Detection and Diagnosis of Stiction in Control Loops: State of the Art and Advanced Methods (Advances in Industrial Control)

Advances in Industrial Control Other titles published in this series: Digital Controller Implementation and Fragility...

Author: Mohieddine Jelali | Biao Huang

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Advances in Industrial Control

Other titles published in this series: Digital Controller Implementation and Fragility Robert S.H. Istepanian and James F. Whidborne (Eds.)

Modelling and Control of Mini-Flying Machines Pedro Castillo, Rogelio Lozano and Alejandro Dzul

Optimisation of Industrial Processes at Supervisory Level Doris Sáez, Aldo Cipriano and Andrzej W. Ordys

Ship Motion Control Tristan Perez

Robust Control of Diesel Ship Propulsion Nikolaos Xiros

Hard Disk Drive Servo Systems (2nd Ed.) Ben M. Chen, Tong H. Lee, Kemao Peng and Venkatakrishnan Venkataramanan

Hydraulic Servo-systems Mohieddine Jelali and Andreas Kroll

Measurement, Control, and Communication Using IEEE 1588 John C. Eidson

Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques Silvio Simani, Cesare Fantuzzi and Ron J. Patton

Piezoelectric Transducers for Vibration Control and Damping S.O. Reza Moheimani and Andrew J. Fleming

Strategies for Feedback Linearisation Freddy Garces, Victor M. Becerra, Chandrasekhar Kambhampati and Kevin Warwick

Manufacturing Systems Control Design Stjepan Bogdan, Frank L. Lewis, Zdenko Kovačić and José Mireles Jr.

Robust Autonomous Guidance Alberto Isidori, Lorenzo Marconi and Andrea Serrani Dynamic Modelling of Gas Turbines Gennady G. Kulikov and Haydn A. Thompson (Eds.) Control of Fuel Cell Power Systems Jay T. Pukrushpan, Anna G. Stefanopoulou and Huei Peng Fuzzy Logic, Identification and Predictive Control Jairo Espinosa, Joos Vandewalle and Vincent Wertz Optimal Real-time Control of Sewer Networks Magdalene Marinaki and Markos Papageorgiou Process Modelling for Control Benoît Codrons Computational Intelligence in Time Series Forecasting Ajoy K. Palit and Dobrivoje Popovic

Windup in Control Peter Hippe Nonlinear H2/H∞ Constrained Feedback Control Murad Abu-Khalaf, Jie Huang and Frank L. Lewis Practical Grey-box Process Identification Torsten Bohlin Control of Traffic Systems in Buildings Sandor Markon, Hajime Kita, Hiroshi Kise and Thomas Bartz-Beielstein Wind Turbine Control Systems Fernando D. Bianchi, Hernán De Battista and Ricardo J. Mantz Advanced Fuzzy Logic Technologies in Industrial Applications Ying Bai, Hanqi Zhuang and Dali Wang (Eds.) Practical PID Control Antonio Visioli (continued after Index)

Mohieddine Jelali • Biao Huang Editors with M.A.A. Shoukat Choudhury, Peter He, Alexander Horch, Manabu Kano, Nazmul Karim, Srinivas Karra, Hidekazu Kugemoto, Kwan-Ho Lee, S. Joe Qin, Claudio Scali, Zhengyun Ren, Maurizio Rossi, Timothy Salsbury, Sirish L. Shah, Ashish Singhal, Nina F. Thornhill, Jin Wang and Yoshiyuki Yamashita

Detection and Diagnosis of Stiction in Control Loops State of the Art and Advanced Methods

13

Mohieddine Jelali, Dr.-Ing Department of Plant and System Technology VDEh-Betriebsforschungsinstitut GmbH (BFI) Sohnstraße 65 40237 Düsseldorf Germany [email protected]

Biao Huang, PhD Department of Chemical and Materials Engineering University of Alberta 536 Chemical and Materials Engineering Building Edmonton, Alberta T6G 2G6 Canada [email protected]

ISSN 1430-9491 ISBN 978-1-84882-774-5 e-ISBN 978-1-84882-775-2 DOI 10.1007/978-1-84882-775-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009936772 © Springer-Verlag London Limited 2010 DuPont™ and Performance Surveyor™ are trademarks of E.I. du Pont de Nemours and Company. http://www2.dupont.com Excel® is a registered trademark of Microsoft Corporation in the United States and other countries. http://www.microsoft.com Genetic Algorithm and Direct Search Toolbox™, MATLAB®, Optimization Toolbox™, Signal Processing Toolbox™, Simulink® and System Identification Toolbox™ are all trademarks or registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, U.S.A. http://www.mathworks.com Intel® and Pentium® are registered trademarks of Intel Corporation in the United States and other countries. http://www intel.com TOMLAB® is a registered trademark of Tomlab Optimization Inc. 1260 SE Bishop Blvd Ste E, Pullman, WA 99163-5451, USA. http://www.tomopt.com Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudioCalamar, Figueres/Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Advances in Industrial Control Series Editors Professor Michael J. Grimble, Professor of Industrial Systems and Director Professor Michael A. Johnson, Professor (Emeritus) of Control Systems and Deputy Director Industrial Control Centre Department of Electronic and Electrical Engineering University of Strathclyde Graham Hills Building 50 George Street Glasgow G1 1QE United Kingdom Series Advisory Board Professor E.F. Camacho Escuela Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos s/n 41092 Sevilla Spain Professor S. Engell Lehrstuhl für Anlagensteuerungstechnik Fachbereich Chemietechnik Universität Dortmund 44221 Dortmund Germany Professor G. Goodwin Department of Electrical and Computer Engineering The University of Newcastle Callaghan NSW 2308 Australia Professor T.J. Harris Department of Chemical Engineering Queen’s University Kingston, Ontario K7L 3N6 Canada Professor T.H. Lee Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576

Professor (Emeritus) O.P. Malik Department of Electrical and Computer Engineering University of Calgary 2500, University Drive, NW Calgary, Alberta T2N 1N4 Canada Professor K.-F. Man Electronic Engineering Department City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong Professor G. Olsson Department of Industrial Electrical Engineering and Automation Lund Institute of Technology Box 118 S-221 00 Lund Sweden Professor A. Ray Department of Mechanical Engineering Pennsylvania State University 0329 Reber Building University Park PA 16802 USA Professor D.E. Seborg Chemical Engineering 3335 Engineering II University of California Santa Barbara Santa Barbara CA 93106 USA Doctor K.K. Tan Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576 Professor I. Yamamoto Department of Mechanical Systems and Environmental Engineering The University of Kitakyushu Faculty of Environmental Engineering 1-1, Hibikino,Wakamatsu-ku, Kitakyushu, Fukuoka, 808-0135 Japan

Für Doris, Yasmin und Dunja M.J.

To Yali and Linda B.H.

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies…, new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. There are two approaches to process performance assessment analysis. One is to base the analysis on a top-down viewpoint that seeks to assess performance through a decomposed structure for the process system. Fig. 0.1 shows such a decomposition comprised of the process, its instrumentation and the controller. Each of these areas can be subdivided further and the lower levels of the decomposition then rigorously assessed to ensure each component is delivering optimum performance – the idea being that achieving optimised performance at component level will translate into good process operation at global level. Process Performance Assessment

Process

Characteristics

Design

Instrumentation

Actuators

Sensors

Controller

Design

Tuning

Fig. 0.1 Process decomposition

A quite different, complementary, bottom-up, approach to process performance assessment which depends on the installation of adequate data acquisition technology, is to pursue the analysis of performance using data-driven methods and algorithms. The data acquisition devices provide access to, and ix

x

Series Editors’ Foreword

measurements of, process parameters, control signals, setpoints, process trajectory information, and process output signals. Features that might be sought in the data include offsets, biases, drifts, oscillations, noise effects, and shape distinctive disturbances. This information then has to be linked to some form of root-cause analysis to determine the possible location in a decomposed process structure (Fig. 0.1) and the cause of the performance deterioration. The techniques of statistical process control and the routines of controller performance assessment (from a field initiated by Professor Thomas Harris) are examples of two different groups of data-driven techniques for investigating process performance issues. Within the decomposition of Fig. 0.1, one large class of actuators is that of pneumatic and hydraulic valves, and one of the possible performance-inhibiting consequences of their use is to introduce stiction (static-friction) into a control loop. However, stiction in a control loop can lead to oscillations or limit cycles. Thus the measurement data from an oscillating control loop can be used as a possible diagnostic signal to (a) detect the presence of valve stiction, and (b) provide an estimate of the stiction magnitude. This Advances in Industrial Control volume aims to establish a comprehensive and fundamental scientific framework for the detection and diagnosis of stiction in a range of industrial control loops. Early chapters of the book concentrate on understanding and modelling the physical origins of stiction. The central part of the book concentrates on the different approaches to constructing automated analysis procedures that provide robust stiction detection and possible stiction magnitude estimates. This is followed by an extended chapter reporting an exhaustive comparative assessment of the different methods and containing guidelines for the performance and use of the routines presented. A chapter on new possible research issues arising from the results reported brings the book to a satisfying conclusion. The Editors of the Advances in Industrial Control series have always been aware of the importance of process performance assessment to industrial process and control engineers, and the series was the first to publish a book-length presentation of the rapidly growing subject of controller performance assessment, namely the monograph Performance Assessment of Control Loops (ISBN 978-185233-639-4, 1999) by Biao Huang and Sirish L. Shah. By way of an update, a second contribution to this particular field was published in 2007, Process Control Performance Assessment (ISBN 978-1-84628-623-0, 2007) edited by Andrzej W. Ordys, Damien Uduehi and Michael A. Johnson. Process nonlinearity analysis, stiction detection and diagnosis are new strands in the development of process performance assessment routines. Recently, M.A.A. Shoukat Choudhury, Sirish L. Shah and Nina F. Thornhill contributed the monograph Diagnosis of Process Nonlinearities and Valve Stiction (ISBN 978-3-540-79223-9, 2008) to the series. Now the Editors are very pleased to add this new and thorough scientific study to the growing book literature of process performance assessment techniques and applications in the series. These are books that can really be said to be essential volumes in every industrial process and control engineer’s library. Industrial Control Centre Glasgow Scotland, UK 2009

M.J. Grimble M.A. Johnson

Preface

Aim of the Book Control performance monitoring (CPM) technology has progressed steadily since the key research step taken by Harris [41]. CPM has emerged as an important area of scientific and technological development in the last decade. Considerable effort has been devoted to the development and application of different CPM methodologies in various industrial fields, such as refining, chemicals, and petrochemicals, mineral processing, mining and metal processing, as well as pulp and paper. The diversity of causes of poor performance in industrial control loops has provided the main motivation for exploring different CPM techniques. The cause of poor performance is not limited to controller design and tuning; other elements in the control systems, such as sensors and actuators, are often responsible for the poor performance. In a recent study by Paulonis and Cox [92], it was reported that the performance of 42% control loops was in the categories of “fair” and/or “poor”. In other studies by Bialkowski [10] and Ender [26], it was found that in 30% of control loops, output cycling and increased output variability was due to instrument performance issues (such as valve backlash, dead time, etc.). Thus, a solution to any control-loop performance problem will all too often involve diagnosis and analysis of oscillation and valve-stiction problems. Besides the detection and diagnosis of sticky control valves, it is also important to be able to quantify stiction so that a list of sticky valves in order of their maintenance priority can be prepared. This book is a collection of contributions from several internationally leading researchers on automatic detection and diagnosis of static friction (stiction) in control loops manipulated by valves, and contains by now the most comprehensive survey of state-of-the-art and advanced techniques. Not only does it present the principles, assumptions, strengths and drawbacks, but also provides guidelines and detailed working procedures for the implementation and application of each method. An exhaustive comparison of the described approaches on nearly 100 control loops from different industrial fields (building, chemicals, pulp and paper, mineral processing, mining, and metal processing) is included.

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Preface

Stiction is a sophisticated non-linear phenomenon. Its detection and quantification has been identified as a highly challenging academic as well as industrial problem. There does not exist a single “perfect” solution; each method has its assumptions, strengths, and weaknesses. Thus, it is imperative for readers to have access to different options, and compare and choose the most appropriate one for a given process-operating condition. In this respect, the book provides a good resource for researchers and industrial practitioners to find a variety of basic as well as advanced methods for solving valve-stiction problems. The methods presented include techniques for diagnosis plus quantification of stiction with both open-loop and closed-loop methods. The reader will learn how the different methods work, what key issues should be considered, and how to parametrise them. The comprehensive comparative study gives readers useful guidelines in choosing the appro® priate method. Moreover, users will have access to MATLAB software associated with the book so that they can directly benefit from the book by applying the methods to their data. The software are available to the public and can be downloaded from http://www.ualberta.ca/˜bhuang/Stiction-Book/.

Readership The algorithms presented in this book are the state-of-the-art, and are demonstrated and compared on industrial case studies from different industrial fields (building, chemicals and petrochemicals, mining, mineral and metal processing). The book will thus serve academic and industrial staff working in all these industries on control systems design, maintenance or optimisation. The book presents – in one place – some material published in the archival literature over the last several years as well as advanced material developed most recently. It is aimed primarily at researchers and engineers in process control and industrial practitioners in process automation, but it is also accessible to postgraduate students and final-year students in process-control engineering.

Outline of the Book The book, which has 14 chapters, is divided into two parts. Chapter 1 is an introduction to the general topics and problems treated in the book. Part I is devoted to reviewing the state-of-the-art in stiction modelling and oscillation detection, and is divided into three chapters. Chapter 2 reviews stiction models with emphasis on simple data-driven relationships between controller output (OP) and valve position (VP), i.e. one- and two-parameter models, and gives examples that investigate the validity and compare the behaviour of three representative stiction models available in the literature. New alternative data-driven models of stiction are proposed in Chap. 3 and compared with earlier modelling approaches. Oscillation detection

Preface

xiii

is one of the first tasks to be carried out when analysing control-loop performance. The different techniques for oscillation detection are described and discussed in detail in Chap. 4. Their applicability to industrial control-loop diagnosis is critically evaluated on various simulation and industrial case studies. Part II of the book presents basic and advanced methodologies for stiction diagnosis and quantification, and contains ten chapters. In Chap. 5, pattern-based stiction-detection and quantification methods that assume the availability of the manipulated variable (MV), i.e. VP, are treated. In all subsequent chapters, the stiction detection is based only on the knowledge of OP and PV, which reflects industrial practice, where MV is usually not known, except for flow control loops, where PV and MV are considered to be coincident. Chapter 6 describes the simplest stictiondetection method based on cross-correlation between OP and PV. Since this method is only applicable to self-regulating processes, another method based on shape analysis (histogram) is also presented in this chapter for detecting stiction in integrating plants. Chapter 7 discusses a method for stiction detection based on curve fitting of the output signal of the first integrating component located after the valve in a control loop, i.e. OP for self-regulating processes or PV for integrating processes. Considering the similarity between the oscillation shapes of PV in loops affected by valve stiction and those obtained by a relay operating in closed loop on a first-order-plus-time-delay process, a technique based on shape determination is described in Chap. 8. Chapter 9 presents a method for detecting stiction-like behaviour in feedback control loops based on extracting features from a time series record of PV, delivering the probability that stiction is present (ranged 0–100 %). Chapter 10 discusses a new procedure for quantifying valve stiction in control loops using separable least-squares and global optimisation algorithms, assuming that the process dynamics can be described by a Hammerstein model. In a similar context, a novel stiction-detection strategy based on closed-loop identification is proposed using closed-loop operating data along with a general analysis on identifiability of the stiction loops in Chap. 11. Chapter 12 presents a novel methodology that bridges gaps in oscillation diagnostic methods. A power spectral density (PSD) based oscillation detection method followed by a model-based approach for identifying and quantifying the root cause of the oscillations is proposed. In all these chapters, the described methodologies are illustrated with simulated data and industrial data. In Chap. 13, an exhaustive comparative study of all methods presented is given, involving a large number of data sets gathered from different industrial plants, assessing the relative efficiency of the techniques, and delivering guidelines for selecting the right method for the application at hand. Chapter 14 provides a summary and a look forward to future research challenges within the topic treated in the book. In Appendix A, some information about the industrial control loops considered throughout the book is given. Appendix B contains a short overview of the two prominent methods for non-linearity detection in the CPM area, i.e. the bicoherence technique and the surrogate analysis method.

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Preface

Acknowledgements This book is an outcome of a project initiated by the principal editor to establish an international database of industrial loops from different fields, to implement a common graphical user interface (GUI) for oscillation and stiction detection, and to compare oscillation and stiction-detection methods. The editors would like to thank all the contributors for their very spontaneous enthusiasm and agreement to contribute to the book and particularly to the comparative study in Chap. 13. For this purpose, industrial data (Appendix A) were provided by courtesy of: Ashish Singhal and Timothy Salsbury for loops BAS 1– 8; Alexander Horch for loops CHEM 1–3 and PAP 1–10; Peter He and Joe Qin for loops CHEM 4–6; Biao Huang for loops CHEM 7–12; Nina Thornhill for loops CHEM 13–17 and CHEM 40–64; Claudio Scali for loops CHEM 18–28 and CHEM 32–39; Shoukat Choudhury and Sirish Shah for loops CHEM 29–31, PAP 11–13, POW 1–5 and MIN 1. As far as possible, we have tried to achieve a common framework for all chapters presented. The contributors were all asked to produce precise algorithms that can be easily implemented by the reader, to demonstrate their methods in simulation and industrial studies, and to use similar notation. Also, overlap in text and figures was minimised, and cross-references between the chapters were created. All this will hopefully help readers to easily understand and compare the techniques proposed. We are grateful to all contributors for their forbearance in meeting the many requests for clarity and consistency. Special thanks are due to Alexander Horch, Claudio Scali, Nina Thornhill and Shoukat Choudhury for numerous discussions and valuable suggestions, which have improved the presentation of many chapters of the book. We also acknowledge Alexander Horch, Claudio Scali, Shoukat Choudhury, Timothy Salsbury and Peter He for providing software (either as m- or p-code) to be included in the oscillation- and stiction-detection GUI implemented by the principal editor. Last but not least, we would like to acknowledge Oliver Jackson, Aislinn Bunning (Springer), Sorina Moosdorf and Katja Röser (le-tex publishing services) for their editorial comments and detailed examination of the book. We also thank Professor Michael A. Johnson for providing many valuable comments and suggestions. Düsseldorf, Germany Edmonton, Alberta, Canada May 2009

Mohieddine Jelali Biao Huang

Copyright Acknowledgements

Parts of the book appeared in the archived literature and the authors gratefully acknowledge permissions to re-use material from the following papers. Chapter 2 contains extracts and figures from Choudhury M.A.A.S., Shah S.L., Thornhill N.F., Shook D., Automatic detection and quantification of stiction in control valves, Control Engineering Practice 14:1395–1412, © 2006, Elsevier Ltd., with permission from Elsevier. Parts of Chap. 6 are reprinted (with revision) from Horch A., A simple method for detection of stiction in control loops, Control Engineering Practice 7:1221–1231, © 1999, Elsevier Ltd., with permission from Elsevier. Chapter 7 is reprinted in large part with permission from He Q.P., Wang J., Pottmann M., Qin S.J., A curve fitting method for detecting valve stiction in oscillating control loops, Ind. Eng. Chem. Res. 2007, 46(13), 4549–4560. Copyright 2007 American Chemical Society. Chapter 8 is reprinted (with revision) from Rossi M., Scali C., A comparison of techniques for automatic detection of stiction: simulation and application to industrial data, Journal of Process Control 15:505–514, © 2005, Elsevier Ltd., with permission from Elsevier. Chapter 9 is an expanded and revised version of Singhal A., Salsbury T.I., A simple method for detecting valve stiction in oscillating control loops, Journal of Process Control 15:371–382, © 2005, Elsevier Ltd., with permission from Elsevier. Chapter 10 is an expanded and revised version of Jelali M., Estimation of valve stiction in control loops using separable least-squares and global search algorithms, Journal of Process Control 18:632–642, © 2008, Elsevier Ltd., with permission from Elsevier.

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Contents

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv Abbreviations and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mohieddine Jelali and Biao Huang 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Typical Valve-controlled Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Stiction Phenomenon and Related Effects . . . . . . . . . . . . . . . . . . . . . 4 1.4 Input–Output Relation of Valves Under Stiction . . . . . . . . . . . . . . . . 6 1.5 Limit Cycles due to Stiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Typical Observations in Control Loops with Sticky Valves . . . . . . . 12 1.7 Industrial Examples of Loops with Stiction . . . . . . . . . . . . . . . . . . . . 15 1.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Part I Stiction Modelling and Oscillation Detection 2

Stiction Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.A.A. Shoukat Choudhury, Nina F. Thornhill, Manabu Kano and Sirish L. Shah 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Physics-based Stiction Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Data-driven Stiction Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 One-parameter Stiction Model . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Two-parameter Stiction Model . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Choudhury’s Stiction Model . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Simulation of the Stiction Model . . . . . . . . . . . . . . . . . . . . . 2.3.5 Kano’s Stiction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Comparison Between Choudhury’s and Kano’s Stiction Models . . 2.4.1 Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 22 24 24 24 25 28 29 32 32 32

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2.5 3

4

2.4.3 Comparisons Using an Industrial Example . . . . . . . . . . . . . 32 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

An Alternative Stiction-modelling Approach and Comparison of Different Stiction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Peter He, Jin Wang and S. Joe Qin 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 He’s Two-parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Three Data-driven Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Implementation of the First-principles Model . . . . . . . . . . 3.3.2 Comparison of Data-driven Models . . . . . . . . . . . . . . . . . . 3.4 Further Investigation of Valve Stiction . . . . . . . . . . . . . . . . . . . . . . . . 3.5 He’s Three-parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 An Industrial Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Appendix: Proof of the Equivalence Between He’s Two-parameter and Three-parameter Model . . . . . . . . . . . . . . . . . . . Detection of Oscillating Control Loops . . . . . . . . . . . . . . . . . . . . . . . . . . Srinivas Karra, Mohieddine Jelali, M. Nazmul Karim and Alexander Horch 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Root-causes for Oscillatory Control Loops . . . . . . . . . . . . . . . . . . . . 4.2.1 Poor Process and Control System Design . . . . . . . . . . . . . . 4.2.2 Aggressive Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Non-linearities in Control-loop Hardware . . . . . . . . . . . . . 4.2.4 External Oscillatory Disturbances . . . . . . . . . . . . . . . . . . . . 4.3 Characterisation of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Auto-covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Strength of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Techniques for Detection of Oscillations in Control Loops . . . . . . . 4.4.1 Detection of Spectral Peaks . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Regularity of Large Enough Integral of Absolute Error . . 4.4.3 Regularity of Upper and Lower IAEs and Zero-crossings 4.4.4 Decay-ratio Approach of the Auto-correlation Function . . 4.4.5 Regularity of Zero-crossings of the Auto-correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Spectral Envelope Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Critical Evaluation of Oscillation-detection Methods . . . . . . . . . . . . 4.5.1 Features of Industrial Control-loop-oscillation Detection . 4.5.2 Detection Test Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Signal with Coloured Noise . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Signal with One Predominant Oscillation . . . . . . . . . . . . . . 4.5.5 Signal with Dampened Oscillation . . . . . . . . . . . . . . . . . . .

37 37 38 40 41 43 45 49 51 56 57 58 61 62 62 62 63 63 64 64 64 65 65 67 67 69 72 74 76 78 80 80 81 83 84 85

Contents

4.6 4.7

4.8

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4.5.6 Signal with Multiple Oscillations . . . . . . . . . . . . . . . . . . . . 87 4.5.7 Signal with Intermittent Oscillations . . . . . . . . . . . . . . . . . . 90 Comprehensive Oscillation Characterisation . . . . . . . . . . . . . . . . . . . 93 Industrial Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.7.1 Oscillating Flow Control Loop . . . . . . . . . . . . . . . . . . . . . . 95 4.7.2 Unit-wide Oscillation Caused by a Sensor Fault . . . . . . . . 96 4.7.3 Plant-wide Oscillation Caused by a Valve Fault . . . . . . . . 97 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Part II Advances in Stiction Detection and Quantification 5

Shape-based Stiction Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Manabu Kano, Yoshiyuki Yamashita and Hidekazu Kugemoto 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2.1 Method A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2.2 Method B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2.3 Method C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Key Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5 Application to Industrial Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6

Stiction Detection Based on Cross-correlation and Signal Shape . . . . 115 Alexander Horch 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 The Cross-correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3 Industrial Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.1 Loop Interaction I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.2 Loop Interaction II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.3.3 Flow Control Loop I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3.4 Flow Control Loop II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3.5 Level Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4 Theoretical Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.1 Correlation for Oscillating External Disturbances . . . . . . . 127 6.4.2 Tight Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.4.3 Correlation in the Presence of Stiction . . . . . . . . . . . . . . . . 129 6.5 Conclusions (Cross-correlation Method) . . . . . . . . . . . . . . . . . . . . . . 131 6.6 Stiction Detection for Integrating Processes . . . . . . . . . . . . . . . . . . . 132 6.7 Detection in Integrating Loops – Basic Idea . . . . . . . . . . . . . . . . . . . 132 6.7.1 Differentiation and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.7.2 Sample Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.7.3 Distribution for the Stiction Case . . . . . . . . . . . . . . . . . . . . 136 6.7.4 Distribution for the Non-stiction Case . . . . . . . . . . . . . . . . 138 6.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.8.1 Level Control Loop with Stiction . . . . . . . . . . . . . . . . . . . . 142

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6.9

6.10

6.8.2 Level Control Loop Without Stiction . . . . . . . . . . . . . . . . . 143 6.8.3 Level Control Loop with Deadband . . . . . . . . . . . . . . . . . . 143 Self-regulating Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.9.1 Flow Control Loop with Stiction . . . . . . . . . . . . . . . . . . . . . 144 6.9.2 Flow Control Loop Without Stiction . . . . . . . . . . . . . . . . . . 145 6.9.3 Loops with Dominant P-control . . . . . . . . . . . . . . . . . . . . . 145 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7

Curve Fitting for Detecting Valve Stiction . . . . . . . . . . . . . . . . . . . . . . . 149 Q. Peter He and S. Joe Qin 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2 Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.2.1 Sinusoidal Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.2.2 Triangular Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2.3 Stiction Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.3 Key Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.5 Application to Industrial Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8

A Relay-based Technique for Detection of Stiction . . . . . . . . . . . . . . . . 165 Claudio Scali and Maurizio Rossi 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.2 Trends of Different Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.3 Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.3.2 Stiction Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.3.3 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.3.4 Fitting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.4.1 Nominal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.4.2 Presence of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.5 Application to Plant Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9

Shape-based Stiction Detection Using Area Calculations . . . . . . . . . . . 183 Timothy I. Salsbury and Ashish Singhal 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.2 Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.2.1 Theoretical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2.2 Stiction Detection Hypothesis Test . . . . . . . . . . . . . . . . . . . 190 9.2.3 Noise Effects and Practical Implementation . . . . . . . . . . . . 191 9.3 Key Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.5 Application to Industrial Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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9.5.1

9.6

Temperature Control Loop with Stiction from a Building Automation System . . . . . . . . . . . . . . . . . . 201 9.5.2 Temperature Control Loop with Stiction from a Pulp and Paper Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

10

Estimation of Valve Stiction Using Separable Least-squares and Global Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Mohieddine Jelali 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.2 Basic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.2.1 Identification Model Structure: Hammerstein Model . . . . 208 10.2.2 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.2.3 Stiction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 10.3 Identification Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 10.3.1 Separable Least-squares Estimator . . . . . . . . . . . . . . . . . . . 210 10.3.2 Global Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 213 10.4 Key Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.4.1 Model Structure Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.4.2 Time-delay Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.4.3 Determination of Initial Parameters and Incorporation of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 10.5 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 10.5.1 First-order-plus-time-delay Process . . . . . . . . . . . . . . . . . . 220 10.5.2 Integrating Process with Time Delay . . . . . . . . . . . . . . . . . 220 10.6 Industrial Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.6.1 Loop CHEM 25: Pressure Control Loop . . . . . . . . . . . . . . 221 10.6.2 Loop PAP 2: Flow Control Loop . . . . . . . . . . . . . . . . . . . . . 224 10.6.3 Loop CHEM 24: Flow Control Loop with Setpoint Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.6.4 Loop POW 2: Level Control Loop . . . . . . . . . . . . . . . . . . . 225 10.6.5 Loop POW 4: Level Control Loop . . . . . . . . . . . . . . . . . . . 226 10.6.6 Loop MIN 1: Temperature Control Loop . . . . . . . . . . . . . . 226 10.6.7 Loop CHEM 70: Flow Control Loop with External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 10.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

11

Stiction Estimation Using Constrained Optimisation and Contour Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Kwan Ho Lee, Zhengyun Ren and Biao Huang 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 11.2 Stiction Model of Control Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.2.1 General Conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.2.2 Physical Model of Valve Sticion . . . . . . . . . . . . . . . . . . . . . 231 11.2.3 Kano’s Valve-stiction Model . . . . . . . . . . . . . . . . . . . . . . . . 232 11.2.4 Choudhury’s Valve-stiction Model . . . . . . . . . . . . . . . . . . . 232

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11.3

11.4

11.5

11.6 11.7 11.8 11.9 12

11.2.5 He’s Valve-stiction Model . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Existing Stiction-detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.3.1 Open-loop Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.3.2 Closed-loop Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.3.3 Discussion of Existing Methods . . . . . . . . . . . . . . . . . . . . . 234 Closed-loop Stiction Detection and Quantification . . . . . . . . . . . . . . 235 11.4.1 Basic Principle and Important Steps . . . . . . . . . . . . . . . . . . 235 11.4.2 Stiction Detection and Quantification Procedure . . . . . . . . 236 11.4.3 Search Space of Stiction-model Parameters . . . . . . . . . . . . 237 11.4.4 Constrained Parameter-search Techniques . . . . . . . . . . . . . 238 11.4.5 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Stiction Detection: Identifiability Analysis . . . . . . . . . . . . . . . . . . . . 241 11.5.1 Heuristic Illustration of Closed-loop Identifiability . . . . . . 241 11.5.2 Identifiability Analysis for Closed-loop Systems with Valve Stiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 11.7.1 Illustrative Industrial Examples . . . . . . . . . . . . . . . . . . . . . . 251 11.7.2 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Oscillation Root-cause Detection and Quantification Under Multiple Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Srinivas Karra and M. Nazmul Karim 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 12.2 Preliminaries and Brief Review of Model-based Oscillation Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 12.2.1 Root-cause for Oscillations and Compensation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 12.2.2 Oscillation Diagnosis and Root-cause Quantification . . . . 269 12.2.3 Challenges to be Addressed . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.3 Overview of the Root-cause Detection and Quantification Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.3.1 Revisiting Control-valve Characteristics Under Stiction . . 270 12.3.2 Oscillation Detection and Diagnosis Methodology . . . . . . 271 12.4 Process-model Identification Under Non-stationary Disturbances . 271 12.4.1 Identification of EARMAX Model . . . . . . . . . . . . . . . . . . . 273 12.4.2 Illustrative Example: Identification Under Non-stationary Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . 275 12.5 Root-cause Detection and Quantification . . . . . . . . . . . . . . . . . . . . . . 278 12.5.1 OP–PV Model Identification Methodology . . . . . . . . . . . . 278 12.5.2 Identification of Controller Transfer Function . . . . . . . . . . 280 12.5.3 Oscillation Root-cause Detection and Quantification Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Contents

12.6

12.7

12.8

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Illustrative Example: Oscillation Diagnosis Under Various Faulty Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 12.6.1 Stiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 12.6.2 Oscillatory External Disturbance . . . . . . . . . . . . . . . . . . . . . 283 12.6.3 Aggressive Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . 284 12.6.4 Stiction and Oscillatory External Disturbance . . . . . . . . . . 285 12.6.5 Stiction and Aggressive Controller Tuning . . . . . . . . . . . . . 286 12.6.6 Oscillatory External Disturbance and Aggressive Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 12.6.7 Stiction, Aggressive Controller Tuning and Oscillatory External Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Industrial Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 12.7.1 Control Loop 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 12.7.2 Control Loop 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 12.7.3 Control Loop 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

13

Comparative Study of Valve-stiction-detection Methods . . . . . . . . . . . 295 Mohieddine Jelali and Claudio Scali 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 13.2 Selected Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 13.3 Industrial Control Loops Involved in the Study . . . . . . . . . . . . . . . . . 298 13.4 Application Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 302 13.4.1 Application Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 13.4.2 Synthesis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 13.4.3 Efficiency of the Techniques, Problems and Countermeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 13.4.4 Comparison on 20 Loops with Known Problems . . . . . . . 317 13.4.5 Selected Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 13.4.6 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 13.5 Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.7 Appendix: Tables of Results of the Comparative Study . . . . . . . . . . 324

14

Conclusions and Future Research Challenges . . . . . . . . . . . . . . . . . . . . 359 Biao Huang, Mohieddine Jelali and Alexander Horch 14.1 Summary of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 14.2 Future Research Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 14.2.1 Stiction Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 14.2.2 Oscillation Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 14.2.3 Stiction Detection and Estimation . . . . . . . . . . . . . . . . . . . . 364 14.2.4 Stiction Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

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Appendix A Evaluated Industrial Control Loops . . . . . . . . . . . . . . . . . . . . . 367 Appendix B Review of Some Non-linearity and Stiction-detection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 B.1 Bicoherence Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 B.1.1 Non-Gaussianity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 B.1.2 Non-linearity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 B.1.3 Total Non-linearity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 B.1.4 Ellipse Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 B.2 Surrogates Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 B.2.1 Surrogate Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 375 B.2.2 Non-linear Predictability Index . . . . . . . . . . . . . . . . . . . . . . 376 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Contributor Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

List of Contributors

M.A.A. Shoukat Choudhury Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh, e-mail: [email protected] Q. Peter He Department of Chemical Engineering, Tuskegee University, 522 B Luther H. Foster Hall, AL 36088, USA, e-mail: [email protected] Alexander Horch Group Process and Production Optimization, ABB Corporate Research Germany, Wallstadter Str. 59, 68526 Ladenburg, Germany, e-mail: [email protected] Biao Huang Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected] Mohieddine Jelali Department of Plant and System Technology, VDEh-Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected] Manabu Kano Department of Chemical Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan, e-mail: [email protected] M. Nazmul Karim Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected]

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xxvi


Srinivas Karra Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected] Hidekazu Kugemoto Sumitomo Chemical Co., Ltd., 5-l, Sobiraki-cho, Niihama City, Ehime 792-8521, Japan, e-mail: [email protected] Kwan Ho Lee Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected] S. Joe Qin Department of Chemical Engineering and Materials Science, Electrical Engineering, and Industrial and Systems Engineering, University of Southern California, 925 Bloom Walk, HED 211, Los Angeles, CA 90089-1211, USA, e-mail: [email protected] Zhengyun Ren Department of Automation, Donghua University, Shanghai, China, e-mail: [email protected] Maurizio Rossi AspenTech - Srl, Pisa, Italy, e-mail: [email protected] Timothy I. Salsbury Controls Research Department, Johnson Controls, Inc, 507 E Michigan Street, Milwaukee, WI 53202, USA, e-mail: [email protected] Claudio Scali Chemical Process Control Laboratory (CPCLab), Department of Chemical Engineering (DICCISM), University of Pisa, Via Diotisalvi, n.2 56126 Pisa, Italy, e-mail: [email protected] Sirish L. Shah Department of Chemical and Materials Engineering, University of Alberta Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected] Ashish Singhal Advanced Control and Operations Research R&D, Praxair, Inc., 175 East Park Dr., Tonawanda, NY 14150, USA, e-mail: [email protected] Nina F. Thornhill Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK, e-mail: [email protected] Jin Wang Department of Chemical Engineering, Auburn University, AL 36849, USA, e-mail: [email protected]


Yoshiyuki Yamashita Department of Chemical Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan, e-mail: [email protected]

xxvii

Abbreviations and Acronyms

AC ACF AMIGO ANSI ARMAX ARX BAS BJ C CC CCF CHEM CHR COC CPM CPU DCS DR EARMAX EWMA FC FOPTD GA GC GUI IAE IPTD LTI MC MET

Analyser control Auto-correlation/covariance function Approximate Ms constrained integral gain optimisation American National Standard Institution Auto-regressive moving average exogenous Auto-regressive exogenous Building automation system Box–Jenkins Controller Cohen–Coon Cross-correlation function Chemicals Chien–Hrones–Reswick Comprehensive oscillation characterisation Control-performance monitoring Central processing unit Distributed control system Decay ratio Extended ARMAX Exponentially weighted moving average Flow control First-order-plus-time-delay Genetic algorithm Gauge control Graphical user interface Integral of absolute error Integrating plus time delay Linear time invariant Monte-Carlo Metal

xxix

xxx

MSE MV NGI NLA NLI NPI ODE OE OP OS P PAP PC P PI PID POW PSD PV QF SI SISO SOPTD SP TDE TF V VP VOC ZN ZOH

Abbreviations and Acronyms

Mean square errors Manipulated variable Non-Gaussianity index Non-linearity analysis Non-linearity index Non-predictability index Ordinary differential equation Output error Controller output Overshoot Process Pulp and paper Pressure control Proportional Proportional-integral Proportional-integral-derivative Power Power spectral density Process variable Quantisation factor Stiction index Single-input single-output Second-order-plus-time-delay Setpoint Time-delay estimation Transfer function Valve Valve position Volatile organic compound Ziegler–Nichols Zero-order hold

Chapter 1

Introduction Mohieddine Jelali and Biao Huang

This introductory chapter explains the role of control valves in the process industry and the motivation for the detection of valve stiction in control loops. It reviews some definitions of the term “stiction” and similar phenomena (deadband, backlash, hysteresis, etc.) that can occur in control valves. Some basic analysis tools will be described, including the input–output relation of valves under stiction and the describing function analysis. Data from typical industrial control loops are presented to illustrate the effects of stiction on the shape of loop signals, such as the process variable and the controller output.

1.1 Motivation Large-scale, highly integrated processing plants, such as oil refineries, ethylene plants, power plants, and rolling mills, include some hundreds or even thousands of control loops. The aim of each control loop is to maintain the process at the desired operating conditions, safely and efficiently. A poorly performing control loop can result in disrupted process operation, degraded product quality, higher material or energy consumption, and thus decreased plant profitability. Therefore, control loops have been increasingly recognised as important capital assets that should be routinely monitored and maintained. The performance of the controllers, as well as of the other loop components, can thus be improved continuously, ensuring products of consistently high quality.

Mohieddine Jelali Department of Plant and System Technology, VDEh-Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected] Biao Huang Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected]

1

2

M. Jelali and B. Huang

Surveys [10, 25, 26, 92] indicate that about 20–30% of all control loops oscillate due to valve problems caused by valve non-linearities, such as stiction, hysteresis, deadband or deadzone. Many control loops in process plants perform poorly due to valve static friction (stiction), as one of the most common equipment problems. It is well known that valve stiction in control loops causes oscillations in the form of periodic finite-amplitude instabilities, known as limit cycles. This phenomenon increases variability in product quality, accelerates equipment wear, or leads to control-system instability.

