 0.32 + 1.6 − 0.8  Km   K m    af1 =

 K 0 ω150 0 20 150 Km 

0.8  K 0  K 0 + 0.5 0.32 + 1.6 150 − 0.8 150  Km   Km

K 0  2 ω150 0 20 150 + 0.5 K m  

Kc =

2

,

  

1

af 2 =

3

   

x1

K m (τCL

2

K150 0

0.32 + 1.6  Km 

K 0 − 0.8 150  Km

   

2

2

.

  

2 −  af1  a f 2 + (a f 1 + x 2 )x1 τCL , , x1 = t − τCL +  , Td = 1 + x1  x1 + a f 1 2(τCL + TCL ) + TCL ) 

2

 with x 2 =  x 1 + τ CL 

−  2 3  τ CL − t  − σ _ τ CL   − t  + , − 2x 1 6x 1 (τ CL + TCL ) 

−

a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ2 = variance of the process impulse response; desired closed loop transfer function = e −sτ CL 1 + sTCL .

b720_Chapter-03

17-Mar-2009

FA


Rule Ramasamy and Sundaramoorthy (2007).

4

4

Kc

Ti

Td

Kc

x3 + af1

Td

Comment

Applies to processes with dominant numerator dynamics.

(

)

Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL 2s + TCL 2 2s 2 . Kc =

 af1  a f 2 + (a f 1 + x 4 )x 3 x3 with 1 +  , Td = K m (τCL + 2ξCL TCL 2 )  x3  x3 + a f1 −

x 3 = t − τCL +

2

2

τCL − 2TCL 2 , 2(τCL + 2ξCL TCL 2 ) 2

 x 4 =  x 3 + τCL 

−   τCL − t  − σ 2 _ τCL 3  , − − t  +  2x 3 6x 3 (τCL + 2ξCL TCL 2 ) 

−

a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ 2 = variance of the process impulse response.

345

b720_Chapter-03

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FA


346



    T s β d K α + Td  c  1+ s  N   E(s) R(s) +

−

Rule Edgar et al. (1997), p. 8–15.

Majhi (1999), pp. 132–137.

1

Kc =

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

    s 

− +

U(s)

Non-model specific

Y(s)

Table 56: PID controller tuning rules – non-model specific Comment Td Kc Ti Minimum performance index: regulator tuning 0.77 K u 0.48Tu 0.11Tu 1

Kc

Minimum IAE; α = 0 , β = 1 , N = ∞ . Minimum ISTE; Ti Td N = 10.

∧  K h + 4.267 A  17.08  K m h − 0.4079A p  m p   , Ti = 0.654 Tu  , K m  K m h + 14.9A p   K m h + 12.15A p  ∧  K h − 0.6498A  m p Td = 2.891Tu   , α = 0 , β =1.  K m h + 10.52A p 

b720_Chapter-03

17-Mar-2009

FA


Rule

Kc

Majhi (1999), pp. 132–137. Shen (2002).

2

Kc

3

Kc

Minimum performance index: servo tuning Minimum ISTE; Ti Td N = 10. Minimum performance index: other tuning ∧ Ti Td Ku > 1 Km

Kc

Direct synthesis: frequency domain criteria 2 0 M s = 1.4 0.90T e −4.4 κ + 2.7 κ

4


Comment

Td

Ti

347

u

2

α = 1 − 1.1e −0.0061κ +1.8κ ; 0 < K m K u < ∞ . 2

0.13K u e1.9 κ −1.3κ 0.90Tu e −4.4 κ + 2.7 κ

0

2

M s = 2.0

2

α = 1 − 0.48e 0.40 κ −0.17 κ ; 0 < K m K u < ∞ .

2

Kc =

∧  K h + 0.1205A  15.75  K m h − 0.3287A p  m p   , Ti = 4.602 Tu  , K m  K m h + 18.04A p   K m h + 16.07 A p  ∧  K h − 0.5917 A  m p Td = 2.93 Tu   , α = 0 , β =1.  K m h + 18.22A p 

3

4

Minimise

∞

∞

0

0

∫ r( t) − y(t) dt + ∫ d( t) − y(t) dt ; N = ∞ , β = 0 , M

 ∧ ^     K c = K u exp 0.17 −  2.62  K m K u   + 1.79       

 2 ^ 2    K m K u   ,      

 ∧  Ti = T u exp  − 0.02 −  2.62   

^     K m K u   + 1.34    

 2 ^ 2    K m K u   ,      

 ∧  Td = T u exp  − 1.70 −  0.59   

^     K m K u   −  0.25    

 2 ^ 2    K m K u   ,      

  α = 1 − exp − 0.30 −  0.48   

^     K m K u   +  0.93    

2

K c = 0.053K u e 2.9 κ − 2.6 κ .

 2 ^ 2    K m K u   .      

s

≤2;

b720_Chapter-03

FA


348 Rule Vrančić (1996), pp. 136–137.

5

17-Mar-2009

5

Kc

Ti

Td

Kc

Ti

0


Mantz and Tacconi (1989).

0.6K u

VanDoren (1998).

0.75K u

0.625Tu

0.1Tu

OMEGA Books (2005).

0.61K u

0.625Tu

0.1Tu

Kc = Kc = x1 =

Ti =

Comment

Quarter decay ratio; α = 0.83 , β = 0.346 , N = ∞ .

A1A 2 − K m A 3 − x1

(1 − [1 − α] )(K 2

2 m

A 3 + A13 − 2K m A1A 2

A1A 2 − K m A 3 + x1

(1 − [1 − α] )(K 2

2 m

A 3 + A13 − 2K m A1A 2

), K

mA3

), K

mA3

− A1A 2 < 0 ; − A1A 2 > 0 ;

(K m A3 − A1A 2 )2 − (1 − [1 − α]2 )A3 (K m 2A3 + A13 − 2K m A1A 2 ) ; A1 2

(

1 K K 2 Km + + c m 1 − [1 − α ] 2K c 2

)

.

0.2 ≤ α ≤ 0.5 – good servo & regulator action (Vrančić, 1996, p. 139).

α = 0 ,β =1, N =∞. α = 0 ,β =1, N =∞.

b720_Chapter-03

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FA



R(s)

+

F1 (s)

U(s)

Y(s)

Model

−

F2 (s) Table 57: PID controller tuning rules – non-model specific Comment Td Kc Ti

Rule

1

Bateson (2002), pp. 629–637.

Direct synthesis: frequency domain criteria Model: Method 1 Ti 2 ω180 0

Kc

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ;

ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = 0.1Td ; a f 5 = 0 .

1

K c = 10

{ (

min  − G p jω 0  140

[

) } or {− G (jω ) − 6} p

]

[

180 0

(

, G p jω1400

]

Ti = min ω82 0 ,0.2ω170 0 . min ω82 0 ,0.2ω170 0 .

)

(

and G p jω1800

)

are given in dB;

349

b720_Chapter-04

04-Apr-2009

FA

Chapter 4

Controller Tuning Rules for Non-Self-Regulating Process Models

4.1 IPD Model G m (s) =

K m e −sτ m s


E(s)

+

−

 1   K c 1 + Ti s  

U(s)

K m e − sτ s

m

Y(s)

Table 58: PI controller tuning rules – IPD model G m (s) =

K m e − sτ m s

Rule

Kc

Ti

Comment

Ziegler and Nichols (1942). Model: Method 2

0.9 K m τm



1.0 K m τm

∞


Two constraints method – Wolfe (1951). Model: Method 2

Also given by Coon (1956), (1964) and NI Labview (2001). Decay ratio = 0.4. 0.6 K m τm 2.78τ m Decay ratio is as small as possible. Minimum error integral (regulator mode).

0.87 K m τm

4.35τ m

350

b720_Chapter-04

04-Apr-2009

FA

Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule

Kc

Ti

Comment

Åström and Hägglund (1995), p. 13. Model: Method 1 Hay (1998), p. 188.

0.63 K mτm

3.2 τ m

0.42 K m τm

5. 8τ m

Ultimate cycle Ziegler–Nichols equivalent. Model: Method 3

Hay (1998), p. 199. Model: Method 1; K c and Ti deduced from graphs.

Kp s(1 + sTp ) n


Minimum IAE – Shinskey (1988), p. 123. Minimum IAE – Shinskey (1994), p. 74. Minimum IAE – Shinskey (1988), p. 148. Model: Method 1

K m τ m = 0. 1

3.5

K m τ m = 0. 2

2.5 2.2

K m τ m = 0. 4

2.0

K m τ m = 0.5

1.7

K m τ m = 0.6

0% overshoot (servo). x 2τm

x 3 K m τm x1

K m τ m = 0.3

4.5K m τ m

x1 K m τ m

Bunzemeier (1998). Model: Method 5 Gp =

7.5

x2

0.18 1.350 0.24 3.050 0.22 1.910 0.21 1.690 0.20 1.584 0.20 1.557 0.44 K m τm

x3

0.23 0.40 0.30 0.28 0.26 0.26

10% overshoot (servo).

Coefficient values: n x1 x2 0 1 2 3 4 5

0.20 0.19 0.19 0.18 0.18

1.477 1.463 1.453 1.385 1.378

x3

n

0.25 0.24 0.24 0.24 0.24

6 7 8 9 10

‘Some’ overshoot. ∞

‘No’ overshoot.

0.9 K m τm

3.33τm


0.4 K m τm

5.33τm


0.24 K m τm

5.33τm

‘No’ overshoot.

0.26 K m τm

Minimum performance index: regulator tuning 0.9524 Model: Method 1 4τ m K τ m m

0.9259 K m τm

4τ m

Model: Method 1

0.61K u

Tu

Also specified by Shinskey (1996), p. 121.

351

b720_Chapter-04

04-Apr-2009

FA


352 Rule

Kc

Ti

Comment

Minimum ISE – Hazebroek and Van der Waerden (1950). Minimum ISE – Haalman (1965). Model: Method 1 Minimum ITAE – Poulin and Pomerleau (1996). Model: Method 1

1.5 K mτm

5.56τ m

Model: Method 2

0.6667 K m τm

∞

Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1 Kuwata (1987). Model: Method 1 Regulator – Gorecki et al. (1989). Model: Method 1 Tyreus and Luyben (1992). Model: Method 1 or Method 10. Fruehauf et al. (1993). Rotach (1995). Model: Method 6 Wang and Cluett (1997).

1

Kc =

M s = 1.9 , A m = 2.36 , φ m = 50 0 .

Process output step load disturbance. Process input step 0.5327 K m τm 3.8853τ m load disturbance. Minimum performance index: other tuning 0.28 K m τm 7τ m M max = 1.4 0.5264 K m τm

4.5804 τ m

0.404 K m τm

7τ m

M max = 1.7

0.49 K m τm

3.77τ m

M max = 2.0

Direct synthesis: time domain criteria Also given by Hay ∞ 0.5 K m τm (1998), p. 188.

e

 τ +T  −  m m   Tm 

∞

K m τm 0.828 K m τm

5.828τm

0.487 K mτm

8.75τ m

0.31K u

2.2Tu

0.5 K m τm

5τ m

0.75 K m τm

2.41τ m

Pole is real and has maximum attainable multiplicity. Maximum closed loop log modulus = 2dB ; TCL = τ m 10 .

Model: Method 2

Damping factor for oscillations to a disturbance input = 0.75. 1 Model: Method 1; Ti Kc ξ = 0.707 .

1 , Ti = (1.4020TCL + 1.2076)τm , τm ≤ TCL ≤ 16τm . K m τm (0.7138TCL + 0.3904)

b720_Chapter-04

04-Apr-2009

FA

Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models

Rule Wang and Cluett (1997) – continued.

Cluett and Wang (1997). Model: Method 1

Kc

Ti

Comment

Kc

Ti

Model: Method 1; ξ =1 .

0.9588 K m τ m

3.0425τ m

TCL = τ m

0.6232 K m τ m

5.2586τ m

TCL = 2τ m

2

0.4668 K m τ m

7.2291τ m

TCL = 3τ m

0.3752 K m τ m

9.1925τ m

TCL = 4τ m

0.3144 K m τ m

11.1637 τ m

TCL = 5τ m

0.2709 K m τ m

13.1416τ m

TCL = 6τ m

1.04Tu

M max = 5 dB

Poulin and Pomerleau 2.13 0.34 K u or (1999). K m Tu Model: Method 1 x1 K m τ m Vítečková (1999), Vítečková et al. OS x1 (2000a). 0.368 0% Model: Method 1 0.514 5% 0.581 10% 0.641 15% Chidambaram and 1.1111 K m τm Sree (2003). Huba and Žáková (2003). Skogestad (2003), (2004b). Model: Method 1

0.696 20% 0.748 25% 0.801 30% 0.853 35% 4.5τ m

0.23 K m τm

2.914 τ m 3.555τ m

1 K m (TCL + τ m )

4ξ 2 (TCL + τ m )

0.5 K mτm 0.4689 K m τm

Sree and Chidambaram (2006). Model: Method 1

2x1 (1 + x1 )K m τm

Kc =

Coefficient values: OS x1

0.281 K m τ m

Benjanarasuth et al. (2005). Model: Method 1

2

∞

0.30K u

x1

OS

0.906 0.957 1.008

40% 45% 50%

Model: Method 1

Model: Method 1

Suggested ξ = 0.7 or 1.

8τ m

‘Good’ robustness – TCL = τ m , ξ = 1 .

∞

‘Stability index’ = 2.5.

0.5(1 + x1 )τm x1 − 1

p. 47. x1 > 1 ; x1 = 1.25 (Chidambaram and Sree, 2003).

1 , Ti = (1.9885TCL + 1.2235)τm , τm ≤ TCL ≤ 16τm . K m τm (0.5080TCL + 0.6208)

353

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354 Rule

Kc

Comment

Ti

Direct synthesis: frequency domain criteria Maximise crossover 3 frequency. ∞ Kc

Hougen (1979), p. 333. Model: Method 1 Buckley et al. (1985), pp. 389–390. Model: Method 1 Chidambaram (1994), Srividya and Chidambaram (1997).

≤

0.67075 Kmτm ωp

Gain and phase margin – Kookos et al. (1999). Model: Method 1

≥ 20τm

‘Well-damped’ response.

3.6547τ m

Model: Method 7; recommended A m = 2.

0.45 K m τm

AmKm

1 ω p 0.5π − ω p τ m

0.942 K m τ m

Representative results: 4.510τ m

(

) A m = 1.5; φ m = 22.5 0 .

Chen et al. (1999a). Model: Method 11

0.698 K m τ m

4.098τ m

A m = 2; φ m = 30 0 .

0.491 K m τ m

6.942 τ m

A m = 3; φ m = 450 .

0.384 K m τ m

18.710τ m

A m = 4; φ m = 600 .

π   φ m + 2 (A m − 1)   K mτm Am2 − 1

(

3

)

Ti

0.5236 K m τ m

8τ m

Values recorded deduced from a graph: 0.1 0.2 0.3 0.4 K mτm 7.0

Kc 4

Ti =

5

Kc =

φm ≥

Ti

Kc

5

Cheng and Yu (2000). Model: Method 1

4

3.5

2.2

1.8

A m = 2.83; φ m = 46.10 .

0.5

0.6

0.7

0.8

0.9

1.0

1.4

1.1

1.0

0.8

0.75

0.7

2  π τm Am − 1 .   4  φm + 0.5π(A m − 1)   φm + 0.5π(A m − 1)  0.5π − φm −  2 Am − 1  

r1π(2r1A m − 1)

(

)

4K m τm r12 A m 2 − 1

, Ti =

(

2

2

)

4τm r1 A m − 1

2

πr1 A m (r1A m − 2 )(2r1A m − 1) 2

π  1.0607  1 −  2  A m 

.

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Rule O’Dwyer (2001a). Model: Method 1

Kc

Ti

Comment

x1 K m τ m

x2τm

6

Representative coefficient values: Am φm x1 x2 Am

x2

0.558

1.4

1.5

46.2

0

0.357

4.3

4.0

60.0 0

0.484

1.55

2.0

45.50

0.305

12.15

5.0

75.0 0

0.458

3.35

3.0

0

O’Dwyer (2001a). Model: Method 1

59.9

x1K u

x 2Tu

7

2.01 − 0.80φ

6.283(1.72φ − 2.55) ωφ

1.920 < φ < 2.356 ; M s = 1.2

0.33

6.61 ω 2.09

φ = 2.09 radians (‘good’ choice).

2.50 − 0.92φ

6.283(1.72φ − 3.05) ωφ

2.094 < φ < 2.443 ; M s = 1.4 .

5.36 ω 2.27

φ = 2.27 radians (‘reasonable’ choice).

( )


G p jωφ

G p ( jω2.09 )

( )

G p jωφ 0.41

G p ( jω2.27 )

6

Am =

π  + 2 

x2 4x 1

π2 π   − 4 x2  

x π 1+ 2  + 4 2 

2

π π   − 4 x2  

2

2

2

, x 2 > 1.273 ;





φ m = tan −1  0.5x 1 2 x 2 2 + 0.5x 1 x 2 x 1 2 x 2 2 + 4  − 0.5x 1 2 + 0.5 

7

Am =



2x 2 πx1 1+

π  + 2 

π2 π   − 4 4x 2  

π  + x12 π2  2  1

φm

x1

x12 x 2 2 + 4 .

2

π2 π   − 4 4x 2  

φ m = tan −1 x 3 − 0.125π2 x12 +

x1 x2

2

; x 3 = 2π2 x12 x 2 2 + 2πx1x 2 4π2 x12 x 2 2 + 4 ;

πx1 4π2 x12 x 2 2 + 4 . 16 x 2

355

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356 Rule

Kc

Ti

Comment

6.283(0.86φ − 1.50) ωφ

2.356 < φ < 2.705 ; M s = 2. 0 .

G p ( jω2.53 )

0.52

4.25 ω 2.53

φ = 2.53 radians (‘good’ choice).


0.35 K m τ m

7τm

Model: Method 2


0.35 K m τ m

13.4τm

Model: Method 2; ‘AMIGO’ tuning rule; M s = M t = 1. 4 .

Clarke (2006). Model: Method 4

 0.222 0.444  ,    K m τm K m τm 

[1.592τm ,3.183τm ]

Hägglund and Åström (2004) – continued. Model: Method 1

Litrico and Fromion (2006). Litrico et al. (2006).

8

Kc =

3.43 − 1.15φ

( )

G p jωφ

Kc

Ti

0.333 K m τm

6τm

8

A m = 12.7dB , φm = 44.50 .


A   − Am   1.571 − 20m 10 sin  0.0175φm + 1.57110 20   , K m τm       Am   − Am   Ti = (0.637 τm )10 20 tan  0.0175φm + 1.57110 20   ; A m in dB, φm in degrees.      

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Rule

Kc

357

Comment

Ti

Robust


1  2λ + τ m K m  [λ + τ m ]2

   

Boudreau and McMillan (2006). Model: Method 1

1 Km

 2λ + τ m   [λ + τ ]2 m  x1 K m τ m

   

Ogawa (1995). Model: Method 1; coefficients of K c and Ti deduced from graphs.

Alvarez-Ramirez et al. (1998). Model: Method 1

9

x1

x2

0.45 0.39 0.34 0.30 0.27

11 12 13 14 15

1 Km

 1 1    + T T m   CL

2λ + τ m

9

4(λ + τm ) 2λ + τ m

2

>

Overdamped response.

x 2τm

20% uncertainty in process parameters. 30% uncertainty in process parameters. 40% uncertainty in process parameters. 50% uncertainty in process parameters. 60% uncertainty in process parameters. Tm + τ m

TCL > τ m

λ ∈ [1 K m , τm ] (Chien and Fruehauf, 1990); λ = 3τ m (Bialkowski, 1996); λ > τ m + Tm (Thomasson, 1997); 1.5τm ≤ λ ≤ 4.5τm (Zhang et al., 1999b).

From a graph, λ = 1.5τ m ….overshoot = 58%, settling time = 6τ m λ = 2.5τ m ….overshoot = 35%, settling time = 11τ m λ = 3.5τ m ….overshoot = 26%, settling time = 16τ m λ = 4.5τ m ….overshoot = 22%, settling time = 20τ m . λ = e max / 100 K m ; e max = maximum output error after a step load disturbance (Andersson, 2000, p. 12); Ti se − τ ms y = (Chen and Seborg, 2002);   2  d  desired K c (λs + 1) λ = τ m (Smith, 2002); λ = 1.65τm , minimum IAE, servo (Panda et al., 2006); λ = 1.15τm , minimum IAE, regulator (Panda et al., 2006); λ = 2.65τm , overshoot (servo) ≈ 20% (Panda et al., 2006); λ = τm , Maximum load rejection capability (Boudreau and McMillan, 2006); λ > τm (Yu, 2006, p. 19).

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358 Rule

Kc

Zhong and Li (2002). Model: Method 2

2 x1 + τm

Smith (2002). Skogestad (2004a). Model: Method 1

Ti

Comment

K m (x1 + τm )2

2 x1 + τ m

x1 not specified.

1 K m τm

τm

Model: Method 1

10

4 K cK m

Kc

Ultimate cycle

Litrico et al. (2007).

  1.234 K u 10   ∧

11

Kc ^

0.37 K u

−

Am 20

A  − m  φm = 900 1 − 10 20      Model: Method 10

∞ Ti ^

1.5 Tu

Representative result; A m = 10 dB; φm = 430 . Other methods

Penner (1988). Model: Method 1

10

0.50 K m τm

∞

Maximum closed loop gain = 1.0.

0.58 K mτm

10τ m


0.8 K mτm

5.9 τ m


K c ≥ ∆d 0 ∆y max , ∆d 0 = maximum magnitude of a sinusoidal disturbance over all frequencies, ∆y max = maximum controlled variable change, corresponding to the sinusoidal disturbance.

11

A   − Am   ∧  − m  K c = 1.234 K u 10 20 sin  0.0175φm + 1.57110 20   ,         Am   − Am ∧   Ti =  0.159 Tu 10 20 tan  0.0175φ m + 1.57110 20      

   ; A m in dB, φm in degrees.   

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  1 4.1.2 Ideal PID controller G c (s) = K c 1 + + Td s    Ti s E(s)

R(s)

+

−

  U(s) 1 K c 1 + + Td s   Ti s 

K m e − sτ s

Y(s)

m

K m e − sτ m s

Table 59: PID controller tuning rules – IPD model G m (s) = Rule

Kc

Callender (1934). τm = 1 ; Model: Method 1 Ziegler and Nichols (1942).

0.2155 K m

x1 K m τ m

2τm

0.5τm

Ford (1953). Model: Method 3

1.48 K m τm

2τ m

0.37τ m

1.36 K m τm

∞

0.265τm

Åström and Hägglund (1995), p. 139. Hay (1998), p. 188.

0.94 K m τm

2τ m

0.5τ m

Hay (1998), p. 199. Model: Method 1; K c , Td deduced from graphs.

10.0


1.1 K m τm

2τm

0.53 K m τm

4τm

0.8τm

0.32 K m τm

4τm

0.8τm

x1 K m τ m

2. 5τ m

0.4τm

Alfaro Ruiz (2005a).

Ti

0.1 K m

Comment

Td

Process reaction 2.529 ∞ 5.0

Decay ratio = 0.11. Decay ratio = 0. 1.2 ≤ x1 ≤ 2.0 ; Model: Method 2 Decay ratio 2.7:1.

Model: Method 1

Ultimate cycle Ziegler–Nichols equivalent. 0.4 K m τm

3.2 τ m

0.8τ m

Model: Method 3

0.5 K m τm

∞

0.5τm

Model: Method 3

0.55τ m

K m τ m = 0. 1

0.30τ m

K m τ m = 0. 2

0.25τ m

K m τ m = 0.3

0.25τ m

K m τ m = 0. 4

0.25τ m

K m τm = 0.5, 0.6

0.5τm

Quarter decay ratio. ‘Some’ overshoot. ‘No’ overshoot.

4.0 2.5 2.0

3. 2 K m τ m

1.8

2

2.0 ≤ x1 ≤ 3.33 ; Model: Method 1

359

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360 Rule

Kc


Paz Ramos et al. (2005).

Åström and Hägglund (2004). Model: Method 1

Alenany et al. (2005). Model: Method 1 Regulator – Gorecki et al. (1989). Model: Method 1 Leonard (1994). Model: Method 1

Ti

Comment

Td

Minimum performance index: regulator tuning x1 K m τ m x 2τm x 3τ m


1

04-Apr-2009

Coefficient values: 1.49 0.59 Minimum ISE. 1.66 0.53 Minimum ITSE. 1.83 0.49 Minimum ISTSE Minimum performance index: servo tuning ∞ x1 K m τ m x 3τ m 1.37 1.36 1.34

Coefficient values: 0.49 0.45 0.45 ∞ 2.1883τm

1.03 0.96 0.90 0.5992 K m τm

Minimum ISE. Minimum ITSE. Minimum ISTSE Minimum IAE; Model: Method 1

Minimum performance index: other tuning x1 K m τ m x 2τm x 3τ m x1

x2

0.139 76.9 0.261 23.3 0.367 12.2 0.460 7.85 0.543 5.78 0.645 K m τm 1.104 K m τm

x3

Coefficient values: M max x1

0.346 0.365 0.378 0.389 0.400

1.1 1.2 1.3 1.4 1.5

x2

x3

M max

∞

0.616 4.58 0.681 3.82 0.740 3.28 0.793 2.89 0.841 2.61 0.357 τm

0.410 1.6 0.418 1.7 0.426 1.8 0.434 1.9 0.440 2.0 Optimum servo.

2.405τm

0.417 τm

Optimum regulator.

Direct synthesis: time domain criteria Pole is real and Tm has maximum ∞ 1 1 + 4(Tm τm ) Kc attainable 2.785 K m τm 3.731τm 0.263τm multiplicity. OS (step input) < 0.74 K m τm 12.2 τ m 0.41τ m 10%; Minimum IAE (disturbance 0.47 K u 3.05Tu 0.10Tu ramp).

τ + 4Tm − Kc = m e K m Tm τm

τ m + 2 Tm Tm

.

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Kc

Ti

Td

2

Kc

Ti

Td

3

Kc

Ti

Td

x1 K m τ m

x 2τm

x 3τ m

Wang and Cluett (1997).

Cluett and Wang (1997). Model: Method 1

Rotach (1995). Model: Method 6

x1

x2

x3


0.9588 3.0425 0.3912 1 0.6232 5.2586 0.2632 2 0.4668 7.2291 0.2058 3 1.21 K m τm 1.60τ m

361

Comment Model: Method 1 TCL = x 4 τ m x2

x3

x4

0.3752 9.1925 0.1702 0.3144 11.1637 0.1453 0.2709 13.1416 0.1269 0.48τ m

4 5 6

Damping factor for oscillations to a disturbance input = 0.75. Chidambaram Model: Method 1 1.2346 K m τm 4.5τ m 0.45τm and Sree (2003). Sree and Chidambaram Model: Method 1 0.896 K m τm 2.5τ m 0.55τm (2005b). 4 Chen and Seborg Model: Method 1 Ti Td Kc (2002).

2

Kc =

1 , Ti = (1.4020TCL + 1.2076)τm , K m τm (0.7138TCL + 0.3904) Td =

3

Kc =

τm ; τm ≤ TCL ≤ 16τm , ξ = 0.707 . 1.4167TCL + 1.6999

1 , Ti = (1.9885TCL + 1.2235)τm , K m τm (0.5080TCL + 0.6208) Td =

4

τm ; τm ≤ TCL ≤ 16τm , ξ = 1 . 1.0043TCL + 1.8194

τ (3TCL + 0.5τm ) T s(1 + 0.5τms )e −sτ m  y , Ti = 3TCL + 0.5τm , = i , Kc = m   3 3 K d K c (TCLs + 1)  desired m (TCL + 0.5τ m ) 2

Td =

2

3

3

1.5TCL τm + 0.75TCL τm + 0.125τm − TCL . τm (3TCL + 0.5τm )

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362 Rule Sree and Chidambaram (2006). Model: Method 1

Kc

Ti

Td

Comment

1 K m τm

∞

0.5τm

p. 46.

4x12 x1 2 K m τm

(1 + )

0.8956 K m τm

Regulator – Gorecki et al. (1989). Chen et al. (1999a). Åström and Hägglund (2006), p. 244. Åström and Hägglund (2006), pp. 263–264.

0.5(1 + x1 ) τm x1 − 1

0.25(1 + x1 ) τm x1

2.5τm

0.5τm

p. 47. x1 > 1 ; x1 = 1.25 (Chidambaram and Sree, 2003). p. 48.

Direct synthesis: frequency domain criteria Model: Method 1 ∞ 0.75 K m τm 0.333τm

Low frequency part of the magnitude Bode diagram is flat. 1.661 AmK mτm

∞

0.4603 K m τm

7.880τm

0.390τm

M s = M t = 1.4 ; Model: Method 1

0.45 K m τm

8τm

0.5τm

M s = M t = 1.4 ; Model: Method 2

5

Td

6

Td

Model: Method 1

Robust λ + 0.25τ m 2 τm Zou et al. (1997), K (λ + 0.5τ ) 2λ + τ m 2λ + τ m m m Zou and Brigham (1998). Model: Method 8 or Method 9; 0.5τ m ≤ λ ≤ 3τ m .

Rice (2004). 2λ + τm Model: Method 1 K (λ + 0.5τ )2 m m

2λ + τ m

0.25τm + λ τm 2λ + τm

7

2 Recommended 1 0.5τm Lee et al. (2006). K (T + τ ) τm ≤ TCL ≤ 2τm . ∞ TCL + τm m CL m Model: Method 1 Desired closed loop response = e −sτ m (1 + sTCL ) .

5 6

7

Td = 1.273τm [1 − 0.6002A m (1.571 − φm )] .

rA   Td = τm 1.2732 − 1 m  . 1.6661   λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’; λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’; λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’; λ = 6.847 τm , ‘very conservative’.

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Rule Shamsuzzoha and Lee (2006a). Shamsuzzoha and Lee (2007a).

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

8

Also given by Shamsuzzoha et al. (2006c); λ , ξ not specified. 9 Model: Method 1 Ti Td K c

Ultimate cycle

Litrico and Fromion (2006).

8

Kc =

10

Ti

Kc

Ti , Td = x1K m (2λξ + τm − x 2 )

Ti = x1 + x 2 −

x1x 2 −

Model: Method 10

Td

0.167 τm 3 − 0.5x 2 τm 2 2 λ2 + x 2 τm − 0.5τm 2λξ + τm − x 2 , Ti 2λξ + τm − x 2

(

)

2 2  λ2 − 2λξx1 + x1 e− τ m λ2 + x 2 τm − 0.5τm , x 2 = x1 1 − 2 2λξ + τm − x 2 x1 

x1

 , 

x1 is a constant of a ‘sufficiently large value’. 9

Kc =

2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 Ti , , Ti = x1 + 2 x 2 − x1K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2

2 x1x 2 + x 2 2 − Td =

λ3 + 0.167 τm 3 − x 2 τm 2 + x 2 2 τm 2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 3λ + τm − 2 x 2 , Ti 3λ + τm − 2 x 2

  , x is a constant of a ‘sufficiently large value’;  1  From graphs: λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 .

 x 2 = x1 1 −  

10

3

 λ 1 −  e − τ m  x  1 

Tm

A  − Am  ∧  − m  K c = 1.234 K u 10 20 sin θ , θ = 0.0175φm + 1.57110 20  ;     A

Am

m ∧  x ∧  10 20   , Ti =  0.159 Tu  1 10 20 tan θ , Td =  0.159 Tu     x1 tan θ  1 + x1

A  − m with 2 ≤ x1 ≤ 6 , φm < 901 − 10 20  

  , A in dB, φ in degrees. m m  

363

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364



E(s)

−

+

  1 K c 1 + + Td s  Ti s  

U(s)

1 1 + Tf s

K m e − sτ s

m


Kc

Ti

Td

Y(s)

K m e − sτ m s Comment

Robust Tan et al. τm 0.463λ + 0.277 (1998b). 0 λ = 0.5 K mτm 0.238λ + 0.123 Model: Method 1 Tf = τm (5.750λ + 0.590) ; labelled H ∞ optimal. Zhang et al. (1999a). Rivera and Jun (2000). Rice and Cooper (2002). Model: Method 1 Ou et al. (2005a). Model: Method 1

1

2

3

4

Kc = Kc = Kc = Kc =

(

1

Kc

2λ + τ m

0

2

Kc

2(τ m + λ )

0

3

Kc

2λ + 1.5τ m

0.5τ m 2 + λτ m 2λ + 1.5τ m

4

Kc

3λ + τm

2λ + τm

K m λ2 + 2λτ m + 0.5τm 2 2(τm + λ ) 2

2

K m ( 2τ m + 4 τ m λ + λ )

(

2λ + 1.5τm

K m λ2 + 2λτ m + 0.5τm 2

)

, Tf =

, Tf =

)

λ2 τm

.

2λ2 + 4λτ m + τm 2 τ m λ2 2

2 τ m + 4 τ m λ + λ2

, Tf =

(6λ + τm )τm 4(3λ + τm )

.

0.5λ2 τ m λ2 + 2λτ m + 0.5τ m 2

1 4(3λ + τm ) 4λ3 , Tf = . 2 2 2 K m 12λ + 6λτ m + τm 12λ + 6λτ m + τm 2

.

λ not specified; Model: Method 1 λ not specified; Model: Method 1 Suggested λ = 3.162τ m .

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Rule Ou et al. (2005b). Model: Method 1 Wang et al. (2005). Arbogast and Cooper (2007). Model: Method 1

Kc

Ti

5

Kc

2λ + 1.5τm

6

Kc

7

365

Comment

Td

(2λ + τm )τm

λ > 0.3034τm

4λ + 3τm

(for stability).

2λ + 1.3866τm

Td

Kc

2TCL + 1.5τm

(TCL + 0.5τm )τm

λ not specified; Model: Method 1

0.465 K m τm

7.82τm

2TCL + 1.5τm 0.468τm

Tf = 0.297 τm

Recommended TCL = 3.16τm .

5

Kc =

6

Kc =

λ2 τ m 1 2λ + 1.5τm , Tf = . 2 2 2 K m λ + 2λτ m + 0.5τm 2λ + 4λτ m + τ m 2

1 2λ + 1.3866τm 0.3866τm (2λ + τm ) , Td = , K m λ2 + 3.2286λτ m + 0.4896τm 2 2λ + 1.3866τm Tf = τm

7

Kc =

(

2(2TCL + 1.5τm ) 2

K m 2TCL + 4τm TCL + τm

2

)

, Tf =

0.3866λ2 − 0.2494λτ m − 0.1247 τm 2

TCL 2 τm 2

2TCL + 4TCL τm + τm 2

λ2 + 3.2286λτ m + 0.4896τm 2 .

.

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366

  Td s 1 + 4.1.4 Controller with filtered derivative G c (s) = K c 1 +  Ti s 1 + sTd  N  R(s)

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

E(s)

−

+

    s 

U(s)

K m e − sτ s


Kc

Ti

m

     

Y(s)


Td

Direct synthesis: frequency domain criteria Kristiansson and Lennartson (2002a).

1


2

Kc

Ti

Kc

2λ + τ m

Yu (2006), p. 19. 2 Model: Method 1 K (λ + 0.5τ ) m m

1

Kc = Ti =

2.5ωu Km

2λ + τ m

   − 0.08 + 0.055 + 0.059  2  K1 K1  

(

)

(

)

Model: Method 1

Td

Robust τ m (λ + 0.25τ m ) 2λ + τ m τ m (λ + 0.25τ m ) 2λ + τ m

(

)

2 2.5  2 0.059 K1 + (0.055 K1 ) − 0.08  − ,  ωu  2.3 − 2K1 0.4(10 + 1 K1 ) 

2

Kc =

(

−1

−1

)

2 Td 0.059 K1 + (0.055 K1 ) − 0.08 , Tf = , 0.5 ≤ K1 ≤ 1 . Tf 0.16ωu (10 + 1 K1 )

2λ + τm

K m (0.5λ + τm )2

N = 10; λ > τm .

 0.059 K 1 2 + (0.055 K 1 ) − 0.08  2  − ,  2.3 − 2 K 1  0.4(10 + 1 K 1 )  

2.5  2 0.059 K12 + (0.055 K1 ) − 0.08   1  Td = −  + 1 ,   0.4(10 + 1 K1 ) ωu  2.3 − 2K1  N  N=

N = 10

; λ ∈ [1 K m , τ m ] (Chien and Fruehauf, 1990).

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Rule

Kc

Ti

Td

Kc

Ti

Td

0.447 K m τm

7.528τm

0.189τm

3

Arbogast and Cooper (2007). Model: Method 1

Comment

N = 0.636

Recommended TCL = 3.16τm .

3

Kc =

(2TCL + 1.5τm )(2TCL 2 + 4τmTCL + τm 2 ) − TCL 2τm

(

2 2 0.5K m 2TCL + 4τmTCL + τm

Ti = 2TCL + 1.5τm − Td =

TCL 2 τm 2

2TCL + 4τm TCL + τm 2

τm (TCL + 0.5τm ) 2

Td TCL 2 τm . , Tf = 2 Tf 2TCL + 4τm TCL + τm 2

,

, 2

TCL τm 2TCL + 4τm TCL + τm

2

−

2

2TCL + 1.5τm −

N=

)

2

TCL τm 2

2TCL + 4τm TCL + τm

2

,

367

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368


   1  1 + sTd  K c 1 + Td Ti s   1 + s N 

E(s)

+

−

     

U(s)

K m e − sτ s

m


Kc

Bunzemeier (1998). Model: Method 5; Gp =

x1 K m τ m

Ti

Y(s)


Td

Process reaction

Kp s(1 + sTp ) n

.

Minimum IAE – Shinskey (1988), p. 143. Minimum IAE – Shinskey (1994), p. 74. Model: Method 1 Minimum IAE – Shinskey (1996), p. 117.

x 2τm

x 4 K m τm x1

x2

0% overshoot (servo). 10% overshoot (servo).

x 3τ m


N

n

0.21 0.68 0.47 0.35 0.30 0.27 0.25 0.23 0.22 0.22

0.730 0.280 0.24 5.60 0 1.570 0.796 0.94 5.99 2 1.200 0.669 0.62 6.03 3 0.981 0.587 0.46 5.99 4 0.866 0.530 0.38 6.02 5 0.828 0.487 0.34 6.01 6 0.801 0.452 0.32 6.03 7 0.782 0.424 0.30 5.97 8 0.767 0.401 0.29 5.99 9 0.756 0.381 0.27 5.95 10 Minimum performance index: regulator tuning Model: 0.93 Method 2; 1 . 57 τ 0 . 58 τ m m K mτm N not specified. 0.93 K mτm 0.93 K mτm

1.60τ m

0.58τ m

N = 10

1.48τ m

0.63τ m

N = 20

1.57 τ m

0.56τ m

Model: Method 2; N not specified.

b720_Chapter-04

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FA


Kc

Minimum IAE – Shinskey (1996), p. 121.

0.56K u

Ti

Comment

Td

Model: Method 1; N not specified. Direct synthesis – frequency domain criteria 1 ∞ 0.45τm Kc

Hougen (1979), pp. 333–334.

0.39Tu

0.15Tu

Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . 2 ∞ 10[log10 τ m +1.65] Five criteria are Kc fulfilled; N = 10. Robust λ ∈ [1 K m , τ m ] 0.5τm Chien (1988). K (λ + 0.5τ )2 0.5τ m 2λ + 0.5τ m (Chien and m Model: Method 1 m Fruehauf, 1990); 2λ + 0.5τm N = 10. 2λ + 0.5τ m 0.5τ m K m (λ + 0.5τm )2

Hougen (1988). Model: Method 1

2λ + 0.5τm Chien (1988). K (λ + 0.5τ )2 m Model: Method 1 m 0.5τm

2λ + 0.5τ m

0.55τm

0.5τ m

2.2λ + 0.55τm

K m (λ + 0.5τm )2

Tuning rules converted to this equivalent controller architecture. 3 0.5τ m 3λ + 0.5τm λ not specified; Kc Model: Method 1 Suggested 4 2λ + τ m 0.5τ m λ = 3.162τ m . Kc

Zhang et al. (1999a). Rice and Cooper (2002). Model: Method 1

1

Values deduced from graph: 0.1 0.2 K mτm

0.3

0.4

0.5

Kc

9

4.2

2.8

2.1

1.7

K mτm

0.6

0.7

0.8

0.9

1.0

Kc

1.45

1.2

1.1

0.95

1 2 Equations deduced from graph; K c ≈ ' 10 Km 3

4

Kc = Kc =

0.5τm

(

K m 3λ2 + 1.5λτ m + 0.25τm 2

(

2λ + τm 2

λ ∈ [1 K m , τ m ] (Chien and Fruehauf, 1990); N = 11.

K m λ + 2λτ m + 0.5τm

2

)

)

, N=

, N=

0.85  τ −  log10  m' T   m

   

, Km =

K 'm Tm'

.

(3λ + 0.5τm )(3λ2 + 1.5λτm + 0.25τm 2 ) . λ3

λ2 + 2λτ m + 0.5τ m 2 λ2

.

369

b720_Chapter-04

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FA


370 Rule

Kc

Ti

Td

2λ + 0.5τm

0.5τm

Kc

2TCL + τm

0.5τm

0.435 K m τm

7.324τm

0.5τm

Rice (2004). 2λ + 0.5τm Model: Method 1 K (λ + 0.5τ )2 m m Arbogast and Cooper (2007). Model: Method 1

6

Comment N=∞ 5

N = 1.684

Recommended TCL = 3.16τm . Ultimate cycle

Luyben (1996). Model: Method 10

0.46K u

2.2Tu

0.16Tu

Maximum closed loop log modulus of +2dB ; N = 10.

Belanger and Luyben (1997). Model: Method 1

3.11K u

2.2Tu

3.64Tu

N = 0.1

5

λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’; λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’; λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’; λ = 6.847 τm , ‘very conservative’.

6

Kc =

(

2(2TCL + τm ) 2

K m 2TCL + 4τm TCL + τm

2

)

, N=

TCL 2 + 2TCL τm + 0.5τm 2 TCL 2

.

b720_Chapter-04

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371



+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 


  2 U(s) K m e − sτ m  b f 0 + b f 1s + b f 2 s 2  1 + a f 1s + a f 2 s s s 


Kc

Ti

Kc

0.667 τm

Y(s)

K m e − sτ m s

Td

Comment

0.25τm

Model: Method 1

Robust Shamsuzzoha et al. (2007). Yu (2006), p. 18. Model: Method 1

1

Kc = −

1

0.46K u


b f 0 = 0 , b f 1 = 0.16Tu , b f 2 = 0 , a f 1 = 0.016Tu , a f 2 = 0 .

3  4τm λ , b f 0 = 1 , b f 1 = x1 1 +  e τ m K m x1 (6τm + 18λ − 6b f 1 ) x1    2

N = ∞ , af1 = af 2 = 0 .

x1

 − 1 , b f 2 = 0 ,  

2b f 1τm + τm + 12λτ m + 18λ2 + x1 ; x1 , λ not specified explicitly; 6τm + 18λ − 6 x1

   

b720_Chapter-04

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372



    T s β d α + K Td  c  1+ s  N   E(s)

−

R(s) +

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

    s 

U(s)

− +

K m e − sτ s

Table 64: PID controller tuning rules – IPD model G m (s) = Rule Minimum IAE – Shinskey (1988), p. 143.

Minimum IAE – Shinskey (1994), p. 74. Minimum IAE – Shinskey (1988), p. 148. Minimum IAE – Shinskey (1996), p. 121.

Kc

Ti

Td

m

Y(s)


Minimum performance index: regulator tuning α = 0 , β =1, 1.28 1.90τ m 0.48τ m N =∞. K mτm Model: Method 2 α = 0 , β =1, 1.28 1.90τ m 0.46τ m N =∞. K mτm Model: Method 1 α = 0 , β =1, 0.82 K u 0.48Tu 0.12Tu N =∞. Model: Method 1 α = 0 , β =1, 0.77 K u 0.48Tu 0.12Tu N =∞. Model: Method 1

b720_Chapter-04

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FA


Rule

Ti

Td

Comment

K m τm

2.3800τ m

0

α =1

K m τm

2.4569 τ m

0.3982τ m

α = 1, β = 1, N =∞.

Kc

Minimum ISE – 0.6330 Arvanitis et al. (2003a). 1.4394 Model: Method 1 0.6114 Arvanitis et al. 1.2986 (2003a). Model: Method 1

K m τm

2.7442 τ m

0

α =1

K m τm

3.2616τ m

0.4234τ m

α = 1, β = 1, N =∞.

∞

∫ [e ( t) + K 2

Minimise performance index

2 m

]

u 2 ( t ) dt .

0

0.6146 K m τm Arvanitis et al. 1.1259 K m τm (2003a). Model: Method 1

2.6816τ m

0

α =1

6.7092τ m

0.4627τ m

α = 1, β = 1, N =∞.

∞

2   du   Minimise performance index e 2 ( t ) + K m 2   dt .  dt    0  Minimum performance index: servo tuning 0 0.6270 K m τm 2.4688τ m α =1

∫

Minimum ISE – Arvanitis et al. (2003a). 1.5221 K m τm Model: Method 1 0.6064 K m τm Arvanitis et al. 1.4057 K m τm (2003a). Model: Method 1

2.1084 τ m

0.3814τ m

α = 1, β = 1, N =∞.

2.8489 τ m

0

α =1

2.5986τ m

0.4038τ m

α = 1, β = 1, N =∞.

∞

∫ [e ( t) + K 2


2 m

]

u 2 ( t ) dt .

0

0.6130 K m τm Arvanitis et al. 1.3332 K m τm (2003a). Model: Method 1

2.7126τ m

0

α =1

3.0010τ m

0.4167τ m

α = 1, β = 1, N =∞.

∞


2  2 2  du  e ( t ) + K m   dt .  dt    0 

∫

373

b720_Chapter-04

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374 Rule

Minimum ITAE – Pecharromán (2000a). Model: Method 11

Minimum ITAE – Pecharromán and Pagola (2000). Model: Method 11

Kc

Ti

x1K u

Comment

Td

Minimum performance index: other tuning 0 x 2 Tu Km = 1


x1

x2

α

x2

α

φc

0.049

2.826

0.506

− 164 0

0.066

2.402

0.512

− 160

0

0.218

1.279

0.564

− 1350

0.250

1.216

0.573

− 130 0

0.099

1.962

0.522

− 1550

0.286

1.127

0.578

− 1250

0.129

1.716

0.532

− 150 0

0.330

1.114

0.579

− 120 0

1.506

0.544

0

0.159

0.351

1.093

0.577

− 118 0

0.189

1.392

0.555

− 145

− 140 0 x 2 Tu

x1K u

x 3Tu

x1

x2

Coefficient values: x3

α

φc

1.672

0.366

0.136

0.601

− 164 0

1.236

0.427

0.149

0.607

− 160 0

0.994

0.486

0.155

0.610

− 1550

0.842

0.538

0.154

0.616

− 150 0

0.752

0.567

0.157

0.605

− 1450

0.679

0.610

0.149

0.610

− 140 0

0.635

0.637

0.142

0.612

− 1350

0.590

0.669

0.133

0.610

− 130 0

0.551

0.690

0.114

0.616

− 1250

0.520

0.776

0.087

0.609

− 120 0

0.509

0.810

0.068

0.611

− 118 0 Km = 1

0 x1K u x 2 Tu Minimum ITAE Coefficient values: – Pecharromán α α x1 x2 φc x1 x2 φc (2000b), p. 142. 0 0.0469 2.8052 0.5061 − 164.1 0.2174 1.2566 0.5642 − 135.00 Model: Method 11 0.0662 2.3876 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.0988 1.9581 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00 0.1271 1.6757 0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00

b720_Chapter-04

04-Apr-2009

FA


Rule Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 11

Comment

Kc

Ti

Td

x1K u

x 2 Tu

x 3Tu

x1

x2


α

φc

1.6695

0.3684

0.1323

0.6007

− 164.10

1.2434

0.4389

0.1507

0.6073

− 160.00

0.9901

0.4846

0.1587

0.6102

− 155.00

0.8378

0.5393

0.1562

0.6163

− 150.00

0.7383

0.5789

0.1548

0.6052

− 145.00

0.6728

0.6283

0.1488

0.6105

− 140.00

0.6196

0.6221

0.1376

0.6120

− 135.00

0.5948

0.6939

0.1326

0.6096

− 130.00

0.5513

0.7050

0.1190

0.6157

− 125.00

0.5271

0.7471

0.0992

0.6088

− 120.00

0.5229

0.8886

0.0862

0.6115

− 118.0 0 α = 0.6810

Taguchi and 0 0.7662 K m τm 4.091τ m Araki (2000). 1.253 K m τm 2.388τ m 0.4137τ m OS ≤ 20% ; α = 0.6642 , β = 0.6797 , N fixed but not specified. Model: Method 1 ∞ 0.82 K m τm 0.40τm β = 0.41 ; Taguchi et al. N = 10. (2002). N = 10; ∞ 0 . 4046 τ 1 m 0.8203 K m τm Model: 0.1 ≤ τm ≤ 1.0 . Method 1; ∞ 0.87 K m τm 0.44τm β = 0.44 ; α=0. N= ∞. Minimum IAE – 0.571 K m τm 0 6.139τm Tavakoli et al. M max = 2 ; α = 0.417 ; Model: Method 1. (2005). Direct synthesis: time domain criteria Kuwata (1987). 0 1 K m τm 3.33τm α =1 Model: Method 1 2 1.067 K m τm 0 . 313 τ α = 1, β = 1, m 6.25τm N =∞.

1

β = 0.3623 +

0.02488 . τm + 0.1035

375

b720_Chapter-04

04-Apr-2009

FA


376 Rule Chien et al. (1999). Model: Method 2

Hansen (2000). Model: Method 1 Chidambaram (2000b). Model: Method 1

Kc

Ti

Td

Comment

2

Kc

1.414TCL 2 + τ m

0

α =1

3

Kc

α = 1, β = 1, N =∞. Underdamped system response: ξ = 0.707 . 1.414TCL 2 + τ m

Td

0.938K m τ m

2.7 τ m

0.313τ m

0.64 K m τ m

3.333τ m

0.487 K m τ m

8.75τ m

0.9425 K m τm

2τ m

α = 0.833 , β =1, N = ∞ . α = 0.65

0

α = 0.557

In general, α = 0.6 (on average) or α = 0.5[1 − τ m Ti ] . 0.5τ m

N = ∞ ; β = 1 ; α = 0.707 or α = 0.65 .


0.35 K mτm

Arvanitis et al. (2003a). Model: Method 1

(8 − x1 )K m τm

4

(

)

16 2ξ2 + 1 32ξ2 + 4 K m τ m

Chidambaram and Sree (2003). Model: Method 1 Sree and Chidambaram (2006). Model: Method 1

Klán (2001). Model: Method 1

2

3

Kc = Kc =

(

(

)

0

4τ m x1

0

α = 1, 4 . x1 = 2 4ξ + 1

16ξ 2 + 4 τm 16 2ξ 2 + 1

α = 1, β = 1, N =∞.

2

(

)

4.5τ m

0.45τ m

0.9 K m τm

3.33τm

0

1.2 ≤ x1 ≤ 2

2τ m

0.5τ m

0.8956 K m τm

2.5τm

0.5τm

x1 K m τ m ,

α = 0.6 , β = 0 , N =∞. α = 0.65 . pp. 37–38. α = 0.65 , β = 1 , N =∞. p. 48. N = ∞ , β = 0 , α = 0.6 .

Direct synthesis: frequency domain criteria 0 α = 0.19 0.29 K m τm 8.9τm

1.414TCL 2 + τm

). 2

1.414TCL 2 + τm 2

)

+ 1 τm

0.3 < α < 0.5 .

1.2346 K mτm

K m TCL 2 2 + 1.414TCL 2τm + τm 2

(

(2ξ

M s = 1.4;

7τ m

K m TCL 2 + 0.707TCL 2 τm + 0.25τm

2

)

, Td =

0.25τm + 0.707TCL 2 τm . 1.414TCL 2 + τm

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Td

Comment

0

α =1

0.25τm + λ τm 2λ + τm

α = 0 ,β =1; N =∞, λ = 3τm .

Ti

377

Robust

Arvanitis et al. (2003a). Model: Method 1

2

4

4π τ m x 1 π 2 − x1

(

Kc

2λ + τm Rice (2006). K m (λ + 0.5τm )2 Model: Method 1

Kc =

[(8 − x )π 1

Ti

Kc

Td

equations for λ deduced from graphs.

4π2 2

2λ + τ m

α = 0.6 , β = 0 , N =∞. λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 ; 5


4

)

]

+ x12 K m τm

with 0 < x1 < π2 ; recommended x1 =

5

Kc =

2

π2  16 1− 1− 2 2  2  π 4ξ + 1

(

2

)

  , ξ > 0.3941 .  

2

Ti 3λ + 2x 2 τm − 0.5τm − x 2 , , Ti = x1 + 2 x 2 − x1K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2 2

2 x1x 2 + x 2 − Td =  x 2 = x1 1 −  

3

2

2

λ3 + 0.167 τm − x 2 τm + x 2 τm 2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 3λ + τm − 2 x 2 − , Ti 3λ + τm − 2 x 2 3

 λ 1 −  e − τ m  x  1 

Tm

  , x is a constant with a ‘sufficiently large value’.  1 

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378


+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

   1 + b f 1s  1 + a f 1s s 

+ U(s)

−

K m e − sτ s

K0


Alenany et al. (2005). Model: Method 1

Kc

Ti

Y(s) m

Td


Minimum performance index: servo tuning 1.104 K m τm 2.405τm 0.417 τm N=∞ a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . a f 2 = 1.866τm : OS ≈ 7% ; a f 2 = 2.38τm : OS ≈ 0% .

Normey-Rico et al. (2000). Model: Method 1

0.563 K m τm

Direct synthesis: time domain criteria 1.5τ m 0.667 τ m

N=∞

a f 1 = 0.13τm , b f 1 = 0 ; a f 2 = 0.75τm , b f 2 = 0.30τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0. 5 K m τ m .

Normey-Rico et al. (2001). Model: Method 1

1

Kc

(1.5 − x1 )τm

Td

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0. 5 K m τ m .

1

Kc =

1.5 − x1  τm 1 − x1 (1.5 − x1 ) 4 x1 + 1 − 4 x12 + 0.5x1  , Td = − x1τm , N = ; (1.5 − x1 )x1 K m τm  1.5 − x1 

Recommended x1 = 0.5 ; non-oscillatory response.

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Benjanarasuth et al. (2005). Model: Method 1

Kc

Ti

Td

0.37 K u

Tu

0

Comment

a f 1 = 0 , b f 1 = 0 ; a f 2 = Tu , b f 2 = 0 ; a f 3 = 0 , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . 0.63K u

0.76Tu

0.078Tu

N=∞

a f 1 = 0 , b f 1 = 0 ; a f 2 = 0.76Tu , b f 2 = 0 ; 2

a f 3 = 0.06Tu , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0 . 2

Huang et al. (2005b). Model: Method 1

Kc

Ti

[

0.6325 τm

]

N=∞

a f 1 = max 0.0063 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.0630τm ; 1.5

a f 3 = 0.6723τm

+ 0.6988K m τm

2.5

, b f 3 = 0.2652τm 2 ; a f 4 = 0 ,

a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .

Robust Lee and Edgar (2002). Model: Method 1

3

Ti

Kc

0

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u . 4

Zhang et al. (2002a). Model: Method 1

2λ 2 + 1.5τm

Kc

Td

N=∞

2

a f1 =

0.5λ 2 τ m 2

λ 2 + 2λ 2 τ m + 0.5τ m 2

, b f 1 = 0 ; a f 2 = 2λ 2 + τ m ,

b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , bf 5 = 0 , K 0 = 0 .

2

K c = 0.3221 + (0.3099 K m τm ) , Ti = τm (1.0630 + 1.1048K m τm ) .

3

Kc =

4

Kc =

2 0.25K u Ti 4 τm − τm + . , Ti = (λ + τm ) KuKm 2(λ + τm )

1 2λ 2 + 1.5τm (2λ 2 + τm )0.5τm , λ ,λ not specified. , Td = 1 2 K m λ 2 2 + 2λ 2 τm + 0.5τm 2 2λ 2 + 1.5τm

379

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380 Rule

Kc

Ti

Td

Comment

5 2λ 2 + (1 + δ)τm Td N=∞ Kc Zhang et al. λ 2 2δτm − r2 τm 2 (2λ + τm ) (2006). af1 = 2 , b f 1 = 0 ; a f 2 = 2λ 2 + τ m , Model: Method 1 λ 2 + 2λ 2δτm − r2 τm 2 − r1τm (2λ + τm )

b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , bf 5 = 0 , K 0 = 0 .

5

Kc =

2 1 2λ 2 + τm + δτm 2λ 2 + δτm T = , , d K m λ 2 2 + 2λ 2δτm − r2 τm 2 − r1τm (2λ + τm ) 2λ 2 + (1 + δ)τm

λ1 , λ 2 , λ , r1 , r2 , δ not specified.

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R(s)

+

F1 (s)

U(s)

Y(s)

Model

−

F2 (s) Table 66: PID controller tuning rules – IPD model G m (s) = Rule

Kc

Ti


Td

Minimum performance index: other tuning 0.45 K m 8τ m 0.5τ m

Åström and α = 1 , τm Tm ≤ 1 ; α = 0 , τm Tm > 1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; Hägglund (2004). bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , Model: Method 1 2

a f 3 = 0.2τm , a f 4 = 0.01τm ; K 0 = 0 .

Kaya and Atherton (1999), Kaya (2003). Model: Method 10

1

Direct synthesis: frequency domain criteria ∧ 0 1 0.313 K u 0.733 T u α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ;

∧  0.567  K m K u K0 = ∧ K m   τm T u

∧

bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 .

    

0.667 ∧

− 0.313 K u , b f 4

 ∧  K K u = 0.753 m ∧  τm T u 

    

0.333

−

 1  τm Tm .  τm  K m 

381

b720_Chapter-04

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382 Rule

Kc

Ti

Kc

Ti

Td

Comment

Robust 2


Td

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 3

Kc

Ti

x1

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .

2

3

Kc =

2 2  0.25K u  4 4 τm τm − τm + , − τm +  , Ti =  (λ + τm )  K u K m 2(λ + τm )  KuKm 2(λ + τm )

Td =

2 4 0.25K u  τm 3 τm τ (1 − 0.25K u K m τm )  2 + + m . 0.25τm − 2 (λ + τm )  6(λ + τm ) 4(λ + τm ) 0.5K u K m (λ + τm ) 

Kc =

2 2  0.25K u  4 τm 4 τm − τm + + x1  , Ti = − τm + + x1  (λ + τm )  K u K m 2(λ + τm ) KuKm 2(λ + τm ) 

2  τm τm (1 − 0.25K u K m τm )  x1 = τm 0.5 − + +  2 6( λ + τ m ) 4( λ + τ m ) 0.5K u K m (λ + τm )  

0.5

.

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4.2 IPD Model with a Zero

G m (s) =

K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s s


R(s)

−

+

 1   K c 1 + Ti s  

U(s) Model

Y(s)

Table 67: PI controller tuning rules – IPD model with a negative zero K (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s) = m or m s s Rule

Kc

2

3

4

Kc = Kc = Kc = Kc =

1.571

3

Kc

(

)

.

(

)

.

K m τm 1 + 3.142 Tm 4 τm 2 2.358

Comment

Direct synthesis: frequency domain criteria τm Tm3 < 1 ∞ 4 τm Tm3 > 1 Kc

Jyothi et al. (2001). Model: Method 1

1

Ti

Direct synthesis: time domain criteria 1 ‘Smaller’ τm Tm 4 . Kc ∞ 2 ‘Larger’ τm Tm 4 . Kc


K m τm 1 + 4.712 Tm 4 τm 2 1.57

K m τm 1 + 9.870(Tm3 τm )2 0.79 K m τm 1 + 2.467(Tm3 τm )2

. .

383

b720_Chapter-04

Rule

Kc


0.5 K m Tm 4

Kc =

FA


384

5

04-Apr-2009

5

Ti

Kc

0.79 K m τm 1 + 2.467(Tm 4 τm )2

.

∞

Comment τm Tm 4 < 1 τm Tm 4 > 1

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4.3 FOLIPD Model G m (s) =

K m e − sτm s(1 + sTm )


E(s)

+

−

 1   K c 1 + Ti s  

U(s)

K m e − sτ s(1 + sTm ) m

Y(s)

Table 68: PI controller tuning rules – FOLIPD model G m (s) = Rule

Kc

Ti

K m e − sτ m s(1 + sTm )

Comment

Process reaction Quarter decay ratio Coon (1956), (1964). criterion. ∞ K m (τ m + Tm ) Model: Method 1; Coefficient values: coefficients of K c τ m Tm x1 τ m Tm x1 τ m Tm x1 deduced from graphs. 0.020 5.0 0.25 2.2 4.0 1.1 0.053 4.0 0.43 1.7 0.11 3.0 1.0 1.3 Minimum performance index: regulator tuning Minimum IAE – 0.556 Shinskey (1994), Model: Method 1 3.7(τ m + Tm ) K m (τ m + Tm ) p. 75. Minimum IAE – 0.952 Shinskey (1994), Model: Method 1 4(Tm + τ m ) K m (Tm + τ m ) p. 158. x1

385

b720_Chapter-04

Rule Minimum ITAE – Poulin and Pomerleau (1996). Model: Method 1 Process output step load disturbance.

Process input step load disturbance.

Velázquez-Figueroa (1997). Model: Method 1

Velázquez-Figueroa (1997). Model: Method 1


Kc =

FA


386

1

04-Apr-2009

1

Kc

Ti

Comment

Kc

x 1 (τ m + Tm )

K c and Ti coefficients deduced from graph.

τ m Tm

x1

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397

0.029  Tm    K m  τm 

Coefficient values: x2 τ m Tm 0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311

1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0

0.157

∞

x1

x2

4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099

0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314

0.1 ≤

τm ≤ 0.33 Tm

0.195 τ τ  0.1 ≤ m ≤ 0.5 6.824Tm  m  Tm  Tm  Minimum performance index: servo tuning 0.528 τ 0.031  Tm  0.1 ≤ m ≤ 0.33 ∞   Tm   K m  τm  0.406 0.119 τ  τm  0.042  τm  0.1 ≤ m ≤ 0.5     8 . 807 T T m T  m K m  Tm   m Direct synthesis: time domain criteria 0.5 ∞ K (τ + T )

0.044  τm    K m  Tm 

m

m

0.122

m

x2 Tm 2 +1 . K m (τm + Tm ) x1 (τm + Tm )2

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Rule 2

Huba and Žáková (2003). Model: Method 1

Kc

Ti

Kc

∞

387

Comment

Representative tuning rules; tuning rules for other τ m Tm values may be obtained from a graph. 0.193 K m τ m

3.717 τ m

τ m Tm = 0.5

0.207 K m τ m

3.386τ m

τ m Tm = 1

0.219 K m τ m

3.127 τ m

τ m Tm = 2

Direct synthesis: frequency domain criteria Maximise crossover 3 1.26x 1 frequency. ∞ Kmτm

Hougen (1979), p. 338. Model: Method 1 Poulin and Pomerleau 2.13 0.34 K u or (1999). K m Tu

Robust ∞

Velázquez-Figueroa 0.833(2λ + Tm + τm ) (1997). K m (λ + τm )2

McMillan (1984). Model: Method 1

4

Kc

Tuning rules developed from K u , Tu . ^

K u tan 2 φ m

Perić et al. (1997). Model: Method 8

− 2Tm + τm + 4Tm

3

4

2

2

2 Tm + τ m − τ m + 4 Tm

K m τ m 2e

0.1592 Tu tan φ m

1 + tan 2 φ m

2

Kc =

Model: Method 1; λ not specified.

Ultimate cycle Ti

^

2

Model: Method 1; M max = 5 dB .

1.04Tu

2

.

2 Tm

x 1 values recorded deduced from a graph:

τ m Tm

1

2

3

4

5

6

7

8

9

10

x1

0.31

0.19

0.14

0.11

0.09

0.078

0.068

0.06

0.053

0.05

2   T 0.65   1.477 Tm  1   m  Kc =   , Ti = 3.33τm 1 +   . τ K m τ m 2 1 + (Tm τ m )0.65    m  

b720_Chapter-04

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FA


388

  1 + Td s  4.3.2 Ideal PID controller G c (s) = K c 1 + Ti s   E(s)

R(s)

+

−

  U(s) 1 K c 1 + + Td s   Ti s 


Y(s)

Table 69: PID controller tuning rules – FOLIPD model G m (s) = Rule

Kc

Ti

Td


Comment

Minimum performance index: regulator tuning Minimum ISE – 0.6667 K m τm Model: Method 1 ∞ Tm Haalman (1965). 0 A m = 2.36 ; φ m = 50 ; M s = 1.9 . 0.717 Velázquez  1 Figueroa (1997). 0.062  Tm  Td Ti   Model: Method 1 K m  τm  Minimum performance index: servo tuning 0.75 0.73 Velázquez   τm  2 Figueroa (1997). 0.064  Tm    1 . 2 T T m i   T  Model: Method 1 K m  τm   m Direct synthesis: time domain criteria Van der Grinten ∞ 1 K m τm Tm + 0.5τ m Step disturbance. (1963). −2 ω d τ m Stochastic e Model: Method 1 3 disturbance. ∞ Td K τ m m

Tachibana (1984).

1

2

λ K m (1 + λτ m )

τ  Ti = 3.537Tm  m   Tm 

0.951

∞

τ  , Td = 1.458Tm  m   Tm 

Tm

0.754

.

Ti = 8.071τm , 0 < τm Tm ≤ 0.0333 ; Ti = 22.183τm − 0.4701Tm , 0.0333 < τm Tm ≤ 0.2121 ; Ti = 8.2076Tm − 18.724τm , 0.2121 < τm Tm ≤ 0.3256 ; Ti = 1.7571Tm + 1.089τm , τm Tm > 0.3256 .

3

  T Td = 1 − 0.5e − ωd τ m + m e − ωd τ m τme − ωd τ m . τ m  

Model: Method 5

b720_Chapter-04

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FA


Rule

Kc

Ti

389

Comment

Td

Vítečková ∞ x1 K m τ m Tm (1999), Coefficient values: Vítečková et al. OS OS OS OS x x x1 x1 1 1 (2000a). 0% 0.641 15% 0.801 30% 0.957 45% Model: Method 1 0.368 0.514 5% 0.696 20% 0.853 35% 1.008 50% 0.581 10% 0.748 25% 0.906 40% 4 Chen and Seborg Model: Method 1 3TCL + τ m Td Kc (2002). Sree and Model: Chidambaram Method 1; T T 5 i d Kc (2006). p. 58. Direct synthesis: frequency domain criteria 6

2

Åström and 0.67e−1.9 τ + 0.2 τ Hägglund (1995), K (T + τ ) m m m pp. 212–213. −9.3τ + 6.6 τ 2 Model: Method 1 4.8e or Method 3. K m (Tm + τm )

0.56τm e1.1τ −1.1τ

ωpTm

Chen et al. (1999a).

4

2

0.13τm e13.6 τ −11.0 τ 14.2τm e −13.3τ + 8.6 τ

K m A mTd r1ωx Tm K mTd

2

0.68τm e 2.26 τ −1.8τ

2

2

M max = 1.4

M max = 2.0

∞

7

Td

Model: Method 1

∞

8

Td

Model: Method 1

(3TCL + τm )se−sτ m , K = (3TCL + τm )(Tm + τm ) ,  y =   c (TCL + τm )3 K c (TCLs + 1)3  d desired 2

Td =

0.895 − 0.100 5

6

7

8

Kc =

K m τm

Tm τm

Tm T 0.450 + 0.960 m τm τm τ , T = τ . , Ti = Tm m d Tm m 0.358 − 0.040 0.895 − 0.100 τm τm 0.895 − 0.100

  1 Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] + + Tds  . Tis   1 . Td = 2 −1 2ωp − 1.273ωp τm + Tm

Td =

1 ωx − 1.273ωx 2 τm + Tm −1

, ωx =

0.25π(2r1A m − 1)

(r A 2 1

2

m

)

− 1 τm

.

3

2

3TCL Tm + 3TCL Tm τm − TCL + Tm τm . (3TCL + τm )(Tm + τm )

b720_Chapter-04

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FA


390 Rule

Kc

Ti

Klán (2001). 0.67e−1.9 τ + 0.2 τ Model: Method 1 K m (Tm + τm )

Comment

Td

2

9

x 1Tm K mτm

O’Dwyer (2001a). Model: Method 1

Td

∞

Tm

A m = 0. 5 x 1 ; φ m = 0.5π − x 1 .

Representative coefficient values: Am φm x1 Am

x1

0.785 Åström and Hägglund (2006), p. 246.

Ti

10

2

45

0.524

0

Ti

Kc

Td

Robust Tm (τ m + 2λ ) τ m + Tm + 2λ τ m + Tm + 2λ

Rivera and Jun τ m + Tm + 2λ (2000). K m (τ m + λ )2 Model: Method 1

φm

3

60 0 M s = M t = 1. 4 ; Model: Method 1

λ not specified.

2 Recommended 1 0.5τm Tm + Lee et al. (2006). K (T + τ ) τ m ≤ TCL ≤ 2τ m . ∞ TCL + τm m CL m Model: Method 1 Desired closed loop response = e −sτ m (1 + sTCL ) .

Shamsuzzoha et al. (2006c). Model: Method 1

9

11

Ti

Kc

Td

λ , ξ not specified.

  1 Tuning rules converted from formulae developed for G c (s) = K c  [1 − α ] + + Tds  . Tis   Ti = 0.13(τm + Tm )e13.6 τ −11.0 τ ; Td = 14.2(τm + Tm )e−13.3τ +8.6 τ . 2

10

11

1 Kc = K m τm Kc =

Td =

2

T T 0.37 + 0.02 m 0.16 + 0.28 m  Tm  τm τm τm , Td = τm . 0.37 + 0.02  , Ti = T T τ m m   0.03 + 0.0012 0.37 + 0.02 m τm τm

Ti , Ti = x1 + Tm + x 2 − x 3 , x1K m (4λξ + τm − x 2 ) x 4 + x 2 (Tm + x1 ) + x1Tm −

0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ 4λξ + τm − x 2 − x3 ; Ti

x1 is a constant with ‘a sufficiently large value’, 2

x2 =

Tm x 5e − τ m

2

[

x 4 = Tm x 5e − τ m

x1

Tm

2

− x1 x 5 e − τ m x1 − Tm

x1

2

+ x1 − Tm

2

2

, x3 =

4λ2ξ 2 + 2λ2 + x 2 τm − 0.5τm − x 4 , 4λξ + τm − x 2 2

 λ2 2λξ  − 1 + x 2Tm , x 5 =  2 − +1 .  T Tm   m

]

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Shamsuzzoha and Lee (2007a).

12

Ti

Td

Comment

Ti

Td

Model: Method 1

0.25Ti


Kc

Ultimate cycle


13

Perić et al. (1997). Model: Method 8

12

Kc =

Td =

Ti

Kc

∧

K u cos φ m

x1Td

14

Td

Recommended x1 = 4 .

Ti , Ti = x1 + Tm + x 2 − x 3 , x1K m (4λ + τm − x 2 ) x 4 + x 2 (Tm + x1 ) + x1Tm −

3

2

0.167 τm − 0.5x 2 τm + x 4 τm + 4λ3 4λ + τm − x 2 − x3 ; Ti

x1 is a constant with ‘a sufficiently large value’, 2

x2 =

[

x1 x 5e − τ m

[

x1

]

2

[

− 1 − Tm x 5e − τ m Tm − x1

Tm

],x

−1

2

=

3

6λ2 − x 4 + x 2 τm − 0.5τm , 4λ + τ m − x 2

4

]

 λ x 4 = Tm x 5e − 1 + x 2Tm ; x 5 = 1 −  ; x 1  From graphs: λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ; 2

− τ m Tm

λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ; λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 . 13

14

2   T 0.65   1.111 Tm  1   m  Kc =   , Ti = 2τm 1 +   . τ K m τ m 2 1 + (Tm τ m )0.65    m  

 Td =  tan φ m +  

4 + tan 2 φ m x1

^

 Tu  .  4π 

391

b720_Chapter-04

04-Apr-2009

FA


392


 1  1 G c (s) = K c 1 + + Td s   1 + Tf s  Ti s E(s)

R(s)

−

+

U(s) K e −sτ m 1 m Tf s + 1 s(1 + sTm )

  1 K c 1 + + Td s  Ti s  

Y(s)


Kc

Kristiansson and Lennartson (2006). Model: Method 1; 1.7 < M max < 1.8 .

1

Kc =

10e

−8 τ m T m

K m Tm

2

Ti

Kc

2

Kc

Ti

Kc

Ti

(1 − 0.35e

4e

−4.5 τ m T m

K m Tm

2

(1.7 − e

(

− 4 τ m Tm

)(2.4τ

m

(

− τ m 1.3Tm

)(1.7τ

m

+ 0.9Tm ) , Td = Tm

)(2.4τ

m

Kc =

1.3Tm

) )

0.25 ≤ 1≤

τm 0.2291τm for stability (Ou et al., 2005b).

Tan et al. (1998a). Model: Method 1; λ not specified. Rivera and Jun (2000). Model: Method 1

4

Kc =

Kc =

Kc

Tm + 8.1633τ m

Tm τ m 0.1225Tm + τ m

Tf = 0.5549τ m

7

Kc

Tm + 5.3677 τ m

Tm τ m 0.1863Tm + τ m

Tf = 0.4482τ m

8

Kc

Tm + 3.9635τ m

Tm τ m 0.2523Tm + τ m

Tf = 0.2863τ m

9

Kc

2( τ m + λ ) + Tm

(0.463λ + 0.277)([0.238λ + 0.123]Tm + τm ) K m τm 2

Ti = Tm + 5

6

K m 3λ2 + 3λτ m + τm

2

)

, Tf =

6

Kc =

 0.0337Tm  τm 1 + . 2  K m τm  0.1225Tm 

7

Kc =

 0.0754Tm  τm 1 + . 2   0 . 1863 T K m τm  m 

8

9

, Tf =

λ3 3λ2 + 3λτ m + τ m 2

.