1.2 Typical Valve-controlled Loop Figure 1.1 shows a simple configuration of control loops actuated with a control valve. A typical example of such a configuration, i.e. a level control loop, is illustrated in Fig. 1.2. In most applications in the process industry, pneumatic control valves are used. The diagram of a typical pneumatic valve is shown in Fig. 1.3. The valve aims to restrict the flow of process fluid through the pipe that can be seen at the very bottom of the figure. The valve plug is rigidly attached to a stem that is attached to a diaphragm in an air-pressure chamber in the actuator section at the top of the valve. When compressed air is applied, the diaphragm moves up and the valve opens. At the same time, the spring is compressed. In process operation, a control valve is subject to the following forces: (i) the valve stem driving force caused by the air pressure, (ii) the spring force associated with the valve travel, (iii) the seal friction of the seals sealing the process fluid and the stem thrust originating in the process fluid passing through the valve body.

Fig. 1.1 Simple feedback scheme for a valve-controlled process with definition of the variables setpoint (SP), controller output (OP), manipulating variable or valve position (MV) and process variable (PV) used throughout the book

Stiction in control valves is thought to occur due to seal degradation, lubricant depletion, inclusion of foreign matter, activation at metal sliding surfaces at high temperatures and/or tight packing around the stem. The resistance offered from the stem packing is often considered as the main cause of stiction. Another very common cause of stiction is indirectly due to regulations on volatile organic compound (VOC) emissions. In many plants, a team monitors each valve for VOC emissions, usually between the packing and the stem. If any minute leakage is detected, packing in the valve body is tightened far more than is necessary. This causes the valve

1 Introduction

3

to stick making the process run less efficiently with increased energy consumption. Stiction often varies over time and operating regimes. Since wear is also nonuniform along the body, frictional forces are different at different stem positions. When the control loop is at steady state, and if a valve exhibits this behaviour, persistent oscillations in PV on either side of the setpoint are observed [115]. u Control valve LC p1 Q

r

h

Q d

Fig. 1.2 Level control loop

Compressed air

Diaphragm

pd

Valve drive

Spring Stiction Valve stroke

Packing

Valve seat Q Exit flow

Entry flow

Valve armature

Plug

Fig. 1.3 Diagram of a pneumatic control valve [74]

Control valves should be maintained to have acceptable values for the parameters given in Table 1.1. In many processes, stiction of 0.5% is considered too much, as

4


stiction can introduce cycling and variability, and is thus more harmful than other valve problems. For instance, hysteresis is also undesirable, but usually not really a problem up to 5% [100]. As stiction is the most severe problem, it is important to detect it early on so that appropriate action can be taken and major disruptions to the operation can be avoided. Owing to the large number of loops in an industrial plant, this analysis should be performed automatically, limiting the possibility of false alerts and performing quantitative evaluation of performance loss. This monograph presents state-of-the-art and advanced methods for automatically detecting and estimating stiction in industrial control systems. The aim of this chapter is to give a short introduction to the subject of the book, including some basic definitions, typical observations in control loops with valve stiction, and industrial examples to illustrate them. Also, a general procedure for detecting and diagnosing stiction is described, in which the topics of the subsequent chapters are embedded. Table 1.1 Ideal values and acceptable ranges of valve parameters given as percentage of the valve travel range [100] Parameter Process gain Noise band Hysteresis Stiction

Ideal 1 0 0 0

Practical < 0.5: too small; 0.5–3.0: acceptable; > 3.0: too high < 0.5%: acceptable < 3%: acceptable ; > 3%: to be checked. 1%: desirable ; > 1%: to be checked.

1.3 Stiction Phenomenon and Related Effects There are some terms such as deadband, backlash and hysteresis, which are often misused and wrongly used in describing valve problems. For example, quite commonly a deadband in a valve is referred to as backlash or hysteresis. The following items review some definitions of terms related to stiction, to differentiate clearly between the key concepts that underline the ensuing discussion of friction in control valves. The sequence of this section is adopted from [15]. These definitions can also be found in [27, 29], which also make reference to [55]: • Backlash. “A relative movement between interacting mechanical parts, resulting from looseness, when the motion is reversed.” • Hysteresis. “That property of the element evidenced by the dependence of the value of the output, for a given excursion of the input, upon the history of prior excursions and the direction of the current traverse.” – “It is usually determined by subtracting the value of deadband from the maximum measured separation between upscale-going and downscale-going indications of the measured variable (during a full-range traverse, unless otherwise specified) after transients have decayed.”; Figs. 1.4a and c illustrate the concept.

1 Introduction

5

– “Some reversal of output may be expected for any small reversal of input. This distinguishes hysteresis from deadband.” • Deadband. “The range through which an input signal may be varied, upon reversal of direction, without initiating an observable change in output signal.” – “There are separate and distinct input–output relationships for increasing and decreasing signals.”; see Fig. 1.4b. – “Deadband produces phase lag between input and output.” – “Deadband is usually expressed in percentage of span.” • Deadzone. “A predetermined range of input through which the output remains unchanged, irrespective of the direction of change of the input signal.” – “There is but one input–output relationship”; see Fig. 1.4d. – “Deadzone produces no phase lag between input and output.” The aforementioned definitions show that the term backlash specifically applies to the slack or looseness of the mechanical part when the motion changes its direction. Therefore, in control valves it may only add deadband effects if there is some slack in rack-and-pinion-type actuators [29] or loose connections in a rotary valve shaft. ANSI [55] definitions and Fig. 1.4 show that hysteresis and deadband are distinct effects. Deadband is quantified in terms of input signal span (i.e. on the x-axis), while hysteresis refers to a separation in the measured (output) response (i.e. on the y-axis). Hysteresis + deadband Output

Hysteresis

a

b

b Deadband

Input

Input (a)

d deadzone

(b)

Input

Input (c)

(d)

Fig. 1.4 Definition of valve non-linearities: a) hysteresis; b) deadband; c) hysteresis + deadband; d) deadzone (redrawn from ANSI/ISA-S51.1-1979)

Also for the term stiction, there exist numerous definitions in the literature. Some of them are given below: • “Stiction is the resistance to the start of motion, usually measured as the difference between the driving values required to overcome static friction upscale and downscale.” [55] • “Stiction is a tendency to stick-slip due to high static friction. The phenomenon causes a limited resolution of the resulting control valve motion. ISA terminology

6

•

•

• •


has not settled on a suitable term yet. Stick-slip is the tendency of a control valve to stick while at rest, and to suddenly slip after force has been applied.” [27] “The control valve is stuck in a certain position due to high static friction. The (integrating) controller then increases the set point to the valve until the static friction can be overcome. Then the valve breaks off and moves to a new position (slip phase) where it sticks again. The new position is usually on the other side of the desired setpoint such that the process starts in the opposite direction again.” [48] “Stiction as a combination of the words stick and friction, created to emphasise the difference between static and dynamic friction. Stiction exists when the static (starting) friction exceeds the dynamic (moving) friction inside the valve. Stiction describes the valves stem (or shaft) sticking when small changes are attempted. Friction of a moving object is less than when it is stationary. Stiction can keep the stem from moving for small control input changes, and then the stem moves when there is enough force to free it. The result of stiction is that the force required to get the stem to move is more than is required to go to the desired stem position. In the presence of stiction, the movement is jumpy.” [100] “Short for static friction as opposed to dynamic friction. It describes the friction force at rest. Static friction counteracts external forces below a certain level and thus keeps an object from moving.” [89] “Stiction is a property of an element such that its smooth movement in response to a varying input is preceded by a sudden abrupt jump called the slip jump. Slip jump is expressed as a percentage of the output span. Its origin in a mechanical system is static friction that exceeds the friction during smooth movement.” [15]

Perhaps the most formal and general definition of stiction and its causing mechanism is that proposed by Choudhury et al. [15]: “The presence of stiction impairs proper valve movement, i.e. the valve stem may not move in response to the output signal from the controller or the valve positioner. The smooth movement of the valve in response to a varying input from the controller or the valve positioner is preceded by a stickband and an abrupt jump termed the slip jump. Its origin in a mechanical system is static friction, which exceeds the dynamic friction during smooth movement.”

1.4 Input–Output Relation of Valves Under Stiction Choudhury’s definition of stiction is based on and can be best explained by the input–output behaviour of a sticky valve illustrated in Fig. 1.5. Without stiction, the valve would move along the dash-dotted line crossing the origin: any amount of OP adjustment would result in the same amount of manipulated variable (MV) change. However, for a sticky valve, static and kinetic/dynamic friction components have to be taken into account. The input–output behaviour then consists of four components

1 Introduction

7

deadband, stickband, slip jump and the moving phase, and is characterised by the three phases [99]: 1. Sticking. MV is constant with the time, as the valve is stuck by the presence of the static friction force Fs (deadband plus stickband). Valve deadband is due to the presence of Coulomb friction Fc , a constant friction that acts in the opposite direction to the velocity. 2. Jump. MV changes abruptly, as the active force Fa unblocks the valve. 3. Motion. MV varies gradually; Fa is opposed only by the dynamic friction force Fd .

Actual valve position / Manipulated variable (MV)

Stick + deadband = S Deadband H I J

K

F

J

J

D

L

J

C M

A

B

fd l1

fs Aƍ l2

ov M

g in

as ph

e

G

J E

Slip jump J = stickband Stickband

Bƍ Dƍ Required valve position / Controller output (OP)

Fig. 1.5 Relation between controller output and valve position under valve stiction [15, 63]

Choudhury et al. [15] proposed to use the quantities S (deadband plus stickband) and J (slip jump) for characterising valve stiction, as shown in Fig. 1.5. Because OP and MV are generally given as a percentage of the valve travel range for simplicity, all other variables, i.e. S, J, u (OP), y (PV), and uv (MV), are translated into a percentage of the valve range so that algebra can be performed among them directly. The parameters fs (normalised Coulomb friction) and fd (normalised Coulomb/kinetic/dynamic friction) can be related to S and J by S = fs + fd , or equivalently

J = fs − fd ,

(1.1)

8


S+J S−J , fd = . (1.2) 2 2 See also Sect. 3.5. To illustrate how OP adjustments drive MV change in a sticky valve with stiction, suppose the valve rests at a neutral position A at the beginning in Fig. 1.5. If the OP adjustment is between A B , the valve will not be able to overcome the static friction band so the MV will not change. However, if the OP moves outside of A B , say D , then the valve is able to overcome the static friction band at point B and jumps to point C. After that, the valve moves from C to D, overcoming the kinetic friction band only. Due to very low or zero velocity, the valve may stick again in between points C and G in Fig. 1.5 while travelling in the same direction [27]. In such a case, the magnitude of deadband is zero and only stickband (DE/EF) is present. This can be overcome if the change of OP is larger than the stickband only. It is uncommon to see features such as the sticking between D to E in industrial data because the valve generally keeps moving while the input is changing. The valve movement stops only when the input signal to the valve stops changing or reverses its direction. The deadband and stickband represent the behaviour of the valve when it is not moving, though the input to the valve keeps changing. Slip jump represents the abrupt release of potential energy stored in the actuator chambers due to high static friction in the form of kinetic energy as the valve starts to move. The magnitude of the slip jump is very crucial in determining the limit cyclic behaviour introduced by stiction [83, 93]. Once the valve slips, it continues to move until it sticks again (point G in Fig. 1.5). In this moving phase, dynamic friction is present which may be much lower than the static friction. The sequence motion/stop of the valve due to stiction is called stick-slip motion [15]. Many models have been proposed in the literature to describe the presence of friction in the actuators. Surveys are reported in [1, 89]. Different static friction models are shown in Fig. 1.6. One standard method is to model friction as a function of velocity, which is referred to as the Stribeck friction curve (after Stribeck [120]); see Fig. 1.6d. For more details, see [59]. In particular, the modelling of static friction is treated in [66]. Even though the number of parameters is not large, compared with other models, the lack of knowledge on values of critical variables is a major problem to describe the situation in a real plant. This problem is simplified with alternative approaches presented recently by Kano et al. [63], Choudhury et al. [15] and He et al. [44]. Data-driven models are adopted to describe the relationship MV = f (OP) illustrated in Fig. 1.5. In particular, only the two parameters S and J are used. Depending on these parameters various characteristic curves of the control valve result. These models will be discussed in Chaps. 2 and 3. fs =

Ff

Ff

Ff

9 Ff

1 Introduction

v

v

(a)

v

v

(b)

(c)

(d)

Fig. 1.6 Examples of static friction models: a) Coulomb friction; b) Coulomb plus viscous friction; c) stiction plus Coulomb and viscous friction; d) Stribeck friction (friction force decreasing continuously from the static friction level) [89]

1.5 Limit Cycles due to Stiction Consider a PI-controlled loop shown in Fig. 1.7. The classical method to investigate if non-linearities lead to regular oscillations, i.e. limit cycles, is the describing function analysis. The describing function of a non-linearity is defined as N(A) =

Y , X

(1.3)

where X is a harmonic input to the non-linearity of angular frequency ω and Y is the fundamental Fourier component of the output of the non-linearity. The quantity N depends upon the magnitude of the input A.

SP

PI Controller

−

OP

Stiction

MV

Process

PV

Fig. 1.7 Control loop with actuator stiction non-linearity

Based on a data-driven two-parameter stiction model (Sect. 2.3.3), Choudhury et al. [15] derived the following expression of the describing function for the stiction non-linearity: N(A) = −

1 (Preal − jPim ), πA

where Preal =

π A sin 2ϕ − 2A cos ϕ − A + ϕ + 2(S − J) cos ϕ , 2 2

A A Pim = −3 + cos 2ϕ + 2A sin ϕ − 2(S − J) sin ϕ , 2 2

(1.4)

10


ϕ = sin−1

A−S . A

(1.5)

The describing function is a complex function, parametrised by A, the amplitude of the input sinusoid.

1

G(jω )

0

−1/N(A)

J=0

Imaginary part

−1

J = S/6 −2

ω −3

J = S/4

−4

A −5 −3

J = S/2

J=S −2.5

−2

−1.5

−1

−0.5

0

0.5

1

Real part

(a) 1

0

−1/N(A) J=0

Imaginary part

−1

G(jω )

ω

J = S/6

−2

−3

J = S/4

−4

A −5 −3

J = S/2

J=S −2.5

−2

−1.5

−1

−0.5

0

0.5

1

Real part

(b) Fig. 1.8 Nyquist plots for closed loops with PI-controllers: a) self-regulating process; b) integrating process

1 Introduction

11

Similar to the study in [15], we investigate the Nyquist plots in Fig. 1.8 for a selfregulating process and an integrating process with PI-controllers, given in Table 1.2. The open-loop frequency response function G(jω ) of the controller and process is parametrised by the frequency ω . Both systems are closed-loop stable and thus intersect the negative real axis between 0 and 1. In Fig. 1.8a, one can see that a limit cycle will occur for the self-regulating control loop if there is a slip jump (J > 0). J forces the −1/N curve onto the negative imaginary axis in the A = S/2 limit. Thus, the frequency response curve of the self-regulating loop and its PI-controller is guaranteed to intersect with the describing function because the integral action means the open-loop phase is always below −π/2, i.e. it is in the third quadrant of the complex plane at low frequency [15]. The −1/N curve for the pure deadband case, i.e. J = 0 is also shown in the figure. In the A = S/2 limit, the curve becomes large, negative and imaginary. Since the frequency response curve does not intersect the −1/N curve, a limit cycle does not occur for the case the non-linearity is a pure deadband; consult also [83, 93]. Table 1.2 Parameters of the simulated control loops Loop

Process Controller

S

Self-regulating

5e−12s 30s+1 1 1 s 0.2s+1

6

Integrating

0.1(1 + 3(1 +

1 20s )

1 30s )

6

The phase of the frequency response G(jω ) for the integrating loop with PIcontroller converges to −π at low frequency. Figure 1.8b shows that G(jω ) will always intersect the −1/N curves irrespective of the values of J and S. Consequently, a limit-cycle oscillation can appear for an integrating process with a PI-controller if a valve with a deadband and no slip jump is used. The frequency of oscillation is higher and the period of oscillation shorter when the slip jump is present because the −1/N curves with the slip jump intersect the frequency response curve at higher frequencies than the −1/N curve for the deadband case. In summary, the occurrence of limit cycles depends on the type of the process and controller and of the presence of deadband and stick-slip; see Table 1.3. From this, one can conclude [21]: • Deadband only cannot produce limit cycles in self-regulating processes. • A short-term solution that may solve the stiction problem is to change the controller to proportional(P)-only. In a self-regulating process, the limit cycle should then disappear. This is not the case for integrating processes, but the amplitude of the limit cycle will probably decrease.

12


Table 1.3 Occurrence of limit cycles in control loops [21] Process Self-regulating Self-regulating Integrating Integrating

Controller P-only PI P-only PI

Deadband (J = 0, S = 0) No limit cycles No limit cycles No limit cycles Limit cycles

Stick-slip (J = 0, S = 0) No limit cycles Limit cycles Limit cycles Limit cycles

1.6 Typical Observations in Control Loops with Sticky Valves In general, oscillations can be caused by any one or a combination of the following reasons: (i) sensor or actuator faults, e.g. control valve stiction, (ii) poor control system design, e.g. aggressive controller tuning, (iii) poor process design, and (iv) external oscillatory disturbances. The basis for most detection techniques is the qualitative illustration of the phenomenon of stiction, and how closed-loop variables change with stiction and process parameters. Due to the integral action of the controller, stiction and other nonlinearities leave distinct shapes in PV and OP data. The oscillating shapes can be generally categorised as square, triangular, saw-tooth or sinusoidal. Figure 1.9 shows a sample industrial data of a flow control with a sticky valve. OP shows a perfectly triangular wave, whereas the flow (PV) resembles a square wave. If the process lag is high, PV is generally close to a sinusoidal pattern. When the stiction is higher, the oscillation amplitude of OP and PV increases, causing the patterns in OP and PV to be sharper and more pronounced [32, 115]. Table 1.4 categorises the observed stiction patterns of OP and PV for slow and fast dynamic processes. When oscillations are not caused by stiction, the shapes of OP and PV are more sinusoidal in nature. However, it should be emphasised that the shape library in Table 1.4 is not complete by itself since the shapes depend on controller parameters and process dynamics. To summarise, the effect of stiction can be distinguished from other oscillation root causes by the following observations: • In general, non-linearity-induced oscillations, which are observed both on OP and PV, contain harmonics. • In the case of poor performance or external disturbance, both OP and PV follow sinusoidal waves; the induced oscillations have low harmonic content. • In the case of stiction, OP usually follows a triangular wave for self-regulating processes (Fig. 1.9); for integrating processes, PV shows a triangular wave (Fig. 1.10) as the “ideal” stiction behaviour. However, these patterns can be modified depending on process dynamics and controller parameters (Table 1.4). Triangular (symmetric) waveforms only contain odd harmonics. Triangular waves that are observed in OP are usually asymmetric and contain both even and odd harmonics.

1 Introduction

13

0.8

10 Power

PV

10.5

9.5 50 OP

40

0.6 0.4 0.2

30 20 10

0

900

1000

1100 1200 Samples

1300

1400

1500

−2

10 Norm. frequency (f/f )

10

−1

s

(a)

(b)

Fig. 1.9 Typical stiction patterns in a flow control loop (CHEM 23): a) PV and OP trends; b) power spectrum of PV

42 PV

40

0.8

38 Power

36 34

0.6 0.4

OP

28 0.2

26 24 200

0

300

400

500 Samples

600

700

800

−2

10 Norm. frequency (f/f )

(a)

10

−1

s

(b)

Fig. 1.10 Typical stiction patterns in a level control loop (PAP3): a) PV and OP trends; b) power spectrum of PV Table 1.4 Typical stiction-pattern shapes for different process types [32, 115] Process type

Fast processes (flow)

OP

Dominant I action Triangular (Sharp) Square

PV

Slow processes Integrating processes Dominant (pressure & (Level) P action temperature) Rectangular Triangular Triangular (Smooth) (Sharp) Rectangular Sinusoidal Triangular (Sharp)

Level with PI control Triangular (Sharp) Parabolic

14


The reason for this behaviour is that while the plant input is continuous for aggressive control (except when the controller output is saturated), valve stiction results in a discontinuous plant input that closely resembles a rectangular pulse signal. Note that flow loops, e.g. steam flow loops, can be integrating. The same applies for pressure loops: gas pressure is self-regulating when the vessel (or pipeline) admits more feed when the pressure is low, and reduces the intake when the pressure becomes high. Integrating processes occur, for example, when there is a pump for the exit stream [107]. Indeed, the aforementioned observations, i.e. signal shapes, are used as the basis for some stiction-detection methods, as described in Chaps. 7–9. However, it will be shown that it may not be advisable to rely on the signal patterns alone, since these may not be always due to stiction, but can be caused by other loop problems, such as sensor faults.

a) TC

Controller Output (OP)

Process Variable (PV)

Time

Time

b) STC

c) LC

d) PC

e) TC

f) TC

g) FC

h) FC

i) LC

j) LC

Fig. 1.11 Typical time trends (measured data) found in the process industries showing poor control performance due different causes: a) and b) external disturbances; c) tight tuning problem; e) to j) stiction

1 Introduction

15

A large number of data sets from various industrial fields, including refining, chemicals, and petrochemicals, mineral processing, mining and metal processing, as well as pulp and paper, and power plants, have been gathered by the contributors of this book. A database containing data from almost 100 control loops has been organised by the principal editor. This database will be used to demonstrate the applicability of the stiction-detection methods presented in the remaining chapters, and particularly for the comparative study in Chap. 13. Figure 1.11 shows the measured controller outputs and process variables for some of the loops suffering from different performance problems or faults. Inspecting the plots reveals that all loops are oscillating. The challenge is then to automatically detect oscillation and find out its root-cause(s) based on the collected routine operation data.

1.7 Industrial Examples of Loops with Stiction The objective of this section is to illustrate effects of stiction from the investigation of data from three selected industrial control loops:

15

15

J

J

5

10 PV [%]

OP; PV [%]

10

J

0

600

0

J 700

800 Time

5

900

S −2

0

J 2 OP [%]

(a)

4

6

(b) 15

J

J

5

10

J

PV

SP; PV [%]

10

0

600

0

J 700

800 Time (c)

J 5

900

J 0

5 SP [%]

10

(d)

Fig. 1.12 Data from a flow loop in a refinery (CHEM 2): a) time trend of PV (solid) and OP (dashed); b) PV–OP plot; c) PV (solid) and SP (dashed) time trend; d) PV-SP plot

16


• Loop CHEM 2 is an inner flow loop cascaded with an outer level control loop. A sampling interval of 2 s was used for the acquisition of the data and a total of 1000 samples for each tag (PV, OP, SP) were collected. Figures 1.12a, b show the presence of stiction with a clear indication of stickband plus deadband (S) and the slip jump (J) phase. The slip jump appears as the control valve just overcomes stiction (indicated with “J” in Fig. 1.12). This slip jump is not very clear in the PV-OP plot of the closed-loop data (Fig. 1.12b), because both PV and OP jump together due to the probable presence of a P-only controller. In contrast, the presence of deadband plus stickband can be clearly seen (indicated with “S” in Fig. 1.12a). Sometimes, it is best to look at the PV-SP plot if it is a cascaded loop and the inner loop is operating under P-only control. Figures 1.12c, d show the time trend and phase plot of SP and PV, where the slip jump behaviour is now clearly visible. Estimates of the stiction parameters are S ≈ 4.5 and J ≈ 3.0.

1.5

1

1

0.5

0.5 PV

OP; PV

1.5

0 −0.5

0 −0.5

−1

−1

−1.5

−1.5 0

2000

4000

Time

6000

8000

−1.5

−1

−0.5

0 OP

0.5

1

1.5

0.5

1

1.5

(b)

(a)

1.5

1 0.5

0.5 PV

OP; PV

1

0 −0.5

−0.5

−1

−1

−1.5 5000

0

−1.5 5500

6000 Time (c)

6500

−1.5

−1

−0.5

0 OP (d)

Fig. 1.13 Data from a flow loop in a refinery (CHEM 23): a) normalised time trend of PV (solid) and OP (dashed); b) PV-OP plot; c) PV and OP trend for a data window; d) PV-OP plot for a data window

• Loop CHEM 23 is a flow control loop in a refinery. A total of 1500 samples for each tag (PV, OP) were collected at a sampling period of 10 s. The setpoint was constant for this loop. The left panels of Fig. 1.13 show the time trends of PV and OP: clear oscillations in both PV and OP can be seen. The presence of distinct cycles is observed in the characteristic PV-OP plot (see Fig. 1.13, right

1 Introduction

17

panels). The flow rate data looks quantised but the presence of the stickband and the slip jump of the valve can be clearly observed. Note that the moving phase of the valve is almost absent in this example: once the valve overcomes stiction, it jumps to the new position and sticks again. • Loop POW 2 is a level control loop with a sticky valve in a turbine. 500 samples for OP and PV, collected at a sampling period of 5 s, and the corresponding PVOP plot is shown in Fig. 1.14b. A subview is depicted in the bottom of this figure. Although it is known that the valve has clear deadband plus stickband and slip jump in this case [15], the latter is not visible neither in the PV trend nor in the PV-OP plot. The reason for this is that the process dynamics, i.e. the transfer function between MV and PV, distorts the pattern. Observe the triangular shape of the PV trend, which is characteristic for integrating processes with stiction.

1.5

1

1

0.5

0.5 PV

OP; PV

1.5

0 −0.5

0 −0.5

−1

−1

−1.5

−1.5 0

500

1000

Time

1500

2000

−1.5

−1

−0.5

(a)

1

0.5

1

1.5

1

1

0.5

0.5 PV

OP; PV

0.5

(b)

1.5

0 −0.5

0 −0.5

−1

−1

−1.5

−1.5

100

0 OP

200

300

400 Time (c)

500

600

−1.5

−1

−0.5

0 OP (d)

Fig. 1.14 Data from a level loop in a refinery (POW 2): a) normalised time trend of PV (solid) and OP (dashed); b) PV-OP plot; c) PV and OP trend for a data window; d) PV-OP plot for a data window

Note that the shape of MV-OP plots (parallelogram), when MV is measured, and PV–OP plots (ellipse), when MV is not measured, can be exploited for stiction detection, as presented in the forthcoming chapters of this book. Indeed, PV–OP plots have been used for detection of valve problems, particularly stiction, for a long time in industrial practice. However, experience shows that this type of method is not always successful, because a PV–OP plot only takes the qualitative trend information

18


of the time series into account. This information can be destroyed due to the presence of process, noise, and disturbance dynamics and tightly tuned controllers [17]. Therefore, it is recommended to combine the use of PV–OP plots with other techniques, such as those for non-linearity detection, to get well-founded diagnosis.

1.8 Summary and Conclusions Some definitions of the term stiction and similar phenomena (deadband, backlash, hysteresis, etc.) in control valves have been reviewed in this chapter. The input– output relation of valves under stiction has been explained, to serve as the basis of data-driven stiction modelling. The describing function analysis provides valuable insights into the behaviour of a valve with stiction. Data from typical industrial control loops have been presented to illustrate the effects of stiction on the shape of the loop signals PV and OP. In the case of stiction, OP usually follows a triangular wave for self-regulating processes; for integrating processes, PV shows a triangular wave as the “ideal” stiction behaviour. However, these patterns can be modified depending on process dynamics and controller parameters. The PV–OP plot of a valve with stiction usually shows an elliptical shape, but may also be affected by the same factors. All this makes the detection and quantification of stiction challenging. To solve this task, different techniques will be presented in the subsequent chapters of the book.

Part I

Stiction Modelling and Oscillation Detection

Chapter 2

Stiction Modelling M.A.A. Shoukat Choudhury, Nina F. Thornhill, Manabu Kano and Sirish L. Shah

Pneumatic control valves are widely used in the process industry. Two types of valve-stiction models have been proposed in the literature. One is a detailed physical model that formulates the stiction phenomenon as precisely as possible. The other is a data-driven model that describes the relationship between the controller output and the valve position. Since a detailed physical model has a number of unknown physical parameters, it is time consuming as well as difficult to simulate an actual control valve using such a model. The value of a physics-based model is that it gives insights into the phenomena that are observed in practice and indicates which effects must be captured in a data-driven model. A data-driven model, on the other hand, is useful because it has only a few parameters and these are easy to define and simple to understand. This chapter briefly discusses a physics-based valve model followed by a detailed description of data-driven stiction models.

M.A.A. Shoukat Choudhury Department of Chemical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh, e-mail: [email protected] Nina F. Thornhill Centre for Process Systems Engineering, Department of Chemical Engineering Imperial College London, South Kensington Campus, London SW7 2AZ, UK, e-mail: n.thornhill@ imperial.ac.uk Manabu Kano Process Control & Process Systems Engineering Laboratory, Dept. of Chemical Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan, e-mail: [email protected]. jp Sirish L. Shah Department of Chemical and Materials Engineering, University of Alberta Edmonton, Alberta T6G 2G6, Canada, e-mail: [email protected]

21

22

M.A.A.S. Choudhury, N.F. Thornhill, M. Kano and S.L. Shah

2.1 Introduction One important moving part in a feedback control loop is the control valve. If the control valve contains non-linearities, e.g. stiction, backlash, and deadband, the valve output may be oscillatory that in turn can cause oscillations in the process output. Among the many types of non-linearities in control valves, stiction is the most common and is one of the long-standing problems in the process industry. It hinders the achievement of good performance of control valves as well as control loops. Many studies [5,35,48,51,52,63,83,100,108,121,132] have been conducted to define and detect static friction or stiction. Choudhury et al. [15] proposed a formal definition of stiction and developed a data-driven stiction model. This chapter addresses the issue of modelling valve friction or stiction. Simulation using a physics-based model gives fundamental insights into the effects of friction on a control loop containing a sticking valve. The disadvantage of a physics-based model is that it requires a knowledge of quantities such as the mass of the moving parts and the friction forces that makes it infeasible for industrial studies or span of the valve input signal. In industrial practice, stiction and other related problems are identified in terms of the percentage of the valve travel or span of the valve input signal. The relationship between the magnitudes of the parameters of a physical model and stiction expressed as a percentage of the span of the input signal is also not known explicitly. On the other hand, a data-driven model is useful because it needs only a few parameters to be specified, which are easy to define in terms of the percentage of valve travel and span of the input signal, and simple to comprehend.

2.2 Physics-based Stiction Modelling The general structure of a pneumatic control valve is shown in Fig. 2.1. This valve is closed by a spring elastic force and opened by air pressure. Flow rate is changed according to the plug position, which is determined by the balance between the spring force and the force on the diaphragm due to air pressure. The plug is connected to the valve stem. The stem is moved against the static or kinetic frictional force caused by packing, which is a sealing device to prevent leakage of process fluid. Smooth movement of the stem is impeded by excessive static friction. The valve position cannot be changed until the controller output overcomes this static friction, and the stem may slip and jump when the static friction is overcome. Once the valve is moving, the movement of the stem is affected by dynamic friction that is generally less than the static friction, and after an initial jump the movement proceeds smoothly. The movement stops when the elastic force of the spring and the force due to friction exactly balance the force due to air pressure.

2 Stiction Modelling

23

Packing

Fig. 2.1 Structure of pneumatic control valve

For a pneumatic sliding stem valve, the force-balance equation based on Newton’s second law can be written as: M

d2 x = Forces = Fa + Fr + Ff + Fp + Fi , dt 2 ∑

(2.1)

where M is the mass of the moving parts, x is the relative stem position, Fa = Au is the force applied by pneumatic actuator, where A is the area of the diaphragm and u is the actuator air pressure or the valve input signal, Fr = −kx is the spring force, where k is the spring constant, Fp = −Ap Δ p is the force due to fluid pressure drop, where Ap is the plug unbalance area and Δ p is the fluid pressure drop across the valve, Fi is the extra force required to force the valve to be into the seat and Ff is the friction force [30, 68, 131]. Following Kayihan and Doyle III [68], Fi and Fp will be assumed to be zero because of their negligible contribution to the model. The friction force (Ff ) includes static and moving friction. The expression for the moving friction is in the first line of Eq. 2.2 and comprises a velocity independent term Fc known as Coulomb friction and a viscous friction term vFv that depends linearly upon the velocity. Both act in opposition to the velocity, as shown by the negative signs: ⎧ ⎨ −Fc sign(v) − v Fv if v = 0 if v = 0 and |Fa + Fr | ≤ Fs . (2.2) Ff = −(Fa + Fr ) ⎩ −Fs sign(Fa + Fr ) if v = 0 and |Fa + Fr | > Fs The second line in Eq. 2.2 is the case when the valve is stuck. Fs is the maximum static friction. The velocity of the stuck valve is zero, therefore the acceleration is zero also. Thus, the right-hand side of Newton’s law is zero, such that Ff = −(Fa + Fr ). The third line of the model represents the situation at the instant of breakaway. At that instant the sum of forces is (Fa + Fr ) − Fs sign(Fa + Fr ), which is not zero if |Fa + Fr | > Fs . Therefore, the acceleration becomes non-zero and the valve starts to move.

24


Many refinements exist for this friction model including the Stribeck effect that gives a detailed treatment of the static friction spike between Fc and Fs , and variable breakaway forces such that the value of Fs depends on how long the valve has been stuck. The classic friction model presented here is sufficient to capture major features and can model deadband and stick-slip effects. In order to use the physical model of valve stiction, knowledge of several parameters such as M, Fs , Fc and Fv , is required. Explicit values of these parameters depend on the size and manufacturer of the valve and are difficult to obtain in practice. Therefore, physical models of valve friction are not easily available for routine use. The alternative choice is to use a data-driven stiction model, which is relatively simple and easy to understand and use in simulation.

2.3 Data-driven Stiction Modelling Two main classes of data-driven stiction models have appeared in the literature – they are termed one-parameter and two-parameter models.

2.3.1 One-parameter Stiction Model Stenman et al. [117] reported a one-parameter data-driven stiction model. The model is described as follows: u(k − 1) if u(k) − y(k − 1) |< d y(k) = , (2.3) u(k) otherwise where, u and y are the valve input and output, respectively, d is the valve-stiction band. The model compares the difference between the current input to the valve and the previous output of the valve with the deadband; see Eq. 2.3. However, the deadband (d) should be compared with the cumulative increment of the input signal since the valve became stuck. The one-parameter model is also not adequate as a dynamic data-driven model of valve stiction as explained in more detail in Choudhury et al. [21].

2.3.2 Two-parameter Stiction Model For a dynamic stiction model, the challenges are (i) to model the tendency of the valve to stay moving once it has started until the input changes direction or the velocity goes to zero, and (ii) to include the effects of deadband and the slip jump. This section describes two-parameter stiction models that can meet these challenges. The two-parameter stiction model first appeared in Choudhury et al. [13]. Parameters of


25

this valve-stiction model can be related directly to plant data and, furthermore, such a model produces the same open-loop and closed-loop behaviour as the physical model. The model only requires the controller output signal and the specification of deadband plus stickband (S) and slip jump (J), as shown and explained in Sect. 1.4 (Fig. 1.5). It overcomes the main disadvantages of physical modelling of a control valve; namely it does not require knowledge of the mass of the moving parts of the actuator, spring constant, and the friction forces.

2.3.3 Choudhury’s Stiction Model 2.3.3.1 Model Formulation The valve sticks only when it is at rest or changing direction. When the valve changes its direction it comes to a rest momentarily. Also, when the velocity of the valve is very small, the valve may stick. Once the valve overcomes stiction, it starts moving and may keep moving for some time, depending on the valve input and the amount of stiction present in the valve. In the moving phase (Fig. 1.5), the valve suffers only from dynamic friction, which is usually smaller than static friction. It continues to move until its velocity is again very close to zero or it changes its direction. In the process industry, stiction is generally measured as a percentage of the valve travel or the span of the control signal [35]. For example, 2% stiction means that when the valve gets stuck, it will start moving only after the cumulative change of its control signal is greater than or equal to 2%. If the range of the control signal is from 4 to 20 mA, then a 2% stiction means that a change of the control signal less than 0.32 mA in magnitude will not be able to move the valve. In the modelling approach described herein, it is assumed that the controller has a filter to remove the high-frequency noise from the signal. Section 2.3.3.2 gives more discussion on this topic. Then, the control signal is translated to the percentage of valve travel with the help of a linear look-up table. The model consists of two parameters – namely the size of deadband plus stickband S (specified in the input axis) and slip jump J (specified on the output axis). Figure 2.2 summarises the model algorithm, which can be described as: • First, the controller output (mA) is converted to valve travel percentage using a look-up table. • If the transformed controller output signal expressed as a percentage is less then 0 or more than 100, the valve is saturated (i.e. fully closed or fully open). • If the signal is within the 0 to 100% range, the algorithm calculates the slope of the controller output signal. • Next, the change of the direction of the slope of the input signal is taken into consideration. If the sign of the slope changes or remains zero for two consecutive instants, the valve is assumed to be stuck and does not move. The sign function of the slope gives the following outputs:

26


– If the slope of the input signal is positive, the sign(slope) returns +1. – If the slope of the input signal is negative, the sign(slope) returns –1. – If the slope of the input signal is zero, the sign(slope) returns 0.

op(k)

Look-up table (converts mA to valve %)

x( k ) no

xss = xss y (k ) = 0

x(k ) > 0? yes

xss = xss y(k ) = 100

no

yes

x(k ) < 100?

v− new = [ x(k ) − x(k − 1)] / Δt

no

no

sign(v− new) = 0?

sign(v− new) = sign(v− old)?

yes

I =0

yes

I =1

yes

| x(k ) − xss |> J ?

I = 1? no

no yes no

y (k ) = y (k − 1) remain stuck

xss = x(k − 1) y(k ) = y (k − 1) Valve sticks

| x(k ) − xss | > S ? yes

I =0

y(k ) = x(k ) − sign(v− new)( S − J ) / 2

Valve slips and moves

y (k ) Valve characteristics (e.g. linear, square root, etc. (converts valve % to mA) mv(k)

Fig. 2.2 Signal and logic flow chart of the two-parameter data-driven Choudhury’s stiction model

Therefore, when sign(slope) changes from +1 to –1 or vice versa, this means that the direction of the input signal has changed and the valve is at the beginning


27

of its stick position (points M and G in Fig. 1.5). The algorithm detects the stick position of the valve at this point. Now, the valve may stick again while travelling in the same direction (opening or closing direction) only if the input signal to the valve does not change or remains constant for two consecutive instants, which is uncommon in practice. For this situation, the sign(slope) changes to 0 from +1 or –1 and vice versa. The algorithm again detects the stick position of the valve in the moving phase and this stuck condition is denoted with the indicator variable I = 1. The value of the input signal when the valve gets stuck is denoted as xss. This value of xss is kept constant and does not change until the valve gets stuck again. The cumulative change of input signal to the model is calculated from the deviation of the input signal from xss. It is noteworthy that this can also be performed by using a velocity threshold below which the valve will come to a rest or stop. • For the case where the input signal changes direction (i.e. the sign(slope) changes from +1 to –1 or vice versa), if the cumulative change of the input signal is more than the amount of the deadband plus stickband (S), then the valve slips and starts moving. • For the case when the input signal does not change direction (i.e. the sign(slope) changes from +1 or –1 to zero, or vice versa), if the cumulative change of the input signal is more than the amount of the stickband (J), then the valve slips and starts moving. This takes care of the case when the valve sticks again while travelling in the same direction [27]. • The output is calculated using the equation: output = input − sign(slope)(S − J)/2

(2.4)

and depends on the type of stiction present in the valve. It can be described as follows: – Deadband: if J = 0, then it represents the pure deadband case without any slip jump. – Stiction (undershoot): if J < S, then the valve output can never reach the valve input. There is always some offset or deviation between the valve input and the valve output. This represents the undershoot case of stiction. – Stiction (no offset): if J = S, the algorithm produces pure stick-slip behaviour. There is no offset between the valve input and the valve output. Once the valve overcomes stiction, the valve output tracks or reaches the valve input exactly. This is the well-known stick-slip case. – Stiction (overshoot): if J > S, the valve output overshoots the valve input due to excessive stiction. This is termed the overshoot case of stiction. • The parameter J is an output quantity measured on the vertical axis. It signifies the slip-jump start of the control valve immediately after it overcomes the deadband plus stickband. It accounts for the offset or deviation between the valve input and output signals.