 0.1344Tm  τm 1 + . 2  K m τm  0.2523Tm  2(τ m + λ ) + Tm τ m λ2 Kc = , Tf = . 2 2 2 K m 2τ m + 4τ m λ + λ 2τ m + 4 τ m λ + λ2

Kc =

(

)

λ not specified.

τm , 5.750λ + 0.590

Tm τm τm , Td = . (0.238λ + 0.123)Tm + τm 0.238λ + 0.123

3λ + Tm + τm

(

2Tm (τ m + λ ) 2( τ m + λ ) + Tm

393

b720_Chapter-04

04-Apr-2009

FA


394


  Td s 1 + K c 1 + T  Ti s 1+ d  N 

E(s)

−

+

    s 

U(s)



Kc

Ti

     

Y(s)


Comment

Td

Direct synthesis: frequency domain criteria Kristiansson and Lennartson (2002a).

1

Td

Model: Method 1; 0.5 ≤ K1 ≤ 1 .

Tm (2λ + τ m ) 2λ + Tm + τ m

λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 10.

Ti

Kc

Robust Chien (1988). 2λ + τ m + Tm Model: Method 1 K m (λ + τ m )2

1

Kc = Ti =

2.5ωu Km

2λ + Tm + τ m

   − 0.08 + 0.055 + 0.059  2  K1 K1  

(

)

(

)

(

)

 0.059 K 1 2 + (0.055 K 1 ) − 0.08  2  − ,  2.3 − 2 K 1  0.4(10 + 1 K 1 )  

2 2.5  2 0.059 K1 + (0.055 K1 ) − 0.08  − ,  ωu  2.3 − 2K1 0.4(10 + 1 K1 ) 

−1

2.5  2 0.059 K12 + (0.055 K1 ) − 0.08   1  Td = −  + 1 ,   0.4(10 + 1 K1 ) ωu  2.3 − 2K1  N  N=

(

)

2 Td 0.059 K1 + (0.055 K1 ) − 0.08 , Tf = . Tf 0.16ωu (10 + 1 K1 )

−1

b720_Chapter-04

04-Apr-2009

FA


     1  1 + Td s   4.3.5 Classical controller G c (s) = K c 1 + Td s  Ti s   1 +  N   R(s)

   1  1 + sTd  K c 1 + Td Ti s   1 + s N 

E(s)

+

−

     

Y(s) K m e − sτ s(1 + sTm )

U(s)

m


Kc

Ti


Comment

Td

Minimum performance index: regulator tuning Minimum IAE – τ m = Tm ; 0.78 Shinskey (1994), K ( τ + T ) 1.38( τ m + Tm ) 0.66( τ m + Tm ) N = 10. m m m p. 75. Model: Method 1 Minimum IAE – Shinskey (1994), pp. 158–159.

1

Minimum ITAE – Poulin and Pomerleau (1996), (1997). Model: Method 1. Process output step load disturbance.

3

2

Kc Kc Kc

τm Tm N 0.2 0.4 0.6 0.8 1.0

τm Tm < 2

0.56τ m + 0.75Tm

Ti

Model: Method 1; N not specified. x 2 (τ m + Tm ) Tm

τm Tm ≥ 2

3.33 ≤ N ≤ 10

Coefficient values (deduced from graphs): τm x1 x2 x1 T N m

5.0728 4.9688 4.8983 4.8218 4.7839

0.5231 0.5237 0.5241 0.5245 0.5249

(

1.2 1.4 1.6 1.8 2.0

[

1

Kc =

0.926 , Ti = 1.57 τm 1 + 1.2 1 − e − Tm K m τm (1.22 − 0.03(Tm τm ))

2

Kc =

0.926 . K m τm (1 + 0.4(Tm τm ))

3

Kc =

τm x2 Tm 2 +1 ; 0 ≤ ≤ 2. (Tm N ) K m (τm + Tm ) x1 (τm + Tm )2

4.7565 4.7293 4.7107 4.6837 4.6669

τm

]) .

x2

0.5250 0.5252 0.5254 0.5256 0.5257

395

b720_Chapter-04

04-Apr-2009

FA


396 Rule

Kc

Poulin and Pomerleau (1996), (1997) – continued. Process input step load disturbance.

Ti

Comment

Td

Coefficient values (continued): 3.9465 0.5320 1.2 4.0337 3.9981 0.5315 1.4 4.0278 4.0397 0.5311 1.6 4.0278 4.0397 0.5311 1.8 4.0218 4.0397 0.5311 2.0 4.0099

0.2 0.4 0.6 0.8 1.0

0.5312 0.5312 0.5312 0.5313 0.5314

Direct synthesis: time domain criteria


Hougen (1979), p. 338.

0.5 K mτm

8τ m

N=∞

Tm

Direct synthesis: frequency domain criteria 0.7 0.3 ∞ 0 . 106 x 4 1 0.135 NTm τm Kmτm

Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . N = 5. Poulin et al. 0.5455 (1996). Tm M max = 1 dB K m (τm + 0.2Tm ) 21(0.2Tm + τ m ) Model: Method 1 Robust Tm λ ∈ [τ m , Tm ] Chien (1988). Tm 2λ + τ m (Chien and K m (λ + τ m )2 Model: Method 1 Fruehauf, 1990); 2λ + τ m N = 10. 2λ + τ m Tm K m (λ + τ m )2


2λ + τ m

K m (λ + τ m )

2

Tm

K m (λ + τ m )2

2λ + τ m

1.1Tm

Tm

2.2λ + 1.1τm

λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 11.

Tuning rules converted to this equivalent controller architecture.

4

x 1 values deduced from graph: τ m Tm

1

2

3

4

5

6

x1

6

3

2.1

1.8

1.6

1.3

τ m Tm

7

8

9

10

11

12

x1

1.1

1.0

0.9

0.8

0.7

0.7

b720_Chapter-04

04-Apr-2009

FA


397



+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 


  2 U(s) K m e − sτ  b f 0 + b f 1s + b f 2 s  1 + a f 1s + a f 2 s 2 s(1 + sTm ) s  m




Kc x1 K m τ m

Direct synthesis: time domain criteria 0 ∞

bf 0 = 1 2

ξ

1.6818 0.0 1.3829 0.1 1.1610 0.2 x1 K m τ m

x1

Coefficient values: ξ x1

0.9916 0.8594 0.7542

0.3 0.4 0.5

ξ

0.6693 0.6 0.6000 0.7 0.5429 0.8 0.35τm

∞

x1

ξ

0.4957 0.4569

0.9 1.0

bf 0 = 1

2

2

b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2τm , a f 2 = 0.01τm . x1

Tsang and Rad (1995). Model: Method 2

Comment

Td

b f 1 = Tm + 0.25τm , b f 2 = 0.25Tm τm , a f 1 = 0.2τm , a f 2 = 0.01τm .

x1


Ti

Y(s)

ξ

1.8512 0.0 1.5520 0.1 1.3293 0.2 0.809 K m τm

x1

Coefficient values: ξ x1

1.1595 1.0280 0.9246

0.3 0.4 0.5

0.8411 0.7680 0.6953

∞

b f 0 = 1 , b f 1 = Tm + 0.5τm , b f 2

0

ξ

x1

ξ

0.6 0.7 0.8

0.6219 0.5527

0.9 1.0

Maximum overshoot = 16%. = 0.5Tm τm , a f 1 = 0.12τm , 2

a f 2 = 0.0036τm .

   

b720_Chapter-04

04-Apr-2009

FA


398 Rule

Kc

Ti

Td

Comment

Direct synthesis: frequency domain criteria τ 1 0 ≤ m ≤ 2.5 ∞ 0.333τm Tm K m (λ + τm )

Shamsuzzoha et al. (2006a). Model: Method 1 b = 1 , b = T , b = 0 , a = 0.167 τ 2λ − τm , a = 0 , N = ∞ . f0 f1 m f2 f2 f1 m λ + τm

λ values deduced from graph: λ = 1.4τm , M s = 1.4 ; λ = 0.7 τm , M s = 1.6 ; λ = 0.4τm , M s = 1.8 ; λ = 0.35τm , M s = 2.0 .

b720_Chapter-04

04-Apr-2009

FA




    T s  α + β d K c T   1+ d s  N   E(s)

−

R(s) +

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

+

Kc

Ti

Y(s)



Td


Comment

Minimum performance index: regulator tuning Minimum IAE – α = 0 ,β =1, 1.16 Shinskey (1994), K ( τ + T ) 1.38( τ m + Tm ) 0.55( τ m + Tm ) N =∞. m m m p. 75. Model: Method 1

Minimum IAE – Shinskey (1994), p. 159.

1

Kc =

(

1

Kc

1.28

0.48τ m + 0.7Tm

Ti

K m τm 1 + 0.24(Tm τm ) − 0.14(Tm τm )2

), T

i

399

(

α = 0 ,β =1, N =∞. Model: Method 1

= 1.9τ m 1 + 0.75[1 − e − (T m

τm

)

)

] .

b720_Chapter-04

04-Apr-2009

FA


400 Rule

Kc 4 (8 − x1 )K m τm

Arvanitis et al. (2003b). Model: Method 1

Ti

Td

Comment

4τm x1

0

α =1

Minimum ISE. 2 Minimum performance index. 3

2

2

3

τ  τ  τ  τ  x1 = 2.6302 − 2.1248 m  + 2.5776 m  − 1.8511 m  + 0.82557 m  T T T  m  m  m  Tm  τ − 0.23641 m  Tm

5

 τ  + 0.044128 m   Tm

τ − 1.7164.10 −5  m  Tm

6

 τ  − 0.0053335 m   Tm

9

 τ  + 3.1685.10 −7  m   Tm

4

7

 τ  + 0.00040201 m   Tm

10

 τ  , 0 < m < 4.1 ; T m 

τ  τ x1 = 1.7110073 − 0.003554 m − 4.1 , m ≥ 4.1 .  Tm  Tm ∞

3

Performance index =

∫ [e ( t) + K 2

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x1 = 2.1054 − 0.76462 m  + 0.79445 m  − 0.51826 m  + 0.21771 m   Tm   Tm   Tm   Tm  τ − 0.06 m  Tm

5

 τ  + 0.010927 m   Tm

τ − 4.132.10 −6  m  Tm

9

6

 τ  − 0.0013006 m   Tm

 τ  + 7.6235.10 −8  m   Tm

7

 τ  + 9.717.10 −5  m   Tm

10

 τ  , 0 < m < 4.6 ; Tm 

τ  τ x1 = 1.69667833406 − 0.00148768 m − 4.6  , m ≥ 4.6 . T  m  Tm

  

4

8

  

8

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Arvanitis et al. (2003b) – continued. Arvanitis et al. (2003b). Model: Method 1

Ti

401

Comment

Td

Minimum performance index. 4 5

0

Ti

Kc

α =1

Minimum ISE. 6

∞

4

Performance index =

2  2 2  du  e ( t ) + K m   dt .  dt    0 

∫

τ  τ x1 = 1.6505623 − 0.0034985 m − 8  , m ≥ 2 ; T  m  Tm 2

3


5

 τ  + 8.2395.10 −5  m   Tm

τ + 4.2603.10 −7  m  Tm 5

Kc =

[

9

6

 τ  + 5.0912.10 −5  m   Tm

7

 τ  − 7.283.10 −6  m   Tm

  

8

10

 τ  τ  − 9.5824.10 −9  m  , m range not defined. Tm   Tm  12.5664(τm + Tm ) − 4

) ]

(

K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1

3.4674τm + 3.1416Tm − 1 12.5664(τm + Tm ) − 4 ]x 2 ; Ti = . x1 τm

with x1 = [0.5708 +

2

6

4

3

τ  τ  τ  τ  x 2 = 0.99315 + 1.9536 m  − 2.5157 m  + 1.9163 m  − 0.8893 m   Tm   Tm   Tm   Tm  τ + 0.26167 m  Tm

5

 τ  − 0.049809 m   Tm

τ + 2.0083.10 −5  m  Tm

9

6

 τ  + 0.0061098 m   Tm

 τ  − 3.7368.10 −7  m   Tm

10

7

 τ  − 0.00046593 m   Tm

 τ  , 0 < m < 6.0 ; Tm 

τ  τ x 2 = 1.96705757 + 0.01308189 m − 6  , m ≥ 6.0 . T  m  Tm

4

  

8

b720_Chapter-04

04-Apr-2009


402 Rule

Kc

Ti

Arvanitis et al. (2003b) – continued.

Td

Comment

Minimum performance index. 7 Minimum performance index. 8

∞

7

FA

Performance index =

∫ [e ( t) + K 2

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x 2 = 0.73114 + 2.3461 m  − 2.8509 m  + 2.0965 m  − 0.95252 m  T T T  m  m  m  Tm  τ + 0.27639 m  Tm

5

 τ  − 0.052089 m   Tm

τ + 2.0611.10 −5  m  Tm

6

 τ  + 0.0063413 m   Tm

9

 τ  − 3.8178.10 −7  m   Tm

4

7

 τ  − 0.00048064 m   Tm

  

8

10

 τ  , 0 < m < 4.5 ; T m 

τ  τ x 2 = 1.916423116 + 0.018175439 m − 4.5  , m ≥ 4.5 .  Tm  Tm ∞

8

Performance index =

2  2 2  du  e ( t ) + K m   dt .  dt    0 

∫

2

3


5

 τ  − 0.072771 m   Tm

τ + 2.8623.10 −5  m  Tm

9

6

 τ  + 0.0088395 m   Tm

 τ  − 5.2941.10 −7  m   Tm

10

7

 τ  − 0.00066864 m   Tm

 τ  , 0 < m < 4.5 ; Tm 

τ  τ x 2 = 1.9283751 + 0.016846129 m − 4.5  , m ≥ 4.5 . T  m  Tm

4

  

8

b720_Chapter-04

04-Apr-2009

FA


Rule

Ti

Td

Comment

4 τm x1

(8 − x1 ) τm 16

α = 1, β = 1, N =∞.

Kc

16 Arvanitis et al. (16 − 3x )K τ 1 m m (2003b). Model: Method 1


2

9

3

τ  τ  τ  τ  x1 = 1.5841 + 0.3101 m  + 0.38841 m  − 0.4622 m  + 0.21918 m   Tm   Tm   Tm   Tm  τ − 0.060691 m  Tm

5

 τ  + 0.010771 m   Tm

τ − 3.8302.10 −6  m  Tm

9

 τ  + 7.0316.10 −8  m   Tm

∞

10

Performance index =

6

 τ  − 0.0012466 m   Tm

∫ [e ( t) + K 2

2 m

4

7

 τ  + 9.1201.10 −5  m   Tm

  

10

  . 

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x1 = 0.066485 + 2.0757 m  + 1.0972 m  − 2.7957 m  + 1.9019 m   Tm   Tm   Tm   Tm  τ − 0.68418 m  Tm

5

 τ  + 0.14763 m   Tm

τ − 7.2368.10 −5  m  Tm

9

6

 τ  − 0.019733 m   Tm

 τ  + 1.397.10 −6  m   Tm

10

7

 τ  + 0.0016016 m   Tm

 τ  , 0 < m < 4.0 ; Tm 

τ  τ x1 = 2.128638 − 0.00560812 m − 4.0  , m ≥ 4.0 . T  m  Tm

4

  

8

8

403

b720_Chapter-04

04-Apr-2009

FA


404 Rule

Kc 11


Kc

Ti

Td

Comment

Ti

Td

α = 1, β = 1, N =∞.


11

Kc =

12.5664(τm + Tm ) − 4 K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ] with x1 = [0.5708 +

Ti =

12.5664(τm + Tm ) − 4 x1 2 . , Td = Tm − Tm x1 12.5664(τm + Tm ) − 4 2

12

3.4674τm + 3.1416Tm − 1 ]x 2 , τm

3


5

 τ  − 0.1458 m   Tm

τ + 5.83.10 −5  m  Tm

6

 τ  + 0.017833 m   Tm

7

4

 τ  − 0.0013561 m   Tm

  

8

10

9

 τ  − 1.0823.10 −6  m   Tm

 τ  , 0 < m < 3.5 , Tm 

τ  τ x 2 = 2.2036 − 0.0033077 m − 6.5  , m ≥ 3.5 . T  m  Tm ∞

13

Performance index =

∫ [e ( t) + K 2

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x 2 = −0.1185 + 4.781 m  − 3.9977 m  + 1.733 m  − 0.42245 m  T T T  m  m  m  Tm  τ + 0.057708 m  Tm

5

 τ  − 0.0038421 m   Tm

6

 τ  + 2.6386.10 −5  m   Tm

τ − 4.2486.10 −7  m  Tm

9

  . 

7

4

 τ  + 1.078.10 −5  m   Tm

  

8

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc


Ti

Comment

Td

Minimum performance index: servo tuning 4 4τm (8 − x1 )K m τm 0 α =1 x 1

Minimum ISE. 14 Minimum performance index. 15 Minimum performance index. 16

2

14

3


5

 τ  + 0.017215 m   Tm

τ − 6.5691.10 −6  m  Tm

6

 τ  − 0.0020706 m   Tm

9

 τ  + 1.2025.10 −7  m   Tm

4

7

 τ  + 0.00015505 m   Tm

  

8

10

 τ  , 0 < m < 5.5 ; T m 

τ  τ x1 = 1.6281745 − 0.001171 m − 5.5  , m ≥ 5.5 . T  m  Tm ∞

15

∫ [e ( t) + K 2

Performance index =

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x1 = 1.5693 + 0.1066 m  − 0.10101 m  + 0.05217 m  − 0.016147 m  T T T  m  m  m  Tm  τ + 0.0030412 m  Tm

5

 τ  − 0.00032187 m   Tm

−7 

τ − 1.2274.10  m T  m

9

6

 τ  + 1.2181.10 −5  m   Tm

 τ  + 3.7365.10 −9  m   Tm

10

 τ  , 0 < m < 4.6 . T m 

∞

16

Performance index =

2  2 2  du  e ( t ) + K m   dt  dt    0 

∫

τ  τ x1 = 0.780651417 + 4.7464131 m − 0.1 , 0 < m < 0.3 ; Tm  Tm  τ  τ x1 = 1.729934 − 0.051535 m − 0.3  , 0.3 ≤ m < 2 . T T m  m 

7

4

 τ  + 9.9698.10 −7  m   Tm

  

8

405

b720_Chapter-04

04-Apr-2009

FA


406 Rule

Kc


17

Kc

Ti

Td

Comment

Ti

0

α =1


17

Kc =

12.5664(τm + Tm ) − 4

[

3.4674τm + 3.1416Tm − 1 12.5664(τm + Tm ) − 4 ]x 2 , Ti = . x1 τm

with x1 = [0.5708 +

2

18

) ]

(

K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1

3


5

 τ  − 0.048068 m   Tm

6

 τ  + 0.0058771 m   Tm

7

 τ  − 0.00044707 m   Tm

  

8

10

9

τ + 1.9231.10 −5  m  Tm

4

 τ  − 3.5718.10 −7  m   Tm

 τ  , 0 < m < 4.5 ; T m 

τ  τ x 2 = 1.903177576 + 0.0181584 m − 4.5  , m ≥ 4.5 . T  m  Tm ∞

19

Performance index =

∫ [e ( t) + K 2

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x 2 = 0.48124 + 2.5549 m  − 2.896 m  + 2.0495 m  − 0.91115 m  T T T  m  m  m  Tm  τ + 0.26094 m  Tm −5 

5

 τ  − 0.048775 m   Tm

τ + 1.9095.10  m T  m

9

6

 τ  + 0.0059064 m   Tm

 τ  − 3.5335.10 −7  m   Tm

10

7

 τ  − 0.00044618 m   Tm

 τ  , 0 < m < 4.0 ; T m 

τ  τ x 2 = 1.87054524 + 0.021802 m − 4  , m ≥ 4.0 . T  m  Tm

4

  

8

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Arvanitis et al. (2003b) – continued. Arvanitis et al. 16 (2003b). (16 − 3x1 )K m τm Model: Method 1

Ti

407

Comment

Td

Minimum performance index. 20 α = 1, β = 1, N =∞.

(8 − x1 ) τm 16

4 τm x1

Minimum ISE. 21

∞

20

Performance index =

2  2 2  du  e ( t ) + K m   dt .  dt    0 

∫

2

3

τ  τ  τ  τ  x 2 = 0.31373 + 3.7796 m  − 5.0294 m  + 3.8589 m  − 1.7956 m  T T T  Tm   m  m  m τ + 0.52906 m  Tm

5

 τ  − 0.10079 m   Tm

τ + 4.0678.10 −5  m  Tm

6

 τ  + 0.012369 m   Tm

9

 τ  − 7.5694.10 −7  m   Tm

7

4

 τ  − 0.00094357 m   Tm

  

8

10

 τ  , 0 < m < 3.5 ; T m 

 τ τ x 2 = 1.86144723 + 0.021989 m − 3.5  , m ≥ 3.5 .  Tm  Tm 2

21

3

τ  τ  τ  τ  x1 = 1.2171 + 0.7339 m  + 0.083449 m  − 0.14905 m  + 0.030257 m  T T T  Tm   m  m  m τ + 0.0045567 m  Tm

5

 τ  − 0.0029943 m   Tm τ + 2.7505.10 −6  m  Tm

6

 τ  + 0.00056821 m   Tm 9

7

4

 τ  − 5.4957.10 −5  m   Tm

 τ  − 5.6642.10 −8  m   Tm

10

  . 

  

8

b720_Chapter-04

04-Apr-2009


408 Rule

Kc

Arvanitis et al. (2003b) – continued. Arvanitis et al. (2003b). Model: Method 1

Ti

Comment

Td

Minimum performance index. 22 23

Ti

Kc

Td

α = 1, β = 1, N =∞.

Minimum ISE. 24

∞

22

FA

∫ [e ( t) + K 2

Performance index =

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x1 = 0.083709 + 2.2875 m  − 0.18739 m  − 1.0059 m  + 0.79718 m   Tm   Tm   Tm   Tm  τ − 0.3036 m  Tm

5

 τ  + 0.067859 m   Tm

τ − 3.5473.10 −5  m  Tm

6

 τ  − 0.0093104 m   Tm

9

 τ  + 6.9487.10 −7  m   Tm

7

 τ  + 0.00077161 m   Tm

  

4

8

10

 τ  , 0 < m < 4.5 Tm 

τ  τ x 1 = 2.444547 − 0.0146274714 m − 4.5  , m ≥ 4.5 . T  m  Tm 12.5664(τm + Tm ) − 4 23 Kc = K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ] with x1 = [0.5708 + Ti =

12.5664(τm + Tm ) − 4 x1 2 . , Td = Tm − Tm x1 12.5664(τm + Tm ) − 4 2

24

3

3.4674τm + 3.1416Tm − 1 ]x 2 τm


5

 τ  − 0.17635 m   Tm

τ + 7.0439.10 −5  m  Tm

9

6

 τ  + 0.02158 m   Tm

 τ  − 1.3056.10 −6  m   Tm

10

7

 τ  − 0.0016402 m   Tm

 τ  , 0 < m < 4.5 Tm 

 τ τ x 2 = 2.124307 − 0.004844182 m − 4.5  , m ≥ 4.5 . T  Tm  m

4

  

8

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Ti

Arvanitis et al. (2003b) – continued.

Comment

Minimum performance index. 25 Minimum performance index. 26

∞

25

Td

Performance index =

∫ [e ( t) + K 2

2 m

]

u 2 ( t ) dt .

0

2

3

τ  τ  τ  τ  x 2 = −0.70861 + 7.4212 m  − 8.1214 m  + 4.8168 m  − 1.7217 m  T T T  Tm   m  m  m τ + 0.38998 m  Tm

5

 τ  − 0.056975 m   Tm

6

 τ  + 0.0053141 m   Tm

7

 τ  − 0.00030233 m   Tm

  

8

10

9

τ + 9.3969.10 −6  m  Tm

4

 τ  − 1.1861.10 −7  m   Tm

 τ  , 0 < m < 4.5 T m 

τ  τ x 2 = 2.1069751341 − 0.0017098 m − 4.5  , m ≥ 4.5 .  Tm  Tm ∞

26

Performance index =

2  2 2  du  e ( t ) + K m   dt .  dt    0 

∫

2

3


5

 τ  − 0.013386 m   Tm

τ − 5.307.10 −6  m  Tm

9

6

 τ  + 0.00059665 m   Tm

 τ  + 1.6437.10 −7  m   Tm

10

7

 τ  + 3.8633.10 −5  m   Tm

 τ  , 0 < m < 6.0 ; Tm 

 τ τ x 2 = 2.1111 − 0.001025 m − 6  , m ≥ 6 . T  Tm  m

4

  

8

409

b720_Chapter-04

04-Apr-2009

FA


410 Rule

Kc

Ti

x1 K m

Taguchi et al. (1987). x1 Model: Method 1 12.06 3.93 2.19 1.50 1.14 0.92 0.77 Minimum ITAE – Pecharromán and Pagola (2000). Model: Method 4; Km = 1;

Comment

Td

Minimum performance index: other tuning x 2Tm x 3Tm N=∞

x2

x3

α

β

0.81 1.55 2.23 2.88 3.50 4.10 4.68 x1K u

0.38 0.62 0.78 0.90 1.00 1.09 1.18

0.67 0.67 0.67 0.66 0.66 0.66 0.66

0.84 0.82 0.80 0.79 0.77 0.76 0.75 x 2 Tu

τm Tm

x1

x2

x3

α

β

τm Tm

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.67 0.59 0.53 0.42 0.36 0.27 0.22

5.25 5.82 6.36 7.74 9.06 11.38 14.76 x 3Tu

1.26 1.34 1.42 1.61 1.82 2.26 2.59

0.66 0.66 0.66 0.66 0.66 0.66 0.68

0.74 0.74 0.73 0.72 0.72 0.70 0.76

1.6 1.8 2.0 2.5 3.0 4.0 5.0

x1

x2


α

φc

1.672

0.366

0.136

0.601

− 164 0

1.236

0.427

0.149

0.607

− 160 0

0.994

0.486

0.155

0.610

− 1550

Tm = 1 ;

0.842

0.538

0.154

0.616

− 150 0

β = 1 , N = 10; 0.05 < τ m < 0.8 .

0.752

0.567

0.157

0.605

− 1450

0.679

0.610

0.149

0.610

− 140 0

0.635

0.637

0.142

0.612

− 1350

0.590

0.669

0.133

0.610

− 130 0

0.551

0.690

0.114

0.616

− 1250

0.520

0.776

0.087

0.609

− 120 0

0.509

0.810

0.068

0.611

− 118 0

Minimum ITAE – Pecharromán (2000a). Model: Method 4; Km = 1; Tm = 1 ; 0.05 < τ m < 0.8 .

x1K u

0

x 2 Tu

x1

x2

α

0.049

2.826

0.066


x2

α

φc

0.218

1.279

0.564

− 1350

− 160 0

0.250

1.216

0.573

− 130 0

0.522

− 1550

0.286

1.127

0.578

− 1250

1.716

0.532

− 150

0

0.330

1.114

0.579

− 120 0

0.159

1.506

0.544

− 1450

0.351

1.093

0.577

− 118 0

0.189

1.392

0.555

− 140 0

0.506

− 164

0

2.402

0.512

0.099

1.962

0.129

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

x1K u Minimum ITAE – Pecharromán x1 x2 (2000b), p. 142. 0.0469 2.8052 Model: Method 4; 0.0662 2.3876 Km = 1; 0.0988 1.9581 Tm = 1 ; 0.05 < τ < 0.8 . 0.1271 1.6757 m

Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 4; Km = 1; Tm = 1 ;

β = 1 , N = 10; 0.05 < τm < 0.8 .

Taguchi and Araki (2000). Model: Method 1

27

Kc =

Ti

Td

x 2 Tu

0

Comment


α

411

x2

α

φc

0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00

0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00 x1K u

x 2 Tu

x 3Tu

x1

x2


α

φc

1.6695

0.3684

0.1323

0.6007

− 164.10

1.2434

0.4389

0.1507

0.6073

− 160.00

0.9901

0.4846

0.1587

0.6102

− 155.00

0.8378

0.5393

0.1562

0.6163

− 150.00

0.7383

0.5789

0.1548

0.6052

− 145.00

0.6728

0.6283

0.1488

0.6105

− 140.00

0.6196

0.6221

0.1376

0.6120

− 135.00

0.5948

0.6939

0.1326

0.6096

− 130.00

0.5513

0.7050

0.1190

0.6157

− 125.00

0.5271

0.7471

0.0992

0.6088

− 120.00

0.5229

0.8886

0.0862

0.6115

27

Kc

Ti

0

− 118.0 0 OS ≤ 20% ; τm Tm ≤ 1.0 .

 0.2839 1  ,  0.1787 +  Km  (τm Tm ) + 0.001723  Ti = 4.296 + 3.794(τm Tm ) + 0.2591(τm Tm ) , α = 0.6551 + 0.01877(τ m Tm ) . 2

b720_Chapter-04

04-Apr-2009

FA


412 Rule

Kc

Taguchi and Araki (2000) – continued.

28

Kc

Ti

Td

Ti

Td

Comment

N=∞ x1 K m x 2Tm x 3Tm Taguchi and α α x1 x 2 x3 β x4 x1 x 2 x3 β x4 Araki (2002). 8.91 1.73 0.46 0.32 0.72 0.1 12.36 0.68 0.39 0.66 0.80 1.5 Model: Method 1 11.13 1.40 0.39 0.45 0.91 0.5 13.00 0.54 0.40 0.67 0.76 2.0 12.04 0.81 0.38 0.67 0.84 1.0 Load disturbance model = K m e −sτ m (1 + sx 4Tm ) ; τm Tm = 0.2 . Taguchi et al. (2002). Model: Method 1; α=0. Taguchi et al. (2002). Model: Method 1; α=0. Taguchi et al. (2002).

28

Kc =

∞

x1Tm K m x1

x2

16.54 0.32 5.33 0.52 1.96 0.76 1.17 0.91 x1Tm K m

N = 10

x 2Tm

β

τm Tm

0.89 0.85 0.78 0.73

0.1 0.2 0.4 0.6

x1

∞

x1

x2

β

τm Tm

x1

x2

26.56 7.75 2.54 1.42

0.29 0.50 0.78 0.95

0.81 0.79 0.74 0.70

0.1 0.2 0.4 0.6

0.98 0.74 0.35 0.15

1.08 1.19 1.59 2.77

29

Kc

∞

β

Td

m

)

)

Tm ) − 1.774(τm Tm ) + 0.6878(τm Tm ) , α = 0.6647 , 2

m

τm Tm

0.66 0.8 0.62 1.0 0.50 2.0 0.37 5.0 N = 10; Model: Method 1

Ti = Tm 0.03330 + 3.997(τm Tm ) − 0.5517(τm Tm ) , 2

τm Tm

0.68 0.8 0.64 1.0 0.50 2.0 0.37 5.0 N= ∞

Td

 1  0.5184 , 0.7608 + 2  K m  ((τm Tm ) + 0.01308) 

( = T (0.03432 + 2.058(τ

β

x2

0.83 1.02 0.65 1.10 0.32 1.47 0.14 1.47 x 2Tm

β = 0.8653 − 0.1277(τ m Tm ) + 0.03330(τ m Tm ) , N not specified.

3

2

29

Kc =

 τ τ Tm  0.2938 , β = 0.9305 − 0.4147 m + 0.1234 m 0.4073 + 2 K m  Tm ((τm Tm ) + 0.03829)   Tm

2

  ,  

2 3   τm    τm  τm τm       Td = Tm 0.08289 + 2.682 − 2.959  + 1.309 T   , α = 0 , 0.1 ≤ T ≤ 1.0 . Tm T  m  m   m 

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc


Comment

Td

Direct synthesis: time domain criteria 3.33(Tm + τm ) 0 K m (Tm + τm )

1

30


Ti

Ti

Kc

4 (8 − x1 )K m τm


31

Kc

32

Kc

33

Kc

34

Kc

35

Kc

0

Ti

(2ξ

2

α = 1, β = 1 , N =∞. α = 1; 4 x1 = 2 . 4ξ + 1

Td

4τm x1

)

+ 1 τm

Td

Ti

Td

α =1

α = 1, β = 1, N =∞.

Direct synthesis: frequency domain criteria 2 M s = 1.4 5.7τ m e1.7 τ−0.69 τ 0 Ti M s = 2.0 0.14 ≤ τm Tm ≤ 5.5 ; Model: Method 3.

Klán (2001).

30

31

1.067(Tm + τm )

33

2

; Ti = 2

2

2

m

m

(Tm + τm )

m

2

m

m 2 m m

2

m m

2.5τm (Tm + 0.5τm )

3

; Td =

(Tm + τm )

3

2

2

2

i

m

m

).

m

2

2

Kc

2

2

Model: Method 1

(1 + 4ξ )(τ + T ) , T = (1 + 4ξ )(τ + T K [τ (0.5 + 8ξ ) + τ T (1 + 16ξ ) + 8ξ T ] 16(2ξ + 1) 16ξ + 4 = (32ξ + 4)K τ , T = 16(2ξ + 1) τ . τ T (1 + 4ξ ) τ + T + 4τ ξ . = , T = τ + T + 4τ ξ , T = τ + T + 4τ ξ K τ (1 + 8ξ )

Kc = Kc

6.25τm (2Tm + τm ) 2

K m τm (2Tm + τm ) 2

0

Ti

Kc

3

Kc =

m

32

36

d

m 2

m

2

m m

2

i

m

m

m

2

m m

d

m

m

2

m

2

2

34

Kc =

2 0.41e−0.23τ + 0.019 τ , α = 1 − 0.33e 2.5τ−1.9 τ . K m (Tm + τm )

35

Kc =

2 2 0.81e−1.1τ + 0.76 τ , Ti = 3.4τm e0.28τ − 0.0089 τ , α = 1 − 0.78e −1.9 τ+1.2 τ . K m (Tm + τm )

2

2

36

2 2 0.41e−0.23τ + 0.019 τ Kc = , Ti = 5.7(τm + Tm )e1.7 τ − 0.69 τ , α = 1 − 0.33e 2.5τ−1.9 τ . K m (Tm + τm )

.

413

b720_Chapter-04

04-Apr-2009

FA


414 Rule

Kc

Wang and Cai 37 Kc (2001). Model: Method 1 0.6068 K m τm Wang and Cai (2002). Xu and Shao (2003a).

0.6189 K m τm

Xu and Shao (2003c).

1.5τ m + 0.5Tm

Xu and Shao (2004a). Model: Method 1

40

Td

Ti

Td

5.9784 τ m

38

Td

5.9112 τ m

39

Td

2(Tm + 1.5τ m )

Kc

2K m τ m

Ti

2

(2λ − 1)τm + Tm K m λ2 τm 2

Tm + τ m

(Tm + 0.5τm )2 (Tm + 1.5τ m ) 1.5Tm τ m 1.5τ m + 0.5Tm

(2λ − 1)Tm τm (2λ − 1)τm + Tm (2λ − 1)τm + Tm

N =∞; α =

Comment β = 0, N = ∞. Model: Method 1 Model: Method 1 41

Model: Method 1 λ not defined.

λτ m λ ; β= . Tm + (2λ − 1)τm 2λ − 1

τm (1.3608M s − 1.2064) 1  1.2064  , 1.3608 −  , Ti = K m τm  Ms  0.29M s − 0.3016 0. 2 M s (T + 0.1τm )(1.450Ms − 1.508) , α = , 1.3 ≤ M s ≤ 2 . Td = m 1.3608M s − 1.2064 1.3608M s − 1.2064

37

Kc =

38

Td = 0.8363(Tm + 0.1τm ) , α = 0.3296 , M s = 1.6 ( A m > 2.66 , φ m > 36.4 0 ) .

39

Td = 0.8460(Tm + 0.1τm ) , N = ∞ , β = 0 , α = 0.3232 , A m = 3 , φ m = 60 0 .

40

Kc =

41

α=

τm 0.5(Tm + 1.5τm ) , N= ∞ , β = 0 , A m > 2 , φ m > 60 0 . , α= Tm + 1.5τ m K m τm (Tm + τm )

τm , β = 0.6667 , N= ∞ , A m > 2 , φ m > 60 0 . 1.5Tm + 0.5τ m

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Ti

Td

Comment

Td

α = 0. 6 , β = 0 .

415

Robust 42


Ti

Kc

λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ; λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ; λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 ;


x 1K u x1

0.12 0.19 0.25

x1

Oubrahim and Leonard (1998). Model: Method 1

42

Kc =

Td =

equations for λ deduced from graphs. Ultimate cycle 0 x 2 Tu

x2

1.14 1.09 0.91 x 1K u x2

0.80 0.54 0.73 0.57 0. 6 K u

α =1

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.08 0.10 0.13

0.30 0.34 0.36 x 2 Tu

0.89 0.86 0.84

0.15 0.18 0.20

0.38

0.83

0.25

x3

τm Tu

0.11 0.15 0.09 0.17 0.5Tu

x 3 Tu x1

x2

0.71 0.61 0.69 0.62 0.125Tu

α = 1, β = 1, N =∞. τm Tu x3 0.08 0.20 0.08 0.22 20% overshoot.

α = 0.9 ; β = 0.99 ; N = ∞ ; 0.05 < τm Tm < 0.8 .

2 Ti 6λ2 − x 4 + x 2 τm − 0.5τm , Ti = x1 + Tm + x 2 − x 3 , x 3 = , x1K m (4λ + τm − x 2 ) 4λ + τm − x 2

x 4 + x 2 (Tm + x1 ) + x1Tm −

0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3 4λ + τ m − x 2 − x3 , Ti

x1 is a constant with a ‘sufficiently large value’;



2  x1 1 −

x2 =

 

τm τm 4 4    λ  − x1 λ  − Tm 2     e   − − − − 1 T 1 e 1 m   Tm   x1  ,   Tm − x1 τm 4   λ  − Tm 2   e − 1 + x 2Tm , N = ∞ . x 4 = Tm 1 −   Tm    

b720_Chapter-04

04-Apr-2009

FA


416


+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

   1 + b f 1s  1 + a f 1s s 

+ U(s)

−


K0


Kc 1

Huang et al. (2005b). Model: Method 1

Kc

Ti

[

Td

Direct synthesis: time domain criteria Ti T m

]

Y(s)


Comment N=∞

a f 1 = max 0.01 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.452τm + Tm ; a f 3 = Ti Tm , b f 3 = 1.452τmTm ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K0 = 0 .

Robust Lee and Edgar (2002). Model: Method 1

2

Kc

Ti

0

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u .

1

Kc =

 2.2221K m τ m  0.2328  T  . 1 + 0.6887 m  + 0.5173 , Ti = (1.452τ m + Tm )1 +     K m τm  τm   1 + 0.6887(Tm τ m ) 

2

Kc =

0.25K u Ti 4 τm , T = . −τ + (λ + τm ) i K u K m m 2(λ + τm )

2

b720_Chapter-04

04-Apr-2009

FA

Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule 3

Liu et al. (2003). Model: Method 1

Kc

Ti

Td

Kc

Ti

Td

417

Comment

a f 1 = 0 , b f 1 = 0 ; a f 2 = 3λ + τm , b f 2 = λ ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 .

Representative λ values (deduced from graph): OS Rise time OS Rise time λ

λ 0.3τ m

≈ 57%

≈ 3τ m

2τ m

≈ 0%

≈ 8τ m

0.5τ m

≈ 33%

≈ 3.5τ m

2.5τ m

≈ 0%

≈ 10.5τ m

τm

≈ 7%

≈ 5τ m

3τ m

≈ 0%

≈ 12.5τ m

1.5τ m

≈ 0%

≈ 8.5τ m

3.5τ m

≈ 0%

≈ 14τ m

a f 1 = 0 , b f 1 = 0 ; a f 2 = 7λ + τm , b f 2 = 3λ ; a f 3 = λ(16λ + 4τm ) ,

b f 3 = 3λ2 ; a f 4 = 4λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 , K 0 = 0 . Representative λ values (deduced from graph): OS Rise time OS Rise time λ

λ 0.3τ m

≈ 43%

≈ 3.5τ m

2τ m

≈ 0%

≈ 16.5τ m

0.5τ m

≈ 15%

≈ 4.5τ m

2.5τ m

≈ 0%

≈ 20τ m

τm

≈ 0%

≈ 8τ m

3τ m

≈ 0%

≈ 24τ m

1.5τ m

≈ 0%

≈ 12.5τ m

3.5τ m

≈ 0%

≈ 28τ m

a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 3λ ; a f 3 = λ (3.25λ + τm ) ,

b f 3 = 3λ2 ; a f 4 = 0.25λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 , K0 = 0 .


λ 0.3τ m

3

Kc = Td =

(

≈ 61%

2τ m

≈ 4%

≈ 6.5τ m

0.5τ m

≈ 40%

≈ 3τ m

2.5τ m

≈ 4%

≈ 8τ m

τm

≈ 15%

≈ 3.5τ m

3τ m

≈ 1%

≈ 10τ m

1.5τ m

≈ 7%

≈ 5τ m

3.5τ m

≈ 0%

≈ 12τ m

Ti 2

K m 3λ + 3λτ m + τm

(

≈ 3τ m

2

)

, Ti =

Tm (3λ + τm ) 3λ + 3λτ m + τm x1 2

x1

(3λ + 3λτ )− 2

2

m

+ τm

2

)

,N=

Tm (3λ + τm ) −1 λ3 x1

3

λ

3λ2 + 3λτ m + Tm 2

,

(

)

x1 = (3λ + τm + Tm ) 3λ2 + 3λτ m + τm − λ3 . 2

b720_Chapter-04

04-Apr-2009

FA


418


R(s)

+

F1 (s)

U(s)

−

Y(s)

Model F2 (s)


Kc

Ti

Td


Comment

Direct synthesis: time domain criteria 1

2 0 Skogestad K m (TCL + τ m ) 4ξ (TCL + τ m ) (2004b). Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1. α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 ,

a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 ; b f 4 = Tm ,

Xu and Shao (2004a). Model: Method 1

(λ − 1)τm + Tm 2

K m λ τm

2

a f 5 = Tm x1 , 2 ≤ x1 ≤ 10 .

(λ − 1)τm + Tm

(λ − 1)Tm τm (λ − 1)τm + Tm

λ not defined.

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 λK m τ m ; b f 3 = 1 ; b f 4 = Tm ; a f 5 = 0 .

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Majhi and Mahanta (2001). Model: Method 6

0.5236 Kmτm

Ti

Td

419

Comment

Direct synthesis: frequency domain criteria A m = 3; 0 2τ m e τ ms φ = 60 0 . m

α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; b f 1 = Tm , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = Tm , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.5 K m τm ; b f 3 = 1 ; b f 4 = Tm ; a f 5 = 0 .

Wang et al. (2001b). Model: Method 7

0.4189 K m τm

4τ m

1.25(Tm + 0.1τ m )

A m = 3;

φ m = 60 0 .

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0. 2 K m τ m ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .

Robust 1


Kc

Ti

Td

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 2

Kc

Ti

x1

α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 .

1

2

Kc =

2 2  0.25K u  4 4 τm τm − τm + , − τm +  , Ti =  (λ + τm )  K u K m 2(λ + τm )  KuKm 2(λ + τm )

Td =

2 3 4 0.25K u  4Tm τm τm τ (1 − 0.25K u K m τm )  2 + 0.5τm − + + m .  2 (λ + τm )  K u K m 6(λ + τm ) 4(λ + τm ) 0.5K u K m (λ + τm ) 

Kc =

0.25K u λ + τm

2 2   4 4 τm τm − τm + + x1 − τm + + x1  , Ti =  2(λ + τm ) KuKm 2(λ + τm )   K u K m

3 4  4Tm τm τm (1 − 0.25K u K m τm )τm 2  2 x1 =  + 0.5τm − + +  2 6( λ + τ m ) 4( λ + τ m ) 0.5K u K m (λ + τm )   K u K m

0.5

.

b720_Chapter-04

04-Apr-2009

FA


420

4.4 FOLIPD Model with a Zero K (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτm G m (s) = m or G m (s) = m s(1 + sTm1 ) s(1 + sTm1 )


R(s)

−

+

  U(s) 1 K c 1 + + Td s  Ti s  

Model

Y(s)

Table 77: PID controller tuning rules – FOLIPD model with a zero K (1 − sTm 4 )e −sτ m K (1 + sTm3 )e−sτ m G m (s ) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule

Kc

Ti

Td

Comment

Direct synthesis: time domain criteria 1

Chidambaram (2002), p. 157. Model: Method 1 Jyothi et al. (2001). Model: Method 1

1

2

3

4

Kc = Kc = Kc = Kc =

1.571

Kc

2

Kc

3

Kc

4

Kc

∞

‘Smaller’ τm Tm 4 . ‘Larger’ τm Tm 4 .

Direct synthesis: frequency domain criteria τm Tm3 < 1 ∞ Tm1 τm Tm3 > 1

(

)

.

(

)

.

K m τm 1 + 3.142 Tm 4 τm 2 2.358

Tm1

K m τm 1 + 4.172 Tm 4 τm 2 1.57

K m τm 1 + 9.870(Tm3 τm )2 0.79 K m τm 1 + 2.467(Tm3 τm )2

. .

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc


0.5 K m Tm 4 5

Kc

Comment

Ti

Td

∞

Tm1

τm Tm 4 > 1

Td

Recommended τm ≤ λ ≤ 2τm .

τm Tm 4 < 1

Robust


5

6

Kc =

6

∞

Kc

0.79 K m τm 1 + 2.467(Tm 4 τm )2

.

T (τ − λ ) + 0.5τm 1 ; , Td = Tm1 + m 4 m K m (λ + τm + 2Tm 4 ) λ + τm + 2Tm 4 2

Kc =

(1 − sTm 4 )e −sτ . (1 + sTm 4 )(1 + sλ ) m


421

b720_Chapter-04

04-Apr-2009

FA


422


E(s)

R(s)

−

+

 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s U(s)

  1 1 K c 1 + + Tds   Tis  Tf s + 1

Model

Y(s)

Table 78: PID controller tuning rules – FOLIPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m G m ( s) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule

Td

Comment

Ti

Td

Model: Method 1

Ti

Td

Model: Method 1

Kc

Ti

1

Kc

2

Kc

Robust Anil and Sree (2005). Shamsuzzoha et al. (2006c).

1

2

Kc =

Km

x1 is a constant of a ‘sufficiently large value’.

[(λ + T

m4

Tm1 + Tm 4 + τm + 3λ

+ τm ) + 2λ2 + λ (Tm 4 + τm ) − Tm 4 τm 2

], T =T i

m1

+ Tm 4 + τm + 3λ ,

Td =

Tm1 (Tm 4 + τm + 3λ ) λ3 − Tm 4 τm (3λ + Tm 4 + τm ) . , Tf = Tm1 + Tm 4 + τm + 3λ (λ + Tm 4 + τm )2 + 2λ2 + λ(Tm 4 + τm ) − Tm 4τm

Kc =

Ti , λ , ξ not specified; Ti = x1 + Tm1 + x 2 − x 3 , Tf = Tm3 , x1K m (4λξ + τm − x 2 )

Td =

x2 = x3 =

x 4 + x 2 (Tm1 + x1 ) + x1Tm1 −

Tm12 x 5e − τ m

Tm1

− x12 x 5e − τ m x1 − Tm1

0.167 τm3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ 4λξ + τm − x 2 − x3 , Ti x1

+ x12 − Tm12

,

[

4λ2ξ2 + 2λ2 − x 4 + x 2 τm − 0.5τm 2 2 , x 4 = Tm1 x 5e− τ m 4λξ + τm − x 2

  λ2 2λξ x5 =  − + 1  T 2 T m1   m1

2

.

Tm1

]

− 1 + x 2Tm1 ,

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423

     1  1 + Td s   4.4.3 Classical controller G c (s) = K c 1 + Td s  Ti s   1 +  N   E(s)

R(s)

−

+

   1  1 + sTd  K c 1 +  Ti s  1 + s Td  N 

     

U(s)

Y(s)

Model

Table 79: PID controller tuning rules – FOLIPD model with a zero K (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule

Kc

Ti

Comment

Td

Direct synthesis: time domain criteria Anil and Sree (2005). Model: Method 1

1

0.5τmTm1 λ3K m 2

Kc

Tm1

2.414λ + 0.5τm

N= 2.414λ + 0.5τm Tm3

Tm1

Td

N=∞

1

λ is determined by solving λ3 − 1.207 τm λ2 − 0.603τm 2λ − 0.125τm 3 = 0 .

2

Kc =

0.5τm Tm1 , Td = 2.414λ + 0.5τm + Tm 4 ; K m λ3 − 0.5τmTm 4 [2.414λ + 0.5τm + Tm 4 ]

(

)

3

2

λ is determined by solving λ − 1.207τm λ − (0.6035τm + 2.414Tm 4 )τm λ 2

2

− τm (0.125τm + 0.5Tm 4 τm + Tm 4 ) = 0 .

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424

4.5 I 2 PD Model G m (s) =

K m e −sτm s2

  1 + Td s  4.5.1 Ideal PID controller G c (s) = K c 1 + Ti s   R(s)

+

E(s)

−

U(s)

  1 K c 1 + + Td s  Ti s  

K m e − sτ s

Table 80: PID tuning rules – I 2 PD model G m (s) =

Rule Åström and Hägglund (2006), pp. 244–245.

Kc

Ti

Td

m

Y(s)

2

K m e − sτ m s2 Comment

Direct synthesis: frequency domain criteria M s = M t = 1. 4 ; 0.02140 17.570τm 14.019τm Model: Method 1 K m τm 2

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   1  1 + Td s    4.5.2 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   E(s) R(s)

+

−

     1  1 + Td s  U(s)  K c 1 + Td  Ti s   s 1 + N  

K m e − sτ

m

s2


Rule

Kc


0.0625

Ti

Y(s)

K m e − sτ m

Td

s2 Comment


K mτm2

8τ m

8τ m

N=∞

425

b720_Chapter-04

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426



    T s β d α + K Td  c  1+ s  N   E(s) R(s) +

−

  Td s 1 + K c 1 + T  Ti s 1+ d  N 

    s 

+

U(s)

Hansen (2000). Model: Method 1

Kc

3.75K m τ m

Ti 2

K me

s2

+


Rule

Y(s) − sτ m

Td


Direct synthesis: time domain criteria 5.4 τ m 2.5τ m N = ∞ ; β =1; α = 0.833 .

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427

4.5.4 Two degree of freedom controller 2     T s 1 d  1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0  Ti s T  1 + a f 1s 1+ d s   N   1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s)

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

+

F(s)

−

   1 + b f 1s  1 + a f 1s s 

+ U(s)

−

Kc

Ti

Kc

Ti

Y(s)

m

s2

K0


Rule

K m e − sτ


Td

Robust 1

Liu et al. (2003). Model: Method 1

Td

a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 2λ ; 2

a f 3 = 6λ + 4λτ m + τm , b f 3 = λ2 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , 2

bf 5 = 0 , K 0 = 0 .

1

Kc = Td =

(

Ti 3

2

2

K m 4λ + 6λ τm + 4λτ m + τm

(6λ + 4λτ 2

2

m

+ τm 4λ3 + 6λ2 τm x1 2

N=

)(

3

)

, Ti =

(4λ + 6λ τ + 4λτ + τ ) − 3

2

2

m

(

x1 m

+ 4λτ m 2 + τm 3

3

),

λ4

m

4λ + 6λ τ m + 4λτ m 2 + τ m 3 3

2

)

6λ2 + 4λτ m + τm 2 3 − 1 , x1 = (4λ + τm ) 4λ3 + 6λ2 τm + 4λτ m + τm − λ4 . λ3x1

;

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428 Rule Liu et al. (2003) – continued.

Kc

Ti

Comment

Td


λ 0.8τ m

≈ 33%

≈ 4τ m

2.5τ m

≈ 0%

≈ 10τ m

τm

≈ 20%

≈ 5τ m

3τ m

≈ 0%

≈ 12.5τ m

1.5τ m

≈ 3%

≈ 7τ m

3.5τ m

≈ 0%

≈ 14τ m

2τ m

≈ 0%

≈ 8τ m

4τ m

≈ 0%

≈ 16τ m

2

3

a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ , b f 4 = 4λ , b f 5 = λ4 ; a f 2 = 8λ + τm , a f 3 = 26λ2 + 8λτ m + τm

(

2

)

2

(

2

)

a f 4 = 4λ 10λ2 + 20λτ m + 4τm , a f 5 = 4λ2 6λ2 + 4λτ m + τm , K 0 = 0 .

λ 0.8τ m

Representative λ values (deduced from graph): OS Rise time OS Rise time λ ≈ 14%

≈ 3τ m

2.5τ m

≈ 0%

≈ 7τ m

≈ 4%

≈ 3.5τ m

3τ m

≈ 0%

≈ 8.5τ m

1.5τ m

≈ 0%

≈ 4.5τ m

3.5τ m

≈ 0%

≈ 10.5τ m

2τ m

≈ 0%

≈ 6τ m

4τ m

≈ 0%

≈ 13τ m

τm

a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ2 , b f 4 = 4λ3 , b f 5 = λ4 ; 2

(

a f 2 = 5λ + τm , a f 3 = 10.25λ2 + 5λτ m + τm , 2

)

(

2

)

a f 4 = λ 7λ2 + 4.25λτ m + τm , a f 5 = 0.25λ2 6λ2 + 4λτ m + τm ,

λ 0.8τ m

K0 = 0 .

Representative λ values (deduced from graph): OS Rise time OS Rise time λ ≈ 46%

≈ 6τ m

2.5τ m

≈ 4%

≈ 20.5τ m

τm

≈ 34%

≈ 13.5τ m

3τ m

≈ 2%

≈ 24τ m

1.5τ m

≈ 16%

≈ 13τ m

3.5τ m

≈ 1%

≈ 28τ m

2τ m

≈ 8%

≈ 16.5τ m

4τ m

≈ 0%

≈ 32τ m

b720_Chapter-04

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FA



R(s)

+

F1 (s)

U(s)

−

Y(s)

Model F2 (s)


Rule

Kc

Ti

K m e −sτm

Td

s2 Comment

Direct synthesis: time domain criteria 0.25 0 Skogestad 2 4ξ 2 (TCL + τ m ) K m (TCL + τ m ) (2004b). Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1; α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 ,

a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = 4(TCL + τm ) , a f 5 = b f 4 x 2 , 2 ≤ x 2 ≤ 10 .

Huba (2005). Model: Method 1

−

0.079 K m τm 2

∞

5.835τm

α = 1; β = 1; χ = 1 ; δ = 0 ; ε = 0 ; φ = 1; ϕ = 0 ; N = ∞ ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 .

429

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430

4.6 SOSIPD Model

G m (s) =

K m e −sτ m s(1 + Tm1s) 2

or G m (s) =

K m e − sτ m 2

s(Tm1 s 2 + 2ξ m Tm1s + 1)


R(s)

+

−

 1   K c 1 + Ti s  

U(s)

K m e − sτ

Y(s)

m

s(1 + sTm1 ) 2

Table 85: PI controller tuning rules – SOSIPD model G m (s) = Rule

Kc

Ti

s(1 + sTm1 )2 Comment

Direct synthesis: time domain criteria Kuwata (1987). Model: Method 1

0.5 K m (2Tm1 + τm )

∞

K m e − sτ m

b720_Chapter-04

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FA




    T s β d α + K Td  c  1+ s  N   R(s)

+

E(s)

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

Rule

+

Minimum ITAE – Pecharromán (2000a). Model: Method 2; Km = 1; Tm1 = 1 ; 0.1 < τ m < 10 .

K m e − sτ m or s(1 + Tm1s) 2

K m e − sτ m s(Tm12s 2

Kc

+ 2ξm Tm1s + 1)

Ti

x1K u

Y(s)

SOSIPD model

Table 86: PID controller tuning rules – SOSIPD model G m (s) = G m (s) =

Comment

Td

Minimum performance index: other tuning 0 x 2 Tu

x1

x2

α

0.049

2.826

0.066


x2

α

φc

0.218

1.279

0.564

− 1350

− 160 0

0.250

1.216

0.573

− 130 0

0.522

− 1550

0.286

1.127

0.578

− 1250

1.716

0.532

− 150

0

0.330

1.114

0.579

− 120 0

0.159

1.506

0.544

− 1450

0.351

1.093

0.577

− 118 0

0.189

1.392

0.555

− 140 0

0.506

− 164

0

2.402

0.512

0.099

1.962

0.129

431

b720_Chapter-04

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FA


432 Rule

Kc

x1K u Minimum ITAE – Pecharromán x1 x2 (2000b), p. 142. 0.0469 2.8052 Model: Method 2; 0.0662 2.3876 Km = 1; 0.0988 1.9581 Tm1 = 1 ; 0.05 < τ < 10 . 0.1271 1.6757 m

Minimum ITAE – Pecharromán and Pagola (2000). Model: Method 2; Km = 1;

Ti

Td

x 2 Tu

0

Comment


α

x2

α

φc

0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00

0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00 x1K u

x 2 Tu

x 3Tu

x1

x2


α

φc

1.672

0.366

0.136

0.601

− 164 0

1.236

0.427

0.149

0.607

− 160 0

0.994

0.486

0.155

0.610

− 1550

Tm1 = 1 ;

0.842

0.538

0.154

0.616

− 150 0

0.1 < τ m < 10 ;

0.752

0.567

0.157

0.605

− 1450

β = 1 , N = 10.

0.679

0.610

0.149

0.610

− 140 0

0.635

0.637

0.142

0.612

− 1350

0.590

0.669

0.133

0.610

− 130 0

0.551

0.690

0.114

0.616

− 1250

0.520

0.776

0.087

0.609

− 120 0

0.509

0.810

0.068

0.611

− 118 0

α

φc

Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 2; Km = 1; Tm1 = 1 ;

β = 1 , N = 10; 0.05 < τm < 0.8 .

x1K u

x 2 Tu

x 3Tu

x1

x2


1.6695

0.3684

0.1323

0.6007

− 164.10

1.2434

0.4389

0.1507

0.6073

− 160.00

0.9901

0.4846

0.1587

0.6102

− 155.00

0.8378

0.5393

0.1562

0.6163

− 150.00

0.7383

0.5789

0.1548

0.6052

− 145.00

0.6728

0.6283

0.1488

0.6105

− 140.00

0.6196

0.6221

0.1376

0.6120

− 135.00

b720_Chapter-04

04-Apr-2009

FA


Rule Pecharromán (2000b) – continued.

Ti

Td

x1K u

x 2 Tu

x 3Tu

x1

x2


α

φc

0.5948

0.6939

0.1326

0.6096

− 130.00

0.5513

0.7050

0.1190

0.6157

− 125.00

0.5271

0.7471

0.0992

0.6088

− 120.00

0.5229

0.8886

0.0862

0.6115

− 118.0 0 Tm1 ≤ 1.0 ;

Taguchi and Araki (2000). Model: Method 1 Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 .

1

Kc

Ti

0

2

Kc

Ti

Td

x1Tm1 K m

∞

Kc =

τm

OS ≤ 20% .

x 2Tm1

x1

x2

β

τm Tm1

x1

x2

β

τm Tm1

1.48 1.04 0.71 0.56

1.35 1.41 1.49 1.57

0.89 0.81 0.72 0.67

0.1 0.2 0.4 0.6

0.46 0.40 0.24 0.12

1.63 1.70 2.02 3.08

0.63 0.60 0.50 0.38

0.8 1.0 2.0 5.0

3

1

Comment

Kc

Kc

∞

Td

 1  0.3840  0.07368 +  , T = 8.549 + 4.029[τm Tm1 ] ,  [τm Tm1 ] + 0.7640  i Km 

α = 0.6691 + 0.006606[τm Tm1 ] . 2

Kc =

 1  0.5667  0.1778 +  , T = T (1.262 + 0.3620[τm Tm1 ]) ,  [τm Tm1 ] + 0.002325  d m1 Km 

(

)

Ti = Tm1 0.2011 + 11.16[τm Tm1 ] − 14.98[τm Tm1 ] + 13.70[τm Tm1 ] − 4.835[τm Tm1 ] , 2

3

α = 0.6666 , β = 0.8206 − 0.09750[τm Tm1 ] + 0.03845[τm Tm1 ] ; N not specified. 2

3

Kc =

 Tm1  0.3084 , 0.1309 + Km  ([τm Tm1 ] + 0.1315) 

[

]

Td = Tm 1.308 + 0.5000[τm Tm1 ] − 0.1119[τm Tm1 ] , 2

β = 0.9231 − 0.6049[τm Tm1 ] + 0.2906[τm Tm1 ] ; 0.1 ≤ τm Tm1 ≤ 1.0 . 2

4

433

b720_Chapter-04

Rule Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 .

Kc

Ti

Td

Comment

x1Tm1 K m

∞

x 2Tm1

ξm = 0.5

x1

x2

β

τm Tm1

x1

x2

β

τm Tm1

0.52 0.43 0.34 0.30

1.11 0.89 0.61 0.50

0.82 0.67 0.26 -0.13

0.1 0.2 0.4 0.6

0.28 0.26 0.21 0.13

0.49 0.53 0.91 2.26

-0.34 -0.39 -0.08 0.23

0.8 1.0 2.0 5.0

Kc

∞

Td

x1Tm1 K m

∞

x 2Tm1

4

Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 .

Kc =

FA


434

4

04-Apr-2009

x1

1.06 0.80 0.57 0.46 x1Tm1 x1

2.18 1.48 0.93 0.69 x1Tm1 x1

2.93 1.90 1.13 0.81 x1Tm1

x2

β

τm Tm1

1.34 1.32 1.30 1.31 Km

0.89 0.81 0.70 0.62

0.1 0.2 0.4 0.6

x2

β

τm Tm1

1.34 1.46 1.64 1.77 Km

0.92 0.87 0.81 0.77

0.1 0.2 0.4 0.6

∞

∞

x2

β

τm Tm1

1.29 1.46 1.71 1.90 Km

0.93 0.89 0.83 0.80

0.1 0.2 0.4 0.6 ∞

x1

ξm = 0.8 x2

0.40 1.33 0.35 1.37 0.23 1.64 0.12 2.74 x 2Tm1 x1

x2

0.55 1.88 0.46 1.98 0.26 2.37 0.12 3.42 x 2Tm1 x1

x2

0.63 2.06 0.51 2.19 0.27 2.68 0.12 3.75 x 2Tm1

β

τm Tm1

0.56 0.8 0.52 1.0 0.40 2.0 0.33 5.0 ξ m = 1. 2 β

τm Tm1

0.73 0.8 0.70 1.0 0.59 2.0 0.43 5.0 ξ m = 1. 4 β

τm Tm1

0.77 0.8 0.74 1.0 0.64 2.0 0.47 5.0 ξ m = 1. 6

x1

x2

β

τm Tm1

x1

x2

β

τm Tm1

3.80 2.37 1.34 0.93

1.24 1.45 1.76 2.00

0.94 0.90 0.85 0.82

0.1 0.2 0.4 0.6

0.70 0.56 0.28 0.12

2.20 2.36 2.95 4.08

0.79 0.77 0.68 0.51

0.8 1.0 2.0 5.0

 Tm1  0.08343 , 0.1904 + Km  ((τm Tm1 ) + 0.1496) 

[

]

Td = Tm1 1.237 − 2.031(τm Tm1 ) + 1.344(τm Tm1 ) , 2

β = 1.048 − 1.637(τm Tm1 ) − 1.676(τm Tm1 ) + 1.898(τm Tm1 ) ; 0.1 ≤ τm Tm1 ≤ 1.0 . 2

3

b720_Chapter-04

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FA


Rule

Kc


Ti

Kc


x 1K u x1

0.16 0.19 0.23

Kc =

x2

Td = 1.25

(2.5T

2 m1

α =1

τm Tu

x1

x2

τm Tu

x1

0.01 0.03 0.05

0.26 0.30 0.32 x 2 Tu

0.89 0.88 0.86

0.08 0.10 0.13

0.35 0.36 0.38

1.05 0.97 0.95 x 1K u

1.067 (2Tm1 + τm )3 K m 2T 2 + 4T τ + τ 2 m1 m1 m m

[

α =1

α = 1, β =1, N = ∞ .

Td

Kuwata (1987); x 1 x 2 x 3 τm x1 x 2 x 3 x 1 , x 2 deduced Tu from graphs; β = 1 , N = ∞ . 0.98 0.45 0.12 0.03 0.79 0.54 0.11 0.90 0.47 0.12 0.05 0.76 0.56 0.10 0.82 0.51 0.11 0.08 0.72 0.58 0.09

5

Comment

Td

Direct synthesis: time domain criteria 1 0 K m (2Tm1 + τm ) 3.33(2Tm1 + τm ) 5


Ti

]

2

; Ti =

)(

[

x 3 Tu τm Tu

x1

x2

x3

τm Tu

0.10 0.70 0.60 0.09 0.17 0.12 0.70 0.62 0.08 0.20 0.15

6.25 2Tm12 + 4Tm1τm + τm 2

(2Tm1 + τm )2

+ Tm1τm + 0.25τm 2 2Tm12 + 4Tm1τm + τm 2

(2Tm1 + τm )3

τm Tu

x2

0.85 0.15 0.84 0.18 0.83 0.25 α =1

).

]

2

;

435

b720_Chapter-04

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FA


436

4.7 SOSIPD Model with a Zero G m (s) =

K m (1 + sTm 3 )e − sτ m s(1 + sTm1 )(1 + sTm 2 )

   1  1 + Td s    4.7.1 Classical controller G c (s) = K c 1 +  Ti s  1 + Td s  N   R(s) E(s) +

−

     1  1 + Td s   K c 1 + Td  Ti s   s 1 + N  

Y(s)

U(s)

G m (s)

Table 87: PID controller tuning rules – SOSIPD model with a zero K (1 + sTm3 )e −sτ m G m (s ) = m s(1 + sTm1 )(1 + sTm 2 ) Rule

Kc

Bialkowski (1996). Model: Method 1

Tm1

Ti

Td

Comment

Robust

K m (λ + τm )2

Tm1

2λ + τ m

N=

2λ + τm ; Tm3

Tm1 > Tm 2 .

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437

4.8 TOSIPD Model

G m (s) =

K m e − sτ m K m e −sτ m or G m (s) = 3 s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) s(1 + sTm1 )



    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

+

−

Rule


  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

Y(s)

U(s) Model

+

Table 88: PID controller tuning rules – TOSIPD model Comment Kc Ti Td Ultimate cycle 0 x 2 Tu

x 1K u x1

0.23 0.26 0.28

x2

0.91 0.89 0.88 x 1K u

α =1

τm Tu

x1

x2

τm Tu

x1

x2

τm Tu

0.01 0.03 0.05

0.31 0.33 0.35 x 2 Tu

0.86 0.86 0.85

0.08 0.10 0.13

0.36 0.37 0.38

0.84 0.84 0.83

0.15 0.18 0.25

x 3 Tu

Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.85 0.50 0.11 0.03 0.74 0.56 0.09 0.10 0.70 0.62 0.08 0.18 α = 1, β = 1, 0.80 0.52 0.11 0.05 0.72 0.58 0.09 0.12 0.70 0.63 0.075 0.22 N =∞. 0.77 0.55 0.10 0.08 0.70 0.61 0.08 0.16

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438

4.9 General Model with Integrator G m (s) =

K G m (s) = m s

∏ (T ∏ (T

s(1 + sTm1 )

)∏ (T s + 1)∏ (T

m1i s

i

K m e − sτ m

+1

m 2i

n

or

2 2

m 3i

i

m 4i

) e s + 1)

s + 2ξ m n i Tm 2i s + 1

i

2 2

s + 2ξ m di Tm 4i

i


−

+

U(s)

 1   K c 1 +  T is  

E(s)

G m (s)

Y(s)

Table 89: PI controller tuning rules – general model with integrator Comment Kc Ti

Rule

Direct synthesis: time domain criteria 0.5 ∞ K m (nTm1 + τm )


Direct synthesis: frequency domain criteria (1) Symmetrical Kc optimum method. α1TQ ( 2) Non-symmetrical Kc optimum method.

1

Loron (1997). Model: Method 1

1

Kc

(1)

=

 TQ =   Kc

( 2)

 1 + sin φ m , α1 =  α1 TQ  cos φ m

1 Km

∑T

=

m 3i

i

+2

∑ξ i

λ K m α 2 TQ

  −  

m d i Tm 4i 

, λ=

  

2

∑T

m1i

i

Mc2 Mc2

+2

∑ξ



m n i Tm 2 i 



i

2

+ τm .

 1 + sin ∆φ  −1 , α 2 =   , ∆φ = cos (1 λ ) , −1  cos ∆φ 

M c = desired closed loop resonant peak.

−sτ m

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439


    T s β d α + K Td  c  1+ s  N   E(s)

R(s)

+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

U(s)

−

K m e − sτ n s(1 + sTm1 ) m

+

Table 90: PID controller tuning rules – general model with integrator Comment Kc Ti Td

Rule

Direct synthesis: time domain criteria Kuwata (1987). 1 Model: Method 1 K (nT + τ ) 3.33(nTm1 + τm ) 0 m m1 m Kuwata (1987). Model: Method 1

1

Kc =

Td

Y(s)

1

Ti

Kc

1.067 (nTm1 + τm )3 K m n (n − 1)T 2 + 2nT τ + τ 2 m1 m1 m m

[ [(nT = 1.25x 1

m1

(

]

2

, Ti =

Td

[

α = 1; β = 1; N =∞.

6.25 n (n − 1)Tm12 + 2nTm1τm + τm 2

+ τm )2 − 0.75 n (n − 1)Tm12 + 2nTm1τm + τm 2

(nTm1 + τm )3

α =1

(nTm1 + τm )2

]

2

,

)] ,

x1 = [n (n − 1)Tm1 + 2nTm1τm + τm ] . 2

2

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440

4.10 Unstable FOLPD Model G m (s) =

K m e −sτ m Tm s − 1


E(s)

+

−

 1   K c 1 + Ti s  

U(s)

Y(s)

K m e − sτ Tm s − 1 m

Table 91: PI controller tuning rules – unstable FOLPD model G m (s) = Rule Minimum ITSE – Majhi and Atherton (2000). 0 < τm Tm < 0.693 . Jacob and Chidambaram (1996). Model: Method 1

Kc 1 2

K m e − sτ m Tm s − 1

Comment

Ti

Minimum performance index: servo tuning Model: Method 1 Ti K c

10 TS

Model: Method 3

Ti

Kc

Direct synthesis: time domain criteria TS is the ±1% 0.4ξ 2TS settling time.

Chidambaram (1997).

1.678  Tm ln K m  τ m

Valentine and Chidambaram (1998). Model: Method 1

2.695 −1.277 τ m e Km

   Tm

0.4015Tm e 5.8τ m

Tm

0.4015Tm e 5.8τ m

Tm

Model: Method 1 3

(

ξ = 0.333 ; τm Tm < 0.7 .

)

1

Kc =

e − τ m Tm − 0.064  2.6316Tm e τ m Tm − 0.966 1  , Ti = . 0.889 + τ T   m m Km  e e − τ m Tm − 0.377 − 0.990 

2

Kc =

 1  1 Tm K (1 + K )  0.889 +  , Ti = , K = − Ap Kmh . K m  K (1 + K )  0.5(1 − 0.6382K ) tanh −1 K

3

±1% Ts = 7.5Tm , τ m Tm = 0.1 ; ±1% Ts = 8.75Tm , τ m Tm = 0.2 ;

(

)

±1% Ts = 11.25Tm , τ m Tm = 0.3 ; ±1% Ts = 15.0Tm , τ m Tm = 0.4 ; ±1% Ts = 17.5Tm , τ m Tm = 0.5 ; ±1% Ts = 20.0Tm , τ m Tm = 0.6 .

b720_Chapter-04

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Rule

Kc

Ti

Model: Method 1; ‘small’ τm Tm .

Chandrashekar et al. (2002).

5

Kc

Ti

Model: Method 1

49.535 − 21.644 τm e Km 0.8668  Tm    K m  τm 


Kc =

Tm Ti 2 2

0.04TS ξ + Ti τm

τm < 0.1 Tm

Tm

0.1523Tm e

0.8288

7.9425τ m Tm

0. 1 ≤

τm ≤ 0. 7 Tm

6.0000 K m

Alternative tuning rules: 0.6000Tm

τ m Tm = 0.1

3.2222 K m

1.4500Tm

τ m Tm = 0.2

2.2963 K m

2.6571Tm

τ m Tm = 0.3

1.8333 K m

4.4000Tm

τ m Tm = 0.4

1.5556 K m

7.0000Tm

τ m Tm = 0.5

1.3265 K m

14.6250Tm

τ m Tm = 0.6

1.1875 K m

31.0330Tm

τ m Tm = 0.7

τm

2.695 −1.277 Tm e K m,

0.4015Tme

5.8

τm Tm

0
0.2(A m − 1) +1.  0.4085(τm Tm )    τ m 1 +   1 − 0.2864(τm Tm ) 

b720_Chapter-04

04-Apr-2009


446 Rule

Kc

Rotstein and Lewin (1991). Model: Method 1

 λ  Tm λ  + 2   Tm  λ2 K m

Ti

Comment

Robust  λ  λ  + 2   Tm 

λ obtained graphically – sample values below.

τ m Tm = 0.2, 0.6Tm ≤ λ ≤ 1.9Tm τ m Tm = 0.2, 0.5Tm ≤ λ ≤ 4.5Tm τ m Tm = 0.4, 1.5Tm ≤ λ ≤ 4.5Tm τ m Tm = 0.6, 3.9Tm ≤ λ ≤ 4.1Tm

Tan et al. (1999b). Model: Method 1 Ou et al. (2005a).

17

18

17

Kc

18

Kc

λ2 + 2λTm + Tm τm K m (λ + τm )

2

, Ti =

λ2 + 2λTm + Tm τm . Tm − τm

K m uncertainty = 50%. K m uncertainty = 30%.

Ti

τm Tm < 0.5

Ti

Model: Method 1

  τ Tm 12.7708 m + 3.8019  Tm 1   . Kc = , Ti = τm   τm 0.1486 + 0.6564 K m  0.1486 + 0.6564  Tm Tm   Kc =

FA

b720_Chapter-04

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R(s)

  U(s) 1 K c 1 + + Td s   Ti s 

−

+

K m e − sτ Tm s − 1

m

Y(s)

Table 92: PID controller tuning rules – unstable FOLPD model G m (s) = Rule

Kc

Minimum ISE – Visioli (2001).

Ti

Td


Comment

Minimum performance index: regulator tuning 1.18 1.37  τ m  τ    Model: Method 1 2.42Tm  m  0.60τ m K m  Tm   Tm 

Minimum ITSE – Visioli (2001).