28


• Finally, the output is converted back to a mA signal using a look-up table based on characteristics of the valve such as linear, equal percentage or square root, and the new valve position is reported. 2.3.3.2 Dealing with Stochastic or Noisy Control Signals The two-parameter stiction model uses the sign function to detect the change of valve direction. In the case of a noisy signal, this may cause numerical problems in the simulation. In real life, a noisy control signal causes the valve unnecessary movements and rapid wear and tear. In order to handle a noisy or stochastic control signal in simulation, a time-domain filter, e.g. an exponentially weighted moving average (EWMA) filter, can be used right after the controller to filter the noise. The work presented in this chapter uses the following filter: G f (z) =

λz . z − (1 − λ )

(2.5)

The magnitude of λ will depend on the extent of noise used in the simulation. A typical value of λ can be chosen as 0.1.

2.3.4 Simulation of the Stiction Model The two-parameter data-driven stiction model has been thoroughly examined using both open-loop and closed-loop simulation in [15,21]. Closed-loop behaviour of the two-parameter stiction model using a concentration control loop is presented here. The concentration loop has slow dynamics with a large dead time. The transfer function, controller and parameters used in simulation are shown in Table 2.1. The magnitudes of S and J are specified as a percentage of valve input span and output span, respectively. Table 2.1 Process models, controllers, and stiction model parameters Loop

Stiction Process Controller Deadband Undershoot No offset Overshoot S J S J S J S J

Concentration

3e−10s 10s+1

0.2( 10s+1 10s ) 5

0

5

2

5

5

5

7

The transfer-function model for the concentration loop was adopted from Horch and Isaksson [51]. This transfer function with a PI controller in a feedback closedloop configuration is used here for simulation. Simulation results for different stiction cases are presented in Figs. 2.3a and b. In both figures, thin lines are the controller output and thick lines are the process output. If an integrator is not present in


29

the process, the triangular shape of the time trend of the controller output is one of the characteristics of stiction [48]. If it is an integrating process with a PI-controller, the valve signal is integrated twice and appears as a series of parabolic segments. Figure 2.3a shows the controller output (OP) and valve position (MV). Mapping of MV–OP clearly shows the stiction phenomena in the valve. It is common practice to use a mapping of PV–OP for valve diagnosis; see Fig. 2.3. However, in this case such a mapping only shows elliptical trajectories with sharp turn-around points. The reason for the latter is that the PV–OP map captures not only the non-linear valve characteristic but also the dynamics of the process. Therefore, if the valve position data are available, one should plot valve position (MV) against the controller output (OP) to diagnose a valve problem. Except in cases of liquid flow loops where the flow through the valve (PV) can be taken to be proportional to valve opening (MV), the PV–OP maps should be used with caution for stiction diagnosis.

2.3.5 Kano’s Stiction Model The two-parameter model as discussed in [13, 15] has been modified by Kano et al. [63] attempted to relate S and J to the elastic force, air pressure and frictional force. In Kano’s model, S corresponds to the summation of static and dynamic friction and J corresponds to the difference between the static and dynamic friction; see Eq. 1.1. Having this in mind, they offered an alternative flow chart for simulating stiction. The flowchart for stiction simulation is presented in Fig. 2.4. The input and output of this valve-stiction model are the controller output u and the valve position y, respectively. Here, the controller output is transformed to the range corresponding to the valve position in advance. The first two branches in the flowchart check if the upper and the lower bounds of the controller output are satisfied. In Kano’s model, two states of the valve are explicitly distinguished, denoted by the variable st p. In the moving state st p has the value 0, while in the resting state st p is 1. In addition, the controller output at the moment the valve state changes from moving to resting is defined as uS . uS is updated and the state is changed to the resting state (st p = 1) only when the valve stops or changes its direction (Δ u(t) − Δ u(t −1) ≤ 0) while its state is moving (st p=0). Then, the following two conditions concerning the difference between u(t) and uS are checked unless the valve is in a moving state. The first condition judges whether the valve changes its direction and overcomes the maximum static friction (corresponding to points C and F in Fig. 1.5). Here, d = ±1 denotes the direction of frictional force. The second condition judges whether the valve moves in the similar direction and overcomes friction. If one of these two conditions is satisfied or the valve is in a moving state, the valve position is updated via the following equation: d(S − J) . (2.6) 2 On the other hand, the valve position is unchanged if the valve remains in a resting state. y(t) = u(t) − d fd = u(t) −

30


mv (thick line) and op (thin line)

mv vs. op linear

pure deadband

stiction (undershoot)

stiction (no offset)

stiction (overshoot) 0

100

200

300

time/s

(a) pv (thick line) and op (thin line)

pv vs. op linear

pure deadband

stiction (undershoot)

stiction (no offset)

stiction (overshoot) 0

100

200

300

time/s

(b) Fig. 2.3 Closed-loop simulation results of a concentration loop using the data-driven stiction model: a) time trends of MV and OP (left), MV–OP mappings (right); b) time trends of PV and OP (left), PV–OP mappings (right)


31

Fig. 2.4 Flow chart of Kano’s valve-stiction model

Fig. 2.5 Comparison between flow-rate measurements and estimates for validating valve-stiction model

To demonstrate the validity of this stiction model, simulation results are compared with operation data of a chemical process that suffers from valve stiction. The flow rate is estimated by calculating the valve position from the controller output with the stiction model and by assuming that the dynamics between the valve position and the flow rate is given by a first-order model:

32


1 . (2.7) 0.2s + 1 Measurements and estimates of the flow rate are shown together with the controller output in Fig. 2.5. The flow-rate measurements proved coincident with that estimated by the stiction model. The valve-stiction model developed here is based on the balance among the elastic force, the air pressure, and the frictional force, and it can describe the behaviour of pneumatic control valves with only two parameters S and J. For example, an ideal situation exists when S = J = 0 and no slip jump when J = 0. PF (t) =

2.4 Comparison Between Choudhury’s and Kano’s Stiction Models 2.4.1 Similarities • Both models use the same parameters, S and J. • Both models can produce the stiction behaviour of a control valve both when the valve sticks during the change in travel direction and when the valve sticks while travelling in the same direction.

2.4.2 Differences • To detect when the valve sticks, Choudhury’s model uses the sign function while Kano’s model uses the product of the incremental valve input change. • To deal with a stochastic control signal, Choudhury’s model uses an EWMA, while Kano’s model uses a first-order transfer-function model of the air chamber of the valve.

2.4.3 Comparisons Using an Industrial Example This example compares the performance of both two-parameter data-driven stiction models described above. It is an industrial example that describes a level control loop that controls the level of condensate in the outlet of a turbine by manipulating the flow rate of the liquid condensate. In total, 8641 samples for each tag were collected at a sampling period of 5 s. Figure 2.6 shows a portion of the time-domain data. The left panel shows time trends for level (PV), the controller output (OP), which is also the valve demand, and valve position (MV), which can be taken to be the same as the condensate flow rate. The plots in the right panel show the char-


33

PV and OP

Process output PV

acteristics PV–OP and MV–OP plots. The bottom plot shows the presence of the stickband plus deadband and the slip jump. The slip jump is large and visible from the bottom plot, especially when the valve is moving in a downward direction. It is marked as “A” in the figure. It is evident from this figure that the valve output (MV) can never reach the valve demand (OP). This kind of stiction is termed as the undershoot case of valve stiction in this study. The PV–OP plot does not show the jump behaviour clearly. The slip jump is very difficult to observe in the PV–OP plot because the process dynamics (i.e. the transfer function between MV and PV) destroy the pattern.

Samples (b)

MV and OP

Valve position MV

Samples (a)

Samples (c)

Samples (d)

Fig. 2.6 Flow control cascaded to level control in an industrial setting: a) PV and OP trend (line with circles: PV, thin line: op); b) PV–OP plot; c) MV and OP trend (line with circles: MV, thin line: OP); MV–OP plot

The stiction models are tested by driving them with the controller output signal of the industrial control loop. When the stiction models are run, they generate a simulated manipulated variable that can be compared to the real manipulated variable captured from the plant. A close match would indicate that the stiction model has captured the dynamic stiction behaviour of the real valve. In Fig. 2.7, the output of the stiction models in response to the controller output signal of this data is compared with the actual valve position data to see how accurately the two-parameter

34


stiction models can reproduce the results. In this figure, only 500 samples are shown to ensure a better visibility of the trends. For both models S = 10 and J = 3 were used. The top panel shows the controller output signal. The middle panel compares the actual valve-positioner data with the predicted valve-positioner data using Choudhury’s model while the bottom panel compares the actual valve-positioner data with the predicted valve-positioner data using Kano’s model. For both cases, the mean-squares errors (MSE) are calculated using:

MSE =

∑Ni=1 (mv − mvpredicted )2 , N

(2.8)

Controller Output, OP

where N is number of data points. In this case, N = 8641. For both models, the MSE is 0.93.

90 80 70 60 1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

(a) MV (Choudhury)

75 actual predicted

70 65 60 1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

(b) MV (Kano)

75 actual predicted

70 65 60 1000

1050

1100

1150

1200

1250 Samples

1300

1350

1400

1450

1500

(c) Fig. 2.7 Comparison of Choudhury’s and Kano’s stiction models using an industrial example (data samples 1001–1500): a) OP trend; b) MV trends (Choudhury’s model); c) MV trends (Kano’s model)

Figure 2.7 shows that both models can reproduce the stiction behaviour of a control valve successfully. However, it is interesting to observe the actual industrial valve moving around, while the model-predicted MV says it should be stuck. This is especially true in the valleys of the MV oscillation in the lower signal part. In the upper signal part, the industrial data also show valve sticking in some peaks; these


35

are marked with dashed ellipses in Fig. 2.8, which shows another data window. The shape of the industrial actual MV is somewhat different from that of the modelpredicted MV. This means that the data-driven stiction model is failing to capture some of the physics. One possible reason is a more complicated friction behaviour. For instance, in addition to the static and dynamic friction, the friction in the moving phase may have regimes with different gradients in the velocity versus frictionforce characteristic. Moreover, it seems that the valve shows some asymmetrical behaviour since it sometimes sticks in one direction but not in the other.

Controller Output, OP

90 85 80 75 70 65 8000

8050

8100

8150

8200

8250

8300

8350

8400

8450

8500

(a)

Valve Position, MV

75

actual predicted

70

65

60 8000

8050

8100

8150

8200

8250

Samples

8300

8350

8400

8450

8500

(b) Fig. 2.8 Comparison of Choudhury’s and Kano’s stiction models using an industrial example (data samples 8001–8500): a) OP trend; b) MV trends

2.5 Summary and Conclusions This chapter has described valve-stiction models. Both physics-based models and data-driven models are discussed. Although a physics-based model gives insights into the stiction phenomena that are observed in practice, it is difficult to use for simulation purposes because it is hard to determine the physical parameters, such as spring constant and static and dynamic friction forces. A data-driven model, on the other hand, is useful because it needs only a few parameters that are easy to define and simple to understand. At the same time, it can replicate the valve-stiction phenomenon quite successfully. Finally, both two-parameter stiction models as developed by Choudhury and Kano are compared using an industrial example. The validation test serves to show that both models are able to satisfactorily simulate the stiction effects.

Chapter 3

An Alternative Stiction-modelling Approach and Comparison of Different Stiction Models Q. Peter He, Jin Wang and S. Joe Qin

Due to the adverse impact of valve stiction in the process industry, both physical models and empirical data-driven models have been developed in the past decade to investigate the valve-stiction behaviour. Chapter 2 presented two data-driven models: Choudhury’s model and Kano’s model. In this chapter, we focus on another data-driven modelling approach (i.e. He’s model). We compare different data-driven models against a well-established physical model. The comparison is carried out in several aspects, including model assumptions, valve signatures generated by the models, and closed-loop behaviour when valve models are included in a closed-loop system. A new data-driven model is proposed based on a thorough analysis of the physical model and its effectiveness in simulating valve stiction is demonstrated.

3.1 Introduction Many control loops in process plants perform poorly simply because of valve stiction and valve stiction is one of the most common equipment problems [2,24,26,37, 96,125,132]. To simulate valve stiction, both detailed physical models and empirical models have been developed in recent years. Physical models [15, 68] describe the stiction phenomenon using force balances based on Newton’s second law of motion. Q. Peter He Department of Chemical Engineering, Tuskegee University, 522 B Luther H. Foster Hall, AL 36088, USA, e-mail: [email protected] Jin Wang Department of Chemical Engineering, 218 Ross Hall, Auburn University, AL 36849, USA, e-mail: [email protected] S. Joe Qin Department of Chemical Engineering and Materials Science, Electrical Engineering, and Industrial and Systems Engineering, University of Southern California, 925 Bloom Walk, HED 211, Los Angeles, CA 90089-1211, USA, e-mail: [email protected]

37

38

Q.P. He, J. Wang and S.J. Qin

The main disadvantage of these models is that they require knowledge of several parameters such as the mass of the moving parts and different type of friction forces that cannot be easily measured and depend on the type of fluid and valve wear. On the other hand, empirical or data-driven models [13, 15, 63] use simple empirical relationships between valve input and output to describe valve stiction, with just a few parameters that can be determined from operating data. Due to their simplicity and easy implementation, data-driven models have gained tremendous research interest in recent years. In Chap. 2, two data-driven models: Choudhury’s model and Kano’s model have been presented. Compared to physical models, these data-driven models simplify the simulation of a sticky valve and have been used by several other researchers for valve-stiction simulation. In this chapter, we focus on another data-driven modelling approach proposed by He et al. [44] (shortened as He’s model). He’s model employs a simpler model structure compared to Choudhury’s or Kano’s model. In addition, He’s model naturally handles stochastic noise and can simulate the sticky-valve behaviour similar to the ones observed in industrial cases [44]. The remainder of this chapter is organised as follows. We first present He’s two-parameter model [44], and examine the fundamental difference between He’s model and Choudhury’s/Kano’s model. We point out that the different valve-stiction behaviours produced by different data-driven models are rooted in the different model assumptions. To examine which assumption resembles an actual valve better, we implement the physical valve-stiction model to provide the basis for a comprehensive comparison of three data-driven models. We show that He’s model can simulate the signature stick-slip behaviour of a stick valve that is also produced by the physical model and observed in practice [35, 100]. Using the physical model, we further investigate the valvestiction behaviour analytically, and derive a three-parameter model that more accurately reproduces the physical model response, without the computationally intensive numerical integration required by the physical model.

3.2 He’s Two-parameter Model The purpose of this section is to briefly review the main characteristics of valve stiction and introduce He’s two-parameter model. Figure 1.5 shows the typical input–output behaviour of a sticky valve. Without stiction, the valve would move along the dash-dotted line crossing the origin, i.e. any amount of OP adjustment would result in the same amount of valve position1 (MV) change. However, for a sticky valve, static and kinetic frictions have to be taken into account. In Fig. 1.5, fs , fd and J denote the static friction band, Coulomb or kinetic friction band and slip jump, respectively, where J is defined as fs − fd . Because stiction is generally measured as percentage of the valve travel range for simplicity, as in Choudhury et al. [13, 15] and Kano et al. [63], all variables such as fs , fd , 1

Note that although MV denotes manipulated variable in general, it is used to denote valve position in this chapter as it has been widely adopted by other researchers.

3 Alternative Stiction-modelling Approach

39

J, controller output u, process output y and valve position uv 2 are normalised as a fraction or percentage of the valve range so that algebra can be performed among them directly. Details on the normalisation can be found in Sect. 3.5. Based on the sticky-valve behaviour, a valve-stiction model is proposed and Fig. 3.1 shows the flowchart of the model. The variable ur is the residual force acting on the valve that has not materialised a valve move. Variable cum u is a temporary variable that is the current net external force acting on the valve that is balanced by friction band. If the magnitude of cum u is large enough to overcome the static friction band fs , the valve position uv (t) will be the controller output u(t) offset by the kinetic or dynamic friction band fd . Otherwise, the valve position will not change and cum u is the residual force on the valve to be used in the next control instant. Controller output u(t)

cum _ u = ur + (u (t ) − u (t − 1))

yes

no

abs(cum−u ) > fs ?

uv (t ) = u (t ) − sign(cum−u − fs ) f d ur = sign(cum−u − fs ) f d

uv (t ) = uv (t − 1) ur = cum−u

Fig. 3.1 Flowchart of He’s two-parameter stiction model

To illustrate how He’s model works in a closed-loop system, a flow control loop with strong stiction ( fs = 0.65 and fd = 0.35) is used and details on the system can be found in Sect. 3.6. The controller output (OP) and process output (PV), which is the flow rate that reflects the valve-stem position, are shown in Fig. 3.2. It can be seen that PV roughly resembles a rectangular wave and the PV–OP plot takes the shape of a parallelogram, which is consistent with what has been observed in industry [35, 47, 100]. In addition, He’s two-parameter model naturally handles both deterministic and stochastic signals, and is flexible in simulating different types of stictions by tuning fs and fd . It is worth noting that He’s two-parameter model assumes that the valve stops at each sampling interval, which is why the total force applied on the valve is always compared to the static friction band fs to determine whether the valve moves or not. This assumption is different from Choudhury’s/Kano’s model, which determines whether the valve stops based on the comparison of the current and previous inputs to the valve.

2 In this chapter, both OP and u stand for controller output. OP is used in descriptions, tables and figures; u is used in mathematical derivations. Similarly, both PV and y denote process variable; both MV and uv denote valve position or valve output.

40


PV

0.5

0

−0.5

1

OP

0.5 0 −0.5 −1

0

100

200

300 Time [s]

400

500

(a) 0.25 0.2 0.15 0.1

PV

0.05 0 −0.05 −0.1 −0.15 −0.2 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

OP

(b) Fig. 3.2 Closed-loop behaviour with a sticky valve simulated by He’s two-parameter model: a) time-series plots of OP and PV; b) PV–OP

3.3 Three Data-driven Models Currently, whether the valve stiction happens (i.e. the valve sticks) at each sampling interval is a grey area in valve-stiction modelling, which is also the fundamental difference between He’s model and Choudhury’s/Kano’s model. It has been agreed that if the valve stops moving before the next control signal comes in, static friction will engage and valve stiction will happen at the next sampling interval. On the other hand, it has been argued that if the valve is sluggish and moves slowly with respect to the sampling interval, the valve may still be moving when the next control signal is executed, hence static stiction may not be in effect. Therefore, the fundamental question boils down to whether the valve stops moving at the end of each sampling interval. Choudhury’s/Kano’s model assumes that the valve moves slowly and will not stop until the control signal changes its direction or the same control signal is applied for two consecutive sampling intervals. Therefore, several logic loops are


41

needed to determine whether the valve stops at the end of each sampling interval. On the contrary, He’s model assumes that the valve stops at each sampling interval. In this section we implement the physical model of valve stiction to explore this grey area. In addition, we compare the valve behaviour generated from three datadriven models with that from the physical model.

3.3.1 Implementation of the First-principles Model The differential equations that describe the valve dynamics are the same as those given in Chap. 2. To facilitate numerical integration, we take the state-space representation of valve dynamics based on Newton’s second law of motion, with the valve-stem position and velocity as two system states: x˙ = v , mv˙ = Fa + Fr + Ff + Fp + Fi ,

(3.1) (3.2)

where x and v are the relative position and velocity of the stem. m is the mass of the moving parts in a valve, including the stem and plug. The definitions of the pneumatic actuator force Fa , spring force Fr , pressure drop caused force Fp , and extra force Fi are the same as given in the previous chapter. Fp and Fi are assumed zero due to their negligible contributions. Among all factors, the friction force Ff is the most important component that determines valve dynamics. Ff combines the effects of different forces depending on the status of the valve: ⎧

⎨ −Fc sign(v) − vFv − (Fs − Fc )exp −(v/vs )2 sign(v), if v = 0 if v = 0 and |Fa + Fr | ≤ Fs , (3.3) Ff = −(Fa + Fr ), ⎩ −Fs sign(Fa + Fr ), if v = 0 and |Fa + Fr | > Fs where Fc is the Coulomb friction, sometimes also referred to as the kinetic or dynamic friction, is a constant force that acts in the opposite direction to motion and is independent of the magnitude of velocity v. The combined term vFv is the viscous friction that acts in the opposite direction to motion and its magnitude increases lin

early with v at a slope of Fv . The term (Fs − Fc )exp −(v/vs )2 sign(v) denotes the Stribeck effect, which is used to account for a valve’s stick-slip (or stiction) behaviour, where vs is an empirical Stribeck velocity parameter. Figure 3.3 shows the relationship between Ff and v based on Eq. 3.3. The Stribeck term addresses the discontinuity of the friction force going from Fs in magnitude right before the stem starts to move, to Fc in magnitude right after the stem starts to move. The set of the parameters used in the physical model are listed in Table 3.1, which are obtained from an actual valve [68]. Dynamic simulation of a sticky valve has been implemented by integrating Eqs. 3.1 and 3.2 for a given input Fa using an ordinary differential equation (ODE) solver for stiff systems. Note that although a stiff ODE solver is used, if Eqs. 3.1 and 3.2 are used directly, difficulties in numerical integration exist due to the hash dis-

42


continuity caused by the sign function at velocity equal to zero as shown in Fig. 3.3. To address this difficulty, we approximate Eq. 3.3 using a piecewise function, where |v| < δ is used to approximate v = 0. Therefore, the ODEs we use to simulate the sticky valve are the following, with δ = 1 × 10−6 in/s: x˙ = v , ⎧ 2 ⎪ ⎪ S , if v > δ u − kx − F − vF − (F − F )exp − vvs a c v s c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sa u − kx − Fs , if − δ ≤ v ≤ δ and (Sa u − kx) > Fs ⎪ ⎪ ⎪ ⎨ if − δ ≤ v ≤ δ and − Fs ≤ (Sa u − kx) ≤ Fs . mv˙ = 0, ⎪ ⎪ ⎪ ⎪ ⎪ Sa u − kx + Fs , if − δ ≤ v ≤ δ and (Sa u − kx) < −Fs ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ Sa u − kx + Fc − vFv + (Fs − Fc )exp − v , if v < −δ vs

(3.4)

(3.5)

400 300 Friction force (lbf)

200 100 0 −100 −200 −300 −400 −0.5

0 Velocity (in/s)

0.5

Fig. 3.3 Friction discontinuity at zero velocity

The corresponding Jacobian matrix is: 0 1 J = ∂f ∂f , 2

∂x

2

∂v

where f2 refers to Eq. 3.5, and

(3.6)


43

⎧ ⎨ 0,

if − δ ≤ v ≤ δ and − Fs ≤ (Sa u − kx) ≤ Fs ∂ f2 = , k ⎩ − m , else ∂x ⎧ 2 ⎪ 2v 1 ⎪ − (F − F )exp − vvs −F , if v > δ ⎪ v s c 2 ⎪ m v ⎪ s ⎪ ⎪ ⎨ ∂ f2 if − δ ≤ v ≤ δ . = 0, ⎪ ∂v ⎪ 2 ⎪ ⎪1 v ⎪ −Fv + 2v , if v < −δ ⎪ 2 (Fs − Fc )exp − vs m ⎪ v s ⎩

(3.7)

(3.8)

Table 3.1 Valve parameters [68] Parameter x v u Sa k m Fc Fs Fv vs xm gc

Description Stem position Stem velocity Actuator air pressure Diaphragm area Spring constant Mass of stem and plug Coulomb friction Static friction Viscous friction Stribeck velocity Valve stroking range Universal gravitational constant

Value/Unit(English) in in/s psi 100 in2 300 lbf/in 3 lb 320 lbf 384 lbf 3.5 lb/s 0.01 in/s 4 in 32.2 lbm·ft lbf·s2

Value/Unit(SI) m m/s Pa 6.45×10−2 m2 5.25 × 104 N/m 1.36 kg 1423 N 1708 N 1.59 kg/s 2.54 × 10−4 m/s 0.1016 m 1 kg·m N·s2

3.3.2 Comparison of Data-driven Models In practice, to evaluate the open-loop behaviour of a valve, one starts with the valve fully closed, i.e. v = 0 and x = 0, then slowly cycles (i.e. ramps up and down) air pressure u, and reads the stem position x. The plot of stem position vs. input (air pressure) generated during this test is called the valve signature. In this section, the valve signature is first generated using the physical model with parameters given in Table 3.1; then the valve signatures generated by three data-driven models are compared with that generated by the physical model. In this section, the air pressure is cycled from 0 to 12 psi at a rate of 0.02 psi/s. Zero-order hold (ZOH) is implemented during the sampling interval which is 1 s. The input signal is shown in Fig. 3.4a, and the corresponding valve signature obtained based on the physical model is shown in Fig. 3.4b. Three cases have been simulated with different levels (i.e. the severity) of stiction: ideal valve with almost no stiction, weak stiction with nominal parameters in Table 3.1, and strong stiction with increased static and Coulomb frictions. Static and kinetic frictions are listed in Table 3.2 for different cases while other model parameters are the same as the

44


ones listed in Table 3.1. The following discussion and comparison are based on the nominal valve with weak stiction. Notice that the stick-slip behaviour, which is the signature behaviour of an actual stick valve [47, 63] and has been observed in actual control valves [35, 100], is successfully simulated by the physical model as shown in Fig. 3.4b for both nominal valve and strong stiction valve. In addition, the theoretical amount of air pressure required to overcome the static friction force, which is 3.84 psi for the nominal valve, is verified in Fig. 3.4b at point A.

12

Actuator Pressure (psi)

10 8 6 4 2 0 0

10

20 Time (min)

30

40

(a) 4 Ideal valve Weak stiction Strong stiction

3.5

Stem Position (in)

3 2.5 2 1.5 1 0.5 0

A 0

2

4

6

8

10

12

Air Pressure (psi)

(b) Fig. 3.4 Physical model: a) cycling of air pressure; b) stem position vs. actuator pressure

Next, we examine the responses (or the valve signatures) generated by different data-driven models given the same input signal shown in Fig. 3.4a. The parameters for different models are set to be consistent with the physical model of a nominal valve and are given in Table 3.3.


45

Table 3.2 Model parameters for different levels of stiction Case Ideal valve Weak stiction Strong stiction

Fs 10 lbf 384 lbf 600 lbf

Fc fs fd 10 lbf 8.3 × 10−3 8.3 × 10−3 320 lbf 0.320 0.267 500 lbf 0.500 0.417

Table 3.3 Data-driven model parameters Model Parameters Choudhury S = 0.587 J = 0.053 Kano S = 0.587 J = 0.053 He fs = 0.320 fd = 0.267

Figure 3.5a compares the valve signatures generated by Choudhury’s model and the physical model, where the dashed line represents Choudhury’s model, while the solid line represents the physical model. From Fig. 3.5a we see that there are two major discrepancies between Choudhury’s model and the physical model: first, with Choudhury’s model, the air pressure required to move the valve initially is the stick and deadband (S = fs + fd = 0.587) instead of the static friction band ( fs = 0.320), and this required air pressure will not change no matter what parameter initialisation of xss is used; second, once the valve starts to move, it does not stick nor slip until the valve changes its moving direction, which is not consistent with the physical model where valve stick-slips several times during the air pressure ramping up or down. The comparison between Kano’s model and the physical mode is shown in Fig. 3.5b. Kano’s model behaves similarly to Choudhury’s model except that Kano’s model predicts the initial valve movement correctly, where the applied air pressure has to overcome Fs in order to move. But like Choudhury’s model, Kano’s model does not capture the stick-slip behaviour either when the valve is moving along the same direction. The comparison between He’s model and the physical model is shown in Fig. 3.5c. It can be seen that although He’s model behaves slightly differently from the physical model in terms of the frequency and magnitude of valve stick/slip, it does capture the qualitative characteristics of a sticky valve, i.e. there is stick-slip not only when a valve changes direction, but also when a valve moves along the same direction, which has been observed in industrial control valves [35, 100]. In addition, it correctly predicts the initial valve movement.

3.4 Further Investigation of Valve Stiction In this section, we examine the dynamic response of a sticky valve using the physical model in order to find out whether the valve stops at each sampling interval. In the following analysis, we ignore the Stribeck-effect term and consider the valve

46

Q.P. He, J. Wang and S.J. Qin 3.5

Stem position (in)

3 2.5 2 1.5 1 0.5 0 0

Physical model Choudhury's model 2

4

6

8

10

12

Air Pressure (psi)

(a) 3.5

Stem position (in)

3 2.5 2 1.5 1 0.5 0 0

Physical model Kano'model 2

4 6 8 Air Pressure (psi)

10

12

(b) 3.5

Stem position (in)

3 2.5 2 1.5 1 0.5 0 0

Physical model He’s model 2

4

6

8

10

12

Air Pressure (psi)

(c) Fig. 3.5 Comparison of data-driven models and physical model: a) Choudhury’s model; b) Kano’s model; c) He’s model


47

described by the following second-order system: mx¨ Fv x˙ = Sa u − Fc − kx − . gc gc

(3.9)

By denoting the valve-stem position x as the system output, and (Sa u − Fc ) as the system input, the system transfer function is obtained as X = U

1 k

Fv m 2 kgc s + kgc s + 1

,

(3.10)

where U = L{Sa u − Fc }. Due to the ZOH of the control signal during sample interval, the input applied to the valve (i.e. applied air pressure) is a series of step changes. Therefore, it is sufficient to examine the step response of the valve described by Eq. 3.10, which is a simple second-order system. Using the model parameters listed in Table 3.1, the step response of the valve is given in Fig. 3.6. The parameters that characterise the step response are listed below: • Damping coefficient: 1 Fv ζ= = 0.003 . 2 kmgc • Overshoot:

OS = exp

−πζ 1−ζ2

(3.11)

= 0.991 .

(3.12)

• Time of the first maximum: tp =

πτ 1−ζ2

= 0.016s .

(3.13)

• Decay ratio: DR = exp

−2πζ 1−ζ2

= 0.982 .

(3.14)

There are several observations we can make based on its transfer function and step response. First, if there is no static friction, the valve will keep oscillating (the dotted line) for a long time, as indicated by its small damping coefficient (ζ = 0.003) and close to 1 decay ratio (DR = 0.982). But such rapid and sustained oscillation caused by the step input is not observed in a functioning valve, which indicates that static friction is indeed engaged whenever v = 0 as described by Eq. 3.3, and the valve stops when the valve-stem position reaches its first peak, i.e. when its velocity reaches the first zero (the solid line). The corresponding response of the valve-stem

48


x [in]

velocity is given in Fig. 3.6b. Second, the valve response to the step input is very fast, as it only takes 0.016 s (the first peak time) for the valve to move to its final position and stops there. Because the sampling interval of most distributed control systems (DCS) are 1 s or longer, it is safe to say that the valve stops at the end of each sampling interval. In other words, valve stiction is in effect at every sampling interval, which is the fundamental reason for the stick-slip behaviour as shown in the valve signature generated by the physical model and He’s model.

v [in/s]

t [s] (a)

t [s] (b) Fig. 3.6 Valve-step response by physical model: a) stem position; b) stem velocity


49

3.5 He’s Three-parameter Model As shown in the previous section, the valve responds to the step input very fast within each sampling interval. Therefore, we can ignore the transient response and use the constant gain to approximate the valve’s step response. As the valve stops at the first peak, the constant gain is K = 1 + OS. Based on this simplification, we can derive a three-parameter data-driven model to reproduce the valve response generated by the physical model, without involving computationally intensive numerical integration. In order to follow the conventional notation in data-driven models, we normalise all parameters into dimensionless quantities. By doing so, algebraic operations can be performed directly among them without unit conversion. First, we introduce a variable, xm , which denotes the entire stoking range of the valve, i.e. 0 ≤ x ≤ xm . As shown below, xm is used as the characteristic length to scale different variables. The normalised variables are: • Valve input, i.e. controller output: u =

Sa u . kxm

(3.15)

To match with the notation used in He’s two parameter model, in the following we abuse the notation and use u to replace u . • Stem position, i.e. the valve output, which is the input to the process: uv =

x . xm

(3.16)

• Static and Coulomb/dynamic frictions: Fs , kxm Fc . fd = kxm fs =

(3.17) (3.18)

It is worth noting that all normalised variables are expressed as the fraction or percentage of the valve-stroke range, and therefore are dimensionless. Using the normalised variables defined above, the flow chart of the three-parameter valvestiction model is given in Fig. 3.7, where the three model parameters are: fs , fd used in the original model and K to account for the overshoot observed in the physical model. Based on the physical model, K can be calculated as below: 2 ⎞ ⎛ Fv −π 4kmg c ⎠ ⎝ K = 1 + OS = 1 + exp . (3.19) Fv2 1 − 4kmgc

50

Q.P. He, J. Wang and S.J. Qin Normalised OP u(t)

e(t ) = u (t ) − uv (t − 1) yes

no

| e(t ) |> fs ?

uv (t ) = uv (t − 1)

uv (t ) = uv (t − 1) + K (e(t ) − sign(e(t )) fd )

Fig. 3.7 Flow diagram of the new model

Sensitivity analysis of K with respect to Fv and km shows that the effects of these parameters on K are not significant. Figure 3.8 plots the value of K for a wide range of Fv and km around their nominal values given in Table 3.1 and denoted as Fv∗ and km∗ . It shows that K is insensitive to Fv , m or k. For example, a ten-fold difference in Fv around the nominal value in Table 3.1 will only result in a 2% difference in K, while a ten-fold difference in m or k will result in less than a 1% difference in K. Fv2 is very small, e.g. 9 × 10−6 for the nominal valve. In other words, K is Usually 4kmg c always close to 2. Therefore, we can set K = 1.99 as a constant for different valves and the three-parameter model reduces to the modified two-parameter model.

2 1.8 1.6

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0

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0

Fig. 3.8 Sensitivity analysis

Next, we verify that the modified He’s two-parameter model (with K = 1.99) can exactly reproduce the valve behaviour generated by the physical model. Fig-


51

ure 3.9 shows the valve signatures generated by the physical model and modified He’s model. We compare the two models for the three different valves listed in Table 3.2. Figure 3.9 shows that for all three cases, i.e. an ideal valve shown in Fig. 3.9a, a valve with weak stiction shown in Fig. 3.9b and strong stiction shown in Fig. 3.9c, the valve signatures generated by the modified He’s model match with the physical model accurately. It is worth noting that if an empirically measured valve signature is available, the physical parameters of the valve (Fs and fc ) can be estimated from it. Fs can be estimated through the air pressure applied on the valve when the valve started to move initially, shown as the point A in Fig. 3.4b. In this case, the air pressure uA = 3.84 psi, if the cross-sectional area is known, which is 100 in2 in this case, then Fs is estimated as Sa uA = 384 lbf. In addition, the average magnitude of the slip jump in Fig. 3.4 is K(Fs − Fc ). From it Fc is estimated to be 316 lbf, which is very close to the true value of 320 lbf. On the other hand, if the valve is installed in a control loop, a non-invasive automatic quantification method that we developed [44] can be applied to estimate Fs and Fc , in which only limited process knowledge and routine operational data OP and PV are required. Finally, it is worth noting that although the three-parameter model seems slightly different from the original two-parameter model, the two-parameter model is actually a special case of the three-parameter model with K = 1. This explains the discrepancy between He’s two-parameter model and the physical model, i.e. the stick-slip predicted by He’s model has half the magnitude and twice the frequency as that predicted by the physical model. The detailed proof is given in Sect. 3.9.

3.6 Simulation Results In the previous sections, He’s two-parameter model based on the assumption that the valve stops at each interval has been thoroughly discussed and examined by comparing to a physical model. Based on the comparison results, the difference between two models was explored and a new data-driven model was proposed to simulate and closely resemble the sticky-valve behaviour. In this section, the closed-loop behaviour of the new three-parameter model is examined using a self-regulating flow control system and an integrating level control system [63]. The transfer functions for the flow and level processes are given by: Gf (s) =

1 , 0.2s + 1

(3.20)

1 −s e . (3.21) 15s The PI-controllers are used for both control systems and their transfer functions are given by: Gl (s) =

52


1 Flow control: Gc = 0.5 1 + , 0.3s 1 . Level control: Gc = 3 1 + 30s

(3.22) (3.23)

Four cases are examined for both systems: no stiction, pure deadband, weak stiction and strong stiction. Valve-stiction model parameters are summarised in Table 3.4. In the case of no stiction, in addition to white measurement noise, an external sinusoidal disturbance is introduced to compare its oscillatory behaviour with the ones causFed by stiction. For the other three cases, no external disturbance is considered but only white measurement noise. Simulation results are shown in Fig. 3.10 for the flow-control system and Fig. 3.11 for the level-control system. They are presented in the order of time-series plots of controller output (OP), valve position (MV) and process output (PV); PV–OP plot; and MV–OP plot for each case listed in Table 3.4. Table 3.4 Valve stiction model parameters [63] Case number Case 1 Case 2 Case 2 Case 3

Degree of stiction No stiction Pure deadband Weak stiction Strong stiction

fd 0 0.05 0.08 0.35

fs 0 0.05 0.05 0.25

From Fig. 3.10, we see that for self-regulating processes such as the flow process in this work, valve position (MV) is the most accurate indicator of stiction and stiction can be visually detected if the valve position is available. For either strong stiction or weak stiction, MV always takes a form of a rectangular wave in time-series plots, and MV–OP plot resembles a parallelogram if there is stiction. In addition, we see that the PV–OP plot also resembles a parallelogram when the stiction is strong. However, this is less obvious when stiction is weak. If MV is not available, there are other ways to detect stiction, as discussed in the later chapters of this book. For example, the triangular wave of OP would indicate valve stiction according to the curve-fitting method described in [44] and Chap. 7 of this book. From Fig. 3.11, we see that for integrating processes such as the level control system in this work, again, the valve position (MV) is the most accurate and reliable indicator. Whenever we see a rectangular wave of MV in its time-series plot, or a parallelogram of MV–OP, we know that there is valve stiction. When MV is not available, it is difficult to detect valve stiction based on the PV–OP plot because there is no characteristic shape that can differentiate stiction from other causes. However, there are other methods available as discussed in the later chapters. For example, the triangular wave of PV would indicate valve stiction based on the curve-fitting method.