1.37  τ m  K m  Tm

  

τ 4.12Tm  m  Tm

  

0.90

Minimum ISTSE – Visioli (2001).

1.70  τ m  K m  Tm

  

τ 4.52Tm  m  Tm

  

1.13

0.55τ m

Model: Method 1

0.50τ m

Model: Method 1

Minimum performance index: servo tuning Minimum ISE – Visioli (2001).

1.32  τ m  K m  Tm

  

0.92

Minimum ITSE – 1.38  τ m  Visioli (2001). K m  Tm

  

0.90

Minimum ISTSE 1.35  τ m  – Visioli (2001). K m  Tm

  

0.95

[ = 3.62T [1 − 0.85(T = 3.70T [1 − 0.86(T

1

Td = 3.78Tm 1 − 0.84(Tm τm )

2

m

τm )

m

τm )

3

Td Td

m

m

0.02 0.02 0.02

](τ ](τ ](τ

τ 4.00Tm  m  Tm

  

0.47

τ 4.12Tm  m  Tm

  

0.90

τ 4.52Tm  m  Tm

  

m

Tm )

.

m

Tm )

.

m

Tm )

.

0.95 0.93 0.97

1

Td

Model: Method 1

2

Td

Model: Method 1

3

Td

Model: Method 1

1.13

447

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FA


448 Rule

Kc

Ti

Comment

Td

Minimum performance index: servo/regulator tuning Jhunjhunwala and Chidambaram (2001).

4

Ti

Kc

Direct synthesis: time domain criteria Ti 1.165  Tm    5 T x1 i Km  τm 

Chidambaram (1997). Model: Method 1

6

Valentine and Chidambaram 1.165  Tm (1997b).  Model: Method 1 K m  τ m

  

0.245

Kc =

τm Tm < 0.6

Ti

4.4Tm + 9.0τm 7

4

0.176Tm + 0.36τm

2.044 τ m Tm

0.6 ≤ 0.8
1 .

(

λ2 τm + λ2 + 2λτ m Tm + 2(λ + τm )Tm λ2 τm λ (λ + 2τm ) , Ti = + + 2(λ + τm ) , 2 K mλ[λτ m + (λ + 2τm )Tm ] Tm Tm

λ2 τm + λ(λ + 2τm )Tm + (2λ + τm )Tm

Td = τm 4

Td

2

λ2 τm + λ(λ + 2τm )Tm + 2(λ + τm )Tm 2

, Tf =

λ2 τm Tm . λ2 τm + λ (λ + 2τm )Tm

2 Ti λ2 + x1τm − 0.5τm , Ti = −Tm + x1 − , − K m ( 2λ + τ m − x 1 ) 2λ + τm − x1

τm (0.167 τm − 0.5x1 ) 2 λ2 + x1τm − 0.5τm 2λ + τm − x1 − ; Ti 2λ + τm − x1 2

− Tm x1 − Td =

Desired closed loop transfer function = λ3 5

Kc =

Tm

2

+

3λ2 + 3λ + τm Tm

 λ3 3λ2  Km  2 + T Tm   m

, Ti =

λ3 Tm

2

+

 λ  , x1 = Tm  + 1 e τ m 2  Tm (λs + 1)   2

e − sτ m

Tm

 − 1 .  

3λ2 + 3λ + τm , Tm λ2 Td =

Tm λ3 Tm

2

2

+

3λ +3 Tm

3λ2 + + 3λ + τm Tm

λτ m , Tf =

λTm . λ + 3Tm

b720_Chapter-04

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458

     1  1 + Td s   4.10.4 Classical controller G c (s) = K c 1 + Td s  Ti s   1 +  N   E(s) R(s)

+

−

    U(s)  1  1 + Td s   K c 1 + Td  Ti s   s 1 + N  


m

Y(s)


Minimum IAE – Shinskey (1988), p. 381. Method: Model 1; N not specified.

Huang and Chen (1997), (1999).

1

Kc 0.91Tm

Ti

Td


Comment

Minimum performance index: regulator tuning K m τm 1.70τ m 0.60τ m τ m Tm = 0.1

1.00Tm K m τm

1.90τ m

0.60τ m

τ m Tm = 0.2

0.89Tm K m τm

2.00τ m

0.80τ m

τ m Tm = 0.5

0.86Tm K m τm

2.25τ m

0.90τ m

τ m Tm = 0.67

0.83Tm K m τm

2.40τ m

1.00τ m

τ m Tm = 0.8

1

Kc

Direct synthesis: time domain criteria Ti Td N =∞; Model: Method 4

0 < τm Tm < 1 ; ‘Good’ servo and regulator response.

Kc =

1.02379    T  0.5   , Ti = K c K m Tm  2 x1 + τm  , 1 + 1.61519 m  T  K K 1 T − Km  τ  c m m   m  m  

 τ τ with recommended x 1 = Tm  − 1.3735.10 −3 + 0.22091 m + 1.6653 m Tm   Tm  2   τm   τm −4  .   Td = Tm 1.5006.10 + 0.23549 + 0.16970 Tm Tm      

  

2

;  

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Ti

Comment

Td

Direct synthesis: frequency domain criteria Paraskevopoulos x1τm Tm N =∞; 2 2 Ti et al. (2006). Ti x 2 1 + x 2 τ > 0.15τm Tm 0.02 < m < 0.9. 3 1 + Ti 2 x 2 2 Model: Method 1 Ti Tm

2

0 ≤ x1 ≤ 0.55 +

τm  0.0098   0.31 −  ; x1 < 0.15 .  Tm  1 − (τm Tm ) 

Ti 2 (1 − (τm Tm ) + Td ) + Ti − (τm Tm ) + Td + x 3

x2 =

2Ti 2 ((τm Tm ) − Td )

,

2

    τ  2  τ τ τ x 3 = Ti 2 1 − m + Td  + Ti − m + Td  + 4 m − Td Ti 1 + Ti − m + Td  ; Tm Tm   Tm    Tm   

(

)

 0.632((τ m Tm ) − Td ) + 0.436 (τ m Tm ) − Td − 0.0153 φ m φ max m Ti = x 4 1 + (τm Tm ) − Td 1 − φ m φ max  m x4 =

0.0029 − 0.0682 (τm Tm ) − Td + 1.4941((τm Tm ) − Td )

(1.003 − (τm

Tm ) + Td )

2

(

;

 T  τ  T max φm > 0.2φmax = − m − Td  m − Td − 1 + tan −1  m − Td − 1  . m , φm  τm   Tm  τm   3

x 2 and x 3 given in footnote 2;  T    Tτ Ti = x 4 1 + m 1 + 0.4 d m − 0.22Td  x 5  , x 4 given in footnote 2; Tm    τm 

(

)1 −(φ(φ φ φ ) ) , τ ) − 1).

x 5 = − 0.0153 − 0.436 τm Tm + 0.632(τm Tm )

φmax = −(τm Tm ) (Tm τm ) − 1 + tan −1 m

( (T

m

m

m

max m max m m

)

 , 

459

b720_Chapter-04

FA


460 Rule Paraskevopoulos et al. (2006). Model: Method 1

4

04-Apr-2009

0 ≤ x1 ≤ 0.55 + Kc =

4

Kc

Ti

Td

Comment

Kc

Ti

x1τm Tm

N=∞

τm  0.0098   0.31 −  ; x1 < 0.15 . 0.02 < τm Tm < 0.9 .  Tm  1 − (τm Tm ) 

2 2 K max 1 + ωmin 1 + ωmax or K min A m , K min = Ti ωmin , K max = Ti ωmax ; 2 2 2 Am 1 + Ti ωmin 1 + Ti ωmax 2

0.03([τm Tm ] − Td ) 1.14 − [τm Tm ] + Td = 1+   0.05  0.973 + Ti − x 2   [ ] − τ + 1 T T m m d   0.006 +

ωmin

x2

T i −([τm Tm ] − Td )(1 + Ti )

,

4 2   x   τ    τ 1.571 ωmax = 1 + 0.22 m − Td   1 +  0.1 − 0.3 m − Td  2     Tm   Tm      Ti   (τm Tm ) − Td      Ti Ti − 0.9463  − 0.5609 .   (Ti + 1)([τm Tm ] − Td )   (Ti + 1)([τm Tm ] − Td )  1+ (τ m Tm )− Td    max     A m − 1  A m − 1    Ti = x 2 1 + 0.65      1 −  Am − 1 A max − 1      m  

[

[

]

]

[

     + 0.01 − 0.18 + 5 τm − T  d  Tm    

2 3 + 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td )

x2 =

0.0029 − 0.0682 [τm Tm ] − Td + 1.4941([τm Tm ] − Td )

(1.003 − [τm

max A m > 0.2(A max = m − 1) +1, A m

]

 3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2  + 0.01 , (1 − [τm Tm ] + Td )3   Tm ] + Td )

2

;

1.571  τm  0.4085[[τm Tm ] − Td ]      T − Td 1 + 1 − 0.2864[[τ T ] − T ]  m m d   m  1

(7.41 − 3.52[τm Tm ])(TdTm τm )3 1+ 3 1 − (0.47 + 0.49[τm Tm ])(Td Tm τm )

.

b720_Chapter-04

04-Apr-2009

FA


Rule Paraskevopoulos et al. (2006) – continued.

5

5

Kc

Ti

Kc

Ti

Comment

Td

>

461

0.15τm Tm

N=∞

K c and x 2 as given in footnote 4, with

0.03([τm Tm ] − Td ) 1.14 − [τm Tm ] + Td = 1+   0.05  0.973 + Ti − x 2   [ ] − τ + 1 T T m m d   0.006 +

ωmin

x2 τ  T i − m − Td (1 + Ti )  Tm   (0.14 + 1.15[τm Tm ])(Td Tm τm )3  x 2     , 1 + 2 T  1 + 2(Td Tm τm )  i   

ωmax

2 4   x 2    τm     τm 1.571         = 1 + 0.22 − Td  1 + 0.1 − 0.3 − Td      [ T T T τ     m  m     i   m Tm ] − Td 

   Ti Ti − 0.9463  − 0.5609   ;  (Ti + 1)([τm Tm ] − Td )   (Ti + 1)([τm Tm ] − Td ) ( 5.53 − 0.41[τm Tm ])(Td Tm τm )4 1+ 3 2 1 − 0.65(Td Tm τm ) 1.27 − 0.4(Ti x 2 )

[

1+ (τ m Tm )− Td    max     A m − 1  A m − 1    Ti = x 3 1 + 0.65      1 −  Am − 1 A max − 1      m  

[

[

]

]

[

][

]

     + 0.01 − 0.18 + 5 τm − T  d  Tm    

2 3 + 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td )

]

2   3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2    Td Tm     + . 1 x4  , + 0.01  3    (1 − [τm Tm ] + Td )    τm   

  T 3 0.367 + 1.78[τ T ] m m  x 3 = x 2 1 +  Td m  ,   τm  1 − (Td Tm τm )2    3τ m    Td Tm  Tm  x 4 =      τm   

 2  (A − 1)3    τ 3  τm  τ m m m   2  − 3.32  , 3   T  + 1.2 T − 0.27  − max   Tm  m  m  A m − 1   

A max as given in footnote 4. 0.02 < τm Tm < 0.9 . m

(

)

b720_Chapter-04

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462



+

−

  Td s 1 K c 1 + +  Ti s 1 + Td  N 

  b + b s + b s 2 U(s) f1 f2  f0 + + 1 a s a s2  f 1 f 2 s 



Y(s)

m


Kc

K m e − sτ m Tms − 1

Td

Comment

0.25τm

Model: Method 1

Ti

Robust 1

Shamsuzzoha et al. (2007).

0.667 τm

Kc

For τm Tm > 0.6 , λ = τm . Otherwise, λ = x1Tm ; x1 given for various τm Tm and M s values:

1

Kc = −

τm Tm

M s = 3. 0

M s = 3. 6

τm Tm

M s = 3. 0

M s = 3. 6

0.05 0.1 0.2 0.3

0.015 0.04 0.09 0.14

0.015 0.04 0.08 0.13

0.4 0.5 0.6

0.21 0.32 0.41

0.18 0.26 0.36

3  4τm λ  τm  e , b f 0 = 1 , b f 1 = Tm 1 + K m (6τm + 18λ − 6b f 1 )  Tm   2

N = ∞ , af1 =

2b f 1τm + τm + 12λτ m + 18λ2 + Tm , a f 2 = 0 . 6τ m + 18λ − 6b f 1

Tm

 − 1 , b f 2 = 0 ,  

   

b720_Chapter-04

04-Apr-2009

FA




    T s  α + β d K c T   1+ d s  N   E(s) R(s) +

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

−

U(s)

+


m

Table 96: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Minimum IAE – Huang and Lin (1995).

1

Kc = −

Kc 1

Kc

Ti

Td

Y(s)


Comment

Minimum performance index: regulator tuning Model: Method 2 Ti Td α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 .

T  1  τ 0.2675 + 0.1226 m + 0.8781 m  Km  Tm  τm  

1.004 

(

,  

Ti = Tm 0.0005 + 2.4631(τm Tm ) + 9.5795(τm Tm )

2.9123

), T

d

463

= 0.0011Tm + 0.4759τm .

b720_Chapter-04

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FA


464 Rule

Kc

Minimum ISE – Paraskevopoulos et al. (2004). Minimum IAE – Huang and Lin (1995). Minimum ISE – Paraskevopoulos et al. (2004).

2

Kc 4

Ti 3

(1)

Ti

Td

Comment

0

Model: Method 1

Minimum performance index: servo tuning Model: Method 2 Ti Td

Kc

α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 . 5

Kc

0

Ti

(1)

Model: Method 1

Minimum performance index: other tuning

Paraskevopoulos et al. (2004). Model: Method 1

2

Kc

(1)

Kc

= Kc

(max)

=

(min)

Kc

ωmax Ti Km

6

(max)

Kc

, Kc

0

Ti

(1)

(min)

=

1 + ω max 2 Tm 2 2

1 + ωmax Ti

2

ωmin Ti Km , ωmin =

1 + ω min 2 Tm 2

α =1

,

1 + ωmin 2 Ti 2

τm  1  Ti −  , Tm  Tm Tm + Ti 

[4.935 τm][(Ti (Ti + τm )) − (τm Tm )] , T ω 2 >> 1 . (τm Tm ) + π[(Ti (Ti + τm )) − (τm Tm )] i max 1.916 + 4.78(τ m Tm ) 3 Ti = τm , α = 1 , 0.05 ≤ τm Tm ≤ 0.85 . 0.88 − (τm Tm ) 1.0251 4 ), K c = −(1 K m )(− 0.433 + 0.2056(τm Tm ) + 0.3135(Tm τm ) 6.6423 ), Ti = Tm (− 0.0018 + 0.8193(τm Tm ) + 7.7853(τm Tm ) 7.6983(τ T ) ). Td = Tm (− 0.0312 + 1.6333(τm Tm ) + 0.0399e 2.725 + 1.71(τ m Tm ) (1) 5 K c given in footnote 2. Ti = τm , α = 1 , 0.05 ≤ τm 0.88 − (τm Tm ) ωmax =

m

6

Kc

(1)

m

∫ [e (t) + (u( t) − u ) ]dt .

given in footnote 2. Minimise performance index =

[

]

∞

2

0

Ti = Tm 0.747 + 3.66(τm Tm ) + 18.64(τm Tm ) , 0.01 ≤ τm Tm ≤ 0.3 . 3

 4.52 + 0.476(τm Tm )2  τm Ti = τm  ≤ 0.85 .  , 0.3 ≤ Tm 0.88 − (τm Tm )  

Tm ≤ 0.85 . ∞

2

b720_Chapter-04

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Rule

Kc

Paraskevopoulos et al. (2004) – continued. Chidambaram (2000c). Model: Method 1

7

Kc

(1)

8

Kc

9

Kc

Ti

Td

Comment

Ti

0

α =1

465

Direct synthesis: time domain criteria Ti ‘Small’ τ m Tm . 0 τ 0.1 ≤ m ≤ 0.6 Ti Tm

10 (1) Chidambaram τm Tm < 0.6 Ti 0.245 (2000c), Srinivas 1.165  Tm  4.4Tm + 9 τ m 0.176T τ and   m 0.6 ≤ m ≤ 0.8 K m  τ m  Chidambaram T +0.36τm m (2001), Sree and ( 2) τ Chidambaram Ti 0.8 ≤ m ≤ 0.9 (2006), Tm pp. 182–184. Model: Method 1 ξ = 0.333 ; N = ∞ , β = 1 , α = 0.74 + 0.42(τm Tm ) − 0.23(τm Tm )2 (Chidambaram, 2000c); 2 τ 1 1  Tm τ  − m  (Srinivas ; β= N =∞, α = m + τm − Ti K c K m Td  K c K m 2Ti  and Chidambaram, 2001); ξ ≥ 1 ; N = ∞ , β = 1 ; α = 1 (Sree and Chidambaram, 2006).

7

Kc

(1)

∞

∫ [e (t) + (du dt) ] dt.

given in footnote 2. Minimise performance index =

2

2

0

 2.58 + 3.60(τm Tm )2  τm Ti = τm  ≤ 0.85 .  , 0.03 ≤ T ( ) 0 . 88 − τ T m m m   8

Kc =

2

Tm Ti 2 2

0.04TS ξ + Ti τm

α = 1−

, Ti =

0.04TS ξ2 + Tm τm + 0.4TSξ 2 , Tm − τm

0.2TS ξ 2 (Tm − τ m ) 0.04TS 2 ξ 2 + Tm τ m + 0.4TS ξ 2

(Chidambaram, 2000c); α = 1 −

0.2TSξ 2 or Ti

α = 0.733 + 0.6(τm Tm ) − 0.36(τm Tm ) (Sree and Chidambaram, 2006). 5.8 τ m Tm Tm , Ti = 0.4015Tm e , 2

9

K c = (2.695 K m )e −1.277 τ m

α = 0.733 + 0.6(τ m Tm ) − 0.36(τ m Tm ) ; ξ = 0.333 . 2

10

Ti (1) =

0.176Tm + 0.36τm

0.179 − 0.324(τm Tm ) + 0.161(τm Tm )2

, Ti

( 2)

=

0.176Tm + 0.36τm . 0.12 − 0.1(τm Tm )

b720_Chapter-04

04-Apr-2009

FA


466 Rule

Kc

Ti

Comment

Td

τm Tm ≤ 0.2

11 Kc 6.461τ m Jhunjhunwala 0.898 and 0.324Tm e Tm  Tm   Chidambaram 0.876   τm  (2001). 12 Ti Model: Method 1 Kc

τm ≤ 0.6 Tm

0

0.2 ≤

Td

β =1, N = ∞ .

Minimum ISE (Chidambaram, 2002, pp. 157–158). Chandrashekar et al. (2002). Model: Method 1

13

0

Ti

Kc

 K c K m (2 − Tm ) + (K c K m − 1)Ti τm  (Jhunjhunwala and 10.7572 − 35.1  ; α = T 2K c K m m  K K (2 + τm ) − (K c K m + 1)Ti Chidambaram, 2001); α = c m (Chidambaram, 2002). 2K c K m

11

Kc =

1 Km

12

Kc =

1  τ  τ 1.397  Tm  13.0 − 39.712 m  , m < 0.2 ; K c =   K m  Tm  Tm K m  τm  2.044 τ m

Ti = 0.856Tm e

Tm

τ , 0 ≤ m ≤ 1 ; Ti = 0.3444Tme Tm

0.769

2.9595 τ m Tm

, 1≤

, 0.2 ≤

τm ≤ 1.4 ; Tm

τm ≤ 1.4 ; Tm

Td = 0.5643τm + 0.0075Tm , 0 ≤ τm Tm ≤ 1.4 ;

α = 0.3137 , τm Tm = 0.1 ; α = 1 − 0.4772e − (1.58τ m 13

49.535 − 21.644 τ m e Km Ti = 0.1523Tm e7.9425τ m

Kc =

Tm Tm

,

Tm )

, 0.2 ≤ τm Tm ≤ 1 .

τm 0.8668  τm    < 0.1 ; K c = K m  Tm  Tm

−0.8288

, 0.1 ≤

τm ≤ 0.7 ; Tm

;


(ηs + 1)e −sτ (TCL2 s + 1)2

m

;

TCL 2 = 0.025Tm + 1.75τm , τm Tm ≤ 0.1 ; TCL 2 = 2 τ m , 0.1 ≤ τm Tm ≤ 0.5 ; TCL 2 = τm (5(τm Tm ) − 0.5) , 0.5 ≤ τm Tm ≤ 0.7 .

α = 0.505 + 3.426(τm Tm ) − 10.57(τm Tm ) , τm Tm ≤ 0.2 ; 2

α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 0.7 (Chandrashekar et al, 2002);

α = 0.67 , τm Tm ≤ 0.1 ; α = 0.6354 + 0.437(τm Tm ) , 0.1 ≤ τ m Tm ≤ 0.7 (Sree and Chidambaram, 2006).

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Chandrashekar et al. (2002). Model: Method 1

14

Kc =

14

Kc

467

Comment

Ti

Td

Ti

0.5τ m

2  τ  τ τ 1  4282 m  − 1334.6 m + 101 , 0 < m < 0.2 ; T Km  T T  m m  m  

Kc =

[

]

1.1161  Tm    K m  τm 

0.9427

, 0.2 ≤

τm ≤ 1.0 ; Tm

Ti = Tm 36.842(τm Tm ) − 10.3(τm Tm ) + 0.8288 , 0 ≤ τm Tm ≤ 0.8 ; 2

Ti = 76.241Tm (τm Tm )

6.77

, 0.8 ≤ τm Tm ≤ 1.0 ;

α = 0.83 , τm Tm ≤ 0.1 ; α = 0.8053 − 0.1935(τm Tm ) , 0.1 ≤ τm Tm ≤ 0.7 (Chandrashekar et al., 2002); α = 0.505 + 3.426(τm Tm ) − 10.17(τm Tm ) , τm Tm ≤ 0.2 ; 2

α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 1.0 (Sree and Chidambaram, 2006).

(ηs + 1)e −sτ ; (TCL2 s + 1)2 2 Tm ) + 2.03(τm Tm ) )Tm , 0 ≤ τm m


(

TCL 2 = 0.0326 + 0.4958(τm TCL 2 = 4.5115Tm (τ m Tm )

Tm ≤ 0.8 ;

5.661

, 0.8 ≤ τm Tm ≤ 1.0 , N = ∞ ; β = 0 (Chandrashekar et al., 2002; Sree and Chidambaram, 2006); [(1 − α )Tis + 1]e−sτm . Desired closed loop transfer function = TCL 2s 2 + 2ξTCLs + 1 If it is desired to minimise the overshoot (servo response), for ξ ≥ 1 , then α = 1 . Otherwise, the optimum value of α is chosen by minimising the ISE (servo response). Then, α = 1 − 2(TCL Ti ) , ξ = 1 ; α = 1 − 2(TCL ξ Ti ) , ξ < 1 ; T α = 1 − CL Ti

2 3   2 2 2       − ξ + ξ − 1  − ξ − ξ − 1  −  − ξ + ξ − 1  + x1       , ξ > 1 with  3    − ξ + ξ2 − 1  − ξ − ξ2 − 1  + x     2       

3

2

x 1 = − − ξ − ξ 2 − 1  +  − ξ + ξ 2 − 1   − ξ − ξ 2 − 1  and       3

2

2

x 2 =  − ξ − ξ 2 − 1  − ξ + ξ 2 − 1  − 2 − ξ + ξ 2 − 1   − ξ − ξ 2 − 1  ;        N= ∞ ; β = 1 (Sree and Chidambaram, 2005a).

b720_Chapter-04

04-Apr-2009

FA


468 Rule Chandrashekar et al. (2002). Model: Method 1

Kc

Ti

Td

Kc

Ti

0

Comment

6.0000 K m

Alternative tuning rules: α Ti 0.6000Tm 0.6667

τ m Tm = 0.1

3.2222 K m

1.4500Tm

0.7246

τ m Tm = 0.2

2.2963 K m

2.6571Tm

0.7742

τ m Tm = 0.3

1.8333 K m

4.4000Tm

0.8182

τ m Tm = 0.4

1.5556 K m

7.0000Tm

0.8571

τ m Tm = 0.5

1.3265 K m

14.6250Tm

0.8974

τ m Tm = 0.6

1.1875 K m

31.0330Tm

0.9323

τ m Tm = 0.7

Kc

Alternative tuning rules; TCL 2 = x 5 Tm : x1 K m

x 2Tm

N = x3 x 4 ,

x 3Tm

β = 0.

Sree and Chidambaram (2006).

x1

x2

10.3662 5.3750 3.2655 2.4516 2.0217 1.7525 1.5458 1.3723 1.2420 1.1400 1.0895 1.0536

0.3874 0.8600 1.8018 3.0400 4.6500 6.7286 10.337 17.693 33.610 80.620 143.6 277.37

x3


0.0435 0.0134 0.10 0.0884 0.0250 0.20 0.1375 0.0435 0.40 0.1868 0.0581 0.60 0.2366 0.0696 0.80 0.2866 0.0784 1.00 0.3380 0.0885 1.30 0.3910 0.1005 1.80 0.4440 0.1124 2.60 0.4969 0.1247 4.20 0.5975 0.1138 5.00 0.6982 0.0957 6.00 pp. 118, 125. Model: Method 1

α

τ m Tm

0.742 0.767 0.778 0.803 0.828 0.851 0.874 0.898 0.923 0.948 0.965 0.978

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Ti > Ti

Paraskevopoulos et al. (2004). Model: Method 1; α = 1.

15

Kc

(1)

>

Comment

Td

(min)

1.35τm

(1 − τm

0

Tm )

2

x1τ m +

x2

Coefficient values: x1 x2 ξ

ξ

0.7269 2.5362 0.5 3.0696 3.7509 1.7016 3.0585 0.75 6.975 5.7053 (1) x 1 τ m + x 2 Tm + K

1 1.5

c

6

x 3 τ m Tm x2

x3

4.59 6.16 8.21

-0.19 0.028 0.37

8.28 9.90 16.88

Kc

16

(1)

x1

x2

x1

x2

1.518 10.54 3.34 16.89

15

Kc

(1)

TCL 2

0.5 0.75

Ti

1 1.5 0

Coefficient values: x1 x2 TCL 2 5.81 13

τm < 0.7 Tm

0.1
1.7 ;

0

0.5τm

τm 1.7

a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = (Tm τm )

2

3

0

Ti

Kc

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,

Ti = −Tm + a f 2 −

Td =

0

Ti

Kc

a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,

Sree and Chidambaram (2006), pp. 27–28. Model: Method 1

Kc =

0 ≤ τm Tm < 2

Td

a f 1 = 0 , bf 1 = 0 ; N = ∞ ; bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 ,


2

Ti

Kc

Km .

2Tm − Tm τm − τm 2λK m

, Ti =

 1  Tm  τ  τ τ 1 − m  sin cos 1 − m  m +  Km  τm  Tm   Tm  Tm 

Tm − Tm τ m Tm τ m − 1

+ 0.5τm .

 τ τ 1 − m  m  T T m  m 

 .  

b720_Chapter-04

04-Apr-2009

FA



R(s)

+

F1 (s)

U(s)

Y(s)

Model

−

F2 (s) Table 98: PID controller tuning rules – unstable FOLPD model G m (s) = Rule

Kc 1

Minimum ITSE – Majhi and Atherton (2000). Model: Method 1

1

Kc =

(

Kc

Ti

K m e − sτ m Tms − 1

Comment

Td

Minimum performance index: servo tuning 0 Ti

α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ; b f 3 = 1 ; b f 4 = 0.5τm Tm ; a f 5 = 0 .

0.8011Tm 1 − 0.9358 τm Tm K m τm

) , T = T  0.1227 + 1.4550 τ i

m



m

Tm

− 1.2711

τm 2  . 2 Tm 

475

b720_Chapter-04

04-Apr-2009


476 Rule Minimum ITSE – Majhi and Atherton (2000). Model: Method 3; Ap K= . K mh

Kc

Ti

Td

2

Kc

Ti

0

3

Kc

Ti

0

Comment 0
> τm (Marchetti et al., 2001).

τm Tm1 = 0.2 , 0.4Tm1 ≤ λ ≤ 4.3Tm1

K m uncertainty = 30%.

τm Tm1 = 0.4 , 1.1Tm1 ≤ λ ≤ 4.3Tm1 τm Tm1 = 0.6 , 2.2Tm1 ≤ λ ≤ 4.3Tm1

1

Kc =

K m Ti (Tm1 − τm )(Tm 2 + τm )

(TCL3 + τm )3

Ti =

TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm ) , (Tm1 − τm )(Tm 2 + τm )

Td =

(Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2 Ti (Tm1 − τm )(Tm 2 + τm )

3

2

,

2

.

  λ     λ Tm1 λ + 2  + Tm 2  λ + 2 Tm 2 T   λ   Tm1    m1   , T = λ  Kc = . i  T + 2  + Tm 2 , Td =  λ  λ2 K m  m1  λ + 2  + Tm 2  Tm1 

b720_Chapter-04

04-Apr-2009

FA


Rule Lee et al. (2000). Shamsuzzoha et al. (2005), Shamsuzzoha and Lee (2007a).

3

4

Kc

Ti

Td

Comment

Kc

Ti

Td

Model: Method 1

Kc

Ti

Td

Model: Method 1; λ values not defined.

0.25Ti


489

Ultimate cycle


3

Kc =

5

Ti

Kc

2 Ti λ2 + x1τm − 0.5τm , , Ti = −Tm1 + Tm 2 + x1 − − K m (2λ + τm − x1 ) 2λ + τm − x1

τm (0.167 τm − 0.5x1 ) 2λ + τm − x1 2

− Tm1x1 + Tm 2 x1 − Tm1Tm 2 − Td =

4

Ti

2

−

λ2 + x1τm − 0.5τm ; 2λ + τm − x1

2  λ  λ = desired closed loop dynamic parameter, x1 = Tm1  + 1 e τ m  Tm1   Ti Kc = − , Ti = Tm 2 − Tm1 + 2 x1 − x 2 , K m (4λ + τm − 2 x1 )

 − 1 .  

0.167 τm 3 − x1τm 2 + x12 τm + 4λ3 4λ + τm − 2 x1 − x2 , Ti

x1 − 2x1 (Tm1 − Tm 2 ) − Tm1Tm 2 − 2

Td =

Tm1

  x1 = Tm1  

 4 τm 2 2  Tm  λ 6λ2 − x1 + 2x1τm − 0.5τm  e − 1 , x 2 =  . 1 +  T  4 λ + τ m − 2 x1  m1   2

5

Kc =

1.111 Tm1Tm 2 K m τm 2

      1 ,  0.65    (Tm1 + Tm 2 )Tm1Tm 2     1 +    (Tm1 − Tm 2 )(Tm1 − τm )τm   0.65   (T + T )T T    m1 m 2 m1 m 2 Ti = 2τm 1 +   .   (Tm1 − Tm 2 )(Tm1 − τm )τm  

b720_Chapter-04

04-Apr-2009

FA


490


 1  1 G c (s) = K c 1 + + Td s   Tf s + 1  Ti s E(s)

R(s)

−

+

  1 K c 1 + + Td s  Ti s  

U(s) 1 (Tf s + 1)

Y(s) K m e − sτ m (Tm1s − 1)(1 + sTm 2 )

Table 105: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule

Kc

Ti

Kc

Ti

Td

Comment

Td

τ m < Tm1

Robust Zhang and Xu (2002). Model: Method 1

1

Kc =

1

For τm Tm1 ≤ 0.5 , λ = 2 − 5(τ m + Tm 2 ) .

Tm 2Tm1 (Tm1 − τm ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12

(

K m λ3 + 3λ2Tm1 + 3λTm1τm + τm 2Tm1 2

Ti = Tm 2 +

Td = Tf =

)

2

λ3 + 3λ2Tm1 + 3λTm1 + τmTm1 , Tm1 (Tm1 − τm )

Tm 2 (λ3 + 3λ2Tm1 + 3λTm12 + τm Tm12 )

Tm1Tm 2 (Tm1 − τ m ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12

(

,

λ3Tm1 (Tm1 − τm )

Tm1 λ + 3λ2Tm1 + 3λTm1τm + τm 2Tm1 3

).

,

b720_Chapter-04

04-Apr-2009

FA


491


E(s)

−

+

    U(s) Y(s)  1  1 + Td s  K m e − sτ m  K c 1 +  T   T s i  1+ d s  (Tm1s − 1)(1 + sTm 2 )   N  

Table 106: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule

Kc

1 Minimum ITAE Kc – Poulin and Pomerleau (1996), (1997). τ m + Tm 2 Tm1

Process output step load disturbance. Process input step load disturbance.

x 2Tm1 1 + 1

Kc =

Ti

Minimum performance index: regulator tuning Model: Method 1 Ti Tm 2

0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25

x1Tm 2

Comment

Td

Coefficient values (deduced from graph): τ m + Tm 2 x1 x2 x1 Tm1 0.9479 1.0799 1.2013 1.3485 1.4905 1.1075 1.2013 1.3132 1.4384 1.5698

2.3546 2.4111 2.4646 2.5318 2.5992 2.4230 2.4646 2.5154 2.5742 2.6381

0.30 0.35 0.40 0.45 0.50 0.30 0.35 0.40 0.45 0.50

1.6163 1.7650 1.9139 2.0658 2.2080 1.6943 1.8161 1.9658 2.1022 2.2379

x2

2.6612 2.7368 2.8161 2.9004 2.9826 2.7007 2.7637 2.8445 2.9210 3.0003

2

4(τm + Tm 2 )2

K m (x1Tm1 − 4[τm + Tm 2 ])

, Ti =

4Tm1 (τm + Tm 2 ) ; x1Tm1 − 4(τm + Tm 2 ) 0≤

T τm ≤ 2 ; 0.1Tm 2 ≤ d ≤ 0.33Tm 2 . N) N

(Td

b720_Chapter-04

04-Apr-2009

FA


492 Rule

Kc

Huang and Chen (1997), (1999). Model: Method 4

2

Ti

Comment

Td

Direct synthesis: time domain criteria Ti Tm 2

Kc

‘Good’ servo and regulator response; N = ∞ . Direct synthesis: frequency domain criteria

Chidambaram and Kalyan (1996). Ho and Xu (1998). Model: Method 1; N =∞.

3

ωpTm1 A m K m

Kc =

4

Ti

Tm 2

Ti

Tm 2


Representative result: −Tm1 Tm 2

0.7854Tm1 K m τm

Paraskevopoulos et al. (2006). Ti x1

2

Kc

Am = 2 ,

φ m = 450 . N =∞; Model: Method 1

2

1 + x1

5

1 + Ti 2 x12

Tm 2

Ti

1.13076    T  τ 0.5   , Ti = K c K m Tm  2 x1 + τm  , 0 < m < 0.8 ; 1 + 1.11416 m    T K K 1 T T − Km  τ  m1 m c m m m      

Recommended x 1 = 0.160367Tm e 3

6.06426

τm Tm 1

.

K c and Ti are defined by De Paor and O’Malley (1989), Rotstein and Lewin (1991) or Chidambaram (1995b), for the ideal PI control of an unstable FOLPD model [see Table 91]. 1 4 . Ti = 2 1.57ωp − ωp τm − (1 Tm1 ) 5

τm Tm1 < 0.9 ; Tm 2 Tm1 > 0.3 .

Ti (1 − (τm Tm1 )) + Ti − (τm Tm1 ) + x 2 2

x1 =

2Ti 2 (τm Tm1 )

{T (1 − (τ T )) + T − (τ T = x (1 + (T τ )[0.632(τ T i

x3 =

i

3

m

m1

m

m1

i

m

}

2

2

x2 =

,

m

m1

Tm1 ) + 4(τm Tm1 )Ti 2 (1 + Ti − (τm Tm1 )) ;

) + 0.436

0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 )

(1.003 − (τm Tm1 ))2 max φm > 0.2φmax = −(τm Tm1 ) (Tm1 m , φm

])1 −(φ(φ φ φ ) ) ,

τm Tm1 − 0.0153

m

max m max m m

,

τm ) − 1 + tan −1

( (T

m1

)

τm ) − 1 .

b720_Chapter-04

04-Apr-2009

FA


Rule Paraskevopoulos et al. (2006). Model: Method 1

6

Kc

Ti

Td

Comment

Kc

Ti

Tm 2

N=∞

6

A m > 0.2(A max m − 1) +1;

493

τm T < 0. 9 ; m 2 > 0. 3 . Tm1 Tm1

K c = K max A m or K min A m ,

K min = Ti ωmin

1 + ωmin 2 1 + Ti 2ωmin 2

K max = Ti ωmax

0.03[τm Tm1 ] 1.14 − [τm Tm1 ] = 1+   0.05  0.973 + Ti − x1  1 − [τm Tm1 ]  

1 + ωmax 2 1 + Ti 2ωmax 2

0.006 +

ωmin

x1

,

Ti −[τm Tm1 ](1 + Ti )

,

4 2   τ    τ  x   1.571Tm1 ωmax = 1 + 0.22 m   1 +  0.1 − 0.3 m  1   Tm1  Ti   τm   Tm1           Ti Ti − 0.9463  − 0.5609 ,  T + 1 τ T T + 1 τ T ( )( ) ( )( ) m m1 m m1  i   i 

x1 =

0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 )

(1.003 − [τm

Tm1 ])2

;

1+ (τ m Tm1 )     max      A m − 1  A m − 1    + Ti = x 2 1 + 0.65     A max − 1   1 A 1 − −   m     m   

[

with A max m

[

]

]

2 3  τ   τ  3(τm Tm1 )4 − 2.3(τm Tm1 )2  τ τ , 0.01− 0.18 + 5 m − 32 m + 75 m  − 51 m  + Tm1 Tm1   (1 − (τm Tm1 ))3  Tm1   Tm1    1.571Tm1 = .  0.4085[τm Tm1 ]   τm 1 +   1 − 0.2864[τm Tm1 ] 

b720_Chapter-04

04-Apr-2009


494 Rule

Kc

Ti

Paraskevopoulos 2 et al. (2006). Ti x 2 1 + x 2 2 2 1 + Ti x 2 Model: Method 1

7

0 ≤ x1 ≤ 0.55 +

(τm + Tm 2 )  0.31 − Tm1

Comment

Td

x1 ( τm + Tm 2 ) Tm1

Ti

N=∞

> 8

7

FA

 

Ti

0.0098

0.15(τm + Tm 2 ) Tm1

  ; x1 < 0.15 .