53

4

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3 2.5 2 1.5 1 0.5 0 0

2


(c) Fig. 3.9 Comparison between the new data-driven model and the physical model: a) ideal valve; b) nominal valve; c) rough valve

54

Q.P. He, J. Wang and S.J. Qin 0.3

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(d) Fig. 3.10 Simulation of valve stiction in closed-loop flow control system: a) no stiction; b) pure deadband; c) weak stiction; d) strong stiction


OP

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(d) Fig. 3.11 Simulation of valve stiction in closed-loop level control system: a) no stiction; b) pure deadband; c) weak stiction; d) strong stiction

56


3.7 An Industrial Example In this section, we apply He’s model to an industrial example to examine its performance. The same industrial level-control loop example used in Sect. 2.4.3 is used here, where the level of condensate in the outlet of a turbine is controlled by manipulating the flow rate of the liquid condensate. In total 8641 samples were collected at a sampling interval of 5 s for the following variables: condensate level (PV), controller output (OP), condensate flow rate, which is viewed as equivalent to valve position (MV). We test He’s model by feeding the controller output signal into the stiction model, and compare the model-predicted valve position with the measured flow rate. Figure 3.12 shows the controller output signal (top), the valve position predicted by He’s model (middle), and the valve position predicted by Kano’s model (bottom). Because Choudhury’s model produces the same result as Kano’s model, we do not include valve position predicted by Choudhury’s model. The model parameters used for both models are listed in Table 3.5. Figure 3.12 shows that He’s model can reproduce the stiction behaviour as good as Kano’s model. However, if we examine the model parameters used by both models, it is interesting to note that the two models indicate different stiction conditions. Table 3.5 shows that both He’s model and Kano’s model indicates the same level of kinetic friction band (i.e. fd = 3.5). However, He’s model assumes that there is little difference between static and kinetic friction bands (i.e. J ≈ 0), while Kano’s model assumes that there is significant slip jump (i.e. J = 3). Table 3.5 Model parameters for the industrial example Model K fs fd S J He’s model 1.9 3.6 3.5 7.1 0.1 Kano’s model N/A 6.5 3.5 10 3

It is worth noting that He’s model, as well as Kano’s model, predicts that a valve sticks whenever the controller output changes direction. However, the measured flow rate indicates that sometimes the valve is not stuck when the controller output changes direction, especially for the low flow-rate region, where valve stiction seems to be insignificant (i.e. fs ≈ fd ≈ 0). This can be seen by comparing the controller output signal with the measured flow rate. In addition, the stochastic fluctuations in the pattern of the measured flow rate, i.e. the valve does not seem to be sticky for every cycle, indicates that stiction associated with an actual valve could be a stochastic process that cannot be exactly captured by a deterministic model with constant fs and fd . This may explain the discrepancy between the model predictions and the measured flow rates.


57

Controller Output

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Fig. 3.12 Comparison of He stiction model and Kano stiction model using an industrial example

3.8 Summary and Conclusions In this chapter, He’s two-parameter model is presented as an alternative modelling approach to Choudhury’s/Kano’s model. To compare the three data-driven models, a well-established physical model is implemented. The differences among the three different data-driven models are revealed by comparing them to the physical model. It is shown that He’s model can best reproduce the signature stick-slip behaviour of a sticky valve that is simulated by the physical model and observed in actual control valves. The different behaviour of He’s model from Choudhury’s/Kano’s model is rooted in its different assumption from the others. He’s model assumes that a valve stops/sticks at each sampling interval, while Choudhury’s/Kano’s model assumes that a valve does not stop until the control signal changes its direction or the same control signal is applied for two consecutive sampling intervals. Analysis of the physical model shows that the valve response to step inputs in air pressure is very fast and usually in a tiny fraction of a second to reach the first peak and stops there due to friction forces that are inevitable in control valves. Because the sampling interval of most DCS are 1 s or longer, it is safe to say that the valve stops at the end of each sampling interval. This explains why the prediction of He’s model is closer to that of the physical model and the actual valve behaviour observed in industry.

58


To resolve the discrepancy between He’s model and the physical model, we conducted further analysis on the physical model. It is found that because He’s model does not consider the dynamic response of the valve, the steady-state value is used for the final stem position without overshoot. Because overshoot accounts for about half of the valve movement, it explains the phenomenon shown in Fig. 3.5c that He’s model slips or jumps only about half of that of the physical model. After taking valve dynamics into account, we propose a three-parameter model by including a constant gain K to address overshoot. We show that the three-parameter model can accurately reproduce the physical model behaviour without involving cumbersome numerical integration. We also show that K is insensitive to valve parameters and in general very close to 2. Therefore, in the case that valve parameters are not available, K can be set as a constant (e.g. 1.99) so the three-parameter model reduces to the modified two-parameter model. Closed-loop simulation examples are given to illustrate the characteristics of valve stiction and some of these characteristics have been explored for valve-stiction detection. The three-parameter model is also tested using an industrial example. It is shown that the model can satisfactorily simulate the industrial case. Minor discrepancies are observed between the model predictions and actual measurements. They may due to the complexity of an actual valve that could have varying fs and fd along the valve’s travel path, and the possible stochastic aspects of the stiction phenomenon. These characteristics, if they exist in an actual valve, cannot be exactly simulated by a deterministic model with constant fs and fd . Nevertheless, the model does capture the essential characteristics of the stiction and therefore should be a useful tool for combating valve stiction in the process industry.

3.9 Appendix: Proof of the Equivalence Between He’s Two-parameter and Three-parameter Model In this appendix, we derive the equivalence between He’s two-parameter model and three-parameter model when K = 1. First, we observe the fact that at the end of each sampling interval, the air pressured applied to the valve is balanced by the spring elastic force (i.e. uv ) and the friction (i.e. ur , which is defined in Sect. 3.2 as the residual force). This fact is given as below: u(t) = uv (t) + ur (t) .

(3.24)

Plugging Eq. 3.24 into the definition of e(t) given in Fig. 3.7, e(t) = u(t) − uv (t − 1) = u(t) − u(t − 1) + ur (t − 1) ,

(3.25)

which is exactly the same as “cum u” defined in the two-parameter model given in Fig. 3.1, where ur is actually ur (t − 1) – the residual force from the previous run.


59

For the case that |e(t) = cum u(t)| ≤ fs , we see that uv (t) = uv (t − 1)

(3.26)

is the same for both two-parameter and three-parameter model, and ur (t) = u(t) − uv (t) = u(t) − uv (t − 1) = e(t) = cum u(t) .

(3.27) (3.28)

For the case that |e(t) = cum u(t)| > fs , and K = 1, from the three-parameter model we have, uv (t) = uv (t − 1) + (e(t) − fd × sign (e(t))) = uv (t − 1) + u(t) − uv (t − 1) − fd × sign (e(t)) = u(t) − fd × sign(cum u − fs )

(3.29)

and ur (t) = u(t) − uv (t) = fd × sign(cum u − fs ) .

(3.30)

Therefore, we see that when we set K = 1 in He’s three-parameter model, it reduces to the original two-parameter model.

Chapter 4

Detection of Oscillating Control Loops Srinivas Karra, Mohieddine Jelali, M. Nazmul Karim and Alexander Horch

Oscillations in process control loops are a very common problem. Oscillations often indicate a more severe problem than irregular variability. The presence of oscillations in a control loop increases the deviations from the setpoint of the process variables, thus causing inferior products, larger rejection rates, increased energy consumption and reduced average throughput. There are several reasons for oscillations in control loops. They may be caused by excessively high controller gains, oscillating disturbances or interactions, but a very common reason for oscillations is friction in control valves. Detection and diagnosis of oscillatory behaviour in a production process is of importance because process variability has an impact on profit. In the context of performance assessment and troubleshooting, the objectives of oscillation diagnostics of control loops can be stated as to i) identify sustained oscillations in a control loop, and ii) distinctly detect and quantify the root-cause(s) once the loop is confirmed to be oscillatory. In this chapter, the methods that are present in the literature to identify oscillations in process variables are discussed in detail. Their applicability to industrial control-loop diagnosis is critically evaluated on various simulation and industrial case studies.

Srinivas Karra Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected] Mohieddine Jelali (corresponding author) Department of Plant and System Technology, VDEh-Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected] M. Nazmul Karim Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected] Alexander Horch Group Process and Production Optimization, ABB Corporate Research Germany, Wallstadter Str. 59, 68526 Ladenburg, Germany, e-mail: [email protected]

61

62

S. Karra, M. Jelali, M.N. Karim and A. Horch

4.1 Introduction Oscillations are a very drastic form of plant-performance degradation, which can, in many cases, be induced by the feedback mechanism itself. Since control loops are not isolated from each other, an oscillation occurring in one loop may propagate to another, often leading to plant-wide oscillations. Therefore, a key topic in control performance monitoring (CPM) is to detect and diagnose oscillations in control systems. This chapter reviews and discusses the techniques for identifying oscillations in process variables. The chapter is organised as follows: in Sect. 4.2, the root-causes for oscillating control loops are explained. Measures to characterise oscillations are given in Sect. 4.3. Section 4.4 presents techniques for the automatic detection of loop oscillations. In Sect. 4.5, the techniques are critically evaluated on different simulated test data sets with typical oscillating behaviour. In Sect. 4.6, a comprehensive oscillation characterisation (COC) procedure is proposed. Some industrial case studies are presented in Sect. 4.7.

4.2 Root-causes for Oscillatory Control Loops The common causes for oscillatory feedback control loops are poor process and control system design, aggressive controller tuning, oscillatory external disturbances and control valve non-linearities [10, 26, 79, 84]. The probable parts of the control loop that may cause sustained oscillations in the controlled and manipulated variables are shown by dotted circles in Fig. 4.1.

4.2.1 Poor Process and Control System Design In the process industry, often systems with opposing factors acting within the process are encountered (inverse acting process). These pose a much more challenging control problem than is usually observed. If controller selection and tuning are inappropriate, it may lead to sustained oscillations in the controlled variable. The other case, where control-structure selection plays an important role, is in the systems with highly interacting dynamics. The control-system structure is to be chosen such that the interaction between one control loop and other control loops is minimised. In practice, however, control loops are often mutually interacting. Therefore, if one loop is oscillating, it will likely affect other loops too. In many cases, the oscillations are in a frequency range such that the controller cannot remove them. The above two cases are associated with control-system topology selection. The third kind of design issue that may cause oscillatory behaviour of process variable is process flow configuration. If any possibility of fluctuating flows of process fluid exists, all the

4 Detection of Oscillating Control Loops

63

downstream processing is affected by these fluctuations. To arrest such fluctuations, it is advisable to use some intermediate storage tanks.

Upstream process 2

Upstream process 1 Input

Control valve

Output

Process

Sensor Transmitter

I/P

D/A

A/D

Controller (DCS computer) Setpoint

Operator console

Air

Fig. 4.1 Schematic of control-system hardware: dotted circles are the probable components that may cause limit cycles in the control loop

4.2.2 Aggressive Controller Tuning One possible reason for oscillatory control loops is poor controller tuning. If the controller is tuned such that the loop is (nearly) unstable, there will be an oscillation due to saturation non-linearity, as control-signal constraints always exist in real systems. If the controller gain is equal to or higher than the ultimate gain of the process, then it results in sustained oscillations in the controlled variables. Excessive integral action of controller may also lead to oscillatory controlled variable. Proper controller tuning is required to obviate these soft oscillations.

4.2.3 Non-linearities in Control-loop Hardware Perhaps the most likely reason for control-loop oscillations is the presence of static non-linearities in the system, such as static friction (leading to the stick-slip effect), deadzone, backlash, saturation, and quantisation in the control valve [50]. Proper maintenance of control-loop hardware in an appropriate time can result in reduced oscillations caused by non-linearities present in the control-loop hardware. To help maintenance personnel perform this task, automatic detection and diagnosis of oscillations is essential.

64


4.2.4 External Oscillatory Disturbances External oscillatory disturbances that are entering a process may result in oscillatory control-loop behaviour. There may be various sources of such oscillatory disturbances as listed here: presence of oscillatory control loop in the vicinity, cyclic events like ambient-temperature fluctuations, fluctuations in raw-material quality, and variations in product demand with time (more of long-term fluctuations than short term). If these disturbances are not measured and not accounted in controller tuning, they adversely affect the process and lead to oscillations. These are a challenge for an automatic CPM system. When having detected an oscillation, it is important to distinguish between internally and externally generated oscillations. At times, it becomes a challenging task to distinguish between oscillatory signals resulting from external disturbances and aggressive controller tuning as they possess identical properties.

4.3 Characterisation of Oscillations There is no clear mathematical definition of oscillation that could be applied. Therefore, oscillation detection is usually done somewhat heuristically. One speaks of oscillations as periodic variations that are not completely hidden in noise, and hence are visible to the human eye [50]. The following are some of the properties of oscillating signals that may be used to identify the periodicity present in the signals.

4.3.1 Auto-covariance Function The statistical dependence of time-series data can be characterised by the sample auto-correlation/covariance function (ACF), and is defined as [84]:

rxx (k) =

N−k (x(t) − x)(x(t + k) − x) ∑t=1 , N (x(t) − x)2 ∑t=1

(4.1)

where x(t) is the measured value at time t, x is the sample mean for the N samples. k denotes the lag number. Each auto-correlation coefficient has a value within the range [−1, 1]. The ACF of the oscillatory signal is also oscillatory.


65

4.3.2 Power Spectrum Power spectral density (PSD) is a positive real function of a frequency variable associated with a stationary stochastic process, which has dimensions of power per Hz or power per rad/s. It is often called simply the spectrum of the signal. Intuitively, the spectral density captures the frequency content of a stochastic process and helps identify periodicities. Mathematically, the PSD is the Fourier transform of the ACF, rxx (k), of the signal if the signal can be treated as a stationary random process. This results in the formula [137]:

Sx ( f ) =

∞

∑

rxx (k)e−2π j f k .

(4.2)

k=−∞

An equivalent definition of PSD is the squared modulus of the Fourier transform of the time series x(k), scaled by a proper constant term. The ACF and PSD of a white-noise signal and a sinusoidal signal are plotted in Fig. 4.2. It can be seen from Figs. 4.2a (middle, right) that a white-noise signal contains a zero ACF at lags other than 0, and the PSD is uniformly distributed over the entire frequency range with no significant peaks in it. Figure 4.2b (left) shows a noisy sinusoidal signal with single frequency of oscillations. The ACF for such a signal is also oscillatory with similar dominant frequency and is shown in Fig. 4.2b (middle). The corresponding PSD is plotted in Fig. 4.2b (right) and it can be seen that the power of the signal is much higher at the dominant frequency (0.1 rad/s) as compared to other frequencies, resulting in a peak in the PSD.

4.3.3 Strength of Oscillations The amplitude specification remains an ambiguous question since small oscillations are usually not a serious problem, but the presence of oscillations itself is indicative of some faulty situation, as explained in Sect. 4.2. However, the knowledge of the quantifiable properties of oscillatory signals is necessary to make an expert assessment of whether the fault is significantly large and demands a corrective action to be taken. The strength of oscillations can be quantified using period, regularity and power [127]: • Period. The reciprocal of the oscillation frequency is termed the period of oscillation. In other words, it is twice the time lapse between two zero-crossings of an oscillatory signal. In reality, this period may vary around a mean value due to the presence of measurement noise and other stochastic components of the process. In some cases, multiple oscillations with different periods of oscillation may exist in the process variables resulting from multiple fault sources and hence the period may vary with the time.

S. Karra, M. Jelali, M.N. Karim and A. Horch 1.2

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Fig. 4.2 ACF and PSD of various signals: a) white-noise signal; b) noisy sinusoidal signal (x = 0.5 sin (0.1t) + 0.2randn)

• Regularity. Regularity of oscillatory signal translates into a quantity that represents the non-randomness behaviour. If the variation in the signal is due to random disturbances, the period of oscillation will hold a wider distribution compared to that of a true oscillatory nature. Let the expectation and standard deviation of the period of oscillation be T¯p and σTp , respectively, the regularity of T oscillation can be defined as r = f σTp . p • Power. Power of oscillations is a means to quantify the amplitude of the oscillatory signal. It is the sum of the spectral power in the selected frequency channels as a fraction of the total power. Another important property of oscillation signals is the vertical/horizontal symmetry. Different types of faults such as hardware failures (e.g. I/P converter), stiction in control valve, external disturbances or aggressive tuning are reported to produce oscillatory signals with special signatures in their shape. The specific (vertical or horizontal) symmetry/non-symmetry of an oscillatory signal forms the basis of many root-cause analysis algorithms that probe into the isolation of faults.


67

4.4 Techniques for Detection of Oscillations in Control Loops For reasons of safety and profitability, it is important to detect and diagnose oscillations. A number of oscillation-detection methods were suggested in the literature. These methods fall into four categories: • Detection of spectral peaks. • Methods based on time-domain criteria like the integral of absolute error (IAE) [33, 37, 101, 103, 125]. • Methods based on the auto-covariance function [84, 127]. • Use of wavelet plots [82].

1

1 0.8

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0 −2 10

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(b)

Fig. 4.3 PSD of PV signal under: a) no-fault case; b) stiction in control valve

4.4.1 Detection of Spectral Peaks Detecting oscillations by looking for peaks in the power spectrum is a classical approach. The amplitude of the highest peak outside the low-frequency area has to

68


be compared to the total energy in this frequency area. As shown in Fig. 4.3a, under normal operation the process variable (PV) contains a random noise that enters the system as the measurement noise propagates through the controller and then through the process, finally manifesting in a low-amplitude random stochastic component having a uniform PSD throughout the frequency range. In the case of stiction non-linearity, if present in the control valve, it causes a limit cycle in the process variable with a specific frequency. The power spectrum is characterised by a peak at the dominant frequency with much larger power compared to other frequencies (Fig. 4.3b). Let Kc and TI be the proportional gain and the integral time for the PI-controller, Gc (s) and Gp (s) be the controller and process transfer functions, respectively. An estimate of the oscillating frequency can √ be obtainedπSby solving for the frequency at π A2 − s2 + j 4J intersects the negative inverse which the Nyquist curve of N(A) = 4J of the describing function of the relay [3]. The oscillation frequency is calculated by solving the following non-linear equation for ω [110]: 4Kc (cos ω + ω TI sin ω ) + 4Kc ω T (− sin ω + ω TI cos ω ) = παω TI (1 + ω 2 T 2 ) , (4.3) where α is the ratio of the deadband (S) to the slip-jump (J) parameters, i.e. α = S/J. If the controller is poorly tuned and the gain of the controller (Kc ) is equal to the the ultimate gain (Ku ) of the process, the closed-loop control system produces sustained oscillations with an oscillating period equal to ultimate period, Tu . This results in a peak at the frequency corresponding to the ultimate period in the power spectrum of PV, as shown in Fig. 4.4a. If an external oscillating disturbance with a specific frequency enters the process, the power spectrum of the process variable contains a maximum power at that frequency as shown in Fig. 4.4b. 4.4.1.1 Methodology From these observations, it can be concluded that if the control loop is affected by oscillatory behaviour due to any of the three reasons: control-valve non-linearity, poor controller tuning or external oscillating disturbances, it can be easily identified by observing the power spectrum of the process variable. A peak can be defined as a point that is more than three times greater than the average of the surrounding, e.g. 10 samples. 4.4.1.2 Practical Considerations Visual inspection of spectra is therefore helpful because strong peaks can be easily seen, but determination of period and regularity from the spectrum is not recommended. The ratio between the position of a peak and its bandwidth, known as the Q-factor, gives a measure of the regularity of the oscillation, but the presence of


69

noise in the same frequency channels causes difficulties with the determination of bandwidth [127]. Also, automating the use of spectra for several hundreds or even thousands of loops is a difficult task, as visual inspection is generally necessary, and the tuning parameters are manually specified. Moreover, the application of spectral analysis becomes difficult if the oscillation is intermittent and periods vary every cycle. If the motive is to detect whether the control loop is affected by limit cycle behaviour, but not to quantify the cycling frequency and amplitude, the distinctive nature of the PSD in the case of an oscillatory signal can be used.

400

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Fig. 4.4 PSD of PV signal under: a) poor controller tuning; b) external sinusoidal disturbance

4.4.2 Regularity of Large Enough Integral of Absolute Error Perhaps the first procedure for detecting oscillations in control loops was presented by Hägglund [37]. This methodology is based on computing the regularity of large enough IAE between zero-crossings of the control error e. IAE is defined as

70


IAE =

ti ti−1

|e(t)| dt ,

(4.4)

where ti−1 and ti are two consecutive instances of zero-crossings. If the controller has integral action, there will be no offset and the control error will be centred on zero. In the cases where there is no integral action present in the controller, it is suggested to use the difference between the measurement signal and its average value obtained from a low-pass filter instead of control error. It might be advantageous to use the second approach also for controllers with integral action, since it provides the possibility to detect oscillations that are not centred on the setpoint. A basic observation during periods of good control is that the magnitude of the control error is small, and the times between the zero-crossings are relatively short. This means that the IAE values calculated according to Eq. 4.4 become small in this case. In the presence of load disturbances, the control error increases with a relatively long period between consecutive zero-crossings, resulting in a large IAE value. When the IAE exceeds a threshold IAElim , it is therefore likely that a load disturbance has occurred. The presence of oscillations can be confirmed if the rate of load-disturbance detections becomes high. For this purpose, the behaviour of the control performance is monitored over a supervision time Tsup : if the number of detected load disturbances exceeds a certain limit, nlim , during this time, it can be concluded that an oscillation is present. 4.4.2.1 Methodology The choice of IAElim is a trade-off between the demand for a high probability of detection and the requirement for a small probability of getting false detections (statistically, trade-off between type I and type II errors in hypothesis testing). Suppose that the control error is a pure sinusoidal wave with the amplitude A and the angular frequency ω , and that this signal is to be detected as a sequence of load disturbances. This means that the integral of each half-period of the oscillation must be greater than IAElim . The following upper limit, IAElim is then obtained:

IAElim ≤

π ω

0

|A sin (ω t)| dt =

2A . ω

(4.5)

This procedure should be able to detect all kinds of possible oscillations, i.e. oscillations with frequencies up to the ultimate frequency ωu . It also means that the oscillations with higher amplitude when compared to a peak-to-peak distance of 2A are critical and to be detected. A reasonable choice of amplitude A is 1%, which implies that the signal to be analysed has to be expressed in %. Such a parameter choice gives: IAElim =

2A . ωu

(4.6)


71

If the ultimate frequency ωu is unknown, it can be replaced by the integral frequency ωI = 2π TI . Now, the IAElim becomes IAElim =

TI . π

(4.7)

The oscillation-detection procedure needs two more parameters that must be chosen a priori, supervision time Tsup , and minimum number of detected load disturbances nlim to classify the signal as oscillating [37]. A general suggestion was nlim = 10 and Tsup = 5nlim Tu , where Tu is the ultimate period obtained from a relay auto-tuning experiment. If such information is not available, TI can be used in place of Tu . The whole procedure for load-disturbance and oscillation detection can be summarised as follows: Step 1. Monitor the IAE, where the integration is restarted every time the control error changes sign. Step 2. If the IAE exceeds IAElim , conclude that a load disturbance has occurred. Step 3. Monitor the number of detected load disturbances (nl ). Step 4. If nl exceeds nlim , conclude that an oscillation is present. Note that, for online applications, it is more convenient to perform the calculations recursively. The number of detected load disturbances is determined by nl (k) = γ nl (k − 1) + l ,

(4.8)

where l = 1 if a load disturbance detected and l = 0 otherwise. γ is a weighting s , Ts is the sampling period of the detection algorithm. factor, to be set as: γ = 1 − TTsup 4.4.2.2 Practical Considerations A modified version of Hägglund’s method was suggested in [125] to eliminate the necessity of knowledge of the signal ranges and integral constant of the controller for the computation of IAElim , Tsup , and nlim . The proposed use of the time between zero-crossings (Δti = ti+1 − ti ) in the criterion for detection of oscillations provides an alternative means to generate a threshold for real-time detection of deviations. The criterion for a 1% deviation (Eq. 4.7) becomes: IAE =

2Δti . π

(4.9)

That is, the local period of oscillation is taken to be 2Δti instead of TI . Note that in this case the calculation of Δti from zero-crossings is sensitive to noise, and countermeasures (filtering, noise band) have to be taken to avoid spurious zero-crossings. The use of 2Δti in place of TI also has benefits when the data are subsampled. Such

72


a case might arise if the real-time oscillation detection were to reside in the plant information system layer of a DCS rather than in the PID layer, because the data may be subsampled to reduce traffic across the communications link. In subsampling, the value captured at the sample instant is held constant until the next sampling instant and the IAE value calculated by integration from subsampled data can therefore be larger than expected. The effect gets worse as the sub-sampling period becomes longer. If the subsampling interval is close to the controller integration time, then a deviation may be detected after just one sample. Therefore, the use of the alternative time-scale parameter Δti is again helpful. When the controller output range is unknown, the detection of oscillations is enhanced by use of the RMS value of the noise as a scaling parameter. Of course, the range is recorded with other loop parameters in the DCS, but assessing a scaling factor from the data reduces the dependence on extraneous information. For online use, the noise assessment would need to be assessed over, say, the past 24 h using a recursive method of filtering [125]. All these observations and enhancements show that Hägglund’s [37] online detection method can be applied even in cases where neither the controller-tuning settings nor the range of the process variables are known, and that it can be used for sub-sampled data. Its applicability has therefore been extended to a wider range of cases often met in the industrial practice. To summarise, Hägglund’s [37] oscillation-detection method is very appealing but has two disadvantages: i) it is assumed that the loop oscillates at its ultimate frequency which may not be true, e.g. in the case of stiction, and ii) the ultimate frequency is seldom available and the integral time (also not always available) may be a bad indicator for the ultimate period. One of the strengths of the method is that it can be applied for online detection of oscillations.

4.4.3 Regularity of Upper and Lower IAEs and Zero-crossings The underlying idea of this method introduced by Forsman and Stattin [33] is that if e is near periodic then the time between successive zero-crossings and the successive IAEs should not vary so much over time. 4.4.3.1 Methodology The IAEs are separated for positive and negative errors (Fig. 4.5) Ai =

t2i+1 t2i

|e(t)| dt

and

to generate the oscillation index

Bi =

t2i+2 t2i+1

|e(t)| dt

(4.10)


h=

hA + hB , N

73

(4.11)

δi+1 1 N Ai+1 1 hA = # i < ; α < < ∧γ < < , 2 Ai α δi γ

(4.12)

εi+1 1 N Bi+1 1 hB = # i < ; α < < ∧γ < < , 2 Bi α εi γ

(4.13)

where #S denotes the number of elements in the set S. The oscillation index can be interpreted in the following way [33]: • Loops having h > 0.4 are oscillatory in nature, i.e. candidates for closer examination. • If h > 0.8, a very distinct oscillation pattern in the signal is expected. • White noise has h ≈ 0.1.

Fig. 4.5 Parameters for the calculation of the oscillations index [33]

4.4.3.2 Practical Considerations Forsman and Stattin [33] suggested to select α =0.5–0.7, γ =0.7–0.8, and stated that the criterion is fairly robust to variations in these tuning parameters. One reason for this is that there is a coupling between the condition on the IAE and the condition of the time between zero-crossings. In practice, the error e should be prefiltered prior to the index calculation, so that high-frequency noise is attenuated. A simple low-pass filter (or exponential filter) may be used:

74


ef (k) = α e(k) + (1 − α )ef (k) ,

ef (1) = e(1) ,

(4.14)

where ef is the filtered control error. Equation 4.14 is the digital version of the firstorder filter T e˙ f (k) + ef (t) = e(t) ,

(4.15)

s where T f denotes the filter time constant. α and T f are related by α = T f T+T , where s Ts is the sampling period. The choice of filter constant 0 < α ≤ 1 represents a trade-off between detecting fast, small oscillations (α = 1) and attenuating highfrequency noise (α → 0), typically α = 0.1. That is, a smaller value of α provides more filtering. For offline analysis, a non-causal filter filtfilt from the Signal Pro® cessing ToolboxTM of MATLAB can also be used.

4.4.4 Decay-ratio Approach of the Auto-correlation Function The ACF of an oscillating signal is itself oscillatory with the same period as the oscillation in the time trend. This distinct feature of ACF can be used to find the oscillations in signals that are noisy. The advantage of using the ACF for oscillation detection over time-trend-based methods is that the ACF in a sense provides a kind of filtering. The impact of noise is reduced because white noise has an ACF that is theoretically zero for lags greater than zero. Figure 4.6 clearly shows an example of the filtering effect of the ACF. 4.4.4.1 Methodology The patented method of Miao and Seborg [84] is based on the analysis of the autocovariance function of normal operating data of the controlled variable or the control error. The approach utilises the decay ratio Racf of the auto-covariance function, which provides a measure of how oscillatory the time trend is. Figure 4.6 illustrates the definition of the oscillation index Racf , described by the equation: Racf =

a , b

(4.16)

where a is the distance from the first maximum to the straight line connecting the first two minima, and b the distance from the first minimum to the straight line that connects the zero-lag auto-covariance coefficient and the first maximum. As the decay ratio of the auto-covariance function is directly related to the decay ratio of the signal itself, it is a convenient oscillation index. A value of Racf smaller than 0.5, corresponding to a decay ratio in the time domain smaller than 0.35, may be con-


75

sidered as acceptable for many control problems. On the other hand, if Racf ≥ 0.5, then the signal is considered to exhibit an excessive degree of oscillation. Therefore, Racf can be used to detect excessive oscillations in control loops according to the following simple procedure: Step 1. Calculate the auto-covariance function of the measured y or e as described in Sect. 4.3.1, and determine the decay ratio Racf (Eq. 4.16). For the case where there are less than two minima, set the index value to zero. Step 2. If Racf is greater than a specified threshold, say 0.5, it is concluded that the considered signal is excessively oscillatory. Note that, as in every method, the selection of the threshold is somewhat subjective and application dependent.

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4.4.4.2 Practical Considerations To calculate the oscillation index Racf from auto-correlation coefficients, it is necessary to have at least two minima and one maximum, i.e. 1.25 cycles, in the correlogram. Because the maximum lag is selected to be one quarter of the number of data points [12], 1.25 cycles in the correlogram corresponds to five cycles in the

76


signal data. Hence, an oscillation can be detected by the decay-ratio method if the data exhibit at least five cycles of a damped oscillation during the data-collection period Tc . This means that Tc must be at least five times the period of the lowest frequency of interest. This frequency range can be specified as in Hägglund’s [37] method (Sect. 4.4.2), i.e. depending on the ultimate frequency ωu when known

ωu ≤ ω ≤ ωu 10

(4.17)

or based on the integration time TI 2π 2π ≤ω ≤ . 10TI TI

(4.18)

This also suggests selecting Tc > 50TI .

(4.19)

However, when the controller is poorly tuned, TI may be too small and the nominal data collection period Tc = 50TI would then be too short to detect low-frequency oscillations. To avoid this potential problem, Miao and Seborg [84] recommended repeating the auto-covariance analysis for Tc = 250TI whenever no oscillation is detected using Tc = 50TI . When the signal-to-noise ratio is small, a large number of local maxima and minima may occur in the correlogram. This would cause problems in the determination of the Racf index. Therefore, it is desirable to remove highfrequency noise by filtering. Again, the selection of the filter time constant is a compromise between detecting fast, small oscillations and attenuating high-frequency noise. Excessive filtering attenuates the considered signal over the frequency range of interest and thus adversely affect the calculated Racf value; consult [84] for further details of this point.

4.4.5 Regularity of Zero-crossings of the Auto-correlation Function A regular oscillation will cross the signal mean at regular intervals. Therefore, the intervals between zero-crossings of an oscillatory time trend can be exploited for offline detection of oscillations. The deviations of the intervals between the zerocrossings are compared to the mean interval length; a small deviation indicates an oscillation. The threshold selection is signal independent, i.e. there is no need for scaling the individual signals. However, noise can cause false crossings; drift and transients will destroy the notion of a signal mean.


77

4.4.5.1 Methodology Instead of looking at the zero-crossings of the time trend, Thornhill et al. [127] suggested to use the zero-crossings of the ACF. The benefit of using ACF for oscillation detection is that the impact of noise is much reduced. The pattern of zero-crossings of the ACF therefore reveals the presence of an oscillation more clearly than the zero-crossings of the time trend. By looking at the regularity of the period, an oscillation can be detected. Regularity is assessed by the use of a statistic, r, termed the regularity factor. It is derived from the sequence of ratios between adjacent intervals Δti at which deviations cross the threshold. Thus, the mean period of the oscillation Tp can be determined from 2 n ∑n Δti T¯p = 2 i=1 = ∑ (ti − ti−1 ) n n i=1

(4.20)

and the dimensionless regularity factor, r is [127] r=

1 T¯p , 3 σTp

(4.21)

where σTp is the standard deviation of Tp . An oscillation is considered to be regular with a well-defined period if r is greater than unity. The regularity factor r can thus be regarded as an oscillation index. Practical considerations require that only signals with significant activity in the chosen frequency band be considered. The regularity test (Eq. 4.21) should thus be only applied if the filtered signal has sufficient fractional power defined as f +Δ f

P=

∑ f00 −Δ f S( f ) f

∑0max S( f )

× 100 ,

(4.22)

where Δ f denotes the filter width around f0 . A simple alternative is to first normalise the spectrum by the total power, to get S ∈ [0, 1], and then take the peak value of S at f0 as the fractional power. For instance, the fractional power of the signal in Fig. 4.2b (right) at ω = 0.1 rad/s is P ≈ 70%. A low value of P indicates that the signal does not have significant activity in the selected frequency, i.e. the behaviour of the signal is dominated by other frequencies. 4.4.5.2 Practical Considerations It is recommended to exclude the interval from zero-lag to the first zero-crossing because it corresponds to only one half of a completed deviation. Also, one should not use the last zero-crossings as they can be spurious in the case of very persistent oscillations. The period of data to be used can be selected according to the recom-

78


mendations in Sect. 4.4.4. Thornhill et al. [127] suggested considering 10 intervals between the first 11 zero-crossings for calculating the oscillation period. They also suggested a threshold of 1% for P, but higher values can be used to avoid detection of insignificant oscillations.

4.4.6 Spectral Envelope Method Recently, Jiang et al. [60] proposed the spectral envelope technique to detect oscillations and categorise variables having similar spectral properties. This method is well suited for the detection and diagnosis of plant-wide oscillations. The spectral envelope method itself was introduced and developed further by Stoffer and co-workers; see, e.g. [118, 119]. The extension of this approach towards the analysis of process data is due to Jian et al. [60]. 4.4.6.1 Methodology The spectral envelope, denoted as λ ( f ), is defined as the largest portion of power (or variance) that can be obtained at the frequency f for any scaled series x(k). This represents a process data set that is normalised to have zero mean and unit variance. For plant-wide oscillation detection, a m × N data matrix X = [x(0), x(1), . . . , x(N − 1)] of m variables and N samples for each variable is considered. A simplified mathematical definition of the spectral envelope estimate is given by [60]:

λˆ ( fk ) = sup

β =0

β ∗ Sˆ x ( fk )β β ∗ Vˆ x β

,

−1/2 < fk < 1/2 ,

(4.23)

where Sˆ x is an estimate the PSD, Vˆ x an estimate of the covariance matrix Vˆ x = Cov(X), β the so-called optimal scaling vector that results in the value λ ( f ), β ∗ the conjugate transpose of β , and fk the normalised frequency fk = k/n for k = 1, . . . , [N/2] with [N/2] the greatest integer less than or equal to N/2. Data normalisation implies that Vˆ x = Vˆ = diag{1, 1, . . . , 1}, thus the constraint β ∗ Vˆ β = 1 so that β ∗ β = 1∀ fk . Then λˆ ( fk ) is the largest eigenvalue of Sˆ x ( fk ) and β ( fk ) is the corresponding eigenvector. For the calculation of PSD, a symmetric moving average of the periodogram can be used: Vˆ x = with

r

∑

j=−r

h j IN ( fk+ j ) ,

(4.24)


1 IN ( f k ) = N

N−1

∑ x(k)e

−2π jt fk

t=0

79 N−1

∑ x(k)e

∗

−2π jt fk

.

(4.25)

t=0

Here, h j are symmetric (h j = h− j ) positive weights and ∑rj=−r h j = 1; r is the degree of smoothness: larger values of r lead to smoother estimates, but may smooth away significant peaks. To detect oscillations, the spectral envelope λˆ ( fk ) is plotted vs. fk and significant peaks are determined. An example from a case study to be discussed later in Sect. 4.7.3, is shown in Fig. 4.7. One can clearly identify the frequency of the main oscillation at f = 0.06 cycles/sample and the existence of higher harmonics. Moreover, low-frequency features can be observed, which is quite common in industrial normal operating data.

900 800

Spectral envelope λ ( f )

700 600 500 400 300 200 100 0 −3 10

−2

−1

10 10 Frequency f [cycles/sample]

10

0

Fig. 4.7 Example of spectral envelope

Once a certain oscillation frequency is identified, the magnitude of the optimal scalings at that frequency is investigated. The time series having large optimal scaling amplitude are the candidates that contribute the most to the spectral envelope, and thus are the candidates oscillating at that frequency. Moreover, a statistical hypothesis (Chi-square distribution) test on β ( fk ) should be performed to check whether or not a particular element of β ( fk ) is zero [60]:

80

S. Karra, M. Jelali, M.N. Karim and A. Horch |β

( f )|2

|β

( f )|2

• If 2 σ1, jj( f k) > χ22 (α ), then the null hypothesis “β1, j ( fk ) = 0” is rejected with k (1 − α ) confidence. Then, one can conclude that, with (1 − α ) confidence, the corresponding time series, i.e. row of X, does have oscillation at that frequency. • If 2 σ1, jj( f k) < χ22 (α ), then the null hypothesis “β1, j ( fk ) = 0” is accepted, and k the corresponding time series can be treated as not showing oscillation in the statistical sense. This statistical procedure is particularly valuable in automating the task of clustering oscillations having similar spectral characteristics. 4.4.6.2 Practical Considerations The main parameters to be selected for oscillation detection using the spectral envelope method are the degree of smoothness r, the weights h j , and the confidence level α . Jiang et al. [60] suggested to use triangular smoothing with r = 1 and weights {h0 = 1/2, h±1 = 1/4}, and α = 0.001 (99.9% confidence) that gives χ22 (0.001) = 13.82. The procedure for oscillation detection and clustering can be summarised as follows: Step 1. Normalise the data sets to get variables having zero mean and unit variance, and build the matrix X. Step 2. Compute the estimate of the spectral envelope using Eq. 4.23. Step 3. Identify the main oscillation frequencies. Step 4. Perform statistical hypothesis test to find out the variables that show oscillation at those frequencies identified in Step 3.