1 − ([τm + Tm 2 ] Tm1 ) 

Ti (1 − ([τm + Tm 2 ] Tm1 ) + Td ) + Ti − ([τm + Tm 2 ] Tm1 ) + Td + x 3 2

x2 = x3 =

2Ti 2 (([τm + Tm 2 ] Tm1 ) − Td )

{T

i

2

(1 − ([τm + Tm 2 ] Tm1 ) + Td ) + Ti − ([τm + Tm 2 ] Tm1 ) + Td }

2

,

+ x4 ,

x 4 = 4(([τm + Tm 2 ] Tm1 ) − Td )Ti (1 + Ti − ([τm + Tm 2 ] Tm1 ) + Td ) ; 2

  τ + Tm 2  τ + Tm 2 φm − Td  + 0.436 m − Td − 0.0153  0.632 m max T T φ  m1 m1   m Ti = x 5 1 + [ ( τm + Tm 2 ] Tm1 ) − Td 1 − φm φmax m  

(

x5 =

0.0029 − 0.0682

)

  ,   

([τm + Tm 2 ] Tm1 ) − Td + 1.4941(([τm + Tm 2 ] Tm1 ) − Td ) . (1.003 − ([τm + Tm 2 ] Tm1 ) + Td )2

0.02 < τm Tm1 < 0.9 ; Tm 2 Tm1 < 0.3 . φm > 0.2φmax m ,

    τm + Tm 2 Tm1 Tm1  − Td − 1 + tan −1  − Td − 1  . φmax − Td  m = −  τm + Tm 2   τ m + Tm 2  Tm1   8

x 2 , x 3 and x 4 given in footnote 7;    Tm1  T (τ + Tm 2 ) 1 + 0.4 d m − 0.22Td  x 6  ; x 5 given in footnote 7; Ti = x 5 1 +  Tm1    τm + Tm 2 

(

)

 τ + Tm 2 τ + Tm 2  φm φmax m x 6 =  − 0.0153 − 0.436 m + 0.632 m ,  Tm1 Tm1  1 − φm φmax m 

 τ + Tm 2 φ max = − m m  Tm1

(

   Tm1 Tm1 −1     τ + T − 1 + tan  τ + T − 1  . m2 m m 2  m  

)

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc 9

Ti

K max A m or

Paraskevopoulos et al. (2006).

K min A m

Comment

Td

x3

Ti

495

τm + Tm 2 Tm1


A m > 0.2(A max m − 1) +1; 0.02 < τ m Tm1 < 0.9 .

9

K min = Ti ωmin

1 + ωmin 2 2

1 + Ti ωmin

2

, K max = Ti ωmax

0.03(x1 − Td ) 1.14 − x1 + Td = 1+  0.05  Ti − x 2  0.973 +  1 x1 + Td  − 

1 + ωmax 2

0.006 +

ωmin

with

1 + Ti 2ωmax 2 x2

Ti −(x1 − Td )(1 + Ti )

, x1 =

τm + Tm 2 , Tm1

2   x   1.571 ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td  2     Ti   x1 − Td      Ti Ti − 0.9463  − 0.5609 ;   (Ti + 1)(x1 − Td )   (Ti + 1)(x1 − Td ) 

[

] (

[

] 

    Am − 1  Ti = x 2 1 + 0.65    1 −  Am   

)

x 1 − Td +1 

 − 1  A max m    − 1  A max − 1   m  

  + 0.01 5 x − T − 0.18 1 d     3(x − T )4 − 2.3(x1 − Td )2  + 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 + 0.01 1 d , (1 − x1 + Td )3  

[

[

]

[

x2 =

0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td )

(1.003 − x1 + Td )2

]

]

;

 T 0.0098   ; x 3 < 0.15 ; m 2 < 0.3 . 0 ≤ x 3 ≤ 0.55 + x1  0.31 −  Tm1 1 − x1  

b720_Chapter-04

FA


496 Rule

Kc

Paraskevopoulos et al. (2006) – continued. N =∞.

10

04-Apr-2009

K min = Ti ωmin ωmin

10

Ti

K max A m or

> Tm 2 Tm1 < 0.3 ; 0.15(τm + Tm 2 ) τ 0.02 < m < 0.9. Tm T

Ti

K min A m

m1

1 + ωmin 2 2

1 + Ti ωmin

, K max = Ti ωmax

2

1 + ωmax 2 2

 (0.14 + 1.15x1 )(Td x1 ) 1 + 2 1 + 2(Td x1 ) 

3

][ (

2

1 + Ti ωmax

0.006 + [0.03(x1 − Td ) (1.14 − x1 + Td )] = 1+ (0.973 + 0.05 [1 − x1 + Td ])Ti − x 2

[

Comment

Td

, x1 =

τm + Tm 2 , with Tm1

x2

T i −(x1 − Td )(1 + Ti )

0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td )  x 2    , x 2 = ; T  (1.003 − x1 + Td )2  i 

]

)

ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td (x 2 T1 )2 [1.571 (x1 − Td )]

   Ti Ti − 0.9463  − 0.5609  ( )( ) ( )( ) T 1 x T T 1 x T + − + − 1 d 1 d  i  i  ; ( 5.53 − 0.41x1 )(Td x1 )4 1+ 1 − 0.65(Td x1 )3 1.27 − 0.4(Ti x 2 )2

[

] 

    Am − 1  Ti = x 3 1 + 0.65    1 −  Am   

[

[

][

]

x 1 − Td +1 

 − 1  A max m    − 1  A max − 1   m  

]

 3(x − T )4 − 2.3(x1 − Td )2  + 0.01 1 d  (1 − x1 + Td )3  

  + 0.01 5 x − T − 0.18 1 d   

[

  T 2  1 +  d  x 4    x1    

[

]

]

+ 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 ,

  T  0.367 + 1.78x  1 x 3 = x 2 1 +  d  ,   x1  1 − (Td x1 )2    3

 (A m − 1)3  ,  2x13 − 3.32 x12 + 1.2x1 − 0.27 − 3   A max m −1  1 1.571 = . 3   (x1 − Td )1 + 0.4085[x1 − Td ]  1 + (7.41 − 3.52x1 )(Td x1 ) 3 1 − (0.47 + 0.49x1 )(Td x1 )  1 − 0.2864[x1 − Td ] 

T  x 4 =  d   x1 

A max m

3x1

(

)

b720_Chapter-04

04-Apr-2009

FA



E(s) +


    T s β d α + K Td  c  1+ s  N  

R(s)

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

    s 

− +

U(s)

Y(s) K m e − sτ (Tm1 s − 1)(1 + sTm 2 ) m

Table 107: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s ) = (Tm1s − 1)(1 + sTm 2 ) Rule

Kc

Huang and Lin (1995) – minimum IAE.

1

Ti

Td

Comment

Minimum performance index: regulator tuning α = 0 , β =1; 1 Ti Td N not specified; Kc Model: Method 2

Note: equations continued onto p. 498. 0.05 ≤ τ m Tm1 ≤ 0.4 ; Tm 2 ≤ Tm1 ; Kc = −

−1.164 τ  τ 1  T  − 174.167 − 31.364 m + 0.4642 m  − 103.069 m 2  Km  Tm1 Tm1   Tm1    2 . 54 1 . 065 T  T   Tm 2 T τ 1   − 83.916 m 2   − − 66.962 m 2 2m + 59.496 m1  T  τ Km  T m 1 m      Tm1 m1 

−

497

Tm 2 τ m Tm 2 τm  2 T 1  − 70.79 m 2 + 23.121e Tm1 + 126.924e Tm1 + 26.944e Tm1 Km  τm 

 ;  

  

1.014 

  

b720_Chapter-04

04-Apr-2009

FA


498 Rule Huang and Lin (1995) – continued.

2

Kc

Ti

Td

Comment

Kc

Ti

Td

α = 0 ,β =1, N not specified; Model: Method 2

2 2  T   τ  T τ Ti = Tm1 0.008 + 2.0718 m + 6.431 m  + 0.4556 m 2 + 0.7503 m 2   Tm1 Tm1   Tm1    Tm1   3 3  T τ 2 τ  T  T τ + Tm1 2.4484 m 2 2m − 18.686 m  − 2.9978 m 2  − 21.135 m 2 m  T 3  Tm1  Tm1   Tm1   m1  4 3       + 39.001 τ m  + 22.848 Tm 2 τ m  T    T 4   m1    m1 

 τ T 2 + Tm1 12.822 m m32  T   m1    T 3τ + Tm1  − 4.754 m 2 4 m  T   m1 

2 2    − 0.527 Tm 2 τ m   T 4 m1  

   + 1.64 Tm 2 T   m1 

  

4

;  

2 2  τ  T   T τ Td = Tm1 1.1766 m − 4.4623 m  + 0.5284 m 2 − 1.4281 m 2   Tm1 Tm1   Tm1   Tm1    3 3  T τ  τ 2T τ  T  + Tm1 4.6 m 2 2m + 11.176 m  + 1.0886 m 2  − 5.0229 m 3m 2  T  Tm1  Tm1   Tm1   m1 

 τ T 2 + Tm1  − 0.0301 − 0.5039 m m32  T   m1 

   

4 3       − 9.8564 τ m  − 7.528 Tm 2 τ m  T    T 4   m1    m1 

 τ T 3  τ 2T 2 + Tm1 − 2.3542 m m42  + 9.3804 m m4 2   T  T  m1  m1    2

   

   − 0.1457 Tm 2 T   m1 

  

4

.  

Note: equations continued onto p. 499. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ; Kc = −

2.1984 τ  1  T  τ 1750.08 + 1637.76 m + 1533.91 m  − 7.917 m 2  Km  Tm1 Tm1   Tm1   

−

T 1  6.187 m 2 T Km   m1 

−

Tm 2 τ m Tm 2 τm  2 T 1  1.3729 m 2 − 1739.77e Tm1 − 0.000296e Tm1 + 2.311e Tm1 Km  τm 

  

0.791

− 6.451

Tm 2 τ m Tm1 2

T  + 0.002452 m1   τm 

3.2927

 Tm 2   Tm1  ;  

  

1.0757 

  

b720_Chapter-04

04-Apr-2009

FA


2 2  T   τ  T τ Ti = Tm1 51.678 − 57.043 m + 1337.29 m  + 0.1742 m 2 − 0.1524 m 2    Tm1 Tm1  Tm1    Tm1   3  τ  T T τ + Tm1 7.7266 m 2 2m − 6011.57 m  + 0.0213 m 2 T  Tm1  m1   Tm1 

 T τ 2 + Tm1  − 65.283 m 2 m  T 3   m1   T 3τ + Tm1 0.003926 m 2 4 m  T   m1 

2    + 0.0645 τ m Tm 2   T 3   m1

3

 τ   + 9135.52 m    Tm1 

3     + 274.851 Tm 2 τ m    T 4    m1 

2 2    − 2.0997 Tm 2 τ m   T 4 m1  

   − 0.001077 Tm 2 T   m1 

Tm 2 τ m τm Tm 2  2 + Tm1  − 49.007e Tm1 + 0.000026e Tm1 + 0.2977e Tm1  

  

4

  

 ;  

2   τm  Tm 2  τm    + Td = Tm1 − 0.0605 + 4.6998 0 . 0117 − 29.478  Tm1 Tm1    Tm1   

 T + Tm1  − 0.0129 m 2   Tm1 

τ   T τ  + 0.6874 m 2 2m + 140.135 m  Tm1  Tm1  

 T + Tm1 0.002455 m 2   Tm1 

 τ 2T   + 1.4712 m 3m 2  T   m1

2

3

3

  

2    − 0.1289 τ m Tm 2  T 3   m1 

   

4 3     Tm 2 τ m 3   τm    + 0.007725 τ m Tm 2     0 . 6884 + Tm1 − 238.864 +   T 4   T 4    Tm1   m1   m1  

  τ 2T 2 + Tm1 − 0.1222 m m4 2  T  m1  

   − 0.000135 Tm 2 T   m1 

  

4

.  

4

  

499

b720_Chapter-04

FA


500 Rule

Kc

Minimum IAE – Huang and Lin (1995).

3

04-Apr-2009

3

Kc

Ti

Comment

Td

Minimum performance index: servo tuning α = 0 , β =1; Ti Td N not specified; Model: Method 2

Note: equations continued onto p. 501. 0.05 ≤ τ m Tm1 ≤ 0.4 , Tm 2 ≤ Tm1 ; 1 Kc = − Km

−1.344    Tm 2  10.741 − 13.363 τm + 0.099 τm  + 727 . 914 T  Tm1 Tm1    m1   

1 − Km

 T  − 708.481 m 2 T   m1 

  

0.995

+ 9.915

Tm 2 τ m Tm1 2

T  + 84.273 m1   τm 

Tm 2 τ m τm Tm 2  2 Tm 2 1  Tm 1 Tm 1 − − 90.959 + 9.034e − 2.386e − 16.304e Tm1  Km τm 

1.031

 Tm 2   Tm1

   

2.12  τ  T  τ Ti = Tm1 − 149.685 − 141.418 m − 88.717 m  − 17.29 m 2  Tm1 Tm1    Tm1   

 T + Tm1 20.518 m 2   Tm1 

0.985

  

− 12.82

Tm 2 τ m Tm1 2

τ  + 3.611 m   Tm1 

0.286

 Tm 2   Tm1

Tm 2 τ m τm Tm 2  2 Tm 2 Tm1 Tm1  + Tm1 0.000805 + 141.702e − 2.032e + 10.006e Tm1  τm 

  

1.988 

  

 ;  

2  τ  T  τ Td = Tm1  − 0.4144 + 15.805 m − 142.327 m  + 0.7287 m 2  Tm1 Tm1    Tm1   

 T + Tm1 0.1123 m 2   Tm1 

2

3

T τ   T τ  − 18.317 m 2 2m + 486.95 m  − 10.542 m 2 Tm1  Tm1  Tm1  

  τ 2T + Tm1 204.009 m 3m 2  T   m1 

2    + 47.26 τ m Tm 2   T 3   m1

  τ 3T + Tm1  − 138.038 m 4m 2  T   m1  T + Tm1 19.302 m 2   Tm1 

4

    + 396.349 τ m  T    m1  

4

  

3  2    2  + 52.155 τ m Tm 2  − 646.848 τ m Tm 2  T 4   T 4  m1   m1   5

 τ 4T τ    − 4731.72 m  + 425.789 m 5m 2  T  Tm1    m1

   

   

  

3

  

  

0.997 

  

b720_Chapter-04

04-Apr-2009

FA


Rule Minimum IAE – Huang and Lin (1995) – continued.

Kc

Ti

Td

Comment

Kc

Ti

Td

α = 0 , β =1; N not specified; Model: Method 2

4

 τ T 4 + Tm1  − 289.476 m m52  T  m1   T + Tm1  − 3.7688 m 2   Tm1 

2   3  − 841.807 τ m Tm 2 5   T m1  

5

3    2  + 1313.72 τ m Tm 2  5   T  m1   

6

 τ 5T τ    + 6264.79 m  − 161.469 m 6m 2  T  Tm1    m1

 τ T 5  τ 4T 2 + Tm1 204.689 m m62  + 25.706 m m6 2  T   T  m1  m1  

 τ T + Tm1 648.217 m m2 2  T   m1  4

501

3

   − 5.71 Tm 2 T   m1 

  

   

4   2  − 791.857 τ m Tm 2 6   T m1  

   

6

.  

Note: equations continued onto p. 502. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ; Kc = −

−0.3055 τ  1  T  τ + 10.442 m 2  − 130.2 + 85.914 m + 34.82 m  Km  Tm1 Tm1   Tm1   

−

T 1   − 22.547 m 2 T Km   m1 

−

 T 1  − 51.47 m 2 Km   τm 

  

0.5174

− 14.698

Tm 2 τ m Tm1 2

τm

T  + 52.408 m1   τm  Tm 2

  + 53.378e Tm1 − 0.000001e Tm1 + 0.286e 

1.0077

Tm 2 τ m Tm 12

 Tm 2   Tm1

  

  ;  

2  τ   T  τ Ti = Tm1 72.806 − 268.746 m − 4.9221 m 2  + 2468.19 m   Tm1   Tm1    Tm1  

 T + Tm1 0.6724 m 2   Tm1 

2

3

T τ   T τ  + 151.351 m 2 2m − 6914.46 m  − 0.0092 m 2 Tm1  Tm1  Tm1  

  τ 2T + Tm1  − 795.465 m 3m 2  T   m1    τ 3T + Tm1 1417.65 m 4m 2  T   m1

2    − 14.27 τ m Tm 2   T 3   m1

3    + 0.4057 τ m Tm 2   T 4   m1

    + 5580.17 τ m  T    m1  

4

2   2  + 55.536 τ m Tm 2   T 4 m1  

      

  

3

  

0.9879 

  

b720_Chapter-04

04-Apr-2009

FA


502 Rule

Kc

Ti

Comment

Td


Sree and Chidambaram (2006).

5

Ti

Kc

 T + Tm1  − 0.001119 m 2   Tm1 

τm

4

Model: Method 1; p. 189; β =1; N = ∞ .

Td

Tm 2

  − 44.903e Tm1 + 0.000034e Tm1 − 15.694e 

19.056  τ   Tm 2  + Tm1 678778 m  T  m1    Tm1 

  

Tm 2 τ m Tm 12

7.3464 

;  

1.1798   τm  Tm 2  τm    − Td = Tm1 175.515 − 86.2 0 . 008207 + 348.727  Tm1 Tm1    Tm1    0.1064 0.0355  T  τ   Tm 2 T τ  + Tm1 − 55.619 m 2  + 0.0418 m 2 2m + 78.959 m  T T  Tm1 m1  m1     Tm1 

 T + Tm1 0.005048 m 2   τm  5

Kc =

   

τm

Tm 2

  − 187.01e Tm1 + 0.000001e Tm1 − 0.0149e 

Tm 2 τ m Tm 12

  

0.0827 

  

 .  

1.13076   T   0.5   , Ti = K c K m Tm1  2 x1 + τm  ; α = 1 + τm ; 1 + 1.11416 m1   T K K 1 T K c K m Ti Km  τ −  c m m1   m   m1  

Recommended x 1 = 0.160367Tm1e

6.06426

τm T m1

; 0
0.9 .

Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or

Ti = Tm1 [1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 ;

Td = 4

K0 =

[

Tm1

x1 0.55 + 1.683(Tm1 τm )

1.090

1 G m ( jωu )(1 + jb f 4ωu ) K m

, x1 = 1 − e −1.149ξ m (τ m

]

Tm1 )1.060

.

(

)

, b f 4 = x 3 + x 4 (τm Tm1 ) + x 5 (τm Tm1 )2 Tm1 , with 2

3

4

x 3 = −0.0030 + 0.6482x 6 − 2.2841x 6 + 2.6221x 6 − 0.9611x 6 , 2

3

4

x 4 = 0.2446 − 1.0410 x 6 − 13.6723x 6 − 16.7622 x 6 + 5.1471x 6 , 2

3

4

x 5 = 0.1685 + 0.8289 x 6 − 9.3630 x 6 + 2.9855x 6 + 7.3803x 6 , x 6 = Tm 2 Tm1 . 5

Kc =

Ti , recommended τm ≤ λ ≤ 2τm ; − K m (2λ + τm − a f 1 ) 2

Ti = −Tm1 + Tm 2 + a f 1 −

λ2 + a f 1τm − 0.5τm , 2λ + τm − a f 1

− Tm1a f 1 + Tm 2a f 1 − Tm1Tm 2 − Td =

2 τm (0.167 τm − 0.5a f 1 ) 2λ + τm − a f 1

Ti


e − sτ m

(λs + 1)2

2

−

λ2 + a f 1τm − 0.5τm ; 2λ + τm − a f 1

(Lee et al., 2006).

b720_Chapter-04

04-Apr-2009

FA


506

4.13 Unstable SOSPD Model (two unstable poles)

G m (s ) =

K m e − sτm (Tm1s − 1)(Tm 2s − 1)


E(s)

−

+

  1 K c 1 + + Td s  Ti s  

U(s)

K m e − sτ (Tm1s − 1)(Tm 2 s − 1) m

Y(s)

Table 109: PID controller tuning rules – unstable SOSPD model (two unstable poles) K m e − sτ m G m (s) = (Tm1s − 1)(Tm 2 s − 1) Rule

Kc

Wang and Jin (2004).

1

Kc =

1

Kc

Ti

K m Ti (Tm1 − τm )(Tm 2 − τm )

(TCL3 + τm )3

Comment

Td

Direct synthesis: time domain criteria Model: Method 1 Ti Td

,

Ti =

TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm ) , (Tm1 − τm )(τm − Tm 2 )

Td =

(Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2 Ti (Tm1 − τm )(Tm 2 − τm )

3

2

.

b720_Chapter-04

04-Apr-2009

FA


Rule

Kc

Ti

Kc

Ti

Td

Comment

Td

Model: Method 1

Robust

Lee et al. (2000).

2

Kc =

2

Ti , λ = desired closed loop dynamic parameter; K m (4λ + τm − x1 ) 2

Ti = −Tm1 − Tm 2 + x1 − x 2 , x 2 =

6λ2 − x 3 + x1τm − 0.5τm ; 4 λ + τ m − x1

x 3 − (Tm1 + Tm 2 ) x1 + Tm1Tm 2 − Td =

3 2 4λ3 + x 3τm + 0.167 τm − 0.5x1τm 4λ + τm − x1 − x2 ; Ti

x1 , x 3 values determined by solving 1 −

(x s 3

2

)

+ x1s + 1 e − τ m s

(λs + 1)4

s=

1 1 , Tm1 Tm 2

=0.

507

b720_Chapter-04

04-Apr-2009

FA


508



+

E(s)

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 


  U(s) 2  b f 0 + b f 1s + b f 2 s Model  1 + a f 1s + a f 2 s 2 s 

Y(s)

Table 110: PID controller tuning rules – unstable SOSPD model (two unstable poles) K m e − sτ m G m (s) = (Tm1s − 1)(Tm 2 s − 1) Rule

Kc

Ti

Kc

Ti

Td

Comment

Td

Model: Method 1

Robust Shamsuzzoha and Lee (2007d).

1

Recommended λ = τm .

4

τm

4

τm

 λ   λ  Tm12  + 1 e Tm1 − Tm 2 2  + 1 e Tm 2 + Tm 2 2 − Tm12 Ti  Tm1   Tm 2  1 Kc = , Ti = K m (4λ + τm − Ti ) Tm1 − Tm 2 

4 τm  Tm1  λ  + 1 e − 1 − Ti Tm  Tm1     , b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0 , N = ∞ , Ti

2  Tm1 

Td =

  0.5τm Ti − Ti Td + 2λτ m + 6λ2 a f 2 = 0 , a f 1 = 0.1 + Tm1 + Tm 2  , for large parametric 4 T τ + λ − m i     0.5τm Ti − Ti Td + 2λτ m + 6λ2 + Tm1 + Tm 2  . uncertainties; otherwise, a f 1 =  τm + 4λ − Ti  

   

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+

−

  Td s 1 K c 1 + +  Ti s T 1+ d  N 

   1 + b f 1s  1 + a f 1s s 

+ U(s)

−

Kc

Ti

Kc

Ti

Y(s)

K0

Table 111: PID tuning rules – unstable SOSPD model G m (s) = Rule

Model

K m e − sτ m (Tm1s − 1)(Tm 2 s − 1)

Td

Comment

Td

N=∞

Robust 1


a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , b f 5 = 0 , K 0 = 0 ; recommended τm ≤ λ ≤ 2τm .

1

Desired closed loop transfer function = e −sτ m (λs + 1)4 (Lee et al., 2006). 2

Kc =

6λ2 − a f 3 + a f 2 τm − 0.5τm Ti , , Ti = −Tm1 − Tm 2 + a f 2 − x1 , x1 = K m (4λ + τm − a f 2 ) 4λ + τ m − a f 2

a f 3 − (Tm1 + Tm 2 )a f 2 + Tm1Tm 2 − Td = a f 2 , a f 3 determined by 1 −

(a

f 3s

2

4λ3 + a f 3τm + 0.167 τm 3 − 0.5a f 2 τm 2 4λ + τm − a f 2 − x1 ; Ti

)

+ a f 2s + 1 e − τ m s

(λs + 1)4

s=

1 1 , Tm1 Tm 2

=0; 0≤

τm 1 , x 7 = 1 ; if φ ≤ 1 , 0.6 ≤ x 7 ≤ 0.7 .

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Chapter 5

Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers

5.1 Introduction This chapter will discuss the compensation of FOLPD processes using PI and PID controllers whose parameters are specified using appropriate tuning rules. The gain margin, phase margin and maximum sensitivity of the compensated system as the ratio of time delay to time constant of the process varies, are used as ways of judging the performance and robustness of the system. This work follows the method of Ho et al. (1995b), (1996), in which good approximations for the gain and phase margins of the PI or PID controlled system are analytically calculated. The method will be extended to determine analytically the maximum sensitivity of the compensated system. Insight will be obtained into the range of time delay to time constant ratios over which it is sensible to apply various tuning rules, to compensate a FOLPD process. The chapter is organised as follows. Formulae for calculating analytically the gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2 and 5.3. The performance and robustness of FOLPD processes compensated with a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5, tuning rules are designed to achieve constant gain and phase margins for all values of delay, for a number of process models and controller structures. Conclusions of the work are drawn in Section 5.6. This work was previously published by O’Dwyer (1998), (2001a).

521

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5.2 The Analytical Determination of Gain and Phase Margin 5.2.1 PI tuning formulae The controller and process model are respectively given by

 1   G c (s) = K c 1 +  Ti s 

G m (s ) =

(5.1)

K m e −sτ m 1 + sTm

(5.2)

K m e − jωτ m K c ( jωTi + 1) 1 + jωTm jωTi

(5.3)

Then G m ( jω)G c ( jω) =

From the definition of gain and phase margin, the following sets of equations are obtained:

[

]

φ m = arg G c ( jωg )G m ( jωg ) + π

(5.4)

1 G c ( jω p )G m ( jω p )

(5.5)

Am =

where ωg and ωp are given by G c ( j ω g ) G m ( jω g ) = 1

[

]

arg G c ( jωp )G m ( jωp ) = − π

(5.6) (5.7)

From Equation (5.3), G m ( jω)G c ( jω) =

K m K c 1 + ω2 Ti ωTi 1 + ω2 Tm

2

2

∠ − 0.5π + tan −1 ωTi − tan −1 ωTm − ωτ m (5.8)

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Chapter 5: Performance and Robustness Issues...

523

Therefore, from Equation (5.4) φ m = π − 0.5π + tan −1 ωg Ti − tan −1 ωg Tm − ωg τ m

(5.9)

with ωg given by the solution of Equation (5.6), i.e. 2

K m K c 1 + ωg Ti 2

ωg Ti 1 + ωg Tm

2

2

=1

(5.10)

Also, from Equations (5.5) and (5.8), Am =

1 G m ( jω p ) G c ( jω p )

=

2

2

2

2

ωp Ti

1 + ωp Tm

KmKc

1 + ωp Ti

(5.11)

with ωp given by the solution of Equation (5.7), i.e. − 0.5π + tan −1 ωp Ti − tan −1 ωp Tm − ωp τ m = − π

(5.12)

From Equation (5.10), ωg may be determined analytically to be ωg =

(

2

2

) (K

Ti K c K m − 1 +

2 c

)

2

2

2

2

2

K m − 1 Ti + 4 K c K m Tm

2Ti Tm

2

2

(5.13)

An analytical solution of Equation (5.12) (to determine ωp ) is not possible. An approximate analytical solution may be obtained if the following approximation for the arctan function is made: tan −1 x ≈

π π π x , x < 1 and tan −1 x ≈ − , x >1 4 2 4x

(5.14)

This is quite an accurate approximation, as is shown in Figures 1a and 1b (next page). Considering Equation (5.12), four possibilities present themselves if the approximation in Equation (5.14) is to be used. These possibilities, together with the formula for ωp that may be determined analytically for each of these cases, are

(i)

1 1   π ± π 2 − 4πτ m  −  Ti Tm  ωp Ti > 1, ωp Tm > 1 : ω p = 4τ m

(5.15)

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π ± π2 − ωp Ti > 1, ωp Tm < 1 : ωp =

(ii)

π (0.25πTm + τ m ) Ti

2(0.25πTm + τ m )

(5.16)

π

(iii)

ωp Ti < 1, ωp Tm > 1 : ωp =

(iv)

ωp Ti < 1, ωp Tm < 1 : ωp =

Tm 4τ m − πTi 2π 4τ m + π(Tm − Ti )

(5.17) (5.18)

The gain and phase margin of the compensated system for each of the tuning rules, as a function of τ m Tm , may be calculated by applying Equations (5.9), (5.11), (5.13) and the relevant approximation for ωp from Equations (5.15) to

(5.18). Figure 1a: Arctan x and its approximation (Equation 5.14)

arctan x approximation

Figure 1b: % error in taking approximation (Equation 5.14) to arctan(x)

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5.2.2 PID tuning formulae

The classical PID controller structure is considered, whose transfer function is given as  1  1 + sTd  G c (s) = K c 1 +  Ti s  1 + sαTd

  

(5.19)

and the process model is given by Equation (5.2). Substituting Equations (5.2) and (5.19) into Equations (5.4) and (5.5) gives φ m = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τ m (5.20) and Am =

(1 + ω T )(1 + ω α T ) (1 + ω T )(1 + ω T ) 2

ωp Ti

p

KcKm

2 m 2 2 p i

2

2

p

2

p

2

d 2

(5.21)

d

From Equation (5.6), ωg is given by the solution of 2


ωg Ti 1 + ωg Tm

2

2

2

1 + ωg Td

2

2

1 + α 2 ωg Td

2

=1

(5.22)

The equation for ωg may be determined to be 6

4

2

ωg + a 1ω g + a 2 ωg + a 3 = 0 2

with

a1 =

2

a2 =

(

2

2

)

2

2

2

Ti Tm + α 2 Td − K c K m Ti Td 2

2

Ti Tm α 2 Td 2

2

2

2

2

2

Ti Tm α 2 Td

2

, 2

2

Ti − K c K m (Ti + Td ) 2

(5.23)

and a 3 = −

Kc Km 2

2

2

Ti Tm α 2 Td

2

.

Following the procedure outlined by Ho et al. (1996), the analytical solution of Equation (5.23) is given as ωg =

3

R + Q3 + R 2 + 3 R − Q3 + R 2 − 3

with Q =

2 9a a − 27a 3 − 2a 1 3a 2 − a 1 and R = 1 2 . 9 54

a1 3

(5.24)

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From Equation (5.7), ω p is given by the solution of 0.5π + tan −1 ωp Td + tan −1 ωp Ti − tan −1 ωp Tm − tan −1 ωp αTd − ωp τ m = 0 (5.25) As with Equation (5.12), an analytical solution of this equation is not possible. An approximate analytical solution may be obtained if the approximation detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve possibilities present themselves if the approximation in Equation (5.14) is to be used; these possibilities are (1) ωp Ti > 1, ωp Tm > 1

(2) ωp Ti > 1, ωp Tm < 1

(a) ω p Td ≤ 1, ω p αTd < 1

(a) ω p Td ≤ 1, ωp αTd < 1

(b) ω p Td > 1, ωp αTd < 1

(b) ω p Td > 1, ωp αTd < 1

(c) ω p Td > 1, ω p αTd > 1

(c) ω p Td > 1, ωp αTd > 1

(3) ωp Ti < 1, ωp Tm > 1

(4) ωp Ti < 1, ωp Tm < 1

(a) ω p Td ≤ 1, ωp αTd < 1

(a) ω p Td ≤ 1, ωp αTd < 1

(b) ω p Td > 1, ωp αTd < 1

(b) ω p Td > 1, ωp αTd < 1

(c) ω p Td > 1, ωp αTd > 1

(c) ω p Td > 1, ωp αTd > 1

The formulae for ωp that may be determined for each of these cases are as follows: (1) ωp Ti > 1, ωp Tm > 1 (a) ωp Td ≤ 1, ωp αTd < 1 : 1 1   π ± π 2 − π(4τ m − πTd (1 − α )) −  Ti Tm  ωp = (5.26) 4τ m − πTd (1 − α )

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527

(b) ωp Td > 1, ωp αTd < 1 :  1 1 1   + − 2π ± 4π 2 − π(4τ m + πTd α )  Td Ti Tm  ωp = (5.27) 4τ m + πTd α

(c) ωp Td > 1, ωp αTd > 1 :  1 1 1 1   π ± π 2 − 4πτ m  + − −  Td Ti αTd Tm  ωp = (5.28) 4τ m

(2) ωp Ti > 1, ωp Tm < 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : 2π ± 4π 2 + ωp =

π (πTd − πTm − παTd − 4τ m ) Ti

πTm + 4 τ m + παTd − πTd

(5.29)

(b) ωp Td > 1, ωp αTd < 1 :  1 1 3π ± 9π 2 − π + (πTm + παTd + 4τ m )  Td Ti  ωp = (5.30) πTm + 4τ m + παTd

(c) ωp Td > 1, ωp αTd > 1 :  1 1 1 2π ± 4π 2 + π − −  αTd Td Ti ωp = πTm + 4τ m

 (πTm + 4τ m ) 

(5.31)

(3) ωp Ti < 1, ωp Tm > 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : ωp =

π Tm (4τ m + π[αTd − Td − Ti ])

(5.32)

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(b) ωp Td > 1, ωp αTd < 1 :  1 1 − π ± π 2 − π −  Tm Td ωp = πTi − 4τ m

(c) ωp Td > 1, ωp αTd > 1 : ωp =

 (πTi − παTd − 4 τ m )  (5.33) − παTd

π π π − − Td Tm αTd πTi − 4τ m

(5.34)

(4) ωp Ti < 1, ωp Tm < 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : ωp =

2π π(αTd − Td ) + π(Tm − Ti ) + 4τ m

(5.35)

(b) ωp Td > 1, ωp αTd < 1 : 2π ± 4π 2 − ωp =

π (πTm − πTi + παTd + 4τ m ) Td

πTm − πTi + 4τ m − παTd

(5.36)

(c) ωp Td > 1, ωp αTd > 1 :  1 1  π ± π 2 − π − (πTi − πTm − 4τ m )  αTd Td  ωp = (5.37) 4τ m − πTm − πTi

The gain and phase margin of the compensated system for each of the tuning rules, as a function of τ m Tm , may now be calculated by applying Equations (5.20), (5.21), (5.24) and the relevant approximation for ωp from Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to tuning rules defined for the classical PID controller structure only; however, Ho et al. (1996) suggest that the method may be applied to tuning rules defined for the ideal PID controller structure, by using equations that approximately convert these tuning rules into their equivalent in the classical controller structure.