4.5 Critical Evaluation of Oscillation-detection Methods 4.5.1 Features of Industrial Control-loop-oscillation Detection If the signal is pure sinusoidal with a single dominant frequency, without any noise and disturbances, it is very easy to identify such oscillations with the naked eye. In practice, the measurements of process variables are corrupted by instrument noise, and unknown disturbances. Moreover, the presence of multiple oscillations is prevalent, caused by multiple faults occurring simultaneously. In some cases, intermittent oscillations are also a possibility, for example, a control valve may stick at a certain range of % opening but stiction may be absent outside this range. Based on the process dynamics, fault strength and nature it may result in different shapes and magnitudes of oscillations. In many data sets obtained from industrial control loops, slowly varying trends, high-frequency noise and multiple oscillations are observed


81

routinely. These effects often destroy the regularity of oscillations, which are then difficult to analyse. A good oscillation-detection method should be robust to such kinds of difficult scenarios to accurately detect the presence of oscillations in the time series. It is a valid question to be asked whether the objective of the oscillation-detection task is to identify the presence of any sort of oscillations in the process or to perform an isolation step as well. Alongside the detection accuracy, a critical judgement is also to be made in order to assess the advantages of online oscillation detection over performing the detection offline. In many process plants oscillatory process variables are persistent and most of them are perceived as being normal. Under such a scenario the oscillation-detection speed is not so critical and hence having an offline oscillation-detection algorithm performed over regular intervals say once an hour or a day (i.e. quasi-online) is equivalent to an online oscillation-detection algorithm running at a higher frequency [50]. Additional requirements of being a good candidate for oscillation detection is that the process-specific knowledge demands should be minimal or null. To be applied on thousands of control loops in a process plant routinely, the method should be based on just the time series of process variables. If it requires more process knowledge such as range of process variable, controller tuning parameters, etc., implementation of such a method may not be feasible. To summarise, a good oscillation-detection methodology for industrial applications should have the following features: • Usage of only time-series information of process variables with limited or no additional process knowledge. • Robustness to the high-frequency measurement noise and disturbances. • Ability to handle the presence of multiple and intermittent oscillations. • Amenablity to complete automation without human intervention.

4.5.2 Detection Test Examples These quantifiable properties of oscillatory signals are used in characterising the strength and periodicity of the oscillations in the signal. Keeping in view the formulated objectives for an industrially-relevant oscillation-detection methodology, five sets of data are generated to test the efficacy of the oscillation-detection methods presented in Sect. 4.4: Signal 1. A coloured noise signal 1 − 0.2q−1 v(t) , x1 (t) = 1 − 0.1q−1 + 0.8q−2 where v(t) is a zero-mean white-noise sequence with a variance of 0.2.

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Signal 2. An oscillating signal with one dominant frequency with measurement noise x2 (t) = 2.5 sin (0.1t) + x1 (t) . Signal 3. Damped oscillations x3 (t) = 10e−0.005t sin (0.1t) + x1 (t) . Signal 4. An oscillating signal with two dominant frequencies with measurement noise x4 (t) = 3.0 sin (0.1t) + 2.5 sin (0.3t) + x1 (t) . Signal 5. A signal with intermittent oscillations of varying characteristics ⎧ t ≤ 300 ⎨ 2.5 sin (0.1t) + x1 (t) 300 < t ≤ 700 . x5 (t) = x1 (t) ⎩ 700 < t ≤ 1000 3.5 sin (0.2t) + x1 (t)

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4.5.3 Signal with Coloured Noise The presence of measurement noise (high-frequency low-amplitude component) and setpoint changes (low-frequency high-amplitude component) deteriorate the PSD and ACF and as a result the strength and periodicity estimates of oscillations will be biased. To eliminate such problems, initially the signal has to be preprocessed to filter the high-frequency measurement noise and very low-frequency step changes in setpoint. As a first step, the actual signal has to be scaled to yield a zero mean and unit-variance signal. Then, the resultant signal is bandpass filtered in the frequency range [0.002, 0.97] Hz or [0.0126, 6.0947] rad/s using a digital filter. This means that we ignore the presence of oscillations that contain a time period less than 1 s, and higher than 500 s. Here, the sampling instant is assumed to be 1 s for convenience. The lower limit of frequency may be set based on the process knowledge. A digital FIR filter is used in the examples shown in this chapter. The filter was implemented using the firpm (Parks–McClellan FIR filter) routine in MATLAB. Figure 4.9a shows the coloured noise signal along with its PSD and ACF. The presence of coloured noise introduces a dubious peak in PSD and oscillatory ACF. Figure 4.9b shows the filtered signal along with its PSD and ACF. It can be seen that the filtered signal’s PSD is uniformly distributed over a range of frequencies. Also, the ACF of the filtered signal is negligible at lags equal or higher than to unity.

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The coloured-noise signal 1 after preprocessing contains uniformly distributed components of power at all frequencies. The PSD and ACF of the preprocessed signal are shown in Fig. 4.10. There are no evident peaks with normalised power higher than 0.1 in the PSD plot. The oscillation indices calculated using Miao and Seborg’s method [84] and Forsman and Stattin’s method [33] were found to be 0.58 and 0.13, respectively. Regularity of the oscillation described was found to be 0.67. Hägglund’s method also indicates no oscillation.

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4.5.4 Signal with One Predominant Oscillation The PSD and ACF of the preprocessed signal 2 are shown in Fig. 4.11. There is an evident peak at an ω = 0.1 rad/s in the PSD plot. The oscillation indices calculated using Miao and Seborg’s method [84] and Forsman and Stattin’s method [33] were found to be 0.952 and 1.0, respectively. The regularity of the oscillation described is found to be 19.88. The time period is identified as 62.88 s with a standard deviation of 1 s. The results of Hägglund’s method are presented in Fig. 4.12. The second subplot shows the IAE values calculated between successive zero-crossings of the control error, as well as IAElim . The period of oscillation is generated from


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zero-crossings (Sect. 4.4.2.2). It is observed that the IAE values are significantly larger than IAElim , indicating clear oscillation. This is confirmed by the third subplot, which shows the rate of load detections nl and the rate limit nlim ; both calculated according to the procedure in Sect. 4.4.2.1. The rate nl exceeds the rate limit nlim after about 365 samples, and the detection procedure gives an alert. 10

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4.5.5 Signal with Dampened Oscillation The PSD and ACF of the preprocessed signal 3 with dampened oscillations are shown in Fig. 4.13. There is an evident peak at ω = 0.1 rad/s in the PSD plot. The decay ratio calculated using Miao and Seborg’s method is 0.85. The regularity of this oscillation is 20.89. The time period is identified as 62.66 ±1 s. Hägglund’s method detects oscillation starting after 366 samples, and would indicate disappearance of the oscillation if the data set were long enough; see Fig. 4.14.

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4.5.6 Signal with Multiple Oscillations Figure 4.15 shows the signal 4 having two superimposed oscillations of different periods. The PSD of the preprocessed signal clearly shows the presence of two peaks at dominant frequencies (0.1 and 0.3 rad/s). The decay ratio of this signal is found to be 0.96. However, the regularity of oscillation is 0.58, indicating non-oscillatory behaviour. The zero-crossings of the fast and slow oscillations each destroy the regularity of the other signal’s pattern and resulted in wrongly classifying the signal as non-oscillatory. This difficulty can be overcome by means of bandpass filtering where the filter boundaries are selected from the inspection of the peaks present in the PSD. Frequency-domain-based filtering, e.g. a Wiener filter, which sets the power in unwanted frequency channels to zero, is one good option. The approximate realisation of a Wiener filter [95] should be used because a true Wiener filter requires an estimate of the noise power within the required frequency channels, which would then be subtracted from those channels. The detailed design algorithm is given in [127], which explains how to deal with aliased frequencies above the Nyquist frequency and constraints on the filter width, and also discusses the automation of the frequency-domain filter. It has been suggested to select the filter width Δ f cenf0 . An alternative is to use an FIR filter as tred at ± f0 so that Δ2f ≤ f50 or Δ f ≥ 2.5 mentioned in Sect. 4.5.3.

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When the signal is filtered using a bandpass frequency filter with the boundaries [0.07, 0.13] rad/s, we get the PSD and ACF as shown in Fig. 4.16a. Now, a distinctively regular oscillation with time period of 62.8 ±1.1 s can be identified from the ACF. The regularity of this oscillation is calculated as 19.1. One should particularly notice the marked regular zero-crossings of the ACF. The decay ratio of these oscillations at ω = 0.1 rad/s is 0.97. The power present in this oscillation is 86% of the total power. 10

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The second oscillation having an angular frequency of ω = 0.3 rad/s can also be accurately quantified using PSD and ACF when the filter boundaries are placed on [0.26, 0.34] rad/s; see Fig. 4.16b. In this case, the calculated regularity, decay ratio and time period are 6.8, 0.97 and 20.9 ±1.02 s, respectively. The power present in this oscillation is 44% of the total power. Hägglund’s technique indicates oscillation starting after 233 samples, where nl exceeds the alert threshold nlim , as shown in Fig. 4.17.

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4.5.7 Signal with Intermittent Oscillations In the case of persistent oscillation in the time trend, the power spectrum gives a clear signature for the oscillation since it has a sharp peak of large magnitude at the frequency of oscillation. However, there are some cases where the oscillation is intermittent, i.e. non-persistent. In a set of such time trends, where the nature of the signal changes over time, the Fourier transform has to be used on subsets of the data to observe the time-varying frequency content. Figure 4.18 shows the oscillation detection applied to signal 5, which consists of multiple intermittent oscillations. The PSD and ACF of the entire signal after preprocessing are also shown in Fig. 4.18. Even though oscillations are not present in certain portions of the data and two different kinds of oscillations with difference in amplitude and frequency are present, as the entire data was used for oscillation detection, only one dominant peak was observed in the PSD. Similarly, the ACF’s regularity is also lost. 10

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In such cases, different segments of the data are to be independently tested for the presence of oscillations. Figures 4.19, 4.20 and 4.21 show the oscillation detection performed on three segments of the data: 1–300 data points (Fig. 4.19), 301–700 data points (Fig. 4.20) and 701–1000 data points (Fig. 4.21), respectively. The first segment is found to contain oscillations with ω = 0.1 rad/s. The time period, decay


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ratio and regularity are 62.5 ±1 s, 0.86 and 20.83, respectively. The properties of the second data segment are as follows: no distinct peaks in PSD, the regularity of oscillations is 0.59 with a time period of 5.4 ±3 s, and the decay ratio is 0.66. Finally, the third segment was found to contain a spectral peak at ω = 0.2 rad/s with a power of 55% of the total power. The time period, regularity and decay ratio of these oscillations are found to be 31.6 ±0.88 s, 11.92 and 0.95, respectively. 10

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At this point, it should be emphasised that the decision of dividing data into different segments is to be done heuristically by visual inspection and there are no apparent scientific methods available to perform such segmentation. Alternatively, wavelet analysis may be used. The technique of wavelet transforms and algorithms for its computation [62] can treat time and frequency simultaneously in a time–frequency domain. This provides signal amplitude as a function of frequency of oscillation (the resolution) and time of occurrence. One method of presentation shows times and resolution plotted on the horizontal and vertical axis, and amplitudes represented by hues in the contour lines corresponding to them on the time–frequency plane. It is then possible to analyse the relation between the timing of frequency emerging and disappearing in the process, thus providing more precise and deeper insights into the process behaviour. Wavelet analysis has been successfully applied to plant-wide disturbance (oscillation) detection or diagnosis by Matsuo et al. [82].

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Signal 5 is an example that shows how Hägglund’s method successfully (online) detects intermittent oscillations; see Fig. 4.22. Alerts are correctly flagged around 326 and 763 samples, although no segementation was used for the analysis.

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4.6 Comprehensive Oscillation Characterisation In our experience, any oscillation-detection method should be combined with a calculation of the regularity factor to avoid possibly false detection or false determination of the oscillation period. This can occur when more than one oscillation is present in the signal. The accurate value of the oscillation period is needed in the stage of the root-cause diagnosis of the oscillation. Also, the inspection of the power spectrum is highly recommended to roughly set the filter boundaries. A COC algorithm involves the following steps to characterise multiple oscillations in a signal if present: Step 1. Normalise the data set(s) to get variable(s) having zero mean and unit variance. Step 2. Apply bandpass filtering for a frequency range of [0.002, 0.97] Hz (or in other words if the frequency is expressed in terms of angular frequency [0.0126, 6.0947] rad/s) to remove the effect of very high (measurement noise) and very low (SP changes, etc.) frequency components from the signal.

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Step 3. Generate normalised PSD and identify dominant frequencies (peaks) with a normalised power higher than 0.1. Alternatively, estimate the spectral envelope and identify the significant oscillation frequencies by statistical hypothesis testing. Step 4. In the case of presence of multiple dominant frequencies, apply bandpass filtering for the frequencies encompassing the vicinity of each dominant frequency, i.e. f0 ± Δ f , to generate different data subsets. Step 5. Generate ACF and calculate the regularity factor and decay ratio for each data subset. Step 6. If plant-wide oscillations are investigated, cluster the signals having regular oscillations and similar time periods. Alternatively, use the spectral envelope and statistical hypothesis testing to perform the clustering. Remark 4.1. The component of the signal with frequencies higher than the sampling frequency (≥ 1 Hz), in most cases, can be considered as a high-frequency noise and are to be filtered out (choice of 0.97 Hz as the higher limit on initial bandpass filtering). The low-frequency content such as outliers or discrete setpoint changes are also to be filtered out by choosing an appropriate lower bound (0.002 Hz here) based on the apparent oscillation cycle. The filter bounds for isolating signals with dominant frequency are chosen according to the directions provided by Thornhill et al. [127]. Indeed, this hybrid approach suggested here is a semi-automated method for oscillation detection. Although the method can be fully automated [127], care must be taken not to get misleading results due to improper selection of the bandpass filter. Therefore, it is recommended to use the method in a semi-automated fashion. In this method both PSD and ACF are used to characterise the oscillations. PSD is used for identifying the dominant frequencies at which oscillations are present. Each oscillation is isolated using bandpass filtering and as a result various data subsets are generated. Further, the ACF of each subset corresponding to a specific dominant frequency is used to quantify the regularity and decay ratio. Ultimately, this method can identify and quantify multiple oscillations if present in the signal. For the detection of plant-wide oscillations, the spectral envelope method seems to be very powerful. It can be fully automated and does not need bandpass filtering that may lead to false detections when the filter boundaries are not properly selected.

4.7 Industrial Case Studies The oscillation-detection procedure proposed above is now demonstrated on three industrial case studies.


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4.7.1 Oscillating Flow Control Loop Figure 4.23 shows OP data from a flow control loop in a refinery. The PSD of the signal clearly shows the presence of two peaks at dominant frequencies around 0.014 and 0.047 rad/s, corresponding to the time periods 483 and 134 s. The decay ratio and the regularity of oscillation were found to be 0.45 and 0.43, respectively. It is observed that, without filtering, the ACF of signal reflects neither oscillation accurately, because the zero-crossings of the fast and slow oscillations each destroy the regularity of the other signal’s pattern. The time period was determined as 202 s with a standard deviation of 15.5 s. These numbers do not represent the actual oscillation properties. 50 OP

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When the signal is filtered using a bandpass frequency filter with the boundaries [0.008, 0.018] rad/s, the resulting PSD and ACF are shown in Fig. 4.24. Now, a distinctively regular oscillation with a time period of 267.5 ±7.1 s can be identified from the ACF. The regularity factor is calculated as r = 18.18, clearly indicating regular zero-crossings of the ACF. The decay ratio of these oscillation is 0.68. The power present in this oscillation is 33% of the total power. The second oscillation can be accurately detected and quantified using the ACF when the filter boundaries are placed on [0.028, 0.066] rad/s; see Fig. 4.25. In this case, the regularity factor, decay ratio, time period and fractional power are 5.3, 0.95, 163 ±0.84 s and 8.5%, respectively.

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4.7.2 Unit-wide Oscillation Caused by a Sensor Fault The data in this study are from a refinery separation unit. The sampling interval was 20 s. The control errors in Fig. 4.27a show the presence of a unit-wide oscillation in the loops FC1, TC1 and AC1. Note that the FC1 and TC1 loops are in a cascade configuration. Measurements from upstream and downstream pressure controllers PC1 and PC2 are also available, and show evidence of oscillation along with other disturbances and noise. It is known that there was a faulty steam sensor in the steam flow loop FC1 [123]. Figure 4.27b shows the ACFs for the data with the zero-crossings. Figure 4.27c contains the normalised power spectra. Table 4.1 gives the results of the oscillation analysis, which indicate the presence of two distinct oscillations. The oscillations of PC1 and PC2 have a period of 381 s (or 19 samples per cycle) and that of the other loops of about 414 s (or 21 samples per cycle). A close examination of the power spectra confirms these results, particularly that the spectral peaks of PC1 and PC2 are at a slightly higher frequency. The results are in agreement with those found by Thornhill [123] using spectral principal components analysis. However, with the spectral envelope method, it is quite difficult to see the second frequency in Fig. 4.26 (marked with dotted ellipse). Also, the spectral envelope method fails to detect that AC1 is oscillating with f = 21; see the corresponding values of the χ 2 -test statistic in Table 4.1.


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4.7.3 Plant-wide Oscillation Caused by a Valve Fault A set of refinery data (courtesy of a SE Asian refinery) are examined in this section. This data set was previously used as a benchmark for oscillation-detection and

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Table 4.1 Results of the oscillation analysis for the separation unit data Tag AC1.e FC1.e PC1.e PC2.e TC1.e AC1.OP FC1.OP PC1.OP PC2.OP TC1.OP

Period [s] 412 ±3.6 416 ±2.1 380 ±1.7 380 ±1.1 412 ±1.9 408 ±4.0 416 ±1.0 384 ±4.4 380 ±4.0 420 ±1.0

Regularity r 1.9 3.2 3.7 6.0 3.6 1.7 6.7 1.4 1.6 6.6

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diagnosis methods, e.g. Thornhill [122, 123, 136]. It was found that the loops were associated with a plant-wide oscillation, and concluded that its source is the presence of non-linearity in one of the tags 13, 33 or 34. Note that the real root-cause for the oscillation was not exactly known, but it has been emphasised that it is a valve fault [122]. Figure 4.28 shows the time trends of control errors (1 min samples), the ACFs including zero-crossings and power spectra. It can be observed that tags with welldefined spectral peaks have oscillatory ACFs and a regular pattern of zero-crossings. Table 4.2 summarises the results of the oscillation analysis. A strong spectral peak at 0.06 (cycles per sample) with high fractional power is observed in all time series, with the exception of tag 14, which shows oscillation with 0.02 (cycles per sample). The average time period is 6.9 min. An important point to note is that the power spectra of tags 13, 33 and 34 contain stronger second harmonics than those of other loops. This typically signals the presence of non-linearity (valve fault) and supports the (mechanical) low-pass filtering effect when a limit cycle due to non-linearity propagates away from its source. This means that the magnitude of the harmonics reduces in measurements further from the oscillation root-cause. The values of the χ 2 -test statistic are also listed in Table 4.2. Values higher than 13.82 mean that the corresponding signals are oscillating with a confidence of 99.9%. A remarkably high value for tag 34 can be seen, which implies that this signal should be the source of the oscillation. These results and findings are in agreement with other previous studies performed on the data set [21, 122, 123]. Table 4.2 Analysis of oscillations in the SE Asian refinery data Tag 2 3 4 8 10 13 14 24 25 33 19 34 11 20

Period [s] 996 ±1.0 996 ±1.0 984 ±1.6 960 ±5.9 996 ±1.0 996 ±1.6 2970 ±1.9 1008 ±1.9 984 ±1.8 996 ±4.9 984 ±2.3 984 ±1.8 984 ±3.1 996 ±1.0

Regularity r 5.7 5.7 3.4 0.9 5.7 3.3 8.6 2.8 2.9 1.1 2.4 2.9 1.7 5.7

Power P [%] χ 2 -test statistic at f = 0.06 94 2 756 86 4 838 67 408 26 81 96 1 688 59 809 44 0 40 0 30 60 43 188 49 492 68 12 135 59 4 739 87 2 490

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25

2.9

30%

33

1.1

43%

19

2.4

49%

34

2.9

68%

11

1.7

59%

20

5.7

87%

ACF

2

24

50

100 150 200 250 Samples

20

40 60 Lags

(a)

80

100

(b)

−2

10−1

10

Normal. frequency f / fs

(c)

Fig. 4.28 Analysis of oscillations of the SE Asian refinery data

4.8 Summary and Conclusions The detection of oscillations in process control loops can now be regarded as a largely solved problem. The most important oscillation-detection methods have been presented and critically evaluated. Key issues, such as slowly varying trends, high-frequency noise and multiple oscillations, to be considered when analysing real data, have been addressed. These effects often destroy the regularity of oscillations, which are then difficult to analyse. A comprehensive, but semi-automated oscillation-characterisation procedure that can handle these problems was suggested and demonstrated on industrial case studies. The challenge is still to reliably detect oscillations without human interaction, i.e. without visual inspection or wavelet plots.

Part II

Advances in Stiction Detection and Quantification

Chapter 5

Shape-based Stiction Detection Manabu Kano, Yoshiyuki Yamashita and Hidekazu Kugemoto

In this chapter, shape-based stiction-detection methods are explained. The shape means the relationship between controller output (OP) and manipulated variable (MV) in the two-dimensional space, which forms a parallelogram when valve stiction occurs. In practice, flow rate is used as MV instead of the valve position whose data are not available in many cases. The shape-based methods utilise only routine operation data for detecting stiction, and they can also quantify the degree of stiction. The usefulness of the shape-based stiction-detection methods is demonstrated by applying them to simulation data generated by using the developed valve-stiction model and real operation data obtained from several chemical processes.

5.1 Introduction Typical stick-slip behaviour in a control valve produces a special shape in the phase plot of OP and MV. On the basis of this special shape in the phase plot, three stictiondetection methods are introduced in this chapter. These three shape-based stictiondetection methods were developed through close collaboration between industry and academia in Japan. The shape-based stiction-detection methods explained here are intuitive, easy to understand, easy to implement, and computationally efficient.

Manabu Kano Department of Chemical Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan, email: [email protected] Yoshiyuki Yamashita Department of Chemical Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan, e-mail: [email protected] Hidekazu Kugemoto Sumitomo Chemical Co., Ltd., 5-l, Sobiraki-cho, Niihama City, Ehime 792-8521, Japan, e-mail: [email protected]

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M. Kano, Y. Yamashita and H. Kugemoto

In addition, they utilise only routine operation data for detecting valve stiction, and they can also quantify the degree of stiction. From the practical viewpoint, the shape-based methods are useful for finding out the valves suspected to have hardware defects from a huge amount of plant data. In other words, they are useful for screening to find suspicious valves. In fact, several petrochemical companies in Japan have routinely used the software tool that can detect valve stiction with these three methods. The usefulness of the shape-based stiction-detection methods is demonstrated by applying them to simulation data generated by using the valve-stiction model and real-operation data obtained from several chemical processes.

5.2 Method Description In this section, three methods for detecting valve stiction are explained in detail. As shown in Fig. 1.5, the following characteristics are observed when stiction occurs in control valves. Characteristic 1. There are sections where the valve position does not change even though OP changes. Stiction is stronger as such sections are longer. Characteristic 2. The relationship between OP and MV takes the shape of a parallelogram if slip jump J is neglected. Stiction is stronger as the distance between l1 and l2 is longer. On the basis of these characteristics, shape-based methods for detecting valve stiction are developed. The first method, referred to as method A, is based on characteristic 1; the second, method B, is based on characteristic 2; and the third, method C, is based on characteristic 1, but also utilising the shape information. The common advantages of shape-based stiction-detection methods are: i) they can quantify the stiction, and ii) they are applicable to situations without periodic oscillation.

5.2.1 Method A The first valve-stiction-detection method, Method A [63, 80], is intuitive, easy to understand, and easy to implement. This method utilises the fact that MV does not change even though OP changes if stiction occurs in control valves. Therefore, we can detect stiction by checking the behaviour of the MV against the OP. If MV stays constant while OP increases or decreases, then we can conclude that stiction occurs. In addition, we can quantify the degree of stiction by checking the length where MV stays constant.

5 Shape-based Stiction Detection

105

The algorithm of Method A is summarised as follows: Step 1. Calculate the difference of MV, i.e. y.

Δ y(t) = y(t) − y(t − 1) .

(5.1)

Step 2. Find time intervals when the following condition is satisfied. |Δ y(t)| < ε ,

(5.2)

where ε is a threshold. Step 3. During each time interval found, calculate the difference between the maximum and the minimum of the OP u and define it as u. ˜ Similarly, calculate the difference between the maximum and the minimum of y and define it as ˜ y. ˜ In addition, determine thresholds εu for u˜ and εy for y. Step 4. Conclude that stiction occurs when u˜ ≥ εu and y˜ ≤ εy . Otherwise, conclude that stiction does not occur. Step 5. Calculate the ratio SIA of the total length of intervals when stiction occurs to the total length of all intervals. In addition, calculate δA that is the mean of u˜ when stiction occurs. There is a higher probability of stiction as the normalised measure SIA is closer to one. On the contrary, it is confirmed that no stiction occurs when SIA is zero. Furthermore, the degree of stiction can be quantified by using δA .

5.2.2 Method B Method B [63, 80] is based on the fact that the relationship between the OP and the MV takes the shape of a parallelogram and the distance between l1 and l2 increases as stiction becomes stronger. This method was developed from the idea that stiction can be detected and quantified by knowing the distance between two lines l1 and l2 along the horizontal axis. The distance corresponds to S−J as shown in Fig. 1.5. The basic idea of Method B is very similar to that of Method A, but their approaches are different. In Method B, to measure the distance between l1 and l2 , the following function F is introduced: F(t) = max {min {F(t − 1) + Δ u(t), Fmax } , 0} ,

(5.3)

F(0) = F0 ,

(5.4)

where Fmax is the maximum value of F and it will correspond to S−J. The function F indicates the difference between the OP u and the value on the line l1 at the same MV. For example, F(t) = 0 when the valve moves along the line l1 because F(t − 1) = 0 and Δ u(t) < 0. When the OP u changes its direction, F(t) = Δ u(t) because F(t − 1) = 0 and 0 < Δ u(t) < Fmax . F(t) is continuously increasing till F(t) becomes Fmax . After that, F(t) = Fmax while the valve moves along the line l2 because F(t − 1) = Fmax and Δ u(t) > 0. This is how the function F works.

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The function F is a measure of the distance between the OP u and the line l1 along the horizontal axis. Therefore, u − F will have a strong correlation with y. Thus, the degree of stiction or deadband can be quantified by Fmax under the assumption that slip jump J is much smaller than S and Fmax is usually close to S. Fmax and its initial value F(0) can be identified from operation data by solving an optimisation problem that aims to maximise a correlation coefficient r between u − F and y. As a result, the possibility of stiction gets larger as Fmax becomes larger. However, the results of Method B are reliable only when the correlation coefficient r is close to one.

5.2.3 Method C Method C [134] is based on the qualitative shape analysis of the characteristics in Fig. 1.5. To capture the typical feature of the valve stiction, this method includes three steps. The first step is to approximate the measured time-series signals by qualitative shape analysis. Time segments of signals can be qualitatively approximated by means of three qualitative symbols: increasing (I), decreasing (D) and steady (S). These categorisations can be easily calculated by using the sign of the time differences between sampling intervals. Combination of the symbols for the OP and the MV provides nine qualitative movement patterns as shown in Fig. 5.1.

DI

OP\ MV D

S

SI

II

I

I

ID IS II

S

SD SS SI

D

DD DS DI

DS

IS

DD

ID SD

Fig. 5.1 Qualitative movement patterns

The second step is to find typical patterns in the sequence of the qualitative representation. Segments of typical patterns for valve stiction are the sequence of IS II, IS SI, DS DD and DS SD, as shown in Fig. 5.2. A stiction index SIC to capture these specific pattern sequences can be defined as: SIC =

τIS II + τIS SI + τDS DD + τDS SD , τtotal − τSS

(5.5)


107

where τtotal is the width of the time window, and τSS is the time period for patterns SS in the window. The value τIS II is the sum of interval times of IS in all the found sequence of IS II patterns. Similarly the values τIS SI , τDS DD and τDS DD are defined. The possibility of stiction gets larger as the index SIC becomes larger. If the value is larger than 0.25, for example, the loop possibly has stiction. On the contrary, it is confirmed that no stiction occurs when SIC is nearly equal to zero. DS

DS

MV

DD

SD SI

II IS

IS

OP

Fig. 5.2 Segments of typical stiction patterns

The third step is to qualify the stiction: the width of the sticky movement is also calculated for the found typical sticky patterns. The width δC is defined by the mean value of the maximum differences of OP values during each IS and DS period in sticky patterns. Before applying this method, it is recommended to select adequate sampling interval so as to reduce the effect of noise. If an oscillating frequency of the signal is available, a sampling interval corresponding to about 16 times of the oscillating frequency is recommended.

5.3 Key Issues The three shape-based stiction-detection methods introduced in this chapter were developed through close collaboration between industry and academia in Japan. The developed methods can detect and quantify valve stiction by using routine plant data. In addition, they are intuitive, easy to understand, easy to implement, and also computationally efficient. From the practical viewpoint, their noteworthy function is to easily find out the valves suspected of having hardware defects from a huge amount of plant data. In other words, they are useful for screening to find suspicious valves. In fact, several petrochemical companies in Japan have routinely used the software tool that can detect valve stiction with these three methods. We do not have to choose only one method for stiction detection, and we can use several methods together for improving the reliability of the conclusion. In addition, the software tool has the function of assessing the controller performance. Therefore, the tool can be used to distinguish poor controller tuning and valve stiction when the controller performance is not acceptable. This is an important feature in industrial use, because actions to take differ with causes.

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Another advantage of the shape-based methods over other conventional methods is that they work even when no periodical oscillation occurs. This is a useful feature in practical use because periodical oscillation does not always occur, as shown in the following application section. The shape-based methods require OP data and MV data to find suspicious movement of valves, because they aim to find particular shapes or relationships between OP and MV from their data. Such particular shapes or relationships are caused by mismatch between input and output signals. Flow rate can be used as MV when valve position data are not available. The difficulty associated with this is (i) noise, and (ii) the flow loop has dynamics that can distort the shape of the stiction pattern. Also, in the developed methods, the time delay from the OP to the MV or the flow rate is not taken into account. Thus, the detection performance may deteriorate when the time delay is significant. In addition, the sampling period affects the detection performance. In general, the corners of the parallelogram are rounded off and the detection performance deteriorates when the sampling is too slow. When Method A or Method C is used, the appropriate interval should be selected with respect to the period of oscillation.

5.4 Simulation Results A flow-control system and a level-control system are investigated here to demonstrate how the shape-based methods work. Block diagrams of both control systems are shown in Fig. 5.3. Process transfer functions of flow PF (s) and level PL (s) are given by 1 , (5.6) 0.2s + 1 1 −s PL (s) = e , (5.7) 15s where the unit of time is minutes. In the level-control system, a flow meter is installed for monitoring and the dynamics from valve opening to flow rate is given by Eq. 5.6. PI-controllers are used in both control systems. Disturbances are added into flow rate. Control parameters and valve-stiction-model parameters are summarised in Tables 5.1 and 5.2, respectively. Under these conditions, simulations were executed and data of 1500 min length were obtained. The sampling interval is 3 s. Simulation results are shown in Figs. 5.4 and 5.5. PF (s) =

Table 5.1 Tuning parameters of PI-controllers in flow and level control systems Proportional gain Integral time [min] Flow control 0.5 0.3 Level control 3 30


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Table 5.2 Parameters of valve-stiction model S [%] J [%] Case 1 (no stiction) 0 0 Case 2 (weak stiction) 1 0.3 Case 3 (strong stiction) 5 1

The shape-based stiction-detection methods were applied to re-sampled data. The sampling interval for flow control and level control is 0.5 min and 2.5 min, respectively. The number of sampling points used for analysis is 300. In all methods, data were mean-centred and scaled to have unit variance. In Method A, ε = 0.1 and εu = εy = 0.5 in all cases. Disturbance Valve position

Controller output

Setpoint

−

Flow rate

C

V

PF

Controller

Valvestiction model

Process

(a) Disturbance Controller output

Setpoint

−

Valve position

Flow rate

C

V

PL

Controller

Valvestiction model

Process

(b) Fig. 5.3 Block diagram of the considered control systems: a) flow control; b) level control

The results are summarised in Table 5.3. LC-F is the case where OP and flow rate are used for detection. On the other hand, LC-L is the case where OP and level are used. Average computational times of Methods A, B and C are 0.07 s, 1.12 s and 0.02 s, respectively. The computational load of Method B is heavier than the others, because it requires solving the optimisation problem. However, the computation of Method B is sufficiently fast in practice. In case FC, Methods A and C can detect stiction successfully. The index SIA of Method A and SIC of Method C become nearly zero under no valve stiction, and they become large when stiction occurs. In addition, the index δA corresponds roughly with S in Table 5.2. This means Method A can quantify the degree of valve stiction. Small δA is derived because its mean is adopted. In fact, δA = 0.00, 0.92, 4.66 when its maximum value is adopted. On the other hand, Method B fails to detect stiction

110


because the relationship between the OP and the flow rate does not take the shape of a parallelogram. The small correlation coefficient r indicates that the assumption of Method B is not satisfied. In addition, another detection method, Horch’s method [47], was applied to the same data for comparison. Horch’s method concludes that stiction occurs in all three cases. It does not function well in this case study because flow rate does not fluctuate persistently.

Fig. 5.4 Operation data generated from simulations of the flow-control system in cases 1 and 3

Fig. 5.5 Operation data generated from simulations of the level-control system in cases 1 and 3


111

In case LC-F, Method A can detect stiction successfully, and δA is coincident with S in Table 5.2. Method B can be used in this case because r is almost one. The index Fmax becomes larger as stiction becomes stronger; thus stiction is successfully detected. In addition, S − J in Table 5.2 is accurately estimated by Fmax . Method C can also detect stiction successfully. That is, Methods A, B and C can detect valve stiction and also quantify the degree of stiction. On the other hand, Horch’s method does not function well. It concludes that stiction occurs in all cases. Table 5.3 Application results of shape-based stiction-detection methods (300 samples of simulation data) Method A Method B SIA δA Fmax r Flow control (FC) Case 1 0.00 0.00 0.00 0.03 Case 2 0.77 0.60 0.00 0.18 Case 3 0.83 3.50 0.00 0.11 Level control - F (LC-F) Case 1 0.00 0.00 0.00 1.00 Case 2 0.56 0.83 0.74 1.00 Case 3 0.79 4.54 4.20 0.99 Level control - L (LC-L) Case 1 0.05 0.54 0.00 0.53 Case 2 0.02 0.68 0.00 0.65 Case 3 0.00 0.00 0.00 0.82

Method C SIC δC 0.04 0.27 0.48 2.49 0.51 2.38 0.01 0.05 0.44 0.54 0.69 1.43 0.03 0.25 0.00 0.08 0.00 0.01

In case LC-L, no method can detect stiction successfully. Methods A, B and C do not work well in such a case where the controlled variable is delayed. These methods should be used only when MV, i.e. flow rate or valve position, is measured. Horch’s method is not applicable because persistent fluctuation is not found in a controlled variable.

5.5 Application to Industrial Loops The results of applying stiction-detection methods to actual plant data are described in this section. Normalised operation data obtained from four chemical processes are shown in Fig. 5.6. In case 1, a level-control loop is investigated, in which valve stiction occurs. In case 2, a flow-control loop suffering from valve stiction is investigated. In case 3, a level-control loop is investigated, and the controlled variable fluctuates due to poor tuning. Finally, in case 4, a flow control loop is investigated, and significant disturbances affect the controlled variable. In each case, operation data are stored in the database with the sampling period of 1 min. Operation data of 24 h are used for analysis. The results of applying the three shape-based methods to the normalised data are summarised in Table 5.4. The settings are the same as before.

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In cases 1 and 2, all methods can detect stiction successfully. In case 3, where there is no stiction but poor controller tuning causes fluctuation, Methods A, B, and C give small indexes SIA , Fmax and SIC , respectively. That is, Methods A, B and C correctly conclude that there is no stiction. However, Horch’s method incorrectly concludes that there is stiction.

Time [min]

MV

Fig. 5.6 Real operation data obtained from four control systems in chemical processes

Table 5.4 Application results of shape-based stiction-detection methods (1440 samples of operation data) Case 1 LC, stiction 2 FC, stiction 3 LC, poor tuning 4 FC, disturbance

A SIA 0.50 0.31 0.13 0.03

δA 1.72 1.21 0.89 0.80

B Fmax 2.12 1.92 0.17 0.00

r 0.98 0.97 0.95 0.13

C SIC 0.62 0.48 0.12 0.14

δC 1.37 1.12 1.01 0.48

In case 4, where no stiction occurs, Methods A and C do reach the right conclusion. Although Method B reaches the right conclusion, the correlation coefficient r is too small and thus the result should not be used. In this case, Horch’s method reaches a wrong conclusion again because the plant is in stable operation when disturbance does not enter. The influence of the number of samples on the stiction-detection performance is investigated. The results of applying five methods to the normalised data of 100 samples are summarised in Table 5.5. In cases 1, 3 and 4, the stiction-detection performance of all methods is similar to that using 1440 samples. However, in case 2, Method B fails to detect stiction because the relationship between the OP and the


113

flow rate does not take the shape of a parallelogram due to the small sample number. That is, Method B requires more data than the other methods. Table 5.5 Application results of shape-based stiction-detection methods (100 samples of operation data) Case 1 LC, stiction 2 FC, stiction 3 LC, poor tuning 4 FC, disturbance

A SIA 0.55 0.28 0.00 0.00

δA 1.77 1.02 0.00 0.00

B Fmax 2.06 0.00 0.00 0.00

r 0.98 0.31 0.98 0.34

C SIC 0.59 0.35 0.06 0.09

δC 1.45 1.05 0.72 2.81

5.6 Summary and Conclusions In this chapter, three types of shape-based stiction-detection methods are introduced. The shape means the relationship between OP and MV in the two-dimensional space, which forms a parallelogram when valve stiction occurs. In practice, flow rate can be used instead of the MV whose data are not available in many cases. The shape-based stiction-detection methods are intuitive, easy to understand, easy to implement, and computationally efficient. In addition, they utilise only routine operation data for detecting valve stiction, and they can also quantify the degree of stiction. Another advantage of the shape-based methods over other conventional methods is that they work even when no periodical oscillation occurs. This is a useful feature in practical use because periodical oscillation does not always occur. The usefulness of the shape-based stiction-detection methods was demonstrated by applying them to simulation data generated by using the valve-stiction model and real operation data obtained from several chemical processes. They work successfully in detecting and quantifying valve stiction. The shape-based stiction-detection methods were developed through close collaboration between industry and academia in Japan, and several petrochemical companies have routinely used these methods for detecting valve stiction. We do not have to choose only one method for stiction detection, and we can use several methods together for improving the reliability of the conclusion.

Chapter 6

Stiction Detection Based on Cross-correlation and Signal Shape Alexander Horch

Among the methods that aim to detect stiction, quite a few focus on analysing the specific signal shapes that loops exhibiting stiction usually show. The idea to use cross-correlation to investigate the signal shape for control loop data has triggered quite some further research in this area. Cross-correlation-based detection is introduced and shown to robustly indicate a stiction behaviour in many typical industrial cases. The chapter then presents a theoretical explanation why cross-correlation can robustly detect more or less typical stiction patterns and also states where it should not be used. There are, however, cases where cross-correlation will give misleading results, the most important being processes that are of an integrating nature (level control). For such loops, an alternative to cross-correlation needs to be offered. Such an approach that is signal-shape based is presented afterwards. Interestingly, the same approach can also be used for self-regulating (i.e. non-integrating) loops as well.

6.1 Introduction Oscillations usually indicate a severe deterioration of performance in process control loops. When an oscillation has been detected, one would like a monitoring system to indicate the possible cause of the problem in order to enable the operators to alleviate the problem. Earlier, only a few methods to solve this task have been proposed, see e.g. [24, 37, 51, 121, 132]. Unfortunately, all the approaches mentioned require either detailed process knowledge, user interaction or rather special process structures. More recent methods are presented in this book. It has been argued [48] that it might be impossible to distinguish between certain oscillation causes based on signal information only. However, in this chapter it will Alexander Horch Group Process and Production Optimization, ABB Corporate Research Germany, Wallstadter Str. 59, 68526 Ladenburg, Germany, e-mail: [email protected]

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116

A. Horch

be shown that the measurements in the case of static friction (stiction) show a unique behaviour. This fact will enable a simple diagnosis procedure. Ideally, the stiction phenomenon results in signals as shown Figs. 6.1a and c. In basically all other cases, the signals are more or less sinusoidal, as indicated in Figs. 6.1b and d.

(a)

(b)

(c)

(d)

Fig. 6.1 Ideal signals: a) control signal for stiction case; b) control signal for no-stiction case; c) process output for stiction case; d) process output for no-stiction case

(a)

(b)

Fig. 6.2 Cross-correlation between process output and control signal: a) stiction; b) no stiction

In practice, of course, the signal shapes do vary quite a lot. This is a dilemma for all methods that use the signal shape itself for diagnosis. A very robust property of the signals in the stiction case is the following: if the left signals in Fig. 6.1 are multiplied for all samples and added up, the sum will be exactly zero. If then the same thing is done for the signals on the right-hand side of Fig. 6.1, the sum will be different from zero (in this case negative). Then, consider the case where the

6 Stiction Detection Based on Cross-correlation and Signal Shape

117

signals are shifted against each other prior to multiplication and summation. This procedure will yield the cross-correlation function between the two signals. Doing this for both cases in Fig. 6.1, the (ideal) cross-correlation functions as shown in Fig. 6.2 are obtained. Now, one can define a simple strategy to diagnose stiction: If the cross-correlation function (CCF) between controller output and process output is an odd function (i.e. asymmetric with respect to (w.r.t.) the vertical axis), the likely cause of the oscillation is stiction. If the CCF is even (i.e. symmetric w.r.t. the vertical axis), then stiction is not likely to having caused the oscillation. Note that the proposed method will work under the following assumptions, which will be discussed later: • • • •

An oscillation has already been detected. The process itself is not integrating (such as level control). The controller has (significant) integral action. The loop does not handle compressible media.

Note that the method may give questionable results if any of these assumptions are violated. The remainder of this part of the chapter is organised as follows: first, the algorithm based on the CCF between process output and control signal is presented in Sect. 6.2. The method has been extensively tested on industrial data. Examples where the cause of the oscillation was known are presented in Sect. 6.3. After a theoretical motivation in Sect. 6.4, some conclusions are given. The rest of the chapter describes a method for stiction detection in integrating loops where the CCF method must not be used.

6.2 The Cross-correlation Function In this section, a simple algorithm is described that allows diagnosis of stiction as motivated in the last section. As mentioned there, this will be done by calculating the CCF between the process output y(t) and the control signal u(t). The CCF between the discrete-time stationary (ergodic) signals u(k) and y(k) is defined as [8] ruy (τ ) = E{u(k + τ )y(k)} .

(6.1)

Assume, furthermore, that the data are pretreated so that it is zero mean. Since the available data sets are of a finite length N, the correlation function cannot be calculated exactly and the computation has to be done from finite data. The estimate of ruy (τ ) can be obtained by N−τ ∑k=0 u(k + τ ) y(k) for τ ≥ 0 ruy (τ ) ≈ . (6.2) for τ < 0 ryu (τ )

118

A. Horch

Note that for the purpose of this, normalisation is not important and can be chosen arbitrarily. For automatic distinction between odd and even correlation functions, it would be sufficient to consider the CCF at lag zero. However, for a practically working algorithm it was found to be better to make use of the CCF up to the first zerocrossing in each direction. For a human, it is simple to tell whether or not a function is odd or even. For automatic diagnosis, one can define different measures to distinguish between odd and even functions. Note that it is important that a procedure for automatic distinction between odd and even functions needs to have a deadzone. This means that the method must not give any indication if the CCF is neither odd nor really even. This may, for example, be the case when there is a compressible medium in the loop. For (one possible) automatic distinction of odd and even CCFs, define the following measures:

τr −τl r0 ropt

= zero-crossing for positive lags , = zero-crossing for negative lags , = CCF at lag 0 , = sign(r0 ) · max |ruy (τ )| , τ ∈[−τl ,τr ]

|τl − τr | , τl + τr |r0 − ropt | ; Δρ = |r0 + ropt |

Δτ =

see also Fig. 6.3. Theoretically, it would be sufficient to consider either Δ τ or Δ ρ . ruy (τ ) ropt r0

−τl

τr

τ

Fig. 6.3 Definition of key variables for the correlation function


119

However, the use of both variables will result in a more reliable procedure. For automatic distinction between the two mentioned oscillation cases, one has to set up limits for Δ τ and Δ ρ . Note that both test variables are bounded in the interval [0, 1], where 0 corresponds to an even CCF and 1 to an odd one. Let now Δ φ denote the deviation of the CCF from the “ideal” position as was shown in Fig. 6.2. Of course, one has to allow a certain deviation range for both cases. In between, however, it is important to have an interval where no decision is taken. This interval corresponds to a CCF that is neither odd nor even. This will typically happen when the oscillation is strongly asymmetrical. It may be an indication of a more unusual problem, like, for example, sensor or other equipment faults. If all three intervals are defined equally large, we allow a deviation of 1/12th of the period (Tp ) of the CCF for each case. See Fig. 6.4. Using these allowed deviation intervals, the values for Δ τ and Δ ρ are related to the diagnosis as follows: √ √3 0.0 < Δ ρ ≤ 2− 2+ 3 ⇒ no stiction , 0.0 < Δ τ ≤ 13 √ 2−√3 1 < Δ ρ ≤ 3 2+ 3 ⇒ no decision , (6.3) 1 2 3 < Δτ ≤ 3 1 3 < Δ ρ < 1.0 ⇒ stiction . 2 3 < Δ τ < 1.0 Note that these results are derived assuming a pure sinusoidal CCF. The rules are also illustrated graphically in Fig. 6.5. It may be interesting to know what the limits of Δ τ and Δ ρ are if no deadzone is used. In that case, the deviation from each ideal CCF is allowed to be 1/8th of a period. The limits for Δ τ and Δ ρ are then as depicted in Fig. 6.5.

(a)

(b)

Fig. 6.4 Allowed deviation (Δ φ = ±Tp /12) of the CCF from the ideal positions: a) stiction case; b) non-stiction case

120

A. Horch 1 3

√ 2−√3 2+ 3

2 3

Δτ

1 3

Δρ 1 2

Δτ

√ √2−1 2+1

0

Δρ 1

Fig. 6.5 The decision rules with (top) and without deadzone (bottom). The limits for non-stiction are marked in light grey, the ones for stiction are marked in dark-grey. The deadzone is the interval between the light and the dark grey bars

6.3 Industrial Examples In this section, the new method will be evaluated on real-world data sets, collected from different pulp and paper mills. In the following, for each data set, OP (dashed) and PV (solid) are plotted (top plot) together with their CCF estimate (bottom plot).

6.3.1 Loop Interaction I Consider two coupled oscillating loops from a paper mill. Figure 6.6 shows a process schematic. Pulp is diluted with water in order to obtain a desired consistency. The water flow is controlled by one control valve and the pulp flow by another. The water valve is known to have static friction that is too high, whereas the pulp valve performs satisfactorily. The measured data and the CCF for the consistency control loop are shown in Fig. 6.7. The CCF here is clearly odd. Note that the data does not have the “ideal” stiction behaviour with the typical rectangular and triangular shapes. Now, consider the flow control loop in Fig. 6.6. Since the consistency loop oscillates, the flow control loop does so too. Thus, there is a situation of an oscillating load disturbance. Data and CCF patterns are shown in Fig. 6.8. As can be seen, the CCF is approximately even and the result is in agreement with the knowledge about the loop.


121

QC

Water Q Pulp F FC

Fig. 6.6 Two coupled loops from a consistency control loop 04 03

Data

02 0.1 0 −0.1 −0 2 200

400

600

800

1000

1200

1400

Time 1

CCF

05

0

−0 5

−1 −200

−150

−100

−50

0

50

100

150

200

Lags Fig. 6.7 Consistency loop (loop interaction I)

6.3.2 Loop Interaction II Consider a two-by-two system similar to the one in Fig. 6.6. It is known from experiments that both loops oscillate because the PI-controller in the consistency loop was too tightly tuned. The oscillations have been eliminated by re-tuning the controller. The oscillation in the flow control loop was also stopped by this. Hence, the correlation functions for both loops are expected to be even functions. Figures 6.9 and 6.10 show that this is actually the case.

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0.1

Data

0.05 0 −0.05 −0.1 −0.15 100

200

300

400

500

600

Time 1

CCF

0.5

0

−0.5

−1 −200

−150

−100

−50

0

50

100

150

200

Lags Fig. 6.8 Flow loop (loop interaction I) 10

Data

5

0

−5 50

100

150

200

250

300

350

400

Time 1

CCF

0.5

0

−0.5

−1 −60

−40

−20

0

Lags Fig. 6.9 Flow loop (loop interaction II)

20

40

60


123

Note that the CCFs at lag zero are positive and negative, respectively. This is because the consistency loop has a negative gain. In this example we obviously have a situation where the root-cause of the problem cannot be diagnosed using the CCF. Another way of attacking this problem – based on some process knowledge – has been described in [48].

6.3.3 Flow Control Loop I Another example of an oscillating flow control loop is considered here. From experiments it is known that interaction with other control loops is not very likely. Also, the controller re-tuning did not remove the oscillation. One likely explanation is a sticking control valve. There are data sets available for this loop that were collected daily over a period of several months. The oscillation is mostly present but it changes shape dramatically on different days. The correlation result for a case that is not easily diagnosed by visual inspection is shown in Fig. 6.11. This can be compared with another data set from the same control loop shown in Fig. 6.12. It can be seen that the CCFs are odd functions in both cases. Even though the signal shapes are different, there is almost no difference in the CCF pattern. Hence, one may (correctly) conclude that the loop exhibits stiction behaviour. 6 4

Data

2 0 −2 −4 50

100

150

200

250

300

350

400

10

20

30

40

Time 1

CCF

05

0

−0 5

−1 −40

−30

−20

−10

0

Lags Fig. 6.10 Consistency loop (loop interaction II)

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Data

5

0

−5

50

100

150

200

250

300

350

400

450

500

10

20

30

40

50

Time 1

CCF

0.5

0

−0.5

−1 −50

−40

−30

−20

−10

0

Lags Fig. 6.11 Flow loop Ia 4

Data

2 0 −2 −4 −6 50

100

150

200

250

300

350

400

Time 1

CCF

0.5

0

−0.5

−1 −50

−40

−30

−20

−10

0

Lags Fig. 6.12 Flow loop Ib

10

20

30

40

50


125

6.3.4 Flow Control Loop II For this example it is known that the loop in question has problems with valve stiction. Apart from that, the flow measurement is very noisy. If one assumes that the oscillation has been detected, the CCF method will easily diagnose stiction. The results from correlation analysis are shown in Fig. 6.13. Note that the measured process output does not have a pronounced rectangular shape as stiction often has. However, it does if one filters the data, but this is not necessary when using the CCF. It will yield the same results in either case. Note also that the length of the data batch can be very short. As an illustration consider Fig. 6.14 where very little data from the same loop was used. The interesting result is that the CCFs are very similar (at least concerning the asymmetry w.r.t. the vertical axis), no matter how much data one uses.

4

Data

2 0 −2 −4 −6

50

100

150

200

250

300

350

400

450

500

20

40

60

80

100

Time 1

CCF

05

0

−0 5

−1 −100

−80

−60

−40

−20

0

Lags Fig. 6.13 Flow loop II: cross-correlation for a long data set

6.3.5 Level Control Consider now an example of a level control loop that is known to oscillate due to stiction. As can be seen in Fig. 6.15, the CCF is even, which would result in a deceiving diagnosis. The problem of diagnosing and detecting stiction in control loops with integrating processes wil be discussed below (see also [48]), where an

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A. Horch

alternative method – based on the detection of abrupt changes in the process variable – will be presented in order to cover the case of integrating plants. 1

Data

0.5 0 −0.5 −1 20

40

60

80

100

120

140

Time 1

CCF

0.5

0

−0.5

−1 −100

−80

−60

−40

−20

0

20

40

60

80

100

Lags Fig. 6.14 Flow loop II: cross-correlation for a short data set

6.4 Theoretical Explanation The aim of this section is to motivate the diagnosis procedure described earlier in this chapter. Consider a control loop that oscillates with constant amplitude and frequency, i.e. all transients have vanished. Assume that the reference value is constant; see Fig. 6.16. Then, the relationship between controller output u(t) and process output y(t) is completely described by the controller F(s), U(s) = −F(s)Y (s) . where U(s) and Y (s) are the Laplace transform of process output y(t) and control signal u(t), respectively. This relation will be used frequently in the following where the theoretical CCF for different possible root-causes of oscillations is discussed.


127

6.4.1 Correlation for Oscillating External Disturbances Assume that the process in question is controlled by a PI-controller, i.e. 1 F(s) = Kc 1 + . TI s Since one is interested in the phase shift between u(t) and y(t), consider the Bode plot of −F(s). The phase curve of −F(s) is obtained as 3 Δ φ = arg{−F(iω )} = − π + arctan(TI ω ) . 2

(6.4)

The Bode plot for −F(s) is shown in Fig. 6.17 where the cut-off frequency is ωc = 1/TI . In general, low-frequency disturbances will be eliminated efficiently by the PI-controller since a controller with integral action yields high loop gain at low frequencies.

4

Data

3 2 1 0 −1 −2 50

100

150

200

250

300

350

400

450

500

550

600

Time 1

CCF

0.5

0

−0.5

−1 −80

−60

−40

−20

0

20

40

60

80

Lags Fig. 6.15 Level control loop: if CCF was used, it would wrongly indicate stiction in this case

On the other hand, if a medium- or high-frequency oscillating disturbance enters the loop, it may not be attenuated sufficiently so that an oscillation will be detected. However, in that case the phase shift will typically be approximately π and thus be

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classified correctly by the algorithm.1 This is because the second term of Eq. 6.4 approaches π/2 for high frequencies. u(t)

r(t)

y(t)

G(s)

F(s) −

Fig. 6.16 A general control loop

2

Magnitude

10

ωc

1

10

0

10

−3

10

−3

10

10

−2

10

−1

10

−2

10

0

10

−1

10

1

0

10

−160 −180

Phase

−200 −220 −240 −260 −280 10

1

log ω Fig. 6.17 Bode diagram for −F(s): exact (solid) and asymptotic (dash-dotted)

It is therefore important that only signals that are clearly oscillating are used. Therefore, one of the detection methods described earlier must have detected an oscillation.

1 If the phase shift if exactly −π, the deviation of the CCF from the ideal position (Fig. 6.2) will be zero.


129

6.4.2 Tight Tuning Oscillation may also be caused by tight controller tuning in connection with some non-linearity (except for stiction), typically backlash or deadzone. In such a case, the CCF will typically be even. The following reasoning can be used: when a loop oscillates due to any of the reasons mentioned above, it will do so at (or close to) its critical frequency ωu , i.e. when the phase shift between control error and process output is (approximately) −π. Hence, the CCF will be approximately even. Therefore, as mentioned before, the CCF method cannot be used to distinguish between external load disturbances and oscillation caused by tight control in connection with some non-linearity.

6.4.3 Correlation in the Presence of Stiction For ideal stiction one can calculate the CCF analytically. Stiction implies that the valve piston periodically pops from one position to another. This, together with the assumption of a PI-controller results in the typical stiction pattern (see Fig. 6.12), i.e. a square wave (process output) and a triangular wave (control signal). Expressions for the CCF in the stiction case will now be derived. Then, let each of the signals y(t) and u(t) be expanded in a Fourier series f (t) =

∞ a0 + ∑ (ak cos(kt) + bk sin(kt)) . 2 k=1

(6.5)

Assume that the signals considered here are described such that they are asymmetric with respect to the origin;2 see Fig. 6.18. This implied that ai = 0, ∀ i in Eq. 6.5. The key point of the new method is to check whether the CCF is odd or even. This is the same as determining whether the phase shift between both signals is −π or −π/2, respectively. Let the signals hence be described by their Fourier series expansion as u(t + τ ) = y(t + Δ φ ) =

∞

∑ bk sin[k(t + τ )]

(6.6)

∑ ck sin[l(t + Δ φ )] ,

(6.7)

k=1 ∞ l=1

where τ is the lag of the CCF. Inserting these in the continuous-time definition of the cross-correlation 1 T →∞ 2T

ruy (τ ) = lim

T −T

u(t + τ )y(t) dt

(6.8)

gives 2

Here it is assumed – without loss of generality – that the signals are periodic with period 2π.

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A. Horch

1 T →∞ 2T

ruy (τ , Δ φ ) = lim

∞

∑

∞

∑ bk cl

k=1 l=1

T −T

sin[k(t + τ )] sin[l(t + Δ φ )] dt .

After some tedious but straightforward calculations, this is evaluated to be ruy (τ , Δ φ ) =

1 ∞ ∑ bk ck cos[k(τ − Δ φ )] , 2 k=1

(6.9)

where

1 π u(t) sin(kt) dt , π −π π 1 ck = y(t) sin(kt) dt . π −π

bk =

(6.10)

Thus, u(t) and y(t) are as shown in Fig. 6.18. y(t)

u(t)

C −π

−π

m

π

(a)

π

(b)

Fig. 6.18 a) Ideal signals in the presence of stiction: b) control signal u(t); b) process output y(t)

For the case of a rectangular signal y(t), 4C if k = 1, 3, 5, . . . . bk = kπ 0 if k = 2, 4, 6, . . . A triangular signal u(t) results in (k+3) 4m 2 if k = 1, 3, 5, . . . . ck = k2 π (−1) 0 if k = 2, 4, 6, . . .

(6.11)

(6.12)

Inserting Eqs. 6.11 and 6.12 into Eq. 6.9 yields ruy (τ , Δ φ ) =

8 ∞ cos[(2i − 1)(τ − Δ φ )] (−1)i+1 . ∑ π2 i=1 (2i − 1)3

(6.13)


131

Figure 6.19 shows3 ruy (τ , − π2 ) in Eq. 6.13 using only the first element and after having used the first 50 elements of the sum in Eq. 6.13. It can be seen that it is sufficient to truncate the infinite sum after the first element. Truncating this series after the first element and with Δ φ = −π/2 gives 8 π ≈ 2 sin(τ ) . ruy τ , − 2 π

(6.14)

Now, it can easily be seen that the CCF for u(t) and y(t) for the case of stiction is in good agreement with the empirical results described above.

0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –3

–2

–1

0

1

2

3

Fig. 6.19 Approximation of ruy (τ , − π2 ); truncation after first term (dashed) and after 50 terms (solid)

6.5 Conclusions (Cross-correlation Method) A simple method for diagnosing the likely cause of a detected oscillation in a control loop is presented. Only routine operating data that are usually available in the control system are needed. A sustained oscillation has to be detected (and thus be confirmed) first; otherwise wrong indications may be given. This method, which is based on cross-correlation, is easy to implement and does not require any process knowledge. It is possible to distinguish between the two most important reasons for 3

Note that Δ φ = −π/2 corresponds to the ideal stiction signals as shown in the left part of Fig. 6.1.

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A. Horch

oscillations in control loops, namely static friction in the actuator and loop interaction. The method has been successfully evaluated on industrial data sets where the actual oscillation cause was known.

6.6 Stiction Detection for Integrating Processes As already pointed out, a weakness of the CCF method described above is that it cannot (and must not!) be used for integrating processes. The reason being the fact that the integrator in the process changes the correlation structure differently for the stiction and the non-stiction case. An ad hoc remedy would be to try to re-calculate the valve position signal, i.e. to differentiate the process output. Unfortunately, this does not work as is motivated in [48], Appendix D. An alternative method will therefore be introduced here. The basic idea is to check whether the process output exhibits regular abrupt changes or not. These abrupt changes will be detected by considering the second (filtered) derivative of the process output y(t). If y(t) is triangular (perfectly and without noise), the second derivative will be a pulse train only. Then, neglecting the pulse train and assuming additional Gaussian noise the second derivative will have a probability distribution that is close to normal. On the other hand, in the non-stiction case, the (oscillating) process output and its second derivative are usually relatively sinusoidal. Hence, the probability distribution has two separate maxima. The proposed detection algorithm tests which of the two distributions is more likely to fit the data. Examples using industrial data are presented in Sect. 6.8. Since one has to be careful when differentiating noisy signals, the signals will be filtered first. The filter constant can be chosen dependent on the oscillation frequency. As a matter of fact, the idea of considering the distribution of the second derivative of the process output can also be used for detection of stiction in self-regulating processes (for which the CCF method usually works well). The only difference is that one has to consider the first instead of the second derivative of the process output because there is no integration in the process.

6.7 Detection in Integrating Loops – Basic Idea In this section, a simple method is described that detects abrupt changes in the process output. The main idea is to fit two different distribution functions (one corresponding to the stiction case and one to the non-stiction case) to the sample histogram of the second derivative of the measured process output. For ideal signals and pure differentiation (without filtering) the second derivatives (of the process output), and their sample histograms, are shown in Fig. 6.20. The decision procedure can now be summarised as follows:


133

Consider the second (filtered) derivative of the process output. Check whether the probability density function fits better to the top or the bottom distribution in Fig. 6.20. A better fit to the first distribution will indicate stiction, a better fit for the second one non-stiction. As a motivation consider Fig. 6.21 where ideal signals for the stiction and the non-stiction case are shown. It can be seen that if the process output is a sinusoid, the second derivative is a (negative) sinusoid as well. For the ideal stiction case, the second derivative is a pulse train with alternating sign. In the following, first the filtering and the differentiation of the process output will be discussed. Then, it will be described how to obtain the sample histogram of the second derivative. Finally, in order to be able to fit distribution functions to the sample histograms, the theoretically expected distributions have to be derived.

(a)

(b) Fig. 6.20 Ideal second derivatives of the process output and their sample histograms for a) the stiction case; b) the non-stiction case

6.7.1 Differentiation and Filtering It is well known that differentiation of noisy signals is a trade-off between accuracy and noise amplification. The choice of the filter bandwidth can be done using information of the oscillation frequency that can easily be obtained, for example, as a by-product from the oscillation detection. The filter constant has to be chosen such that the noise is filtered out without affecting the shape of the oscillation too much. In this work, a first-order discrete-time low-pass filter,

134

A. Horch

y(t)

t

dy(t) dt

d2 y(t) dt 2

(a)

y(t)

t

t

dy(t) dt

t

t

d2 y(t) dt 2

t

(b)

Fig. 6.21 Motivation of the proposed detection mechanism: the process output and its first and second derivatives are shown for a) the stiction case; b) the non-stiction case

H1 (q−1 ) =

1−α , 1 − α q−1

(6.15)

was used before each differentiation. The cut-off frequency of the filter is chosen as ωc = 3ωosc , where ωosc is the oscillation frequency. Note that the choice of the filter cut-off frequency is very important. If it is too small, the form of the oscillating signal will become too smooth (yielding almost surely a Gaussian distribution). If it is too large the noise will be amplified too much by the differentiation. The filter pole in Eq. 6.15 is determined as

α = e−ωc Ts ,

(6.16)

where Ts is the sampling interval. The whole filtering and differentiation procedure can be summarised in the filter 2 (1 − α )(1 − q−1 ) , (6.17) H2 (q−1 ) = (1 − α q−1 ) yielding the second (filtered) derivative of the process output as yd f (t) = H2 (q−1 ) y(t) .

(6.18)


135

6.7.2 Sample Histogram The sample histogram of the second derivative of the process output can easily be obtained. The only choice one has to make is the number of classes. In the statistical literature guidelines for this choice are given when one wants to perform a goodness-of-fit test. A suitable number of classes K for data batches up to N = 2000 samples has been proposed by Williams [133]. For batches with more samples, [8] proposes to use K = 1.87 (N − 1)0.4 . The combination of both suggestions is shown in Fig. 6.22.4

# of classes K

50

40

30

20

10

0

500

1000

1500

# of samples N

2000

2500

3000

Fig. 6.22 Recommended number of histogram classes K for use in goodness-of-fit test

The values in Fig. 6.22 have been found to fit well for the purpose of this chapter. Of course, the raw histogram has to be normalised for comparison with theoretical probability density functions. This is achieved as follows. Let Ni , N and Δ x denote the number of elements in class i, the number of samples of data and the width of each class, respectively. Then, the probability pi (x) that a data sample is an element of class i is obtained as

Δ x → 0N → ∞

Ni . Δ xN

(6.19)

In practice, one cannot reach the limit and once N is determined, one has to divide Ni by the number of samples N and the class width Δ x in order to obtain an estimate of pi (x). A note has to be made here about the length of the data batch. Assuming a reasonable sampling frequency (i.e. fast enough to cover the oscillation well), one has to make sure that the data contain only a small amount of oscillation periods. This is especially important if the signal shape changes a lot over time. It has been found that a value of 5–10 oscillation periods is a reasonable value.

4

These rules were developed for a χ 2 goodness-of-fit test with a 5% level of significance.

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A. Horch

6.7.3 Distribution for the Stiction Case As can be seen from Fig. 6.20, for ideal stiction the sample histogram of the second derivative of the process output will be purely white noise with a pulse train (assuming Gaussian measurement noise). The distribution will thus mainly be Gaussian if we assume that the number of oscillation periods is small compared to the number of samples.5 The ideal probability distribution for the stiction case is hence normal, i.e. 1 −(x − μ )2 exp . fG (x) = √ 2σ 2 2πσ

(6.20)

This distribution can easily be fitted to the sample histogram since the data mean μ and standard deviation σ can be computed directly from the data. However, for real data, the peaks in the second derivative will not be infinitely narrow. The sample histogram will therefore have significant tails on both ends, which will typically increase the variance. The same effect has the – already mentioned – filtering that is necessary prior to differentiation. The choice of the filter bandwidth will of course influence the resulting distribution. Therefore, the distribution (Eq. 6.20) will now be modified. One alternative is the computation of the theoretical probability distribution when filtering and differentiating a triangular signal with additional white noise. This may be possible but will lead to very complicated expressions. Therefore, another approach is chosen here. Consider a noise-free triangular signal and its second (filtered) derivative. Let the filter pole be chosen from Eq. 6.16. Figure 6.23 shows the (noise-free) data and the resulting sample histogram. It can clearly be seen that the peaks in the second derivative have now a significant contribution to the sample histogram (compare with Fig. 6.20!). A simple way of approximating the histogram in Fig. 6.23 would be to combine a Gaussian distribution with a uniform distribution, i.e. fE (x) = (1 − ε ) fG (x) + ε fu (x) ,

(6.21)

where fu (x) =

1 2A

|x| ≤ A , 0 |x| > A

where A is the amplitude of d2 y(t)/dt 2 . Note that the integral of Eq. 6.21 is ∞ −∞

fE (x) dx = 1

5 This is an implicit assumption that the data has to be sampled sufficiently fast such that the oscillation has a reasonable (e.g. > 30) amount of samples per period.


137

(a)

(b) Fig. 6.23 Noise-free (ideal) signals in the stiction case: a) process output; b) second (filtered) derivative with its sample histogram

as required. It was found that this modification is sufficient for real data. Finally, see Fig. 6.24 for a plot of fE (x) as a function of x. Of course, one has to choose the parameter ε . It could be estimated together with the variance of the distribution. However, as will be shown later, for the non-stiction case, there is only one parameter to be estimated. We therefore chose to fix the parameter ε to a predefined value of 0.3. This choice has been found by extensive tests on both real and simulated data. Note that ε = 0 corresponds to a pure Gaussian distribution and ε = 1 to a pure uniform distribution. In no case was it found that the diagnosis was changed for reasonable variation of ε . Even a choice of ε = 0 has been tested on data without problems. However, using the proposed choice will make the decision unique. Although modified, the distribution (Eq. 6.21) will be referred to as the Gaussian distribution in the following. 1.5

1

0.5

0

−1

−0.8

−0 6

−0.4

−0 2

0

0.2

04

0.6

0.8

Fig. 6.24 Combined Gaussian and uniform distribution fE (x) from Eq. 6.21

1

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The variance σ 2 in Eq. 6.20 will hence be estimated using a non-linear leastsquares algorithm. If it is assumed that de-trended data is used such that we have μ = 0. The estimation of the standard deviation may improve the fit compared to using the variance as calculated directly from data. In order to avoid local minima, the standard deviation obtained directly from data is used as an initial guess. The estimation will be done in the same way as for the non-stiction case described below.

6.7.4 Distribution for the Non-stiction Case Consider the bottom plot of Fig. 6.20. As motivated earlier, in the non-stiction case, the signal in question is assumed to be a sinusoid with additional white noise. It is assumed that the noise is independent of the sinusoid. In order to justify describtion of a sinusoid in probabilistic terms, the initial phase can be thought of as being a random variable. The problem is then to find the probability function for the signal z(t) = x(t) + e(t) = A sin[ω t + y(t)] + e(t), where e(t) is zero mean white noise and y(t) is uniformly distributed. The resulting probability density function when considering the sum of two stochastic variables x(t) and e(t) is given by the convolution of their individual density functions, i.e. fZ (z) =

∞ −∞

fX (x) fG (z − x) dx .

(6.22)

The density function for a sinusoid with random initial phase can be found using the following result from the statistical literature; see for example [45]. Theorem 6.1. If Y is a continuous random variable with probability density function fY (y) that satisfies fY (y) > 0 for a < y < b, and x = H(y) is a continuous strictly increasing or strictly decreasing function of y, then the random variable X = H(Y ) has the density function fX (x) = fY (H −1 (x))

d −1 H (x) . dx

For the case of a sinusoid, x(t) = A sin[ω t + y(t)], we have H −1 (x) = −ω t + arcsin x(t) A and 1/π for H(− π2 ) < x < H( π2 ) fY (H −1 (x)) = , 0 else and for |x| < A, d −1 1 H (x) = √ . 2 dx A − x2 Then, it follows that


139

√ −1 2 − x2 A for |x| < A , π fX (x) = 0 else

(6.23)

!

∞ fX (x) dx = 1 is satisfied. Figure 6.25 shows a plot of Eq. 6.23 as a funcwhere −∞ tion of x.

1.2 1 0.8 0.6 0.4 0.2 0 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

Fig. 6.25 Probability density function fX (x) of a pure sinusoid with stochastic initial phase (A = 1)

Then, with fG (x) from Eq. 6.20, the convolution integral (Eq. 6.22) becomes 2 A exp − (z−x−2μ ) 1 2σ √ dx . (6.24) fZ (z) = √ A2 − x2 σ 2π3 −A Figure 6.26 shows a plot of the distribution fZ (z). 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –2

–1.5

–1

–0.5

0

0.5

1

1.5

2

Fig. 6.26 Probability density function of a sinusoid (A = 1) with additional zero-mean white noise (σ = 0.1)

The distribution (Eq. 6.24) exhibits a certain amount of camelicity.6 It will therefore be referred to as the “camel distribution” in the following. Unfortunately, there is no analytic solution to Eq. 6.24, and it has therefore to be evaluated numerically. There are three unknowns in Eq. 6.24, two of which can easily be determined. Since 6

This term was coined by Dr. Krister Forsman from Perstorp, Sweden.

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A. Horch

the data used is assumed to be de-trended we have μ = 0. The amplitude A can easily be estimated from data. Then, the parameter σ is used to fit the distribution (Eq. 6.24) to the sample histogram:

σˆ = arg min ∑ [ fZ (z, σ ) −Y ]2 ,

(6.25)

σ

where Y contains the data from the sample histogram. The optimisation can be done using a standard non-linear least-squares fit algorithm (i.e. lsqcurvefit ® in MATLAB ). A disadvantage is that the integral (Eq. 6.24) has to be evaluated numerically during the fitting procedure. This will, however, be very fast since the limits of the integral are finite. Note also that one has to make sure that the fitting algorithm does not find a solution that is very similar to a normal distribution (i.e. that the two maxima in Fig. 6.26 are merged into one). This may happen when the estimated value of σ is “too large” compared to A. The disappearance of the two maxima can be avoided by, for example, constraining the admissible values for σ to be significantly smaller than A when running the non-linear least-squares algorithm (here we chose σ < 0.4 A). To summarise: with some additional white noise, one may have a situation as shown in Figs. 6.27 and 6.28. 1 0.5

d2 y(t)/dt 2

y(t)

0.5

0

−0.5

0 −0.5 −1

−1

100

200

300

400

500

600

100

200

Time (a)

400

500

1.5

fZ (z)

fE (x)

1.5

1

0.5

0

300

Time (b)

1

0.5

−2

−1

0

1

2

0

−2

−1

0

Classes

Classes

(c)

(d)

1

2

Fig. 6.27 The stiction case, ideal data: a) PV; b) second derivative of PV; c) fit of distribution to normal; d) fit to camel distribution


141

1

d2 y(t)/dt 2

1

y(t)

05 0 −0 5

05

0

−0 5

−1 200

300

Time

400

−1

500

08

08

06

06

fZ (z)

fE (x)

100

04 02 0 −4

100

200

300

400

Time

04 02

−2

0

Classes

2

4

0 −4

−2

0

Classes

2

4

Fig. 6.28 The non-stiction case, ideal data

There, the measured process outputs are shown in the top left plot, the (filtered) second derivative in the top right plot and the sample histogram with the best fit of Gaussian and camel distributions in the bottom left and right plot, respectively. The results from all examples will be shown in plots with the same layout. The algorithm for diagnosing stiction in integrating control loops is summarised as: Step 1. Check if the loop is oscillating and determine the frequency of oscillation [33]. Step 2. Compute the filtered second derivative yf (t) using Eq. 6.17. Step 3. Compute the sample histogram (Eq. 6.19) of the second derivative. Step 4. Estimate the standard deviation in Eq. 6.21 in order to fit the sample histogram as well as possible. Step 5. Determine the (approximate) amplitude of the second filtered derivative yf (t). Step 6. Estimate the standard deviation in Eq. 6.24 in order to fit the sample histogram as well as possible. Step 7. If the fit for the Gaussian distribution is significantly better than the fit of the camel distribution, conclude stiction. In the opposite case, conclude no stiction. If both fits are approximately equal (e.g. difference smaller than 10%), no decision is made. It is worth mentioning that for each distribution, one single parameter is estimated (standard deviation of the noise). It is therefore fair to compare the resulting MSEs directly. The MSE is calculated from MSE = ∑ [ f (σˆ ) −Y ]2 ,

(6.26)

142

A. Horch

where f (σˆ ) is the distribution function for the case in question using the standard deviation that gives the best fit to the data. The MSE can be obtained as the value of the loss function from the numerical optimisation (Eq. 6.25).

6.8 Examples The method is now demonstrated on different industrial examples.

6.8.1 Level Control Loop with Stiction Consider a level control loop that is known (from experiments) to oscillate due to stiction. When applying the decision algorithm to the process output, the results shown in Fig. 6.29 are obtained. 1 4

d2 y(t)/dt 2

y(t)

3 2 1 0 −1

0.5 0 −0.5 −1

−2 200

Time

300

400

100

15

1.5

1

1

fZ (z)

fE (x)

100

05

0 −2

200

Time

300

0.5

−1

0

Classes

1

2

0 −2

−1

0

Classes

1

2

Fig. 6.29 Level control loop with stiction

The MSE for the Gaussian distribution is 0.97 compared to 2.01 for the camel distribution. It can hence be (correctly) concluded that the oscillation is due to stiction.


143

6.8.2 Level Control Loop Without Stiction Consider a level control loop that is known to oscillate for a reason other than stiction. The oscillation could in this case be removed by re-tuning the controller. The results when applying the diagnosis algorithm are shown in Fig. 6.30. The fit for the Gaussian distribution is 1.17 compared to 0.46 for the camel distribution. It can hence be (again correctly) concluded that the oscillation is not due to stiction.

6.8.3 Level Control Loop with Deadband Consider another level control loop that is known to oscillate due to deadband in the controller software. The oscillation could in this case be removed by re-tuning the controller. The results when applying the diagnosis algorithm are shown in Fig. 6.31. The fit for the Gaussian distribution is 1.52 compared to 0.70 for the camel distribution, therefore yielding a correct diagnosis, namely non-stiction. 1

2

0.5

d2 y(t)/dt 2

3

y(t)

1 0 −1

−1 −1.5

−2 200

400

Time

600

800

200

1

1

08

0.8

fZ (z)

fE (x)

0 −0.5

06 04 02 0

400

Time

600

800

0.6 0.4 0.2

−2

0

Classes

2

0

−2

0

Classes

2

Fig. 6.30 Level control loop without stiction

6.9 Self-regulating Processes As mentioned above, oscillation in self-regulating processes due to stiction can very conveniently be detected using the CCF method. However, the same idea as presented in the previous section can also be used on self-regulating processes. The only difference is that one makes use of the first derivative of the process output instead of the second. The method is now demonstrated on two examples.