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5.3 The Analytical Determination of Maximum Sensitivity

The maximum sensitivity is the reciprocal of the shortest distance from the Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows: M max = Max all ω

1 1 + G m ( jω)G c ( jω)

(5.38)

For a FOLPD process model controlled by a PI controller, G m ( jω)G c ( jω) =

1 + ω2 Ti

2

1 + ω2 Tm

2

KcKm ωTi

(5.39)

and arg[G m ( jω)G c ( jω)] = −0.5π − tan −1 ωTm + tan −1 ωTi − ωτ m

(5.40)

For a FOLPD process model controlled by a PID controller, 2

G m ( jω)G c ( jω) =

2

(1 + ω2 Ti )(1 + ω2 Td )

KcKm 2 2 (1 + ω2 Tm )(1 + α 2 ω2 Td ) ωTi

(5.41)

and arg[G m ( jω)G c ( jω)] = − 0.5π − tan −1 ωTm − tan −1 ωαTd + tan −1 ωTi + tan −1 ωTd − ωτ m

(5.42)

The maximum sensitivity may be calculated over an appropriate range of frequencies corresponding to phase lags of 100 0 to 260 0 . 5.4 Simulation Results

Space considerations dictate that only representative simulation results may be provided; an extensive set of simulation results covering 103 PI controller tuning rules and 125 PID controller tuning rules (for the ideal and classical controller structures) are available (O’Dwyer, 2000). The MATLAB package has been used in the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical controller structure (with α = 0.1) are used (Figures 5 to 7). In these results, Z-N refers to the process reaction curve method of Ziegler and Nichols (1942); W-W refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg, ISE reg and ITAE reg refer to the tuning rules for regulator applications that minimise the integral of absolute error criterion, the integral of squared error criterion

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530

and the integral of time multiplied by absolute error criterion, respectively, as defined by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning rules based on the classical controller structure. Figures 8 to 15 show gain and phase margin comparisons between corresponding PI and PID controller tuning rules. It is clear that the gain margin is generally less when the PID rather than the PI tuning rules are considered, over the ratios of time delay to time constant taken; the difference between the phase margins is less clear cut. This suggests that these PID tuning rules should provide a greater degree of performance than the corresponding PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is striking that the ISE based tuning rules have generally the smallest gain margin and have also a small phase margin, suggesting that this is a less robust tuning strategy. The results in Figures 4 and 7 confirm these comments. Figure 2: Gain margin

Figure 3: Phase margin

Figure 4: Max. sensitivity

120

4

- = Z-N * = IAE reg + = ITAE reg o = ISE reg

3.5

3

8

7

100

6

80 5

2.5

60

2

40

4

3

20

1.5

1

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2

0

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1

0

Ratio of τ m to Tm

Ratio of τ m to Tm Figure 5: Gain margin - = W-W * = IAE reg + = ITAE reg o = ISE reg

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ratio of τ m to Tm Figure 7: Max. sensitivity

Figure 6: Phase margin

2.5

2

0.2

100

6

90

5.5

80

5

70

4.5

60

4

50

3.5

40

3

30

2.5

1.5

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ratio of τ m to Tm

1.8

2

20

2

10

1.5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ratio of τ m to Tm

1.8

2

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Ratio of τ m to Tm

2

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No general conclusion can be reached as to the best tuning rule (as expected); it is interesting, though, that a full panorama of simulation results (O’Dwyer, 2000) show that many tuning rules may be applied at ratios of time delay to time constant greater than that normally recommended. One example may be seen in Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity (associated with the use of the PID tuning rule for obtaining minimum IAE in the regulator mode) tends to level out when the ratio of time delay to time constant is greater than 1; normally, the tuning rule is used when the ratio is less than 1 (Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a significant degradation of performance when using the PID tuning rule of Witt and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for large ratios of time delay to time constant, which is compatible with application experience. The decision between the use of a PI and PID controller to compensate the process depends on the ratio of time delay to time constant in the FOLPD model, together with the desired trade-off between performance and robustness, as expected. It turns out, however, that the analytical method explored allows the calculation of a far wider range of gain and phase margins for PI controllers; it is also true that stability tends to be assured when a PI controller is used (O’Dwyer, 2000). Thus, a cautious design approach is to use a PI controller, particularly at larger ratios of time delay to time constant. Finally, the volume of tuning rules and data generated means that the use of an expert system to recommend a tuning rule based on user defined requirements is indicated; work is ongoing on such an implementation (Feeney and O’Dwyer, 2002).

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Figure 8: Gain margin comparison

Figure 9: Phase margin comparison

4

120

3.5

100

- = W-W PID o = Z-N PI

3

80

2.5

60

2

40

1.5

20

1

FA

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

Ratio of τ m to Tm

0.8

1

1.2

1.4

1.6

1.8

2

Ratio of τ m to Tm Figure 11: Phase margin comparison

Figure 10: Gain margin comparison 100

3

90

2.8

- = IAE reg PID o = IAE reg PI

2.6

80

2.4

70

2.2

60

2

50

1.8

40

1.6

30

1.4

20

1.2

10

1

0.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ratio of τ m to Tm

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ratio of τ m to Tm

1.8

2

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Figure 12: Gain margin comparison

533


2.5

100

- = ISE reg PID o = ISE reg PI

90 80 70

2

60 50 40

1.5 30 20 10

1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

2

0.2

0.4

Figure 14: Gain margin comparison 100

2.8

90

- = ITAE reg PID o = ITAE reg PI

70

2.2

60

2

50

1.8

40

1.6

30

1.4

20

1.2

10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

80

2.4

0

1


3

1

0.8

Ratio of τ m to Tm

Ratio of τ m to Tm

2.6

0.6

1.2

1.4

Ratio of τ m to Tm

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ratio of τ m to Tm

1.8

2

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5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for All Values of Delay

Normally, the gain and phase margins of the compensated systems tend to increase as the time delay increases, supporting the common view that PI and PID controllers are less suitable for the control of dominant time delay processes. However, for the PI control of a FOLPD process model, O’Dwyer (2000) shows that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and Pemberton (1972a), among others, facilitate the achievement of a constant gain and phase margin as the time delay of the process model varies. All of these tuning rules have the following structure: K c = aTm K m τ m , Ti = Tm . Following this observation, original approaches to the design of tuning rules for PI, PD and PID controllers are proposed, for a wide variety of process models, which allow constant gain and phase margins for the compensated system. This work is organised as follows. In Section 5.5.1, PI controller tuning rules are specified for processes modelled in FOLPD form and IPD form. In Section 5.5.2, PID controller tuning rules are described for processes modelled in FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3 deals with the design of PD controller tuning rules for the control of processes modelled in FOLIPD form. 5.5.1 PI controller design

5.5.1.1 Processes modelled in FOLPD form For such processes and controllers, Equations (5.1) and (5.2) apply, i.e., G m (s) =


 1   G c (s) = K c 1 +  Ti s 

with From Section 5.2.1,

φ m = π − 0.5π + tan −1 ω g Ti − tan −1 ω g Tm − ω g τ m

(Equation 5.9)

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535

with ω g given by the solution of 2


ωg Ti 1 + ωg Tm

2

2

=1

(Equation 5.10)

and Am =

2

2

2

2

ωpTi

1 + ωp Tm

K mK c

1 + ωp Ti

(Equation 5.11)

with ωp given by the solution of − 0.5π + tan −1 ω p Ti − tan −1 ω p Tm − ω p τ m = −π

(Equation 5.12)

If K c and Ti are designed as follows: Kc =

aTm K mτm

(5.43)

and Ti = Tm

(5.44)

Then Equation (5.12) becomes − 0.5π − ωp τm = −π

i.e.

ω p = π 2τ m

(5.45) (5.46)

Substituting Equation (5.46) into Equation (5.11) gives A m = πTm 2K m K c τ m

(5.47)

Substituting Equation (5.44) into Equation (5.10) gives

i.e.

K m K c ω g Tm = 1

(5.48)

ω g = K m K c Tm

(5.49)

Substituting Equations (5.44) and (5.49) into Equation (5.9) gives φ m = 0 . 5 π − K m K c τ m Tm

(5.50)

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536

Finally, substituting Equation (5.43) into Equation (5.47) gives A m = π 2a

(5.51)

and substituting Equation (5.43) into Equation (5.50) gives φ m = 0 .5 π − a

(5.52)

Some typical tuning rules are shown in Table 117. Table 117: Typical PI controller tuning rules – FOLPD process model a

Kc

Ti

Am

φm π6

π3

1.047Tm K m τ m

Tm

1.5

π 4

0.785Tm K m τm

Tm

2.0

π 4

π6

0.524Tm K m τm

Tm

3.0

π3

These rules are also provided in Table 9. It may also be demonstrated that, if K c and Ti are designed as follows:

Kc =

aTm τm

Tu K u 2

Tu + 4π 2 Tm

2

(5.53)

and Ti = Tm

(5.54)

where K u and Tu are the ultimate gain and ultimate period, respectively, then the constant gain and phase margins provided in Equations (5.51) and (5.52) are obtained. 5.5.1.2 Processes modelled in IPD form A similar analysis to that of Section 5.5.1.1 may be done for the design of PI controllers for processes modelled in IPD form. The process is modelled as follows: G m (s) = K m e −sτ m s

(5.55)

Corresponding to Equations (5.4) and (5.9), the phase margin is φ m = π − 0.5π + tan −1 ω g Ti − 0.5π − ω g τ m

(5.56)

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537

with ωg given by the solution of 2

K m K c 1 + ω g Ti

2

=1

2

ω g Ti

(5.57)

If K c and Ti are designed as follows: Kc =

a K m τm

(5.58)

and Ti = bτ m

(5.59)

then it may be shown that  a2 a ωg =  ± 2 2 2bτ m  2τ m

 a b + 4  2

0.5

2

(5.60)

and, substituting Equations (5.59) and (5.60) into Equation (5.56), φ m = tan −1 0.5a 2 b 2 + 0.5ab a 2 b 2 + 4   

0.5

− 0.5a 2 + 0.5

a a 2b2 + 4 b

(5.61)

Corresponding to Equations (5.5) and (5.11), the gain margin, 2

Am =

ω p Ti 2

K m K c 1 + ω p Ti

2

(5.62)

with ω p given by the solution of − 0.5π + tan −1 ω p Ti − 0.5π − ω p τ m = −π

(5.63)

i.e. ω p is given by the solution of tan −1 ω p Ti = ω p τ m

(5.64)

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538

An analytical solution to this equation is not possible, though if the π (when ω p Ti > 1 ) is used, simple approximation tan −1 ω p Ti ≈ 0.5π − 4ω p Ti calculations show that the following analytical solution for ω p may be obtained: 0.5  π 2 0.5π + 0.25π −  τ m  b 

ωp =

(5.65)

The inequality ω p Ti > 1 may be shown to be equivalent to b > 1.273. Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations show that: b 4a

Am =

1+

π π2 π   + −  4 b  2 

2

b π π π  + −  4 2 4 b   2

2

(5.66)

2

Some typical tuning rules are shown in Table 118. Table 118: Typical PI controller tuning rules – IPD process model Kc

Ti

Am

φm

0.558 K m τm

1.4τ m

1.5

46.2 0

0.484 K m τm

1.55τm

2.0

45.5 0

0.458 K m τm

3.35τ m

3.0

59.9 0

0.357 K m τm

4.3τ m

4.0

60.0 0

0.305 K m τm

12.15τ m

5.0

75.0 0

It may also be shown that the maximum sensitivity is a constant if K c and Ti are specified according to Equations (5.58) and (5.59). The maximum sensitivity may be calculated to be: M max

 a2  = 1 − 2a cos(ω r τ m ) − 2 2  ω r τ m  

−0.5

(5.67)

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539

with ω r τ m being a constant obtained numerically from the solution of the 2

2

equation (a 2 ω r τ m ) + cos ω r τ m = sin(ω r τ m ) / ω r τ m . It may also be shown that, if K c and Ti are designed as follows: K c = aK u

(5.68)

Ti = bTu

(5.69)

and

then the following constant gain and phase margins are determined: 2b  π π2 π   + −  4 4b  πa  2  

Am =

1+

2

π π2 π   + −  4 4b  a 2 π 2  2 

1

[

and φ m = tan −1 2π 2 a 2 b 2 + 2πab 4π 2 a 2 b 2 + 4 − 0.125π 2 a 2 +

(5.70)

2

]

0 .5

πa 4π 2 a 2 b 2 + 4 16b

(5.71)

5.5.2 PID controller design

5.5.2.1 Processes modelled in FOLPD form – classical controller The classical PID controller is given by  1  1 + sTd  G c (s) = K c 1 +  Ti s  1 + sαTd

  

(Equation 5.19)

and the process model is given by G m (s) =


(Equation 5.2)

From Section 5.2.2, φm = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τm (Equation 5.20)

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540

with ωg given by the solution of 2

K m K c 1 + ω g Ti 2

ω g Ti 1 + ω g Tm Also Am =

ω p Ti KcK m

2

2

2

1 + ω g Td 2

2

1 + α 2 ω g Td

=1

2

(Equation 5.22)

(1 + ω T )(1 + ω α T ) (1 + ω T )(1 + ω T ) 2

p

2

2

m

2

p

2

p

2

i

2

p

2

d

2

(Equation 5.21)

d

with ω p given by 0.5π + tan −1 ω p Td + tan −1 ω p Ti − tan −1 ω p Tm − tan −1 ω p αTd − ω p τ m = 0 (Equation 5.25) If K c , Ti and Td are designed as follows: aTm K mτm Ti = αTm Kc =

(5.72) (5.73)

and Td = Tm

(5.74)

then, Equation (5.25) becomes 0.5π − ω p τ m = 0

(5.75)

ω p = π 2τ m

(5.76)

i.e.

and Equation (5.21) becomes A m = α πTm 2K m K c τ m

(5.77)

K m K c ω g Tm = 1

(5.78)

ω g = K m K c Tm

(5.79)

Equation (5.22) becomes

i.e.

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541

Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20) gives φ m = 0 .5 π − K m K c τ m α T m

(5.80)

Substituting Equations (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives A m = πα 2a

(5.81)

and substituting Equation (5.72) into Equation (5.80) gives φ m = 0.5π − a α

(5.82)

This design reduces to the PI controller design when α = 1 .

5.5.2.2 Processes modelled in SOSPD form – series controller For such processes and controllers, K m e − sτ m G m (s) = (1 + sTm1 )(1 + sTm 2 )

(5.83)

 1  (1 + sTd ) G c (s) = K c 1 +  Ti s 

(5.84)

with

Therefore, G m (s)G c (s) =

K m K c e −sτ m (1 + sTi )(1 + sTd ) Ti s(1 + sTm1 )(1 + sTm 2 )

(5.85)

If Ti and Td are designed as follows: Ti = Tm1

(5.86)

Td = Tm 2

(5.87)

and

then, from Equation (5.85) G m (s)G c (s) =

K m K c e −sτ m Ti s

(5.88)

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542

which is equal to G m (s)G c (s) in Section 5.5.1.1 when Ti = Tm . Therefore, designing aTm Kc = (5.89) K mτm will allow A m = π 2a and φ m = 0.5π − a , as before. 5.5.2.3 Processes modelled in SOSPD form with a negative zero – classical controller

For such processes, K m e −sτ m (1 + sTm 3 ) G m (s) = (1 + sTm1 )(1 + sTm 2 )

(5.90)

with G c (s) given by Equation (5.19). Therefore, G m (s)G c (s) =

K m K c e −sτ m (1 + sTm 3 )(1 + sTi )(1 + sTd ) Ti s(1 + sTm1 )(1 + sTm 2 )(1 + αsTd )

(5.91)

If Ti , Td and α are designed as follows: Ti = Tm1

(5.92)

Td = Tm 2

(5.93)

α = Tm 3 Tm 2

(5.94)

and therefore, designing Kc =

aTm K mτm

(5.95)

will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1 and 5.5.2.2. 5.5.3 PD controller design

In this case, the process is modelled in FOLIPD form, i.e. G m (s) =

K m e −sτ m s(1 + sTm )

(5.96)

with G c (s) = K c (1 + Td s )

(5.97)

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543

Therefore, G m (s)G c (s) =

K m K c e −sτ m (1 + sTd ) s(1 + sTm )

(5.98)

Therefore, designing Kc =

aTm K mτm

(5.99)

and Td = Tm

(5.100)

will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1, 5.5.2.2 and 5.5.2.3. 5.6 Conclusions

This chapter has considered the performance and robustness of a PI and PID controlled FOLPD process, with the parameters of the controllers determined by a variety of tuning rules. The original contributions of this work are as follows: (a) an expansion of the analytical approach of Ho et al. (1995b), (1996) to determine the approximate gain and phase margin analytically under all operating conditions, (b) the analytical determination of the approximate maximum sensitivity of the compensated system and (c) the application of the algorithms using a wide variety of PI and PID tuning rules. The implementation of autotuning algorithms in commercial controllers means that choice of a suitable tuning rule is an important issue; the techniques discussed allow an analytical evaluation to be performed of candidate tuning rules. Finally, the chapter discusses an original approach to design tuning rules for both PI and PID controllers, for a variety of delayed process models, with the objective of achieving constant gain and phase margins for all values of delay. In one of the cases discussed (PI control of an IPD process model), an analytical approximation is used in the development; this approximation may also be used to determine further tuning rules for other process models with delay, and for other PID controller structures.

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Appendix 1

Glossary of Symbols and Abbreviations

a f 1 , a f 2 , a f 3 , a f 4 , a f 5 , a f 6 = Denominator parameters of a filter associated with some PID controller structures a 1 , a 2 , a 3 , b1 , b 2 , b 3 = Parameters of a third order process model a 1 , a 2 , a 3 , a 4 , a 5 , b1 , b 2 , b 3 , b 4 , b 5 = Parameters of a fifth order process model A 1 , A 2 , A 3 , A 4 , A 5 = Areas calculated from the process open loop step response (see, for example, Vrančić et al., 1996) A m = Gain margin

A p = Peak output amplitude of limit cycle determined from relay autotuning b f 1 , b f 2 , b f 3 , b f 4 , b f 5 , b f 6 = Numerator parameters of a filter associated with some PID controller structures D R = Desired closed loop damping ratio d(t) = Disturbance variable (time domain) du dt = Time derivative of the manipulated variable (time domain) e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain) E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model F1 (s), F2 (s) = Transfer functions of components of some PID controller structures G c (s) = PID controller transfer function G CL (s) = Closed loop transfer function

544

b720_Appendix-01

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Appendix 1: Glossary of Symbols and Abbreviations

545

G CL ( jω) = Desired closed loop frequency response G m (s) = Process model transfer function G p (s) = Process transfer function G p ( jω) = Process transfer function at frequency ω G p ( jω) = Magnitude of G p ( jω) G p ( jωφ ) = Magnitude of G p ( jω) , at frequency ω , corresponding to phase lag of φ ∠G p ( jω) = Phase of G p ( jω) ± h = Relay amplitude (relay autotuning)

IE = Integral of Error =

∞

∫ e( t )dt 0

∫

IAE = Integral of Absolute Error =

∞

0

e( t ) dt

IMC = Internal Model Controller IPD model = Integral Plus time Delay model I 2 PD model = Integral Squared Plus time Delay model ISE = Integral of Squared Error =

∫

∞

0

e 2 ( t )dt

ISTES = Integral of Squared Time multiplied by Error, all to be Squared =

∫

∞

0

[ t 2 e( t )]2 dt

ISTSE = Integral of Squared Time multiplied by Squared Error =

∫

∞

0

t 2 e 2 ( t )dt

ITSE = Integral of Time multiplied by Squared Error =

∫

ITAE = Integral of Time multiplied by Absolute Error = K c = Proportional gain of the controller KH =

9 2

2τ m K m

τm2 ξ T τ  2 − Tm1 − m m1 m  (Hwang, 1995)  9   18

∞

0

∫

te 2 ( t )dt ∞

0

t e( t ) dt

b720_Appendix-01

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546

K H1 =

 τ m 2 τ m Tm 1.884K c K u τ m  − +   18 9ω u  18 

9 2K m τ m +

2

2 2 2 2 9  τm 49Tm 7T τ 4.71K u K c (τm + 1.6Tm ) 1.775K c K u    − + + m m− 2 2K m τm  324 324 162 81ωu 81ωu  

(Hwang, 1995) K H2 =

9 2K m τ m

 τm 1.884K u K c τ m  ξ τ T 9 2 − Tm1 − m m m1 + x1  + 2 18 9 9 ω u   2K m τ m 2

2

with 4 2 2 2 2 2 3 3   τ 49τm ξm Tm1 τm Tm1 7Tm1ξm τm 10τm Tm1 ξ m 4 − + + − x2  x1 =  Tm1 + m + 324 81 9 81 9  

and x2 =

2 2 2 3 0.471K u K c 10τ m 4T τ (8ξ m τ m + 9Tm1 )  1.775K u K c τ m + m1 m −  2 81 ωu 81ω u   81

(Hwang, 1995) K i = Integral gain of the controller K L = Gain of a load disturbance process model K m = Gain of the process model K p = Gain of the process K u = Ultimate proportional gain ∧

K u = Ultimate proportional gain estimate determined from relay autotuning K φ = Proportional gain when G c ( jω)G p ( jω) has a phase lag of φ K 0 = Weighting parameter used in some two degree of freedom PID controller structures K x % = Proportional gain required to achieve a decay ratio of 0.01x

K φ0 = Controller gain when G c ( jω)G p ( jω) has a phase lag of φ 0 min = Minimum max = Maximum

b720_Appendix-01

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547

M s = Closed loop sensitivity M max = Maximum value of closed loop sensitivity, M s M t = Complementary closed loop sensitivity n = Order of a process model with a repeated pole N = Parameter that determines the amount of filtering on the derivative term on some PID controller structures OS = Closed loop response overshoot (often a percentage) PI controller = Proportional Integral controller PID controller = Proportional Integral Derivative controller r = Desired variable (time domain) 2

r1 =

0.7071M max − M max 1 − 0.5M max

2

2

M max − 1

(Chen et al.,1999a) R(s) = Desired variable (Laplace domain) s = Laplace variable SOSPD model = Second Order System Plus time Delay model SOSIPD model = Second Order System plus Integral Plus time Delay model t = Time Tar = Average residence time = time taken for the open loop process step response to reach 63% of its final value Td = Derivative time of the controller TCL = Desired closed loop system time constant TCL 2 = Desired parameter of second order closed loop system response (1)

( 2)

( 3)

TCL , TCL , TCL = Desired dynamic parameters of the closed loop system response Tf = Time constant of the lag in some PID controller structures Ti = Integral time of the controller TL = Time constant of a load disturbance FOLPD process model Tm = Time constant of a FOLPD process model Tm1, Tm2, Tm3, Tm4 = Time constants of second, third or fourth order process models, as appropriate

b720_Appendix-01

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548

Tm1i , Tm2i , Tm3i , Tm4i = Time constants of a general process model Tp = Time constant of a FOLPD process Tp1, Tp2, Tp3 = Time constants of second or third order process, as appropriate TR = Closed loop rise time TS = Closed loop settling time Tu = Ultimate period ∧

T u = Ultimate period estimate determined from relay autotuning Tx % = Period of the waveform with a decay ratio of 0.01x , when the closed loop system is under proportional control TOSPD model = Third Order System Plus time Delay model u(t) = Manipulated variable (time domain) u ∞ = Final value of the manipulated variable (time domain) U(s) = Manipulated variable (Laplace domain) V = Closed loop response overshoot (as a fraction of the controlled variable final value) y(t) = Controlled variable (time domain) y ∞ = Final value of the controlled variable (time domain) Y(s) = Controlled variable (Laplace domain) α , β , χ , δ , ε , φ , ϕ = Weighting factors in some PID controller structures 2

ε=

6Tm1 + 4ξ m Tm1 τ m + K H K m τ m

ε1 =

6Tm1 + 4Tm1 ξ m τ m + K u K m τ m 2

2τ m Tm1 ω u

2

(Hwang, 1995)

2Tm ωu + K H1 K m τ m ωu − 1.884 K u K c (Hwang, 1995) 0.471K c K u ω H1 τ m 2

ε2 =

(Hwang, 1995)

2

2Tm1 τ m ω H 2

ε0 =

2

2

6Tm1 ωu + 4Tm1ξ m τ m ω u + K H 2 K m τ m ωu − 1.884 K u K c τ m 2

2

(0.471K u K c τ m + 2 τ m Tm1 ωu )ω H 2 (Hwang, 1995)

b720_Appendix-01

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549

± ε 4 = Relay hysteresis width (relay autotuning) φ = Phase lag φc = Phase of the plant at the crossover frequency of the compensated system, with a ‘conservative’ PI controller (Pecharromán and Pagola, 2000) φ m = Phase margin φω = Phase lag at an angular frequency of ω κ = 1 Km Ku λ = Parameter that determines robustness of compensated system. τ = τ m (τ m + Tm ) τ CL = Desired closed loop system time delay τ L = Time delay of a load disturbance process model τ m = Time delay of the process model τ am = τ 'm − τ m , τ 'm is obtained using the tangent and point method of Ziegler and Nichols (1942). Subsequently, τ m is estimated from the process step response (Schaedel, 1997). ω = Angular frequency ω bw = − 3 dB closed loop system bandwidth (Shi and Lee, 2002) ω−6dB = Frequency where closed loop system magnitude is –6dB (Shi and Lee, 2004) ωc = Maximum cut-off frequency ωd = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963) ω CL = 2π TCL ωg = Specified gain crossover frequency (Shi and Lee, 2002)

1+ KHKm

ωH = Tm1

ω H1 =

2

2T τ ξ K K τ + m1 m m + H m m 3 6

2

(Hwang, 1995)

1 + K H1 K m 2

0.942 K c K u τ m τ m Tm K H1 K m τ m + − 3 6 3ωu

(Hwang, 1995)

b720_Appendix-01

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550

1 + K H2 K m

ωH 2 = 2

Tm1 +

2 0.942 K c K u τ m 2ξ m τ m Tm1 K H 2 K m τ m + − 3 6 3ωu

(Hwang, 1995) ω M max = Frequency where the sensitivity function is maximised (Rotach, 1994) ωn = Undamped natural frequency of the compensated system ωp =

A m φ m + 0.5πA m (A m − 1)

(A

2 m

)

− 1 τm

(see, for example, Hang et al. 1993a, 1993b)

ω pc = Phase crossover frequency ωr = Resonant frequency (Rotach, 1994) ωu = Ultimate frequency ^

ω u = Ultimate frequency estimate obtained from relay autotuning ω φ = Angular frequency when G c ( jω)G p ( jω) has a phase lag of φ ^

ω φ = Estimate of angular frequency when G c ( jω)G p ( jω) has a phase lag of φ ξ = Damping factor of the compensated system ξ m = Damping factor of an underdamped process model ξ des m = Desired damping factor of an underdamped process model ξ m ni , ξ mdi = Damping factors of a general underdamped process model

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Appendix 2

Some Further Details on Process Modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables in Chapters 3 and 4. Some outline details of these modelling strategies are provided, together with references that describe the modelling method in detail. The references are given in the bibliography. For all models, the label ‘Model: Method 1’ indicates that the model method has not been defined or that the model parameters are assumed known; interestingly, this is the modelling method assigned to a majority of tuning rules. A2.1 FOLPD Model A2.1.1 Parameters estimated from the open loop process step or impulse response Method 2:

Parameters estimated using a tangent and point method (Ziegler and Nichols, 1942). Method 3: K m , τ m determined from the tangent and point method of Ziegler and Nichols (1942); Tm determined at 60% of the total process variable change (Fertik and Sharpe, 1979). Method 4: K m , τ m assumed known; Tm estimated using a tangent method (Wolfe, 1951).

551

b720_Appendix-02

552

Method 5: Method 6: Method 7: Method 8:

Method 9:

Method 10:

Method 11:

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Parameters estimated using a second tangent and point method (Murrill, 1967). Parameters estimated using a third tangent and point method (Davydov et al., 1995). τ m and Tm estimated using the two-point method; K m estimated from the open loop step response (ABB, 2001). τ m and Tm estimated from the process step response: Tm = 1.4( t 67% − t 33% ) , τ m = t 67% − 1.1Tm ; K m assumed known (Chen and Yang, 2000). 1 τ m and Tm estimated from the process step response: Tm = 1.245( t 70% − t 33% ) , τ m = 1.498t 33% − 0.498t 70% ; K m assumed known (Vítečková et al., 2000b). 2 τ m and Tm estimated from the process step response: Tm = 0.910(t 75% − t 25% ) , τ m = 1.262 t 25% − 0.262 t 75% ; K m is estimated from the process step response (Arrieta Orozco and Alfaro Ruiz, 2003; Arrieta Orozco, 2003). 3 K m estimated from the process step response; τ m = time for which the process variable does not change; Tm determined at 63% of the total process variable change (Gerry, 1999).

Method 12: K m estimated from the process step response; τ m = time for which the process variable has to change by 5% of its total value; Tm determined at 63% of the total process variable change (Kristiansson, 2003). Method 13: K m estimated from the process step response; τ m is the time for which the process variable has to change until it is outside a predefined noise band; Tm = 0.25(t 98% − τ m ) (McMillan, 2005). 4

1

2 3

4

t 67% , t 33% are the times taken by the process variable to change by 67% and 33%, respectively, of its total value. t 70% is the time taken by the process variable to change by 70% of its total value. t 75% , t 25% are the times taken by the process variable to change by 75% and 25%, respectively, of its total value. t 98% is the time taken by the process variable to change by 98% of its total value.

b720_Appendix-02

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Appendix 2: Some Further Details on Process Modelling

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Method 14: Parameters estimated using a least squares method in the time domain (Cheng and Hung, 1985). Method 15: The model parameters are estimated by minimising a function of the error between the process and model open loop step response (Zhuang, 1992). Method 16: Parameters estimated from linear regression equations in the time domain (Bi et al., 1999). Method 17: Parameters estimated using the method of moments (Åström and Hägglund, 1995). Method 18: Model parameters estimated using the area method (Klán, 2000). Method 19: Parameters estimated from the process step response and its first time derivative (Tsang and Rad, 1995). Method 20: Parameters estimated from the process step response using numerical integration procedures (Nishikawa et al., 1984). Method 21: Parameters estimated from the process impulse response (Peng and Wu, 2000). Method 22: Parameters estimated using a ‘process setter’ block (Saito et al., 1990). Method 23: Parameters estimated from a number of process step response data values (Kraus, 1986). Method 24: Parameters estimated in the sampled data domain using two process step tests (Pinnella et al., 1986). Method 25: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm . Subsequently, τ m is estimated from the process step response; then, a parameter labelled τ am = τ 'm − τ m is defined (Schaedel, 1997). Method 26: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm . Subsequently, τ m is estimated from the process step response (Henry and Schaedel, 2005). Method 27: Parameters estimated from the open loop step response (Clarke, 2006).

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A2.1.2 Parameters estimated from the closed loop step response

Method 28: K m estimated from the open loop step response. T90% and τ m estimated from the closed loop step response under proportional control (Åström and Hägglund, 1988). Method 29: Parameters estimated using a method based on the closed loop transient response to a step input under proportional control (Sain and Özgen, 1992). Method 30: Parameters estimated using a second method based on the closed loop transient response to a step input under proportional control (Hwang, 1993). Method 31: Parameters estimated using a third method based on the closed loop transient response to a step input under proportional control (Chen, 1989; Taiwo, 1993). Method 32: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Åström and Hägglund, 1995). Method 33: Parameters estimated from the closed loop step response when process is in series with a PID controller (Morilla et al., 2000). Method 34: Parameters estimated by modelling the closed loop response as a second order system (Ettaleb and Roche, 2000). A2.1.3 Parameters estimated from frequency domain closed loop information

Method 35: Model parameters estimated using a relay autotuning method (Yu, 2006). Method 36: Parameters estimated from two points, determined on process frequency response, using a relay and a relay in series with a delay (Tan et al., 1996). Method 37: Tm and τ m estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; K m assumed known (Hang and Cao, 1996). Method 38: Tm and τ m estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; K m estimated from the process step response (Hang et al., 1993b).

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Method 39: Tm and τ m estimated from data obtained when a relay is introduced in series with the process in closed loop; K m assumed known (Padhy and Majhi, 2006a). Method 40: Parameters estimated from data obtained when a relay is introduced in series with the process in closed loop (Padhy and Majhi, 2006b). Method 41: Tm and τ m estimated from the ultimate gain and period, determined using the ultimate cycle method of Ziegler and Nichols (1942); K m estimated from the process step response (Hang et al., 1993b). Method 42: Tm and τ m estimated at ω1800 ; K m estimated at ω 0 0 (Tavakoli et al., 2006). Method 43: Tm and τ m estimated from relay autotuning method (Lee and Sung, 1993); K m estimated from the closed loop process step response under proportional control (Chun et al., 1999). Method 44: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995). Method 45: K u and Tu estimated from relay autotuning method; τ m estimated from the open loop process step response, using a tangent and point method (Wojsznis et al., 1999). Method 46: Parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Method 47: Parameters estimated from measurements performed on the manipulated and controlled variables when a relay with hysteresis is introduced in place of the controller (Zhang et al., 1996b). Method 48: Non-gain parameters estimated using a relay, with hysteresis, in series with the process in closed loop; K m is estimated with the aid of a small step signal added to the reference (Potočnik et al., 2001). Method 49: Parameters estimated using a relay in series with the process in closed loop (Perić et al., 1997). Method 50: Parameters estimated using an iterative method, based on data from a relay experiment (Leva and Colombo, 2004).

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Method 51: K m estimated from relay autotuning method; τ m , Tm estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005a). Method 52: Parameters estimated from data obtained when an asymmetrical relay is introduced in series with the process in closed loop (Majhi, 2005). Method 53: Parameters estimated from data obtained when an asymmetrical relay is introduced in series with the process in closed loop (Prokop and Korbel, 2006). A2.1.4 Other methods

Method 54: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 55: Parameter estimates determined graphically from a known higher order process (McMillan, 1984). Method 56: Parameter estimates determined from a known higher order process (Skogestad, 2003). Method 57: The parameters of a second order model plus time delay are estimated using a system identification approach in discrete time; the parameters of a FOLPD model are subsequently determined using standard equations (Ou and Chen, 1995). Method 58: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968). Method 59: Parameter estimates determined using a recursive least squares method (Bai and Zhang, 2007).

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A2.2 FOLPD Model with a Zero

Method 2:

Method 3:

Non-delay parameters are estimated in the discrete time domain using a least squares approach; time delay assumed known (Chang et al., 1997). Parameters estimated graphically from the open-loop step response (Slätteke, 2006).

A2.3 SOSPD Model A2.3.1 Parameters estimated from the open loop process step response

Method 2:

Method 3:

Method 4: Method 5:

Method 6:

5

K m , Tm1 and τ m are determined from the tangent and point method of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm 2 assumed known. FOLPD model parameters estimated using a tangent and point method (Ziegler and Nichols, 1942); corresponding SOSPD parameters subsequently deduced (Auslander et al., 1975). Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Tm1 = Tm 2 . τ m , Tm1 estimated from process step response: Tm1 = 0.794( t 70% − t 33% ) , τ m = 1.937 t 33% − 0.937 t 70% . Km 5 assumed known (Vítečková et al., 2000b). Tm1 = Tm 2 . K m , τ m and Tm1 estimated from the process step response; τ m = time for which the process variable does not change; Tm1 determined at 73% of the total process variable change (Pomerleau and Poulin, 2004).

t 70% , t 33% are the times taken by the process variable to change by 70% and 33%, respectively, of its total value.

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Method 7:

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Parameters estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi, 1997).


Method 8: Method 9:

Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Åström and Hägglund, 1995). Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995).

A2.3.3 Parameters estimated from frequency domain closed loop information

Method 10: In this method, Tm1 = Tm 2 = Tm . Tm and τ m estimated from K u , Tu determined using a relay autotuning method; K m estimated from the process step response (Hang et al., 1993a). Method 11: Parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a proportional controller, in series with the process in closed loop (Sung et al., 1996). Method 12: Parameters estimated using a two-stage identification procedure involving (a) placing a relay in series with the process in closed loop and (b) introducing an excitation to the relay feedback system through an external DC input at the output of the relay (Huang et al., 2003). Method 13: Parameters estimated from data determined by inserting a relay in series with the process in closed loop (Lavanya et al., 2006a). Method 14: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).

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Method 15: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Method 16: Parameters estimated from data obtained when the process phase lag is − 90 0 and − 180 0 , respectively (Wang et al., 1999). Method 17: Model parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Method 18: K m estimated from a relay autotuning method; τ m , Tm1 and ξ m estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005a). Method 19: Model parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a relay which operate on the integral of the error, in series with the process in closed loop (Naşcu et al., 2006). Method 20: Model parameters estimated using a relay autotuning method (Thyagarajan and Yu, 2003). A2.3.4 Other methods

Method 21: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 22: Parameter estimates back-calculated from a second discrete time identification method (Wang and Clements, 1995). Method 23: Parameter estimates back-calculated from a third discrete time identification method (Lopez et al., 1969). Method 24: Model parameters estimated assuming higher order process parameters are known (Skogestad, 2003). Method 25: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968).

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A2.4 SOSPD Model with a Zero

Method 2: Method 3:

Parameters estimated from a closed loop step response test using a least squares approach (Wang et al., 2001a). Tm1 = Tm 2 . K m , τ m , Tm1 and Tm 3 (or Tm 4 ) are estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).

A2.5 TOSPD Model

Method 2: Method 3:

Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Model parameters estimated using least squares regression in the time domain; the model input is a step signal (Yu, 2006, p. 107).

A2.6 General Model

Method 2:

A FOLPD model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm .

(

Method 3: Method 4:

)

Then, n = 10 τ 'm Tm' + 1 . Subsequently, τ m is estimated from the open loop step response, and Tar is estimated using ‘known methods’ (Schaedel, 1997). Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Model parameters Taa , τ m and Tar are estimated using the area method (Gorez and Klàn, 2000).

A2.7 Non-Model Specific

Modelling methods have not been specified in the tables in most cases, as typically the tuning rules are based on K u and Tu . Modelling methods have been specified whenever relevant data have been estimated using a relay method.

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Method 2: Method 3: Method 4:

561

K u and Tu estimated from an experiment using a relay in series with the process in closed loop (Jones et al., 1997). K u and Tu estimated with the assistance of a relay (Lloyd, 1994). ω900 and A p estimated using a relay in series with an integrator (Tang et al., 2002).

Method 5:

Method 6:

∧

∧

K u and Tu estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Parameters estimated from an experiment using a relay in series with the process in closed loop (Chen and Yang, 1993).

A2.8 IPD Model A2.8.1 Parameters estimated from the open loop process step response


τ m assumed known; K m estimated from the slope at start of the process step response (Ziegler and Nichols, 1942). K m , τ m estimated from the process step response (Hay, 1998). K m , τm estimated from the process step response (Clarke, 2006). K m , τ m estimated from the process step response using a tangent and point method (Bunzemeier, 1998).


Method 6:

Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995).

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Parameters estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram, 1997). Parameters estimated from the closed loop response, under the control of an on-off controller (Zou et al., 1997). Parameters estimated from the closed loop response, under the control of an on-off controller (Zou and Brigham, 1998).

A2.8.3 Parameters estimated from frequency domain closed loop information

Method 10: Parameters estimated from K u and Tu values, determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben, 1992). Method 11: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Method 12: Parameters estimated from data determined when a relay is placed in series, in closed loop, with the closed loop compensated process, under PI or PID control (NI Labview, 2001). A2.9 FOLIPD Model


Method 5:

Parameters estimated from the open loop process step response and its first and second time derivatives (Tsang and Rad, 1995). Parameters estimated using the method of moments (Åström and Hägglund, 1995). Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Parameters estimated from the open loop response of the process to a pulse signal (Tachibana, 1984).

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Method 6:

Method 7:

Method 8:

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Parameters estimated from the waveform obtained by introducing a single symmetrical relay in series with the process in closed loop (Majhi and Mahanta, 2001). K m estimated from the response of the process to a square wave pulse; other process parameters estimated from the waveform obtained by introducing a relay in series with the process in closed loop (Wang et al., 2001a). Parameters estimated using a relay in series with the process in closed loop (Perić et al., 1997).

A2.10 SOSIPD model

Method 2:

Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999).

A2.11 Unstable FOLPD Model

Method 2:


Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Parameters estimated using a relay feedback approach (Majhi and Atherton, 2000). Parameters estimated using a biased relay feedback test (Huang and Chen, 1999). Parameters estimated using a single symmetric relay feedback feedback test (Vivek and Chidambaram, 2005). Tm and τ m estimated from data obtained when a relay is introduced in series with the process in closed loop; K m assumed known (Padhy and Majhi, 2006a).

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A2.12 Unstable SOSPD Model (one unstable pole)

Method 2:

Method 3:

Method 4:

Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Parameters estimated by minimising the difference in the frequency responses between the high order process and the model, up to the ultimate frequency. The gain and delay are estimated from analytical equations, with the other parameters estimated using a least squares method (Kwak et al., 2000). Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).

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FA

b720_Index

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FA

Index

Åström and Hägglund (2004), 20, 98, 174, 360, 381 Åström and Hägglund (2006), 21, 22, 72, 74, 76, 109, 110–111, 145, 177, 227, 276, 321, 326, 356, 362, 390, 424 Atherton and Boz (1998), 175, 274, 477, 503–504 Atherton and Majhi (1998), 476–477 Atkinson (1968), 341 Atkinson and Davey (1968), 324 Auslander et al. (1975), 206

ABB (1996), 325 ABB (2001), 9, 32, 52, 152, 157, 319 Abbas (1997), 57, 100 Aikman (1950), 321 Alenany et al. (2005), 360, 378 Alfa-Laval, 7, 322 Alfaro Ruiz (2005a), 80, 122, 124, 207, 326, 359 Allen Bradley, 5, 8 Alvarez-Ramirez et al. (1998), 357 Andersson (2000), 74, 357 Anil and Sree (2005), 422, 423 Araki (1985), 156 Arbogast and Cooper (2007), 365, 367, 370 Arbogast et al. (2006), 74 Argelaguet et al. (1997), 156 Argelaguet et al. (2000), 156 Arrieta and Vilanova (2007), 125–127 Arrieta Orozco (2003), 39, 43, 47, 49, 122, 123, 124 Arrieta Orozco and Alfaro Ruiz (2003), 39, 43, 47, 49, 122, 123, 124 Arrieta Orozco (2006), 43, 49, 86, 87–88, 91, 94 Arvanitis et al. (2000a), 221, 232, 261, 275 Arvanitis et al. (2000b), 473, 477 Arvanitis et al. (2003a), 373, 376, 377 Arvanitis et al. (2003b), 400–409, 413 Åström (1982), 322, 325 Åström and Hägglund (1984), 325 Åström and Hägglund (1988), 79, 325 Åström and Hägglund (1995), 21, 56, 108, 164, 248, 319, 322, 325, 331, 347, 351, 359, 389, 413 Åström and Hägglund (2000), 19

Bai and Zhang (2007), 70 Bailey, 5, 6, 7, 8, 10 Bain and Martin (1983), 46, 53 Barberà (2006), 49, 94 Bateson (2002), 322, 327, 349 Bekker et al. (1991), 55 Belanger and Luyben (1997), 370 Benjanarasuth et al. (2005), 353, 379 Bequette (2003), 21, 24, 120 Bialkowski (1996), 74, 283, 287, 357, 436 Bi et al. (1999), 57 Bi et al. (2000), 57, 221 Blickley (1990), 324 Boe and Chang (1988), 77 Bohl and McAvoy (1976b), 210, 230, 241, 250 Borresen and Grindal (1990), 32, 79 Boudreau and McMillan (2006), 223, 357 Brambilla et al. (1990), 74, 112, 203, 227 Branica et al. (2002), 109 Bryant et al. (1973), 20, 52, 197–198 Buckley (1964), 21

599

b720_Index

600

17-Mar-2009

FA


Buckley et al. (1985), 354 Bunzemeier (1998), 351, 368 Calcev and Gorez (1995), 319, 325 Callender (1934), 359 Callender et al. (1935/6), 18, 23, 30, 78 Camacho et al. (1997), 100 Carr (1986), 324 Chandrashekar et al. (2002), 441, 455–456, 466–468 Chang et al. (1997), 182 Chao et al. (1989), 189, 190, 194, 239, 240, 241–242, 242–243, 244, 245, 246–247 Chau (2002), 326 Chen and Seborg (2002), 58, 103, 221–222, 357, 361, 389 Chen and Yang (1993), 318, 325, 330 Chen and Yang (2000), 68 Chen et al. (1997), 75 Chen et al. (1999a), 64, 354, 362, 389 Chen et al. (1999b), 64, 226 Chen et al. (2006), 283 Cheng and Hung (1985), 86, 93 Cheng and Yu (2000), 354 Chesmond (1982), 322, 327 Chidambaram (1994), 354 Chidambaram (1995a), 107–108 Chidambaram (1995b), 443, 452 Chidambaram (1997), 440, 448, 474 Chidambaram (1998), 443, 452 Chidambaram (2000a), 161, 176 Chidambaram (2000b), 376 Chidambaram (2000c), 441, 465 Chidambaram (2002), 28, 33, 58, 80, 103, 162, 181, 222, 280, 383, 420, 441–442, 449, 466 Chidambaram and Kalyan (1996), 492 Chidambaram and Sree (2003), 353, 361, 362, 376 Chidambaram et al. (2005), 59 Chien (1988), 74, 130, 146, 285, 287, 357, 366, 369, 394, 396 Chien and Fruehauf (1990), 130, 146, 285, 287, 357, 366, 369, 394, 396 Chien et al. (1952), 31, 78 Chien et al. (1999), 160–161, 376 Chien et al. (2003), 277, 282, 284, 286, 290, 292 Chiu et al. (1973), 52, 220 Chun et al. (1999), 75

Clarke (2006), 72, 356 Cluett and Wang (1997), 66–67, 101, 353, 361 Cohen and Coon (1953), 31, 78 Concept, 6 Connell (1996), 79 Controller structures Classical controller, 6–7, 25, 134–148, 238–250, 286–287, 341–342, 368–370, 395–396, 423, 425, 436, 458–461, 491– 496 Controller with filtered derivative, 6, 122– 133, 236–237, 284–285, 298, 305–307, 316, 336–340, 366–367, 394 Generalised classical controller, 8, 26, 149–151, 251–252, 288, 343–345, 371, 397–398, 462, 483, 508, 516–517 Ideal controller in series with a first order lag, 6, 24, 118–121, 182, 232–235, 282– 283, 297, 315, 332–335, 364–365, 392– 393, 422, 455–457, 481–482, 490, 513– 515 Ideal PI controller, 5, 18–22, 28–29, 30– 77, 180–181, 183–205, 277–278, 293– 295, 310–311, 318–323, 350–358, 383– 384, 385–387, 430, 438, 440–446, 480, 486–487, 511–512 Ideal PID controller, 4, 5, 23, 78–117, 206–231, 279–281, 296, 303–304, 312– 314, 324–331, 359–363, 388–391, 420– 421, 424, 447–454, 488–489, 506–507 Two degree of freedom controller 1, 8–9, 27, 152–167, 253–263, 289–291, 299– 301, 308–309, 317, 346–348, 372–377, 399–415, 426, 431–435, 437, 439, 463– 472, 484–485, 497–502, 518–519 Two degree of freedom controller 2, 9, 168–169, 378–380, 416–417, 427–428, 473–474, 509–510 Two degree of freedom controller 3, 10– 11, 170–179, 264–276, 292, 302, 349, 381–382, 418–419, 429, 475–479, 503– 505, 520 ControlSoft Inc. (2005), 80 Coon (1956), 350, 385 Coon (1964), 350, 385 Cooper (2006a), 165, 179 Cooper (2006b), 120, 179 Corripio (1990), 324, 341 Cox et al. (1994), 323

b720_Index

17-Mar-2009

Index Cox et al. (1997), 67 Cuesta et al. (2006), 60 Davydov et al. (1995), 57, 128, 311, 316 De Oliveira et al. (1995), 74 De Paor (1993), 325 De Paor and O’Malley (1989), 442, 452 Desbiens (2007), 181, 203, 278 Devanathan (1991), 45, 90 Dutton et al. (1997), 330 ECOSSE Team (1996a), 45 ECOSSE Team (1996b), 322, 327 Edgar et al. (1997), 38, 137–138, 153, 319, 341, 346 Eriksson and Johansson (2007), 174, 178 Ettaleb and Roche (2000), 57 Faanes and Skogestad (2004), 33, 58, 74, 136, 147 Farrington (1950), 324 Fanuc, 5 Fertik (1975), 19, 42, 49, 136, 140 Fertik and Sharpe (1979), 32 Fischer and Porter, 6, 7, 10 Fisher–Rosemount, 10 Foley et al. (2005), 59 Ford (1953), 359 Foxboro, 6, 7, 10, 134 Frank and Lenz (1969), 33, 40, 41, 43 Friman and Waller (1997), 64, 109, 319 Fruehauf et al. (1993), 56, 99, 352 Fukura and Tamura (1983), 50, 197 Gaikward and Chidambaram (2000), 443, 452 Gain margin, analytical calculation, 522– 528 Gain margin, figures, 530, 532–533 Gain and phase margins, constant, 534–543 PI controller, FOLPD model, 534–536 PI controller, IPD model, 536–539 PD controller, FOLIPD model, 542–543 PID controller, FOLPD model, classical controller, 539–541 PID controller, SOSPD model, series controller, 541–542 PID controller, SOSPD model with a negative zero, classical controller, 542

FA

601 Gallier and Otto (1968), 45, 90, 191, 214 García and Castelo (2000), 328 Genesis, 9 Geng and Geary (1993), 77, 116 Gerry (1998), 19, 89 Gerry (1999), 74, 113 Gerry (2003), 45 Gerry and Hansen (1987), 19 Gong et al. (1996), 130 Gong et al. (1998), 23, 131 Gong (2000), 113 Gonzalez (1994), 56, 100 Goodwin et al. (2001), 129 Gorecki et al. (1989), 55, 62, 99, 105, 352, 360, 362 Gorez (2003), 58, 104, 109 Gorez and Klàn (2000), 220, 312, 313 Gu et al. (2003), 117 Haalman (1965), 19, 40, 239, 352, 388 Haeri (2002), 163 Haeri (2005), 59 Hägglund and Åström (1991), 322 Hägglund and Åström (2002), 68, 165, 376, 471 Hägglund and Åström (2004), 21, 70–71, 202–203, 205, 321, 355–356 Hang and Åström (1988a), 167 Hang and Åström (1988b), 325 Hang and Cao (1996), 167 Hang et al. (1991), 55, 167 Hang et al. (1993a), 63, 223, 233 Hang et al. (1993b), 63, 134, 318, 341 Hang et al. (1994), 236 Hansen (1998), 275 Hansen (2000), 21, 376, 426 Harriott (1964), 327 Harriott (1988), 39, 47, 184 Harris and Tyreus (1987), 146, 178 Harrold (1999), 135, 342 Hartmann and Braun, 6 Hartree et al. (1937), 25 Hassan (1993), 189, 210–211 Hay (1998), 32–33, 79–80, 322, 327, 351, 352, 359 Hazebroek and Van der Waerden (1950), 30–31, 39–40, 352 Heck (1969), 32 Henry and Schaedel (2005), 65 Hill and Adams (1988), 77

b720_Index

602

17-Mar-2009

FA


Hill and Venable (1989), 36–37, 83–85 Hiroi and Terauchi (1986), 152, 170 Ho and Xu (1998), 443, 492 Ho et al. (1994), 223 Ho et al. (1995a), 223–224 Ho et al. (1997), 224 Ho et al. (1998), 87, 92 Ho et al. (1999), 63, 87, 92 Ho et al. (2001), 178 Honeywell, 5, 7, 9, 11, 56, 248 Horn et al. (1996), 120, 151 Hougen (1979), 61, 146, 199, 247, 294, 354, 369, 387, 396 Hougen (1988), 199, 248, 369 Huang and Chao (1982), 136, 138, 139, 140, 141, 142, 143, 238, 240, 242, 243, 244, 245, 246 Huang and Chen (1997), 458, 492 Huang and Chen (1999), 458, 492, 502 Huang and Jeng (2002), 19, 46 Huang and Jeng (2003), 97–98, 218–219 Huang and Jeng (2005), 71, 146, 232, 249 Huang and Lin (1995), 463, 464, 497–502 Huang et al. (1996), 38, 46, 185–187, 191– 193, 264–268, 270–273 Huang et al. (1998), 74, 228 Huang et al. (2000), 165, 176, 261, 276, 290, 292 Huang et al. (2002), 121 Huang et al. (2003), 146, 251 Huang et al. (2005a), 71, 77, 146, 148, 232, 235 Huang et al. (2005b), 379, 416 Huba (2005), 429 Huba and Žáková (2003), 353, 387 Hwang (1995), 44, 48, 89, 96–97, 190, 195– 196, 213, 215–217 Hwang and Chang (1987), 318 Hwang and Fang (1995), 44, 51, 89, 97 Idzerda et al. (1955), 318 Intellution, 5 Isaksson and Graebe (1999), 75 Jacob and Chidambaram (1996), 440, 473 Jahanmiri and Fallahi (1997), 251 Jhunjhunwala and Chidambaram (2001), 448, 466 Johnson (1956), 318 Jones and Tham (2006), 296, 302

Jones et al. (1997), 323, 329 Juang and Wang (1995), 100 Jyothi et al. (2001), 29, 182, 291, 383–384, 420–421 Kamimura et al. (1994), 21, 56 Kang (1989), 208 Karaboga and Kalinli (1996), 326 Kasahara et al. (1997), 160 Kasahara et al. (1999), 131–132 Kasahara et al. (2001), 161 Kaya (2003), 381 Kaya and Atherton (1999), 381 Kaya and Scheib (1988), 136, 138, 139, 140, 142, 171, 172 Keane et al. (2000), 218 Keane et al. (2005), 331 Keviczky and Csáki (1973), 19, 20, 53, 194 Khan and Lehman (1996), 48, 55, 57 Khodabakhshian and Golbon (2004), 278 Kim et al. (2004), 163 King (2006), 73, 112 Kinney (1983), 341 Klán (2000), 58, 312 Klán (2001), 376, 390, 413 Klán and Gorez (2000), 319 Klán and Gorez (2005a), 59 Klán and Gorez (2005b), 59 Klein et al. (1992), 32, 61 Kookos et al. (1999), 354 Kosinsani (1985), 34, 82 Kotaki et al. (2005a), 153–154, 164 Kotaki et al. (2005b), 153–155 Kraus (1986), 134 Kristiansson (2003), 69, 149–150, 204, 228, 234, 251, 320, 332, 344 Kristiansson and Lennartson (1998a), 337– 338 Kristiansson and Lennartson (1998b), 338 Kristiansson and Lennartson (2000), 320, 338–339 Kristiansson and Lennartson (2002a), 320, 340, 366, 394 Kristiansson and Lennartson (2002b), 69, 119, 334 Kristiansson and Lennartson (2003), 320, 321 Kristiansson and Lennartson (2006), 69–70, 119, 120, 321, 335, 392 Kristiansson et al. (2000), 343

b720_Index

17-Mar-2009

FA

Index Kuhn (1995), 57, 128 Kukal (2006), 60, 104 Kuwata (1987), 20, 27, 53, 77, 159, 160, 166–167, 198, 204, 262–263, 294, 295, 296, 300, 317, 352, 375, 386, 413, 415, 437, 438, 439 Kwak et al. (2000), 504–505

116, 231, 301, 430,

158, 259, 311, 435,

Landau and Voda (1992), 231 Larionescu (2002), 314, 316 Latzel (1988), 61, 129–130 Lavanya et al. (2006a), 237 Lavanya et al. (2006b), 73 Lee and Edgar (2002), 121, 168–169, 178– 179, 233, 379, 382, 416, 419, 474, 479 Lee and Shi (2002), 147, 151 Lee and Teng (2003), 313 Lee et al. (1992), 63, 106, 164, 177 Lee et al. (1998), 75, 113, 228 Lee et al. (2000), 454, 474, 489, 505, 507, 509 Lee et al. (2005), 323 Lee et al. (2006), 113, 227, 281, 362, 390, 421, 457, 505 Leeds and Northup, 5, 79 Lennartson and Kristiansson (1997), 337 Leonard (1994), 360 Leva (1993), 322, 336 Leva (2001), 65, 74, 112, 132 Leva (2005), 165 Leva (2007), 74 Leva and Colombo (2000), 132 Leva and Colombo (2004), 165 Leva et al. (1994), 70, 224–225 Leva et al. (2003), 68–69, 132, 202, 226 Leva et al. (2006), 132 Li et al. (1994), 105 Li et al. (2004), 283 Lim et al. (1985), 118, 232 Lipták (1970), 327 Lipták (2001), 33, 80 Litrico and Fromion (2006), 356, 363 Litrico et al. (2006), 356 Litrico et al. (2007), 358 Liu et al. (2003), 417, 427–428 Lloyd (1994), 328 Lopez (1968), 81, 86, 87, 183, 188, 207, 208–209 Lopez et al. (1969), 188, 210

603 Loron (1997), 438 Luo et al. (1996), 326 Luyben (1993), 77 Luyben (1996), 370 Luyben (1998), 443 Luyben (2000), 278 Luyben (2001), 74, 120 Luyben and Luyben (1997), 326 MacLellan (1950), 321, 327 Madhuranthakam et al. (2008), 155, 156, 256, 257, 269, 289–290 Maffezzoni and Rocco (1997), 130 Majhi (1999), 323, 329, 346, 347 Majhi (2005), 50 Majhi and Atherton (1999), 175, 274, 477, 503–504 Majhi and Atherton (2000), 440, 475–476 Majhi and Mahanta (2001), 419 Majhi and Litz (2003), 250 Mann et al. (2001), 54, 102 Mantz and Tacconi (1989), 348 Manum (2005), 21, 222, 442 Marchetti and Scali (2000), 181, 203, 228, 278, 280, 295, 296 Marchetti et al. (2001), 453, 488 Marlin (1995), 38, 46, 81, 91 Mataušek and Kvaščev (2003), 69 Matsuba et al. (1998), 77, 116 Maximum sensitivity, analytical calculation, 529 Maximum sensitivity, figures, 530 Maximum sensitivity, constant, 538 McAnany (1993), 56 McAvoy and Johnson (1967), 187, 240, 324 McMillan (1984), 76, 116, 387, 391, 487, 489 McMillan (1994), 32, 318, 322, 325, 327 McMillan (2005), 22, 59, 322, 325, 335 Mesa et al. (2006), 60 Modicon, 6 Modcomp, 8 Mollenkamp et al. (1973), 52, 220 Moore, 10, 11 Morari and Zafiriou (1989), 130 Morilla et al. (2000), 102 Moros (1999), 31, 80 Murata and Sagara (1977), 34, 46, 183, 191, 293, 310 Murrill (1967), 32, 33, 40, 41, 79, 81, 86, 87

b720_Index

604

17-Mar-2009

FA


Naşcu et al. (2006), 235 NI Labview (2001), 323, 329–330, 350, 351, 359 Normey-Rico et al. (2000), 168, 378 Normey-Rico et al. (2001), 378 Nomura et al. (1993), 20, 23, 27, 56, 99– 100, 160 O’Connor and Denn (1972), 90 O’Dwyer (2001a), 63, 146, 249, 287, 355, 390 O’Dwyer (2001b), 135–136, 342 Ogawa (1995), 75, 357 Ogawa and Katayama (2001), 162 Okada et al. (2006), 54 OMEGA Books (2005), 152, 348 Omron, 8 Oppelt (1951), 31 Ou and Chen (1995), 139, 142 Ou et al. (2005a), 364, 446 Ou et al. (2005b), 365, 393 Ou et al. (2005c), 132 Oubrahim and Leonard (1998), 263, 415 Ozawa et al. (2003), 163 Padhy and Majhi (2006a), 177, 478 Padhy and Majhi (2006b), 73 Pagola and Pecharromán (2002), 323 Panda et al. (2004), 204, 228, 232, 234, 249 Panda et al. (2006), 357 Paraskevopoulos et al. (2004), 464–465, 469 Paraskevopoulos et al. (2006), 444–445, 459–461, 492–496 Park et al. (1997), 212, 214 Park et al. (1998), 478 Parr (1989), 32, 79, 318, 322, 323, 324, 327, 328 Paz Ramos et al. (2005), 360 Pecharromán (2000a), 257, 258, 374, 410, 431 Pecharromán (2000b), 258–259, 374–375, 411, 432–433 Pecharromán and Pagola (2000), 257–258, 374, 410, 432 Pemberton (1972a), 34, 220 Pemberton (1972b), 219–220 Peng and Wu (2000), 81 Penner (1988), 358 Perić et al. (1997), 77, 116, 387, 391 Pessen (1953), 341

Pessen (1954), 341 Pessen (1994), 86, 148, 318 Pettit and Carr (1987), 324 Phase margin, analytical calculation, 522– 528 Phase margin, figures, 530, 532–533 Pinnella et al. (1986), 54 PMA (2006), 33, 80 Polonyi (1989), 296 Pomerleau and Poulin (2004), 199, 277, 282 Potočnik et al. (2001), 66 Poulin and Pomerleau (1996), 352, 386, 395–396, 486, 491 Poulin and Pomerleau (1997), 395–396, 491 Poulin and Pomerleau (1999), 353, 387 Poulin et al. (1996), 201, 249, 286, 396, 487 Pramod and Chidambaram (2000), 449 Prashanti and Chidambaram (2000), 470– 471 Process models Delay model, 12, 18–27 Delay model with a zero, 12, 28–29 Fifth order system plus delay model, 13, 303–309 FOLIPD model, 14, 385–419 FOLIPD model with a zero, 14, 420–423 FOLPD model, 13, 30–179 FOLPD model with a zero, 13, 180–182 General model, 14, 310–317 General model with integrator, 15, 438– 439 IPD model, 14, 350–382 IPD model with a zero, 14, 383–384 I 2 PD model, 14, 424–429 Non-model specific, 14, 318–349 SOSIPD model, 14, 430–435 SOSIPD model with a zero, 14, 436 SOSPD model, 13, 183–276 SOSPD model with a zero, 13, 277–292 TOSIPD model, 14, 437 TOSPD model, 13, 293–302 Unstable FOLPD model, 15, 440–479 Unstable FOLPD model with a zero, 15, 480–485 Unstable SOSPD model (one unstable pole), 15, 486–505 Unstable SOSPD model (two unstable poles), 15, 506–510 Unstable SOSPD model with a zero, 15, 511–520

b720_Index

17-Mar-2009

Index

Prokop et al. (2000), 57 Prokop et al. (2005), 319 Pulkkinen et al. (1993), 75 Ramasamy and Sundaramoorthy (2007), 330–331, 333, 344–345 Rao and Chidambaram (2006a), 516 Rao and Chidambaram (2006b), 516, 520 Ream (1954), 23, 52, 197 Reswick (1956), 31 Rice (2004), 362, 370 Rice (2006), 377 Rice and Cooper (2002), 364, 369 Rivera et al. (1986), 74, 112 Rivera and Jun (2000), 120, 228, 233, 364, 390, 393 Robbins (2002), 319, 326 Rotach (1994), 327–328 Rotach (1995), 199, 220, 352, 361 Rotstein and Lewin (1991), 446, 453, 488 Rovira et al. (1969), 45, 48, 90, 93 Rutherford (1950), 321, 327 Sadeghi and Tych (2003), 91, 92, 93 Sain and Özgen (1992), 79 Saito et al. (1990), 99 Sakai et al. (1989), 32 SATT Instruments, 11 Schaedel (1997), 65, 118, 128, 144, 248, 297, 298, 311, 315 Schlegel (1998), 21 Schneider (1988), 55 Seki et al. (2000), 325 Setiawan et al. (2000), 158 Shamsuzzoha and Lee (2006a), 151, 288, 363 Shamsuzzoha and Lee (2006b), 454 Shamsuzzoha and Lee (2007a), 115, 229–230, 262, 363, 377, 391, 415, 489 Shamsuzzoha and Lee (2007b), 150 Shamsuzzoha and Lee (2007c), 169 Shamsuzzoha and Lee (2007d), 508 Shamsuzzoha et al. (2004), 151 Shamsuzzoha et al. (2005), 454, 489 Shamsuzzoha et al. (2006a), 398 Shamsuzzoha et al. (2006b), 510 Shamsuzzoha et al. (2006c), 114, 229, 390, 422 Shamsuzzoha et al. (2007), 371, 462

201,

252,

166, 454,

363,

FA

605 Shen (1999), 254 Shen (2000), 255–256 Shen (2002), 158, 347 Shi and Lee (2002), 147 Shi and Lee (2004), 252 Shigemasa et al. (1987), 159 Shin et al. (1997), 329 Shinskey (1988), 19, 35, 39, 137, 138, 153, 156, 208, 238, 351, 368, 372, 458 Shinskey (1994), 19, 39, 138, 157, 184, 239, 254, 351, 368, 372, 385, 395, 399 Shinskey (1996), 19, 38, 39, 137, 152, 184, 187, 239, 253, 254, 368–369, 372 Shinskey (2000), 33, 135 Shinskey (2001), 33 Shinskey (2003), 39, 138 Siemens, 6 Sklaroff (1992), 56, 248 Skoczowski (2004), 59, 249 Skoczowski and Tarasiejski (1996), 314 Skogestad (2001), 20, 352 Skogestad (2003), 20, 21, 22, 58, 248, 352, 353, 396, 425 Skogestad (2004a), 358 Skogestad (2004b), 221, 353, 418, 429 Skogestad (2004c), 222 Skogestad (2006a), 61 Skogestad (2006b), 61, 249 Slätteke (2006), 180, 181 Smith (1998), 67 Smith (2002), 50, 56, 74, 357, 358 Smith (2003), 93, 228, 326 Smith and Corripio (1997), 34, 46, 53, 138, 140, 144 Smith et al. (1975), 53, 220, 247 Somani et al. (1992), 62, 106, 200–201 Square D, 9 Sree and Chidambaram (2003a), 181, 182 Sree and Chidambaram (2003b), 481, 484– 485 Sree and Chidambaram (2003c), 480, 485 Sree and Chidambaram (2004), 511, 513, 518–519 Sree and Chidambaram (2005a), 467 Sree and Chidambaram (2005b), 361 Sree and Chidambaram (2006), 353, 362, 376, 389, 442, 444, 450–452, 456, 465, 466–467, 468, 474, 477, 480, 481–482, 483, 485, 502, 511, 514–515, 517, 519 Sree et al. (2004), 59, 104, 442, 449–450

b720_Index

606

17-Mar-2009

FA


Srinivas and Chidambaram (2001), 162, 465 Srividya and Chidambaram (1997), 354 St. Clair (1997), 32, 135, 341 Streeter et al. (2003), 331 Sung et al. (1996), 212, 214 Suyama (1992), 53, 99, 220 Suyama (1993), 159 Syrcos and Kookos (2005), 90 Tachibana (1984), 388 Taguchi and Araki (2000), 157, 260, 299– 300, 375, 411–412, 433 Taguchi and Araki (2002), 157, 412 Taguchi et al. (1987), 156, 257, 410 Taguchi et al. (2002), 375, 412, 433–434 Tan et al. (1996), 70, 108, 201 Tan et al. (1998a), 393 Tan et al. (1998b), 120, 364, 393 Tan et al. (1999a), 135, 326, 342 Tan et al. (1999b), 446, 453 Tan et al. (2001), 326, 342 Tang et al. (2002), 323, 330 Tang et al. (2007), 121, 133, 147–148 Tavakoli and Fleming (2003), 47 Tavakoli and Tavakoli (2003), 123 Tavakoli et al. (2005), 158, 375 Tavakoli et al. (2006), 47 Taylor, 11 Thomasson (1997), 74, 75, 357 Thyagarajan and Yu (2003), 205 Tinham (1989), 324 Toshiba, 7, 9, 159 Trybus (2005), 60, 222–223, 313–314 Tsang and Rad (1995), 144, 397 Tsang et al. (1993), 144, 149, 397 Tuning rules: Direct synthesis, frequency domain criteria, 21, 29, 61–73, 105–112, 118– 120, 129–130, 146, 149–150, 164–165, 177–178, 181, 182, 199–203, 223–227, 232, 236, 249–250, 261–262, 276, 278, 280, 286–287, 290–291, 294, 297, 298, 301, 303–304, 305–307, 308–309, 311, 314, 315, 319–321, 331, 334–335, 336– 340, 347–348, 349, 354–356, 362, 366, 369, 376, 381, 383–384, 387, 389–390, 392, 394, 396, 398, 413–414, 419, 420– 421, 424, 438, 442–445, 452, 459–461, 470–472, 473, 477–478, 487, 492–496, 504–505

Direct synthesis, time domain criteria, 20– 21, 23, 27, 28, 52–61, 98–105, 118, 128–129, 144–145, 149, 158–164, 168, 175–176, 181, 182, 197–199, 219–223, 232, 247–248, 251, 259–260, 274–276, 277, 280, 282–283, 284, 286, 290, 292, 294, 300, 311, 312–314, 316, 317, 330– 331, 333, 344–345, 352–353, 360–362, 375–376, 378–379, 383, 386–387, 388– 389, 396, 397, 413, 416, 418, 420, 423, 425, 426, 429, 430, 435, 438, 439, 440– 442, 448–452, 455–456, 458, 465–469, 473, 476–477, 480, 481–482, 483, 484– 485, 488, 492, 502, 503–504, 506, 511– 512, 513–515, 516, 518–519, 520 Minimum performance index, regulator tuning, 19, 33–45, 81–90, 122–123, 136–139, 152–155, 171, 180, 183–190, 207–213, 238–243, 253–256, 264–269, 289, 293, 310, 341, 346, 351–352, 360, 368–369, 372–373, 385–386, 388, 395– 396, 399–404, 447, 458, 463–464, 486, 491, 497–499 Minimum performance index, servo tuning, 19, 45–51, 90–97, 123–124, 140–143, 156, 172–173, 191–197, 213– 217, 243–247, 257, 269–273, 279, 290, 293, 310, 347, 360, 373, 378, 386, 388, 405–409, 440, 448, 464, 475–476, 500– 501 Minimum performance index, servo/ regulator tuning, 125–127, 257–259, 299–300, 448 Minimum performance index, other tuning, 19–20, 51–52, 97–98, 156–158, 174, 218–219, 296, 332, 343–344, 347, 352, 360, 374–375, 381, 410–412, 431– 434, 464–465 Other tuning rules, 22, 205, 321–323, 327–330, 358 Process reaction curve methods, 18–19, 23, 25, 30–33, 78–80, 134–136, 152, 170, 206–207, 350–351, 359, 368, 385 Robust, 21–22, 23, 24, 74–76, 112–115, 120–121, 130–133, 146–148, 151, 165– 166, 168–169, 178–179, 181, 182, 203– 204, 227–230, 233–235, 236–237, 251– 252, 278, 280–281, 283, 285, 287, 288, 295, 296, 302, 314, 316, 357–358, 362– 363, 364–365, 366–367, 369–370, 371,

b720_Index

17-Mar-2009

Index

377, 379–380, 382, 387, 390–391, 393, 394, 396, 415, 416–417, 419, 421, 422, 427–428, 436, 446, 453–454, 457, 462, 472, 474, 478–479, 482, 488–489, 490, 505, 507, 508, 509–510, 517 Ultimate cycle, 22, 26, 76–77, 116–117, 148, 151, 166–167, 169, 204–205, 230– 231, 235, 250, 262–263, 278, 283, 295, 296, 301, 318–319, 324–326, 335, 341– 342, 348, 358, 363, 370, 371, 387, 391, 415, 435, 437, 487, 489 Turnbull, 7 Tyreus and Luyben (1992), 326, 352 Urrea et al. (2006), 22 Valentine and Chidambaram (1997a), 57, 102 Valentine and Chidambaram (1997b), 448 Valentine and Chidambaram (1998), 440 Van der Grinten (1963), 20, 98, 219, 388 VanDoren (1998), 152, 348 Velázquez-Figueroa (1997), 386, 387, 388 Venkatashankar and Chidambaram (1994), 443 Vilanova (2006), 121 Vilanova and Balaguer (2006), 133 Visioli (2001), 360, 447 Visioli (2002), 179 Visioli (2005), 74 Visioli (2006), 61 Vítečková (1999), 53, 220–221, 353, 389 Vítečková (2006), 61, 105, 199, 223 Vítečková and Víteček (2002), 54, 103 Vítečková and Víteček (2003), 54, 103 Vítečková et al. (2000a), 53, 220–221, 353, 389 Vítečková et al. (2000b), 53, 221 Vivek and Chidambaram (2005), 450 Voda and Landau (1995), 53, 64 Vrančić (1996), 294, 301, 305, 319, 336– 337, 348 Vrančić and Lumbar (2004), 307, 308–309 Vrančić et al. (1996), 294, 319 Vrančić et al. (1999), 303–304, 306, 337 Vrančić et al. (2004a), 294, 307, 308–309 Vrančić et al. (2004b), 294 Wade (1994), 322, 327 Wang (2000), 226

FA

607 Wang (2003), 23, 75–76, 113–114, 237 Wang and Cai (2001), 414 Wang and Cai (2002), 414, 472 Wang and Clements (1995), 220 Wang and Cluett (1997), 352–353, 361 Wang and Cluett (2000), 66–67, 101 Wang and Jin (2004), 488, 506 Wang and Shao (1999a), 225 Wang and Shao (1999b), 68 Wang and Shao (2000a), 68 Wang and Shao (2000b), 226 Wang et al. (1995a), 91, 93, 279 Wang et al. (1995b), 319, 334 Wang et al. (1999), 225 Wang et al. (2000a), 67 Wang et al. (2000b), 280 Wang et al. (2001a), 280 Wang et al. (2001b), 419 Wang et al. (2002), 144–145 Wang et al. (2005), 365 Wills (1962a), 214, 219, 230 Wills (1962b), 207, 213 Wilton (1999), 51–52, 218 Witt and Waggoner (1990), 134–135, 137, 139, 141, 143 Wojsznis and Blevins (1995), 116 Wojsznis et al. (1999), 116–117, 326 Wolfe (1951), 19, 31, 350 Xing et al. (2001), 92, 95, 96 Xu and Shao (2003a), 414 Xu and Shao (2003b), 472 Xu and Shao (2003c), 414 Xu and Shao (2004a), 414, 418 Xu and Shao (2004b), 71 Xu et al. (2004), 71 Xu et al. (2005), 60 Yang and Clarke (1996), 63, 223, 249 Yi and De Moor (1994), 336 Yokogawa, 5, 8 Young (1955), 322 Yu (1988), 35, 40–41, 42 Yu (1999), 326 Yu (2006), 21, 22, 24, 26, 130, 151, 295, 357, 366, 371 Zhang (1994), 56, 286 Zhang (2006), 457 Zhang and Xu (2002), 472, 490

b720_Index

608

17-Mar-2009

FA


Zhang et al. (1996a), 328 Zhang et al. (1996b), 147 Zhang et al. (1999a), 364, 369 Zhang et al. (1999b), 357, 393 Zhang et al. (2000), 457 Zhang et al. (2002a), 379 Zhang et al. (2002b), 121, 132, 147, 233 Zhang et al. (2002c), 169 Zhang et al. (2003), 169 Zhang et al. (2005), 93

Zhang et al. (2006), 380 Zhong and Li (2002), 75, 236, 358 Zhuang (1992), 41, 44, 48, 50, 51, 86, 88, 92, 95, 172–173 Zhuang and Atherton (1993), 41, 44, 48, 50, 51, 86, 88, 92, 95, 107, 172–173 Ziegler and Nichols (1942), 30, 78, 318, 324, 350, 359 Zou and Brigham (1998), 362 Zou et al. (1997), 112, 362

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Handbook Of Pi And Pid Controller Tuning Rules