144

A. Horch 3

1

d2 y(t)/dt 2

2

y(t)

1 0 −1

05

0

−0 5

−2 −3

0

50

100

150

Time

−1

200

1

0.8

08

0.6 0.4

100

150

Time

200

06 04

0.2 0 −1.5

50

12

1

fZ (z)

fE (x)

1.2

0

02 −1

−0.5

0

Classes

05

1

0 −1.5

1.5

−1

−0 5

0

Classes

05

1

1.5

Fig. 6.31 Level control loop with deadband

6.9.1 Flow Control Loop with Stiction Consider a flow control loop that is known to oscillate due to stiction. The results of the diagnosis algorithm are shown in Fig. 6.32. The mean-squares error for the Gaussian distribution is 0.80 compared to 1.36 for the camel distribution. It can hence be concluded that the oscillation is due to stiction. 1 10 0.5

dy(t)/dt

y(t)

5 0 −5

0 −0.5

−10

−1 50

100

Time

150

200

50

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

150

1.2

1

fZ (z)

fE (x)

1.2

100

Time

−1

0

Classes

1

Fig. 6.32 Flow control loop with stiction

0

−2

−1

0

Classes

1

2


145

6.9.2 Flow Control Loop Without Stiction Consider a flow control loop that is known to oscillate due to interaction with another loop. The results when applying the diagnosis algorithm are shown in Fig. 6.33. The fit for the Gaussian distribution is 2.44 compared to 1.47 for the camel distribution. It can hence be concluded that the oscillation is not due to stiction. 1 4 0.5

dy(t)/dt

y(t)

2 0 −2

0 −0.5

−4

−1 100

150

Time

200

50

1

1

08

0.8

fZ (z)

fE (x)

50

06

0.4

02

0.2 −2

0

Classes

150

0.6

04

0

100

Time

0

2

−2

−1

0

Classes

1

2

Fig. 6.33 Flow control loop without stiction

6.9.3 Loops with Dominant P-control There may be cases where the CCF method gives no (or possibly a wrong) indication. One of them is when the controller is basically proportional. Then, the control signal has a very steep initial phase after each peak; see Fig. 6.34. These potential problems were also discussed in [32]. In such a case the CCF is neither even nor odd, as illustrated in Fig. 6.35.

control signal

22.1 22 21.9 21.8 21.7 21.6 21.5 20

40

60

80

100

Time

120

140

160

180

200

Fig. 6.34 Typical control signal from a loop exhibiting stiction when the proportional part dominates the controller

146

A. Horch

2

data

1 0 −1 −2 −3 50

100

150

200

Time

250

300

1

CCF

0.5

0

−0.5

−1 −50

−40

−30

−20

−10

0

Lags

10

20

30

40

50

Fig. 6.35 Example of a flow control loop with stiction for which the CCF method is used: control signal (solid) and process output (dash-dotted)

Figure 6.36 shows the results of the histogram-based method on the above data. The MSEs are 0.77 and 1.87 for the Gaussian and the camel distribution, respectively. It can hence (correctly) be concluded that the loop oscillates due to stiction. 1 2 0.5

dy(t)/dt

y(t)

1 0 −1 −2

0

−0.5

−3 50

100

150

Time

200

250

−1

300

1

08

0.8

06

150

Time

200

250

0.6

04

0.4

02

0.2

0

100

1.2

1

fZ (z)

fE (x)

12

50

−1

0

Classes

1

0

−2

−1

0

Classes

1

Fig. 6.36 Example of a flow control loop with stiction and dominant P-control

2


147

6.10 Summary and Conclusions Two methods have been proposed that allow detection of stiction in both selfregulating and integrating processes. The cross-correlation-based algorithm will work for self-regulating processes only. Therefore, an alternative has been presented that is based on the fact that stiction involves abrupt changes in process variables. This shape-based method (histogram method) can be used for both integrating and self-regulating processes. Hence, for self-regulating processes, there are two different methods presented. The cross-correlation method seems to be more convenient since it does not require any parameters. On the other hand, the histogram-based method does cover more cases. For more automatic use (e.g. in a monitoring tool), both methods should be combined in order to get redundancy, and thus a more reliable diagnosis. The histogram-based method has been patented by ABB [49].

Chapter 7

Curve Fitting for Detecting Valve Stiction Q. Peter He and S. Joe Qin

In this chapter, a curve-fitting method is introduced for detecting valve stiction. It is assumed that the loop in question is known to be oscillating by using any of the oscillation detection methods presented in Chap. 4. The stiction-detection method is based on the curve-fitting results of the output signal of the first integrating component located after the valve, i.e. the controller output for self-regulating processes or the process output for integrating processes. In this chapter, we answer important questions such as which signal should be looked after, and why that signal takes a certain shape in the presence of valve stiction. A metric referred to as the stiction index (SI), is introduced based on the proposed method to facilitate the automatic detection of valve stiction. The effectiveness of the proposed method is demonstrated using both simulated and real industrial data sets.

7.1 Introduction In the process industry, oscillations are commonly caused by any one or a combination of the following reasons: (i) limit cycles caused by valve stiction or other process non-linearities, (ii) poor controller tuning, (iii) poor process and controlsystem design, and (iv) external oscillatory disturbances [10, 26, 84]. Simple and efficient methods have been developed to detect oscillating control loops automatically [33, 37, 84], as has been discussed at length in Chap. 4. In this chapter, we

Q. Peter He Department of Chemical Engineering, Tuskegee University, 522 B Luther H. Foster Hall, AL 36088, USA, e-mail: [email protected] S. Joe Qin Department of Chemical Engineering and Materials Science, Electrical Engineering, and Industrial and Systems Engineering, University of Southern California, 925 Bloom Walk, HED 211, Los Angeles, CA 90089-1211, USA, e-mail: [email protected]

149

150

Q.P. He and S.J. Qin

focus on diagnosing valve stiction from other causes given that oscillation has been detected. In the last decade, several methods [24, 51, 121, 132] have been developed to detect valve stiction for automated monitoring systems [47]. Most of these methods are discussed extensively in the other chapters of this book so only a short review is given here. Horch [47] presents the first automatic detection method based on the cross-correlation function (CCF) between the controller output (OP) and the process variable (PV), which is applicable to non-integrating processes. Later, Horch [48] proposed another method to detect valve stiction in integrating processes by considering the probability distribution of the second-order derivative of the controlled variable. Singhal and Salsbury [110] propose a valve-stiction-detection method based on the comparison of areas before and after the peak of an oscillating control-error signal, i.e. the difference between the setpoint (SP) and the controlled variable (PV). Kano et al. [63] proposed two valve-stiction-detection methods, one method requires knowledge of the valve position (MV), and the other method is based on detecting characteristic parallelogram shapes within the PV–OP plots. He and Pottmann [42] developed a stiction-detection technique for self-regulating processes in which the OP is fitted piecewise to both triangular and sinusoidal waves using least-squares estimation. A better fit to a triangular wave indicates valve stiction, while a better fit to a sinusoidal wave indicates non-stiction. Rossi and Scali [98] propose a similar technique independently, in which the PV signal is fitted using three different models: a relay wave, a triangular wave and a sinusoidal wave. Srinivasan et al. [114,116] present a qualitative pattern-matching approach for simultaneous oscillation detection and valve-stiction detection using dynamic time warping, as well as a Hammerstein model identification approach for diagnosis and quantification of valve stiction. Yamashita [134] proposes another pattern-matching method based on the typical patterns from valve input and valve output in a control loop. Choudhury et al. [17] proposed a method for detecting and quantifying stiction in a control valve. The method consists of three steps. First, it detects non-linearity in a control loop. Next, if non-linearity is detected, an ellipse is fitted to the filtered PV and OP signals to detect valve stiction. Finally, the stiction is quantified by clustering or fitted-ellipse techniques. The curve-fitting method presented in this chapter is based on the following qualitative analysis of the control signals: in the case of control-loop oscillations caused by controller tuning or external oscillating disturbances, the OP and PV typically follow sinusoidal waves for both self-regulating and integrating processes. In the case of stiction, the valve-position signal usually takes the form of a rectangular wave [37, 47] and the reason has been explained in Chap. 2. Because the valveposition signal is usually unmeasured, instead of looking at the valve position signal, we examine the measured output of the first integrating element after the valve, which is either OP or PV. The integrating element converts the rectangular valve position moves into a triangular wave. For self-regulating processes, the PI-controller acts as the first integrator and the OP’s move follows a triangular wave, whereas for integrating processes such as level control, the integrator in the process integrates the rectangular waves and the PV signal follows a triangular wave. The above anal-

7 Curve Fitting for Detecting Valve Stiction

151

ysis answers the questions of which signal to look after and why it takes a triangular shape in the presence of valve stiction. The basic idea of the new detection method is to fit two different functions, triangular wave and sinusoidal wave, to the measured oscillating signal of the first control-loop component containing an integrator after the valve (i.e. OP for self-regulating processes or PV for integrating processes). A better fit to a triangular wave indicates valve stiction, while a better fit to a sinusoidal wave indicates non-stiction.

7.2 Method Description In this section, a simple curve-fitting method is proposed for valve-stiction detection based on the original work of He and Pottmann [42]. The key idea of the proposed method is based on the following analysis of the control signal flow: • In the case of stiction-induced oscillations, the valve position switches back and forth periodically, which results in a rectangular wave. The first integrator after the valve in the control loop (i.e. the PI-controller if it is a self-regulating process, or the process if it is an integrating process) converts it into a triangular wave. • A sinusoidal external disturbance1 results in sinusoidal OP and PV signals because the integration of a sinusoidal wave results in a sinusoidal wave with phase shift. • A marginally stable control loop also results in smooth sinusoidal-shaped OP and PV signals for the same reason as for a sinusoidal external disturbance. To identify stiction-induced oscillations from others, we fit two different functions, triangular wave and sinusoidal wave, to the output signal of the first integrating component located after the valve, i.e. OP for self-regulating processes or PV for integrating processes. A better fit to a triangular wave indicates valve stiction, while a better fit to a sinusoidal wave indicates non-stiction. It is assumed that the loop in question is known to be oscillating by using oscillation-detection methods discussed earlier at length in Chap. 4. After the detection of the oscillation, the signal is de-trended and mean-centred. Time-varying SP is subtracted from PV. The location of each zero-crossing is automatically detected, and determined by linear interpolation of two points on both sides of the axis. Detailed implementation of the method involves isolating half-periods of the oscillating signal and applying piecewise least-squares fitting with noisy data as presented in the following two sections.

1 Disturbances will eventually become more sinusoidal as they propagate away from the source due to low-pass plant dynamics [122, 126].

152


7.2.1 Sinusoidal Fitting The OP or PV signal is fitted piecewisely for each half-period of oscillation (Fig. 7.1a), which means each fitting piece may have different amplitude and/or frequency. This consideration is reasonable considering the presence of noise in the signal. Besides, in real processes, the oscillation magnitude may change from time to time and other factors (e.g. external disturbances) may result in an unsymmetrical signal with respect to its mean.

S(t)

S(t) Sin-fit

S(t)

S(t) Tri-fit

Fig. 7.1 Schematic of the curve fittings: (a) sinusoidal fitting; (b) triangular fitting

Denoting the signal to be fitted as S(t), and defining two vectors a ≡ sin (ω (ti : ti+1 − ti ) + φ ) ,

b ≡ S(ti : ti+1 ) ,

(7.1)

the objective function for the sinusoidal fitting is J = min xa − b 2 , x,ω

(7.2)

where x is the amplitude, ω the frequency and φ the phase shift of the sinusoid. (ti : ti+1 ) is the time range of fitting as in Fig. 7.1a. In our case, because the curve is fitted piecewisely, we have φ = 0. For simplicity, we fix ω to be


ω=

π . ti − ti+1

153

(7.3)

By using the least-squares method, we have x = (aT a)−1 (aT b) .

(7.4)

After the optimal x is determined, the mean-squared error for the sinusoidal fitting during the time period (ti : ti+1 ), denoted as MSEsin (i), is calculated. The overall mean-squared error for sinusoidal fitting MSEsin is the average of MSEsin (i) over all time periods.

7.2.2 Triangular Fitting Triangular fitting as shown in Fig. 7.1b is not as straightforward as sinusoidal fitting because it is a piecewise curve fitting with two degrees of freedom: the location and the magnitude of the maxima. We use a numerical iterative method to find the best fitting. The algorithm is described below: Step 1. For each half-period of signal S (e.g. ti to ti+1 ), set the minimum MSE: MSEtri (i) = ∞. Step 2. For a peak location tp between ti and ti+1 , first find the linear least-squares fit for (ti : tp ) with the constraint that the line has to pass the first zerocrossing point at ti , and then the linear least-squares fit for (t p+1 : ti+1 ) with the constraint that the line has to pass the second zero-crossing point at ti+1 . Finally, calculate MSE between ti and ti+1 . If MSEtri (i) > MSE, set MSEtri (i) = MSE. Step 3. Repeat step 2 for different peak locations. Step 4. The overall MSEtri is the average of MSEtri (i).

7.2.3 Stiction Index The stiction index (SI) is defined as the ratio of the MSE of the sinusoidal fit to the summation of the MSEs of both sinusoidal and triangular fits: SI =

MSEsin . MSEsin + MSEtri

(7.5)

Note that the SI is bounded to the interval [0, 1]. SI = 0 indicates non-stiction where S(t) fits a sinusoidal wave perfectly (MSEsin = 0), while SI = 1 indicates stiction where S(t) fits a triangular wave perfectly (MSEtri = 0). For real process data, an SI close to 0 would indicate non-stiction while an SI close to 1 would indicate stiction. When SI is around 0.5, which means MSEsin = MSEtri , it is undetermined. Based

154


on our experience, we recommend the following rules: SI ≤ 0.4 =⇒ No stiction 0.4 < SI < 0.6 =⇒ Undetermined .

(7.6)

SI ≥ 0.6 =⇒ Stiction

7.3 Key Issues There are several advantages associated with the developed curve-fit method. One advantage is that it is applicable to both self-regulating and integrating processes, because for both type of processes, the same idea applies, i.e. after one integration, the rectangular wave (valve position) becomes a triangular wave, while the only difference is where the first integration component after the valve is located in the control loop. Another advantage is its industrial practicability due to the following reasons: (i) the methodology is straightforward and easy to implement. Fundamentally, the curve fit is a simple least-squares regression problem, which also provides its robustness against noise and outliers, (ii) the detection is fully automatic and does not require user interaction, and (iii) because of the piecewise fit, it is flexible in handling asymmetric or damped oscillations. As with any other method, there are some limitations associated with the developed method. First, it requires information on the process type (self-regulating or integrating) and does not guarantee detection of valve stiction in all cases. In addition, sufficient data resolution is required for reliable diagnosis. Since the proposed stiction-detection method is essentially based on shape detection of a signal, the results tend to be biased towards stiction if the data resolution is low. In practice, 15 to 20 points per cycle is sufficient to obtain reliable diagnosis. The proposed method requires simple data processing prior to curve fitting. In general, after the detection of the oscillation, the signal is mean-centred. If necessary, detrend is applied to compute the least-squares fit of a straight line to the data and subtracts the resulting linear trend from the data. If SP is time varying, it is subtracted from PV prior to analysis. More general discussions are provided in Sect. 7.6.

7.4 Simulation Results In this section, the proposed valve-stiction-detection method is applied to simulation examples, in which stiction is simulated using He’s stiction model. Details can be found in Chap. 3.


155

To test how the proposed method is affected by process dynamics, the method is applied to a FOPTD process with stiction and evaluated for different process conditions. As in [99], we vary the ratio θ /τ , and the ratio fs /(2( fs − fd )) that is equivalent to the ratio of S/2J used by Rossi et al. fs , fd and J denote a static friction band, Coulomb or kinetic friction band and slip jump, respectively; J is defined as fs − fd ; see Sect. 3.2. The result is shown in Fig. 7.2 as the contour plot of SI with varying process and stiction parameters. It can be seen that the grey area of the proposed method is small and stiction is detected under almost all test conditions.

7

6 0.8

0.9

0.9

4

0.8 0.6

fs /2( fs − fd )

5

3

2 09

1 0.1

0.5

1

09

0.9 0.8 0 9

0.9 1.5

2

2.5

0.80 9

2.9

θ /τ Fig. 7.2 The proposed method applied to a FOPTD process with stiction, by varying process and stiction parameters

To evaluate the proposed valve-detection method, the same flow control and level control systems used in Sect. 3.6 are considered. Three cases are examined for both systems: no stiction, weak stiction and strong stiction. Valve-stiction-model parameters are summarised in Table 7.1. In the case of no stiction, because both systems are closed-loop stable, there is no oscillation and the proposed method is not applicable. To test the capability of the proposed method on distinguishing valve stiction from external disturbances, for the case of no stiction in the flow control system, an external sinusoidal disturbance is introduced at the PV: d = 5 sin(30t) .

(7.7)

Simulation results for the flow control are shown in Figs. 7.3–7.5, and detection results are summarised in Table 7.2. As we can see, the stiction index based on the

156


controller output (SIOP ) successfully detected valve stiction in the flow control and distinguished it from the external disturbance. Here, we use a sinusoidal disturbance because non-linear disturbances will eventually become more sinusoidal and more linear as they propagate away from the source due to low-pass plant dynamics [122, 126]. However, it is worth noting that if the external oscillatory disturbance is highly non-linear, such as a triangular wave, the proposed curve-fitting method may not be able to diagnose it correctly. Table 7.1 Valve-stiction-model parameters [63] Case number Case 1 Case 2 Case 3

Degree of stiction No stiction Weak stiction Strong stiction

fd fs 0 0 0.35 0.65 2 3

PV

5

6

4

0

2

PV

−5 0

4 −2

OP

2 0

−4

−2 −4

0

100

200

300

Time [s]

400

500

(a)

600

−6 −3

−2

−1

0

OP

1

2

3

(b)

Fig. 7.3 Flow control, case 1 – no stiction, but external sinusoidal disturbance: a) PV and OP trends; b) PV–OP plot

Table 7.2 Flow control case study Control system Flow control Flow control Flow control

Case number Case 1 Case 2 Case 3

SIOP 0.06 0.94 1.00

For the cases where there might be multiple causes of oscillation, the calculated SI may not be able to exclusively indicate whether there is a valve stiction or not. However, the SI can tell us what is the dominant factor that causes the oscillation. In the flow control, the mixed case of the external disturbance and weak valve stiction, i.e. case 1 coexisting with case 2, as well as the mixed case of the external disturbance and strong valve stiction, i.e. case 1 coexisting with case 3, are studied and the results are given in Table 7.3. As we can see, for the case where both external


157

disturbance and weak stiction exist, the SI indicates the dominant factor is external disturbance, while for the case where both external disturbance and strong stiction exist, the SI indicates the dominant factor is valve stiction.

0.5

0.25

0

0.15

PV

0.2

0.1 −0.5

PV

0.05 0

1

−0.05

0.5

−0.1 −0.15

OP

0 −0.5 −1

0

100

200

300

Time [s]

400

−0.2 −0.8

500

−0.6

−0.4

(a)

−0.2

OP

0

0.2

0.4

0.6

(b)

Fig. 7.4 Flow control, case 2 – weak stiction: a) PV and OP trends; b) PV–OP plot

1

1

PV

0.5

0.8

0 0.6

−0.5 −1

PV

0.4 0.2

3 2

0

OP

1 0

−0.2

−1 −2 −3

0

100

200

300

Time [s]

400

500

−0.4 −3

(a)

−2

−1

0

OP

1

2

3

(b)

Fig. 7.5 Flow control, case 3 – strong stiction: a) PV and OP trends; b) PV–OP plot

Table 7.3 Flow control with mixed cases Control system Case number SIOP Flow control Case 1 coexisting with case 2 0.17 Flow control Case 1 coexisting with case 3 0.75

Next, the proposed method is applied to the level control system described earlier with different degrees of stiction listed in Table 7.1. To test the capability of this method on distinguishing valve stiction from bad tuning, for the case of no stiction in a level control system, the controller gain is increased from 3 to 8.4 to make the

158


system marginally stable. Because it is an integrating process, PV is fitted. Simulation results are shown in Figs. 7.6–7.8 and detection results are summarised in Table 7.4. The calculated SI based on the PV (SIPV ) detects stiction successfully and distinguishes it from bad tuning.

5

4

PV

3 0

2 1

PV

−5

60 40 20

OP

0 −1 −2

0 −20

−3

−40 −60

0

500

1000

1500

Time [s]

−4 −60

2000

−40

−20

(a)

0

20

OP

40

60

(b)

Fig. 7.6 Level control, case 1 – no stiction, but aggressive tuning: a) PV and OP trends; b) PV–OP plot

0.5

0.25

0

0.15

−0.5

0.05

PV

0.2

PV

0.1

OP

1

0 −0.05

0.5

−0.1

0

−0.15

−0.5 −1

−0.2 0

2500

5000

7500

Time [s]

(a)

10000

12500

15000

−0.25 −0.8

−0.6

−0.4

−0.2

0

OP

0.2

0.4

0.6

0.8

(b)

Fig. 7.7 Level control, case 2 – weak stiction: a) PV and OP trends; b) PV–OP plot

It is worth noting that limit cycles can also be caused by non-linearities other than valve stiction. This aspect has not been addressed by other curve-fitting methods. In this work, limit cycles caused by valve saturation are tested using the proposed method. Figure 7.9a shows the PV and OP time series plots of a FOPTD process with valve saturation. Because it is a self-regulating process, OP is fitted to a sinusoid and a triangle as shown in Fig. 7.10. The SI is 0.31, which indicates that the oscillation is not caused by valve stiction. Notice that if PV is fitted to a relay, the valve saturation would have been diagnosed as valve stiction.


159

1.5

1

1

0.8

PV

0.5

0.6

0 −0.5

0.4

−1 0.2

OP

PV

−1.5

0

4

−0.2

2

−0.4

0

−0.6

−2 −4

−0.8 0

2500

5000

7500

10000

Time [s]

12500

15000

−1 −4

17500

−3

−2

(a)

−1

OP

0

1

2

3

(b)

Fig. 7.8 Level control, case 3 – strong stiction: a) PV and OP trends; b) PV–OP plot Table 7.4 Level control case study Control system Level control Level control Level control

Case number Case 1 Case 2 Case 3

SIPV 0.00 0.93 0.99

1

0.9

PV

0.8

0.8

0.6

0.7

0.4 0.2

0.6

PV

0

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0 −0.5

0

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30

40

Time [s]

50

60

70

80

(a)

0.1 −0.5

0

0.5

OP

1

1.5

(b)

Fig. 7.9 Valve saturation of a FOPTD process: a) PV and OP trends; b) PV–OP plot

7.5 Application to Industrial Loops The proposed stiction-detection method has been successfully used within Performance SurveyorTM , DuPontTM ’s web-based controller performance monitoring tool [46]. As referred to in Sect. 7.2.3, the stiction index is only calculated if a control loop is known to be oscillating, as indicated by a normalised oscillation index that is determined from the extrema of the error auto-correlation function [84]. Because SP excitation could result in OP patterns similar to the ones observed when

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stiction is present, therefore leading to incorrect diagnosis, data sets in which the oscillations are mainly due to SP variability (as indicated by an oscillation index and period of the SP signal), are also excluded from this analysis. Finally, sufficient data resolution is required for reliable diagnosis in practice. Since the proposed stictiondetection method is essentially based on shape detection of a signal, the results tend to be biased towards stiction if the data resolution is low. A requirement of at least 15 points per cycle is used by the Performance SurveyorTM implementation. Note that we deliberately do not make any assumptions on sampling intervals, because stiction cycles can vary significantly in duration from case to case. The stiction index is reported along with oscillation index, oscillation period, and PV–OP plots to facilitate a final diagnosis by the user.

OP

(a)

OP

Time [s] (b)

Fig. 7.10 Curve fitting of OP for the valve saturation case: a) sinusoidal fit; b) triangular fit

Three cases from DuPont’s chemical processes are given as examples in this section: case 1 is a level control loop that is over aggressively tuned; cases 2 and 3 are flow control loops and it is known that these loops have valve-stiction problems. Figures 7.11, 7.12 and 7.13 show normalised operation data and Table 7.5 summarises the detection results by stiction indices. The proposed method successfully detects valve stiction in case 2 and case 3 and indicates that the level oscillation in case 1 is not caused by stiction. Table 7.5 Industrial examples Control system Level control Flow control Flow control

Case number Case 1 (CHEM 4) Case 2 (CHEM 5) Case 3 (CHEM 6)

SIOP – 0.77 0.75

SIPV 0.34 – –


161

20

15

PV

10 10

0 −10

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PV

−20 0

0.2 −5

OP

0.1 0

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100

Time [s]

120

140

160

−15 −0.15

180

−0.1

−0.05

0

(a)

OP

0.05

0.1

0.15

0.2

(b)

Fig. 7.11 Industrial example, case 1 (CHEM 4) – level control with aggressive tuning: a) PV and OP trends; b) PV–OP plot 30

30

20

25

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10 20

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200

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−0.3

−0.2

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0

OP

0.1

0.2

0.3

0.4

(b)

Fig. 7.12 Industrial example, case 2 (CHEM 5) – flow control with valve stiction: a) PV and OP trends; b) PV–OP plot

7.6 Summary and Conclusions In this chapter, a stiction-detection method is proposed to fit two different functions, triangular wave and sinusoidal wave, to the measured output signal of the first integrating component located after the valve, i.e. controller output for self-regulating processes or process output for integrating processes. A better fit to a triangular wave indicates valve stiction, a better fit to a sinusoidal wave non-stiction. The proposed stiction-detection method is evaluated on simulated examples and industrial data sets where the actual oscillation causes are known, and the results show that the proposed method and metric successfully detect valve stiction in both self-regulating and integrating processes.

0.5

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Q.P. He and S.J. Qin 60

50

PV

40 0 20 −50

PV

0 −20

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−40 0 −60 −0.5

0

100

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Time [s]

(a)

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800

900

−80 −0.4

−0.3

−0.2

−0.1

0

OP

0.1

0.2

0.3

0.4

(b)

Fig. 7.13 Industrial example, case 3 (CHEM 6) – flow control with valve stiction: a) PV and OP trends; b) PV–OP plot

Another advantage of the proposed curve-fitting method lies in its industrial practicability, including straightforward methodology, fully automatic execution with no user interaction, robustness to noise, flexibility in handling asymmetric or damped oscillations, and applicability to both self-regulating and integrating processes. On the other hand, the proposed method does not guarantee detection of valve stiction in all cases. The following remarks discuss some limitations and applicationrelated issues of the proposed method. • It is worth noting that there is a grey area where neither a sinusoid nor a triangle fits the signal well. In this case, the proposed method cannot provide a meaningful detection result. • Limit-cycle oscillations can also be caused by process non-linearities other than valve stiction. In this work, we have shown that the proposed method can distinguish valve stiction from valve saturation in a FOPTD example. However, the capability of the proposed method in distinguishing valve stiction from other non-linearities needs further investigation. • In the proposed method, we assume that the external disturbance is sinusoidal. This is true for most of the cases because disturbances will eventually become more sinusoidal as they propagate away from the source due to low-pass plant dynamics. However, if the disturbance source that leads to an oscillation that is close to the valve being diagnosed, the proposed method may fail or lead to wrong diagnosis. • An exact triangular wave will be obtained only if there is a pure integrator in the controller or process. However, in the proposed method, a clear triangle is not required in order for the method to work. Therefore, the proposed method can handle most of the processes with weak integration action where clear triangular waves are not expected. • When the SP is constant and there are no load disturbances or the load disturbances are stationary time series, de-trending is not needed. For the case of varying load, such as step changes, the user has the choice to select the data seg-


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ment to avoid the transient due to the step changes. Because we use piecewise sinusoidal or triangular fitting on half-periods, the impact of non-stationary disturbances or noise is minimised. In the Performance SurveyorTM implementation at DuPont, de-trending and mean-centring are done automatically for incoming data. • The low-pass filter associated with a PI-controller has no significant impact on the proposed method. For valve stiction in integrating processes, although the triangular PV signal can be smoothed by the filter, because PV is fitted, the smoothing effect does not matter. For valve stiction in self-regulating processes, the rectangular wave will be smoothed by the filter, but after the integration action of the PI-controller, the OP fitting still favours a triangle if the stiction is not too weak, i.e. in the grey area. While the proposed method works well for single-loop controllers, it is of interest to apply the method to interacting processes with multi-loop controllers where oscillations may be caused by one sticky valve and propagated to other loops through interaction. It is important in this case to correctly identify which valve among the oscillating loops is the cause and which are effects.

Chapter 8

A Relay-based Technique for Detection of Stiction Claudio Scali and Maurizio Rossi

A technique for automatic detection of stiction in actuators of basic control loops in industrial plants is illustrated, and its efficiency is demonstrated by application on simulated and industrial data. The basic idea is the similarity of shapes of the oscillatory controlled variable (PV) between loops affected by valve stiction and loops under relay control. A relay scheme is used to generate data to perform a fitting of acquired data of the controlled variable (PV) and to reproduce the effect of process lag in modifying the shape of the PV(t) trends. Each half-period of oscillation is analysed and the same data are also fitted by means of sinusoidal and triangular primitives. Approximation errors are compared, and minimum error obtained by relay or triangular signals is associated with the presence of stiction, while a sinusoid is associated with disturbance. An appropriate index is defined and, according to its value, (normalised in [–1,+1]), it is possible to reach a verdict about the cause of oscillations (disturbance or stiction); also the presence of asymmetric stiction can be put into evidence. Results are validated in simulation for a wide range of variation of process parameters, and confirmed the validity of this approach in detecting the presence of stiction. They also put into evidence that for lag dominant processes, an uncertainty region appears. In applications to industrial data, the effect of measurement noise, sampling period and length of data, number of observation windows is examined, in order to check its suitability for online and offline applications. The uncertainty region appears also in industrial data, depending on process characteristics. This drawback, which is also common to other stiction-detection techniques, suggests the implementation of multiple techniques in automatic monitoring systems. Finally, a simple test to be performed directly on the plant to detect the presence of stiction is suggested for the remaining uncertain situations. Claudio Scali CPCLab, Department of Chemical Engineering (DICCISM), University of Pisa, Via Diotisalvi, n.2 - 56126 Pisa, Italy, e-mail: [email protected] Maurizio Rossi AspenTech - Srl, Pisa, Italy, e-mail: [email protected]

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C. Scali and M. Rossi

8.1 Introduction Good performance of basic control loops is a primary requirement for the process industry, owing to tight specifications on product quality, which force operation as close as possible to optimal conditions, and require causes of scarce performance to be detected promptly [37, 54, 96]. Advanced control, which takes care of optimisation of the global plant, is based on this fundamental control layer and overall performance strongly depends on its efficiency. These loops are generally under PID control and their performance may be far from optimal as a result of various factors: inadequate tuning or design of the control system, sensor failures, presence of friction in actuators, or external perturbations. The most popular problem addressed to date is certainly the incorrect tuning of controllers, which can be handled by process identification and auto-tuning [135]. Instead, a more common cause should be sought in the presence of static friction (stiction) in actuators, which causes a delayed and sluggish actuation of changes in manipulated variables, required by the control system [22]. This fact can be counteracted only by performing a valve maintenance (more frequent action), or by adopting friction compensators [39, 68]. In other cases, the origin of scarce performance may derive from external perturbations and in this case the action to be performed regards design or layout of upstream equipments. Very often the symptom is the presence of oscillations in the output response; the presence of significant perturbations can be detected by the application of the Hägglund test [37]. Therefore it is of great importance that the monitoring tool is able to distinguish automatically different types of oscillations, indicating their origin and thus action to be performed to suppress them. This calls for the adoption of tools that are able to do a complete performance monitoring from the problem (plant data) to the solutions (causes and remedies). Also, the analysis should be based on available data, recorded during routine plant operation, without requiring additional tests to be performed on the plant. The monitoring system should be as automatic as possible, leaving the interaction with the operator to very few crucial decisions. These considerations put into evidence that the technique adopted for automatic detection of stiction should be very reliable in order to reduce to a minimum the possibility of missed alerts (stiction not detected when present) or false alerts (stiction indicated when not present). d

OP

SP

−

C

PD

MV V

PV P

Fig. 8.1 Reference scheme for a single-input single-output (SISO) system

8 A Relay-based Technique for Detection of Stiction

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The basic control system is depicted in Fig. 1.1, where SP, OP, MV and PV represent setpoint, control signal and manipulated and controlled variables. Usually SP, OP, PV are recorded by distributed control systems, and then readily available, while the manipulated variable MV is not. This is a key factor in the choice of the more suitable automatic detection technique. The presence of stiction in the actuator introduces a delay and a non-linear behaviour between the control signal OP and the manipulated variable MV, as illustrated in previous chapters of the book. The relationship MV–OP changes from linear (without stiction, Fig. 1.2), to nonlinear (with stiction, Fig. 1.3).

Fig. 8.2 MV–OP plot without stiction

Fig. 8.3 MV–OP plot with stiction

In synthesis, when the active force Fa , proportional to the controller output, is less than the static friction force Fs the valve does not move. This can cause an offset

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between PV and SP. The constant error, in the presence of an integral component in the controller, produces an increase of OP and then of the active force. When Fa overcomes Fs the valve starts to move. At this point, the motion of the valve is influenced by dynamic friction Fd , depending on the velocity. Changes in MV values may become larger than necessary and, as a consequence, the error PV − SP changes sign. The controller decreases its action and the active force becomes again smaller than the static friction force: the valve gets stuck in a new position. The sequence motion/stop of the valve due to stiction is called stick-slip motion and causes oscillations in all loops variable: while the trend of the manipulated variable MV is very close to a square-wave, the shape of PV and OP changes, owing to process and controller dynamics. This fact can be easily realised by performing some simulations by means of stiction models. Models of different complexity have been introduced in the literature to study the phenomenon of stiction in valves and mechanical devices. A complete review can be found in Choudhury et al. [15], where a new model (adopted also in this study), has been proposed. This is a data-driven model that describes the relationship between MV and OP in the presence of stiction by means of only two parameters: S (representing the range of OP, where the valve is stuck) and J (representing the jump of MV, once the valve changes position). It has been shown [15] that this model is able to reproduce with acceptable accuracy trends observed in industrial data. The need for only two parameters is certainly an advantage with respect to physical models (for instance Karnopp [66]), which require a larger number of parameters, whose values are generally not known in a real plant. Similar results can obtained by the two models, as shown in [97].

8.2 Trends of Different Variables The analysis of the main features of the stiction phenomenon is carried out by observing trends of the loop variables (OP, MV, PV), by varying some important parameters. The limit cycle assumes different shapes depending on the values of process, stiction and controller parameters. As an illustrative case, results for a first-order-plus-time-delay (FOPTD) process with a PI-controller (with Ziegler and Nichols tuning) are presented. In Fig. 1.4, the effect of θ and τ is studied, while the stiction parameters are fixed (J = S/2 = 5). The ratio θ /τ is varied from 0.1 to 10 to analyse both lag and time-delay-dominant processes. In Fig. 1.5, the ratio θ /τ is fixed (equal to 1), while stiction parameters vary from (J = 1, S = 5) to (J = 5, S = 5). In Fig. 1.6, θ /τ and S/2J are fixed and the controller gain is reduced from KC to 2/3KC to 1/2KC . Three main features can be pointed out:

8 A Relay-based Technique for Detection of Stiction θ /τ = 10

θ /τ = 1

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θ /τ = 0.1

OP

MV

PV

Fig. 8.4 OP, MV and PV trends as a function of the ratio θ /τ

• MV always maintains typical square-wave elements; the almost perfect square wave shape, shown for θ / τ 1, can be slightly modified to a saw-tooth shape, but the discontinuity on the derivative is maintained. • PV presents also the limit cycle but the effect of the process modifies the typical square-wave shown in MV. For decreasing values of S/J and θ /τ , the shape changes from square-wave to triangular. This effect is even more pronounced when higher-order processes with underdamped elements are analysed. (Further simulations are not shown here for brevity sake.) • A change in controller parameters has the peculiarity of affecting the amplitude and the frequency of the oscillation frequency, without influencing its shape. In Fig. 1.6, a decrease from KC to 2/3KC to 1/2KC , causes a decrease of oscillation frequency from ω0 to 0.58ω0 , to 0.38ω0 . An intuitive explanation of this behaviour can be given by considering that a lower value of KC causes a lower rate of increase of the active force Fa . Therefore, longer times occur to overcome the static friction force Fs and, consequently, a lower frequency of oscillation is shown. A detailed explanation by means of the describing function analysis can be found in [16].

S = 5, J = 1

S = 5, J = 2.5 S = 5, J = 5

OP

MV

PV

Fig. 8.5 OP, MV and PV trends as a function of the ratio S/J

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C. Scali and M. Rossi KC

1.5 KC

0.5 KC

PV

MV

OP

Fig. 8.6 OP, MV and PV trends as a function of KC

8.3 Method Description In the previous section, trends for OP, PV and MV were shown; the typical square wave on MV, and the different types of waves on PV were observed. In a different scenario (closed-loop identification for auto-tuning purposes [75, 78, 129]) wave shapes similar to the ones reported in Figs. 1.4 and 1.5 have been observed for loop variables. In particular, for the feedback scheme under relay control (depicted in Fig. 1.7), in the case of a FOPTD process, the shapes of the controlled output (PV) change with the ratio θ /τ , as illustrated in Fig. 1.8. For low values of the ratio θ /τ , triangular shapes are observed, while for large values almost square waves appear: the effect of process dynamics in modifying the shape of signals is evident.

r −

G(s)

y

Fig. 8.7 Feedback loop under relay control

8.3.1 Basic Idea Previous considerations suggest the basic idea of the technique proposed here. Every significant half-cycle of the recorded oscillation of the output variable PV is fitted by using three different primitives obtained by varying parameters, namely: a sine wave, a triangular wave and a relay-generated wave. The last primitive is the output response of a FOPTD under relay control, by varying the ratio θ /τ . Following previous remarks about wave shapes, relay and triangular waves are associated with the presence of stiction, while sinusoidal shapes reveal the presence of external perturbations. By comparing the error between real and fitted data, an evaluation of


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the accuracy of approximation and then an indication of the underlying phenomenon observed in the analysed variable can be obtained. θ /τ = 0.1

θ /τ = 1

θ /τ = 10

Fig. 8.8 OP, MV and PV trends as a function of the ratio θ /τ

8.3.2 Stiction Index To quantify the accuracy of the approximations, a stiction index (SI) can be defined. With ES being the minimum square error obtained by the sinusoidal approximation and ERT the one obtained by the better approximation between the relay and the triangular waves, SI can be defined as: . ES − ERT , SI = ES + ERT

(8.1)

where the generic symbol X indicates the average value of X on all the examined half-cycles. SI varies in the range [–1, +1]: negative values indicate a better approximation by means of sinusoids, positive values by means of relay or triangular approximations. Values close to zero indicate that the two approximations have similar errors and the procedure gives an uncertain answer. By considering that real data

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can be affected by several factors (noise, irregular perturbations, etc.), which change the shape of the curves with respect to nominal profiles, an uncertainty region has been defined for values of the stiction index |SI| falling below a threshold SI0 , i.e. SI < −SI0 : SI > +SI0 : |SI| < +SI0 :

disturbance , stiction , uncertain .

(8.2)

Following applications in simulation and on first industrial data sets the value SI0 = 0.21 has been assumed as threshold, which corresponds to a ratio ES /ERT = 0.66 [98]. SI can be applied separately either on positive half-cycles (PV > SP, thus obtaining an index SI+ ) or negative half-cycles (PV < SP, thus obtaining an index SI− ). A direct comparison between values of SI+ and SI− helps in identifying asymmetric stiction, where opening and closing phases of the valve generate different wave shapes on positive and negative half-cycles (nearer to sinusoids where the effect of stiction is weaker).

8.3.3 Fitting Procedure The fitting performed by different primitives is illustrated for the two limit cases of evident presence of stiction and evident presence of a disturbance. In the first case (Fig. 1.9), the better approximation given by the square-wave is supported by values of the stiction indexes SI+ = 0.786 (for positive half-cycles) and SI− = 0.782 (for negative half-cycles), with an average value SI = 0.784, which indicates clearly the presence of stiction. In the second case (Fig. 1.10), the better approximation given by the sinusoidal approximation against the triangular one (which is the best of the two primitives associated to stiction) is evident (overlapping curves). This originates values of the stiction index SI+ = SI− = −1, to indicate the presence of a sinusoidal disturbance. For cases where the underlying phenomenon is not evident the error with different approximating curves will be comparable, falling into an uncertainty region, thus not permitting a clear indication.

8.3.4 Fitting Algorithm The fitting algorithm for the three primitives is illustrated in the following. The sinusoidal approximation CS is: CS = AS sin(ωSt + φS ) .

(8.3)


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Fig. 8.9 Application of relay technique in the case of stiction: approximating curves for three half-cycles: collected data (black thin), relay curve (grey thick), sinusoidal curve (black dots)

Fig. 8.10 Application of relay technique for a sinusoidal disturbance: approximating curves for three half-cycles: collected data (black thin), relay curve (grey thick), sinusoidal curve (black dots)

®

The parameters AS , ωS and φS are obtained by implementing the MATLAB function fminsearch, which uses the Simplex method. The starting point is calculated on the basis of recorded data: amplitude and frequency are the same of the recorded oscillation; the initial phase is chosen equal to zero. The triangular approximation CT is also very simple: AT (m1t + q1 ) n≤n CT = , (8.4) AT (m2t + (m1 − m2 )t(n) + q1 ) n > n where n is the vertex of the triangle. For points ni ≤ n the approximation curve is a straight line with coefficient m1 > 0 and for ni > n the approximation curve is a straight line with coefficient m2 < 0. The intercept of the second straight line is chosen to have the same value for the two straight lines on the vertex of the triangle. The term AT is only a scale factor; all the parameters AT , m1 , m2 , q1 are easily obtained with a linear least-square technique (LLS).

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More complex is the relay wave approximation: it starts with the division of the recorded oscillation into three intervals defined by two numbers n1 and n2 . The (1) corresponding three parts of the recorded curve will be called in the following yre , (2) (3) yre , yre . The division into three intervals allows a good algorithm to be built for the generation of the best approximating relay wave-shape: Step 1. Step 2. Step 3.

The procedure chooses a value for n1 and n2 subject to n2 > n1 . The optimisation modulus (based again on the Simplex method) for each iteration fixes a value for the two parameters that characterise the relay wave: the time constant τ and the delay θ . The use of these parameters allows the approximating curve to be obtained with unitary amplitude in the first interval: yˆ1 = 1 − e−

Step 4.

t−θ τ

.

(8.5)

The amplitude A1 that allows the best approximating curve to be obtained in the first interval (y1 ) is calculated with the LLS technique: (1)

A1 = (yˆT1 yˆ1 )−1 yˆ1 yre → y1 = A1 yˆ1 . Step 5.

The approximating curve in the second interval has the same τ of y1 . The delay in this case (θ2 ) is evaluated in order to obtain yˆ2 (n1 ) ≡ yˆ1 (n1 ) with this imposition the second part of the approximating curve with unitary amplitude is: yˆ2 = −1 + e−

Step 6.

t−θ2 τ

.

(2)

(2)

(8.8)

(2)

y˜re = yre − yre (n1 ) , (2)

A2 = (y˜T2 y˜2 )−1 y˜2 y˜re → y2 = A2 · yˆ2 .

Step 8.

(8.7)

The calculation of the best-fitting amplitude A2 in this case must be subjected to y2 (n1 ) ≡ y1 (n1 ); this problem is resolved by Eqs. 1.8 and 1.9. The value of A2 is then evaluated again with a LLS method (Eq. 1.10): y˜2 = yˆ2 − yˆ2 (n1 ) ,

Step 7.

(8.6)

(8.9) (8.10)

The third part of the approximating function is evaluated in the same manner: it is fixed a value θ3 as at step 5 and an amplitude A3 as at step 6. In this way, we can obtain also y3 . The global approximating curve is the union of the three approximating parts: yap = [yT1 yT2 yT3 ]T . A global scale factor ATot is evaluated with LLS between yap and the recorded curve in order to make the best possible approximation.


175

Step 9.

If yap , now evaluated, generates a lower error than yap at the previous iteration, the system uses the new curve as the best-fit approximation CR ; else the CR at the previous iteration is maintained. Step 10. The Simplex modulus makes a first check: if it has not reached the convergence, it changes values for τ and θ and the procedure re-starts from step 3. Step 11. If the Simplex method has converged or reached the maximum number of allowed iterations, the procedure makes the second check: if n1 and n2 have not assumed all the possible values the procedure restarts from step 1, else the best approximating curve is chosen as CR .

8.4 Simulation Results The procedure has been applied in simulation in the nominal case and in the presence of noise to check its robustness. Output responses for a FOPTD process with a PI-controller (Z–N tuning) have been simulated for varying values of the ratio θ /τ . The effect of stiction is reproduced by means of the Choudhury model [15], for varying values of the two parameters S and J. The case of disturbance on the process output has also been simulated, for varying values of the ratio between the disturbance frequency and the critical frequency of the process ωd /ωu .

8.4.1 Nominal Case Results for the case of stiction are synthesised in Fig. 1.11. The technique correctly indicates the presence of stiction for a large part of the investigated range, (θ /τ : 0.1– 10; S/J: 1–10), where the stiction index is larger than the assumed threshold (SI0 = 0.21). For lag-dominant processes (ratio: θ /τ < 1), values of the stiction index may fail below the threshold to indicate uncertainty. The effect is even more pronounced for very low values of the ratio, requiring very large values of stiction parameters (large ratio S/J) to be detected. This is not surprising, as the smoothing effect of the process capacity changes the wave’s shape and may make the approximation errors comparable. This is a common drawback of different techniques; for comparison, results for the cross-correlation technique (CORR) [47] are reported in Fig. 1.12: here, also a region of wrong indications (disturbance instead of stiction may appear for very low values of the ratio θ /τ . A more general comparison is reported in [99], where also the bicoherence technique is analysed [14]. For disturbance detection, throughout the investigated range of variation of the ratio ωd /ωu : 0.1–10, indications are not affected at all: the stiction index remains in the range [−0.9, −1], to indicate an evident presence of sinusoidal perturbations.

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S / 2J

SI

θ /τ Fig. 8.11 Simulation results for the relay technique in the case of stiction by varying process and stiction parameters

S / 2J

Δτ

θ /τ Fig. 8.12 Simulation results for the cross-correlation technique in the case of stiction by varying process and stiction parameters

This is certainly a favourable feature of the relay technique, while other techniques may be affected and give wrong indications; examples can be found in [99].

8.4.2 Presence of Noise In the case of stiction, the same analysis was repeated adding a noise on the signal of PV and OP to investigate the robustness of the the techniques. A noise able to


177

change the shape of the oscillation was created by varying its amplitude and period in three steps: • Adding a sinusoidal signal with a variation in the amplitude. • Modifying the previous signal by the use of a first-order filter with a variation in the time constant. • Overlapping a Gaussian random noise. The algorithm receives as inputs PV values and the desired ratio R between noise and signal amplitude. All variations depend on a randomly generated parameter rand(x); the output is represented by PVN , signal corrupted by noise. The first step of the used algorithm is the evaluation of frequency of the oscillation ωD , used to built a sinusoidal function with the same amplitude and frequency of PV: (1)

SN = sin(ωD · tK ) · max {PV} · δ (t) ,

(8.11)

t

(x)

where the suffix SN indicates the sinusoidal noise signal at step x of the described algorithm. δ (t) is a function, whose value is updated randomly every half-cycle (detected by a change in PV sign): rand(x) · R i f sign(PV(tK )) = sign(PV(tK−1 )) δ (tK ) = . (8.12) δ (tK−1 ) i f sign(PV(tK )) = sign(PV(tK−1 )) With this definition of δ (t), the original signal is corrupted by a sinusoid composed by half-cycles with different amplitudes, randomly varied. In the second step of the algorithm, the time constant τ for the filtering action is calculated to obtain a phase shift Δ φ = (π/ωD )R. A parameter a(t) is defined in Eq. 1.13 and a further corruption of signal is performed (Eq. 1.14). Finally, in step three of the algorithm, a random noise is added by using a normal distribution ψ (0, R) with zero mean and variance equal to R (Eq. 1.15): −1 a(t) = exp , (8.13) τδ (t) (2)

(2)

(1)

SN (tK ) = −a(tK )SN (tK−1 ) + (1 − a(tK ))SN (tk−1 ) , (3)

(8.14)

(2)

SN = SN + ψ (0, R) .

(8.15) (3)

The original signal is therefore corrupted by adding the sinusoid SN to the orig(3) inal signal: PVN = PV + SN . An example of the corruption of signal for different level of noise is depicted in Fig. 1.13 for different values of parameter R. It is evident that this noise is more severe than a Gaussian noise and that, increasing the value of N, the corrupted signal becomes very different from the original, thus affecting the efficacy of the detection techniques. Results are reported in Fig. 1.14. It is to be noted that the effect of the noise is more evident for lower values of S/2J and θ /τ : robustness can be considered to be maintained up to a value of N/A = 20%.

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C. Scali and M. Rossi N

N

N

Samples

Samples

Samples

(a)

(b)

(c)

Fig. 8.13 Effect of simulated noise: a) R = 5%; b) R = 15%; c) R = 30% – original signal PV (grey), corrupted signal PVN (black)

Also for disturbance detection the presence of noise has a similar effect of extending the grey zone, where no decision can be taken. Therefore, even though the simulated noise can be considered more severe with respect to the one affecting plant data, a pretreatment may be necessary for situations of very noisy data.

S / 2J

N /A

θ /τ Fig. 8.14 Increase of uncertainty with the ratio noise/signal (N/A)

8.5 Application to Plant Data To verify the efficiency of the proposed technique, in particular the effect of plant dynamics and data noise, in detecting the presence of stiction in valves operating in industrial plants, data recorded from eight loops of a refinery were analysed. This application was also oriented towards practical issues, both for online and offline


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implementation. Among them are the effect of the number of oscillations and of the chosen time window on the efficiency of the techniques, related computation time and ease of application. For these reasons, a comparison with CORR was carried out. In this case, all loops were known to be affected by stiction; recorded oscillation of control action (OP) and error (e = SP − PV) are illustrated in Fig. 1.15. OP

Samples

Samples

Loop 8

Loop 7

Loop 6

Loop 5

Loop 4

Loop 3

Loop 2

Loop 1

SP − PV

Fig. 8.15 Recorded values of SP − PV and OP as a function of time samples for eight industrial loops in the presence of stiction

It can be seen that loops from 1 to 5 show very regular and clean signals (almost no noise); loop 6 is characterised by noise and irregular amplitude, caused by valve saturation; in loop 8, the amplitude is small and the ratio noise to signal is high. Loop 7 has an intermediate behaviour (low noise, but less regular oscillations). Raw data, without any pretreatment were used; only the Hägglund test [37] to classify each perturbation as significant, according to its amplitude and frequency, was performed.

180


A total of 30 half-cycles was analysed for each loop, by grouping data in 4 different sets made, respectively, by 6 time windows of 5 half-cycles, 3 of 10, 2 of 15, 1 of 30. The minimum number of oscillations was found to be equal to 5 for relay techniques: by performing the analysis on a smaller number of cycles, results may depend on the chosen time window. On the contrary, for CORR in most cases two cycles may be enough. ® ® About the computation time (on a 1 GHz Intel Pentium 4 Processor): the cross-correlation method is the fastest (negligible time also for the largest set of data), while the relay method shows a longer time (up to 4 min for the largest set). The ease of implementation is also favourable for CORR (few lines of any programming language), while the relay method involves more complex procedures to implement the detection algorithm. Concerning the efficiency of detection, results are reported in Table 1.1 for the two techniques: “Yes” indicates a correct detection of stiction, “No” an incorrect interpretation, and “Unc” stands for uncertainty. Where only one answer appears, it means that it was confirmed from all the 4 sets of data; where two answers appear, it means that results changed with the observed set of data. Table 8.1 Results from the application to industrial data Loop 1 2 3 4 5 6 7 8 CORR No Yes No Yes No/Unc Yes/Unc Yes Yes RELAY Yes Yes/Unc Yes Unc Yes Yes Yes Unc

The indications from this application to industrial data are: • CORR indicates stiction on four loops, there is a conflict for two loops and it is not able to detect stiction on two loops (disturbance). The relay method assigns five verdicts of stiction on five loops; on the remaining three loops it gives two uncertainties and one conflict. • Whatever the reason (plant dynamics or presence of noise, which changes the shape of PV signals), both techniques are not able to detect stiction in all the cases and a combined use of more techniques is suggested for applications in industrial monitoring systems; see also Chap. 13. • From a closer examination of output results, the relay technique allows detection the presence of asymmetric stiction in loops 1 and 5. The phenomenon depends on the fact that in these two cases stiction is present only for positive values of PV (valve opening, or vice versa, according to the gain sign).

8.6 Summary and Conclusions The technique proposed in this chapter makes use of signals generated by simulating the controlled output from a feedback loop with a FOPTD process under relay control. These signals are used to fit time trends of controlled variables PV(t) in control loops affected by valve stiction. By changing process parameters, the signal is able


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to reproduce the effect of process lag in modifying the shape of the PV(t) trends. This allows detection of the presence of stiction when the approximation error is smaller with respect to sinusoidal signals, which are associated with disturbances. Simulation results confirm the validity of this approach in distinguishing the presence of stiction from disturbances. They also put into evidence that for lag-dominant processes, an uncertainty region appears (where no decision can be taken). Robustness to noise has been tested in conditions more severe with respect to industrial situations: the uncertainty region increases for large values of the noise to signal ratio and filtering may be necessary for the case of very noisy data. Applications to industrial data show that the minimum number of cycles to analyse can be rather small (five). This also would permit an online application of the technique, even though the fitting algorithm is relatively slow and seems to be more appropriate for offline applications. The close examination of a single half-cycle allows also to put into evidence the presence of asymmetric stiction. The uncertainty region may appear also in industrial data, depending on process characteristics, a drawback that has been found common to other stiction-detection techniques. For this reason, the implementation of multiple techniques in automatic monitoring systems is suggested, in order to decrease the number of uncertain cases. A simple test to detect the presence of stiction, to be performed directly on the plant and consisting in a decrease of the controller gain, is finally advised for the remaining unresolved cases.

Chapter 9

Shape-based Stiction Detection Using Area Calculations Timothy I. Salsbury and Ashish Singhal

The chapter describes a method for detecting stiction-like behaviour in feedback control loops based on extracting features from a time-series record of the plant output. The method only requires measurements of the plant output and does not require tuning or any expert engineering knowledge for configuration. The features that are extracted are the areas before and after the peak between zero-crossing events of the oscillating control error signal, i.e. the difference between the setpoint (SP) and the controlled variable (PV). The output of the method is the probability that stiction is present (0–100%). The chapter describes the method and the basis for its development and presents results from tests on a wide range of data sets. Also described are key assumptions and issues for practical implementation of the method.

9.1 Introduction Despite the increasingly advanced nature of today’s process-control systems, the single-input single-output (SISO) feedback loop still represents a fundamental building block that is consistent in its form across many applications. These loops normally act as the final interface between the control logic and the physical systems being controlled. Monitoring these loops and measuring their performance thus holds the potential of catching a large percentage of problems in process plants [90]. This chapter focuses on the stiction problem that arises in these kinds of SISO loops

Timothy I. Salsbury Controls Research Department, Johnson Controls, Inc, 507 E Michigan Street, Milwaukee, WI 53202, USA, e-mail: [email protected] Ashish Singhal Advanced Control and Operations Research R&D, Praxair, Inc., 175 East Park Dr., Tonawanda, NY 14150, USA, e-mail: [email protected]

183

184

T.I. Salsbury and A. Singhal

whereby a final controlled element such as a valve sticks and slips in response to changes in position demanded by a controller. The action of feedback on a sticking valve can cause oscillations in the controlled variable, which increases variance and is undesirable from a control perspective. Stiction could be detected simply by monitoring variance, but there are many other problems in control loops that can lead to this same symptom. Detection of stiction therefore requires additional analysis in order to distinguish this case from other types of problems. The assumption throughout the book is that stiction results in oscillations or periodic behaviour of the controlled variable. A first step therefore is usually to detect the existence of periodic behaviour. This problem has been addressed elsewhere and several methods have been reported in the literature; see, e.g. Chap. 7 in [90] or Chap. 4 of this book. The method reported in this chapter was originally developed for use in building automation systems. These systems typically have low-cost valves and actuators and stiction is common. The controllers are mostly PI and in many cases it can be difficult to get access to data in order to monitor loops. The control signal, or output from the controller, is sometimes not easy to access and other variables such as valve-position feedback, which would be very useful in detecting stiction, are very rarely available. The method reported here was therefore developed under the constraint of assuming only the controlled variable and its setpoint would be available. The method also had to be robust to irregular sampling caused by data-access problems, which can be common in distributed control networks. Another goal was to keep computational requirements to a low level so that the method could be implemented in a low-cost monitoring device. In addition to being restricted in terms of the quantity of measurements, the quality of measurements in building automation systems is also usually poor. For a method to be successful in this application area, it is therefore critical that it be robust. A common scenario is that a method is proven theoretically and rigorously tested in simulation, but then fails to get adopted in practice because of robustness issues. The main problem is that it is never possible to anticipate all of the uncertainties and unusual phenomena encountered in practical systems. One way to minimise robustness risks is to follow the age-old adage and keep the method as simple as possible. Minimisation of the number of parameters required by a method and the number of processing steps can bolster robustness, but the sensitivity of the method to small changes in assumptions also needs to be taken into account. It is interesting to note that many of the methods that have been successfully adopted by industry for fault detection and advanced control tend to contain a lot of heuristics [71]. These are often included through necessity in order to provide robustness against unusual phenomena but in some cases, the core idea becomes clouded by the additional heuristics. Ideally, the core idea should be developed from the outset with robustness in mind so that the heuristic parts can be kept to a minimum. The method presented here follows this concept and is based on characterising the shape of an oscillating signal using features that are inherently designed for robustness. For example, integration methods are favoured as these naturally smooth out signal anomalies and are also easy to implement. The idea of analysing the shape

9 Shape-based Stiction Detection Using Area Calculations

185

of the controlled variable in a feedback loop is not new and has been used before to detect tuning problems [37, 38, 101]. The method described here is based on the work originally reported in [110] and includes two additional elements: i) improved robustness to noise via estimation of a noise band using a band-stop filter, and ii) hypothesis testing for stiction within a probabilistic framework. Section 9.2 describes the basic idea behind the method and provides some theoretical basis. Section 9.2.2 describes a statistical hypothesis test for transforming information obtained from the control loop into a diagnosis. Section 9.2.3 describes the method for handling noise and determining whether observed behaviour is suitable to be evaluated. Section 9.3 summarises some of the key issues associated with the method. Section 9.4 provides results and analyses from simulation tests and Sect. 9.5 shows two test results from industrial control loops.

9.2 Method Description Figure 9.1 depicts the type of loop that is considered, which contains a digital controller, Gc (z), interfacing through D/A and A/D converters to a continuous-time plant, Gp (s). Disturbances to the plant, w(t), are shown passing through a disturbance model Gd (s). The proposed method analyses samples of the error signal ek , which is given by: ek = rk − yk ,

(9.1)

where rk is the setpoint and yk is the sampled feedback signal. In modern building automation systems, these signals would be accessible either directly from each controller or via a node on the communication network. w(t) Controller

rk

ek

−

d(t) Gd(s)

Gc(z)

uk

D/A

u(t) Gp(s) y(t)

yk

Fig. 9.1 Feedback control loop

A/D

186


The basic idea of the method is to extract features that characterise the behaviour of the control loop between zero-crossing events. A zero-crossing event refers to the time when the error signal crosses zero, as shown in Fig. 9.2. The error signal will deviate from zero when a disturbance of some sort enters the loop. Hägglund [37] also advocated using zero-crossing events for feature extraction in his oscillationdetection method. The features that are extracted are the areas before and after the peak between zero-crossing events. These areas are denoted A1 and A2 , respectively in Fig. 9.2. A normalised index, R, is derived from these features where R

A1 . A2

(9.2)

The areas under the curve can be calculated from samples of the error signal using a standard numerical integration procedure, such as trapezoidal integration. Irregularity in the period between samples can be handled by most integration procedures. An important advantage of using areas is that the effect of noise is reduced, which makes the index more robust to measurement uncertainty. The ratio of areas is also inherently normalised in terms of static and dynamic scales.

peak value

e(t)

A1 A2

+ 0 − zero−crossing events Time Fig. 9.2 Zero-crossing events on the setpoint error signal

The R index condenses the behaviour of a control loop between zero-crossing events into a single numeric value. This index is the basis of the stiction detection procedure. The results in [110] showed that the R index became much larger than unity when stiction was present and was closer to unity when the oscillations were due to other problems such as an overly aggressive controller. The difference in


187

behaviour is illustrated in Fig. 9.3. Section 9.2.1 provides some theoretical basis for the claim that an R value greater than unity is indicative of stiction. Aggressive control

Valve stiction A1

1

Control error

2

Control error

A2

A

A

0

A

1

____

A

>1

0

A

____1

A

Time

~1

2

2

Time

Fig. 9.3 Shape of oscillatory responses and the effect on the R index

9.2.1 Theoretical Basis The stiction phenomenon is quite complex from a physics-based modelling perspective and has been analysed by several researchers [1, 66, 89, 130] who have proposed detailed models. Because of the difficulty in estimating the parameter values in physics-based models, Choudhury et al. [13] proposed an empirical model that captures the main attributes of stick-slip behaviour with a minimal number of parameters. Although both the physical and empirical models are extremely useful in a simulation environment, they are not very amenable to theoretical analyses. An alternative model that can be used quite easily in a theoretical analysis is a relay model. It is shown that this model is a reasonable approximation to stiction behaviour under certain conditions and it also enables some theoretical basis to be put behind the claim that the R index is greater than one when stiction is present. Figure 9.4 shows the Choudhury model and a simple relay model on the same plot. At first glance the relay appears to have a quite different characteristic from the more detailed model. However, further analysis reveals that the actual realised input–output characteristic of the stiction model depends on the nature of the plant to which it is coupled. The time delay of the plant is particularly important and affects the relative sizes of the stick-slip and stickband portions of the input–output characteristic. We found that the detailed model becomes more similar to the simple relay as the ratio of the time delay relative to the dominant time constant of the plant becomes small. This effect is shown in Fig. 9.5 where the time delay to time constant ratio is denoted by λ . For example, the difference is very small when λ is less than 22%, which covers a large range of practical systems. Hence, the relay model,

188


although a gross simplification, is a useful starting point for providing theoretical justification behind the claim about the R index. Once the assumption of a relay model is made, theoretical evaluation of the R index involves the following steps:

Valve output (x)

Step 1. Use the closed-loop transfer function (including plant and controller) to estimate the oscillation frequency for a relay non-linearity based on the describing function method [65]. Step 2. Assume the input to the plant is a square wave with the frequency that is estimated and analytically determine the steady-state (periodic) plant response equations. Step 3. Calculate the R index value from steady-state plant output equations via analytical determination of the peak location and zero-crossing points.

plant with time delay (Choudhury model) plant with no time delay (relay)

slip jump

deadband+stickband Valve input/controller output (u) Fig. 9.4 Input–output characteristic of stiction models

The calculations involved in each of the steps above are somewhat tedious and their complexity of course depends on the type of plant model assumed and also the controller. The work reported in [110] investigated the behaviour of the R index for several different types of plant models, controller tuning rules, and stiction models and one set of results is shown in Fig. 9.6. The top graph shows the analytically determined R index value on the y-axis plotted against different λ values on the x-axis. Each of the curves in the top graph represent different tuning rules used to determine the parameters of a PI-controller that was controlling a secondorder-plus-time-delay (SOPTD) plant model having the following continuous-time transfer function:

Gp (s) =

exp(−Ls) . (τ s + 1)2

(9.3)


189

Each of the tuning rules required the plant parameters to be in the form of an apparent time delay, time constant and gain. These three plant parameters were determined analytically for the SOPTD model. The tuning rules were: approximate Ms constrained integral gain optimisation (AMIGO), Chien–Hrones–Reswick (CHR) for SP change with zero per cent overshoot, Ziegler–Nichols (ZN) and Cohen–Coon (CC). Details of these tuning rules can be found in [88] and [4]. The two lower plots in Fig. 9.6 show the sensitivity function (Ms ) and complementary sensitivity function (MT ) and provide an insight as to how aggressive the tuned controllers were for each case.

λ = 0.45

Valve output (x)

λ = 0.22

relay (λ = 0)

Valve input (u) Fig. 9.5 Stiction model input–output characteristics for different amounts of time delay

The top graph in Fig. 9.6 clearly shows that the R index value is above unity for most of the cases throughout the range of considered λ values. The exceptions are the CC- and ZN-tuned controllers, which do not have R index values above one when the time delay is relatively small. The reason for this is that these two rules lead to an overly aggressive controller for small λ values, as confirmed by the Ms and MT values compared to the other cases. However, the results do show that the R index will indeed exceed unity when oscillations occur for the simple relay approximation to stiction that was considered. The only caveat is that the R index may be close to one or even below it if the controller is tuned too aggressively.

190


10 CHR AMIGO

R

5

CC ZN

1 0.2

0.3

0.4

0.5

0.6

0.7

MS

5 AMIGO

3 1 0.2

0.3

ZN

CHR

0.4

0.5

CC

0.6

0.7

MT

5 AMIGO

3 1 0.2

ZN

CHR

CC

0.3

0.4

λ

0.5

0.6

0.7

Fig. 9.6 R index values for PI-controllers tuned using different tuning rules and controlling an SOPTD plant model

9.2.2 Stiction Detection Hypothesis Test Calculation of R index values from an oscillating control loop will produce a set of numerical values that contain information about the shape of the responses. Inferring whether or not stiction exists requires further analysis of the set. One way to test for stiction is to perform a statistical hypothesis test on the set of R values. A suitable test is the single sample t-test. Because values of R greater than one have been found to correspond to stiction, the null hypothesis can be: H0 : mR > 1 ,

(9.4)

where mR is a measure of location of the R values. The alternative hypothesis is: H1 : mR ≤ 1 .

(9.5)


191

Conventionally, the mR value would be the arithmetic sample mean, but this would only yield an accurate estimate of location if the distribution of R values were symmetrical. In fact, we can quickly deduce that the distribution will not be symmetrical because R cannot be less than zero. A log-normal distribution is often a reasonable assumption for ratios of positive real numbers [23] and empirical results have shown that the distribution of R-values conforms quite well to this type. One way to handle data from this type of distribution is to perform a log transformation and then treat the transformed variables as normal. To guard against outliers and other anomalous events corrupting the estimates, a robust estimate of location is used: mR˜ = med[ ln(R1 ), . . . , ln(RN )] ,

(9.6)

where med denotes a median calculation, and mR˜ is thus the sample median of the log transformed values. To test the hypothesis, a measure of scale is needed. A robust measure is also adopted here to protect against outliers. The median absolute deviation (MAD) about the median is used, which is given by: sR˜ = k med |ln(Ri ) − mR˜ | ,

(9.7)

where k is a constant that puts the MAD on the same scale as the standard deviation. For a normal distribution the k value is 1.4826. The t-score calculation to test the hypothesis about the sample is then: t=

m˜ √R . sR˜ / N − 1

(9.8)

Note that the threshold of one that the sample mean is tested against does not appear in the above equation because its value is zero, i.e. ln(1) = 0. This t-score can then be converted into a probability using the standard student-t cumulative distribution function with N − 1 degrees of freedom. The null hypothesis can then be accepted or rejected when this probability is greater than a threshold. It is also useful in a practical implementation to simply report the probability measure as an index for each considered loop. This approach can be useful in troubleshooting a large plant as loops can then be ranked based on their probability of having stiction.

9.2.3 Noise Effects and Practical Implementation In a practical situation, the error signal monitored from a control loop will be corrupted by noise. Two problems arise with the basic method as described so far when the signal is noisy. The first problem is that the location of the peak values on the

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oscillating signal might be affected leading to errors in the R index values. Although the effect on the R values may possibly get averaged out over a large number of cycles, the effect is nevertheless undesirable, especially when only a few cycles are available for analysis. The most logical solution to the problem of noise is to use a low-pass filter, but this approach has to be used with caution because many filters will detrimentally modify the shape of the signal. Several types of smoothing filters can be used to remove noise without significantly changing the time-domain shape of the underlying signal. One example is the class of FIR filters known as Savitzky–Golay smoothing filters [104]. These fit polynomial curves through a moving window of points and preserve features such as peaks and troughs quite well. Although these filters worked well over many data sets that we tried, one problem was that they were not robust to spikes in the data. We found that spikes were common when a signal flattened near its peak due to quantisation effects in some data sets. An alternative smoothing filter that is robust to spikes is the moving window median filter [94] and this is what was adopted in the method described here. The filter is given by: ef (i) = med(e(i − n), . . . , e(i + n)) ,

(9.9)

where n determines the size of the moving window over which the filter is applied. For most data sets, the sampling period relative to the frequency of oscillations in data sets tested so far has been such that a window size of five samples has proven a good design parameter for the filter. The second problem arising because of noise is that additional zero-crossing events will occur when the error signal is close to zero. These zero-crossing events will not relate to the underlying periodic variation caused by stiction and the R index should not be computed between these events. Figure 9.7 shows how noise on data collected from a control loop with stiction led to an additional zero-crossing event that would have corrupted the computation of the R index value in that half-period. An additional procedure is therefore required to handle the inevitable case of noise corruption. Smoothing or low-pass filtering the signal does not really help with avoiding the additional zero-crossings, especially when the stiction-induced oscillation is very slow. One solution to the problem is to implement the method in the basic form of calculating the areas before and after the peak between every two zero-crossing events, but only calculate the R index when the period is deemed legitimate based on some other criteria. The idea of a noise band is adopted here to develop the more stringent criteria. The concept of having a noise band for the setpoint error signal is shown in Fig. 9.7 as symmetrical limits around zero. A method for estimating this band is described in the next section.


193

1.2 1 0.8 0.6

e(t)

0.4 0.2

additional zero−crossing due to noise

Lim

0 −0.2 −0.4 −0.6

Time Fig. 9.7 Data from a control loop with stiction showing how noise triggers additional zero-crossing events

9.2.3.1 Noise-band Estimation The noise band was denoted as Lim in Fig. 9.7 and is constructed as a band around the zero line of the error signal so that the band is drawn as ±Lim. Estimation of the underlying noise variance could be approached in many different ways. But the main objective is to estimate the variance of the signal excluding the part caused by the stiction-induced oscillation. Put in this way, the most obvious approach is to apply a filter to the signal that blocks a small frequency range corresponding to where the oscillations are occurring. The variance of the filtered signal would then represent the total variance minus the portion due to the oscillations. The idea described above was adopted for the method described in this chapter and has proven to yield good results over a wide range of data. If the oscillation frequency is known, a band-stop filter should be used that blocks a small window of the spectrum around the specified frequency. The application of a band-stop filter to data obtained from a control loop oscillating due to stiction is shown in Fig. 9.8. The figure shows that the filter can be used to remove the portion of the signal spectrum corresponding to where the oscillations are occurring, while leaving the rest of the spectrum intact. Many different types of filters could be used to perform the band-stop function. In tests, we have found that a simple second-order (IIR) filter is sufficient in most cases. For example, one realisation of a second-order filter is: ef,k = −a1 ef,k − a2 ef,k + ek + b1 ek−1 + b2 ek−2 ,

(9.10)

2 where b1 = −2 cos(2π fc ); b2 = 1; fa1 = −2r cos(θ ); a2 = r , r = exp(−π fc /Q), θ = (π fc /Q) 4Q2 − 1, and fc = f0s [31]. This filter has two tunable parameters:

194


f0 , which is the band-stop frequency and Q, which is the quality factor. Note that fs is the sampling frequency of the data.

0.7

band−stop filter

0.6

Normalised power (−)

magnitude (dB) 0 −1 −2

0.5 0.4 0.3

normalised frequency (−)

original signal

0.2 0.1 0 0

filtered signal 0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normalised frequency (−)

0.4

0.45

0.5

Fig. 9.8 Band-stop filter

Once the filter is applied to the error signal, an estimate of the underlying noise variance can be determined from the variance of the filter output. For the work described here, a robust statistical measure of scale was used as a proxy for variance to again circumvent data problems and outliers. The noise variance is then:

σN2 ≈ [1.4826 × MAD(ef )]2 ,

(9.11)

where ef is the filtered error signal and MAD denotes the median absolute deviation about the median. The other part to the filtering is to determine the frequency specification, f0 , for the filter. Ideally, this should be set equal to the frequency of the oscillation. If an oscillation-detection method has first been used, the oscillation frequency should be available as a by-product. However, here we present a method for determining the frequency to use in the filter that can allow use of the method even if an oscillation detection procedure is not available. The idea is to simply estimate the average frequency exhibited in a given data set and use this to configure the filter. This average frequency can be determined by calculating the average period between zero-crossing events: 1 P¯ = N

N

∑ pzc, j ,

j=1

(9.12)


195

where pzc, j is the period between zero-crossings. The average angular signal frequency is then: π ω0 = ¯ , P

(9.13)

which corresponds to the centroid frequency of the signal spectral density [64]. In the case where an oscillation exists, the frequency will be close to the correct value and filtering with a window centred on this frequency will lead to a filtered signal containing most of the residual noise. If the signal does not contain an oscillation and has a relatively flat spectrum, the filter will remove a portion of the variance but will still retain most of the noise component and thus serve as a reasonable estimate of the overall noise. The noise estimate could also be modified to compensate for the window that is removed as follows:

σN2 =

[1.4826 × MAD(ef )]2 |H(z)|2

,

(9.14)

where H(z) is the filter transfer function. The error band, Lim, for the error signal should be set based on the estimated standard deviation, σN . If we assume that the noise comes from a Gaussian distribution, limits can be defined as ±Z σN , where Z is a z-score limit. A value of two has been used in the tests reported here. In applying this band-stop filtering procedure to a data set without oscillations, the noise band will contain the majority of the underlying noise. Hence, use of an oscillationdetection procedure as a first step in processing a data set is optional for the reported method. The procedure for using the noise band to test for legitimate zero-crossing events in described in the next section. 9.2.3.2 Zero-crossing Test Procedure The noise band can be used to test for a legitimate zero-crossing event by evaluating whether the signal has crossed from one side of the band to the other as well as crossing the zero line. In encountering a new zero-crossing point, the approach therefore requires knowing what side of the band the signal was last on, and then looking forward to check whether the signal will be on the other side in the future. Having to look forward in time is not a problem for offline batch processing of data. But it might be considered a drawback for an online implementation because a data buffer would need to be used. However, the size of the data buffer would not need to be longer in time than a half-period of the slowest anticipated oscillation frequency. A state diagram is shown in Fig. 9.9 to illustrate the main features of the algorithm and the test for zero-crossing events. Each of the states and associated transitions are explained in more detail below: • Re-setting State. This state is entered to re-set the integration variables. Two integration variables would need to be reset: the total area under the response

196

T.I. Salsbury and A. Singhal mode ≠ 0

Characterise • calculate and store new R index

Re-setting • re-set variables • set mode = 0 mode = 0

sign(e(t )) ≠ sign(e(t − 1))

Computing • perform integrations

e(t ) < Lim

(max(e(i = t:N )) > Lim AND mode = −1) OR (min(e(i = t:N )) < Lim AND mode = 1)

t=N e(t ) > Lim

Mode change ( 1) • set mode = −1

mode = −1

Mode change (1) • set mode = 1 mode = 1

Fig. 9.9 State diagram for robust R index calculation

between subsequent zero-crossing events and the area after the peak up to the current zero-crossing. • Computing State. This state performs the integrations and also implicitly re-sets the area after the peak calculation whenever the signal reaches a new peak level. The handling of the peak integration is shown as a substate diagram in Fig. 9.10. Two integration variables are shown: A, which is the total integrated area between zero-crossing events and Ap , which is the area after a peak has been reached. • Mode-change States. The mode-change states switch the mode variable between negative one and plus one depending on which side of the noise band the signal travels. • Characterise State. This state is triggered only when a zero-crossing event has occurred, the last mode was not zero, and the future signal travels to a different side of the noise band compared to what is indicated by the current mode variable. These conditions mean that characterisation will occur at the first zerocrossing after the signal has been outside the noise band and is travelling to the opposite side of the noise band in the future. Given that the total area under the response is A and the area after the peak is Ap , the R index value is thus given by: R=

A −1. Ap

(9.15)


197

After characterisation the state changes to resetting. The exit condition for the algorithm is that the position counter, in this case t, is at the end of the data set, i.e. t = N. Note that the algorithm presented in Fig. 9.9 is intended to be applied to the error signal data record after the application of the median filter described earlier.

Computing | e(t ) | > emax

Integration A(t ) = A(t − 1) + ΔA(t )

Ap (t ) = Ap (t − 1) + ΔA(t )

Re-set peak variables emax = | e(t ) |

Ap (t ) = 0 Ap (t ) = 0

Fig. 9.10 Computing state expanded

9.3 Key Issues The proposed method is simple to implement, computationally undemanding, and uses minimal information from the control loop. However, the method has certain limitations that should be considered before using it. First, the shape-based concept at the core of the method will only have the potential to identify stiction when the following conditions are met: • The loop should be tuned more toward the conservative side. Aggressive tuning will modify the shape of stiction-induced oscillations and make them more sinusoidal. The method will then not be able to distinguish stiction oscillations from aggressive control. • Enough samples need to be available within each oscillation cycle. This is of course important for any kind of method. But the proposed method requires that sampling be fast enough to allow a good estimate of the peak location and also the zero-crossing points. • The expected value of the error signal should be zero, which should be true for all systems having controllers that contain integral action. • The plant should be self-regulating with monotone open-loop step responses. The plant should thus not contain any integrating elements and all poles should be on the negative real axis.

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In addition to the above, it should be noted that erroneous results might be obtained when applying the method to data without first checking for the presence of oscillations. This is because the noise band and robust zero-crossing procedure will still possibly identify valid responses for cases such as load changes that occur on both sides of the noise band. However, Salsbury [101] showed that the shape of these load changes can also be evaluated in order to determine whether the controller has been tuned properly. Figure 9.11 shows the continuum of response shapes as the behaviour transitions from sluggish to aggressive to stiction. sluggish load changes, where A1 < A2

R=

A2

A2

e(t)

A1

A1

oscillatory controller, where A1 = A2 increasing

A2

e(t)

A1

sticking valve, where A1 > A2

A2

e(t)

A1

Time

Fig. 9.11 Variation in shape of responses and the effect on the R index

The method is also described as a batch-processing procedure and results obtained for a given data set will therefore reflect the average behaviour over the data set. Time-variant behaviour is not explicitly treated and the method would need to be adapted for this case. Possible ways of handling changing behaviour are: (i) break up the data into smaller subbatches and track changes between the subbatches, or (ii) modify the method to use time-weighted averages instead of straight averages and track changes within the data set.

9.4 Simulation Results Simulation was used as a tool to analyse the behaviour of the stiction-detection method and to supplement the theoretical work. In particular, simulation was useful


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in testing the method with stiction models that were more complex than the simple relay model that was assumed in the theoretical work. In this section, we use the Choudhury model in a simulation of a closed-loop system and calculate the R index numerically. Figure 9.12 shows the variation of R with λ (the ratio of the apparent time delay to time constant) for the Choudhury and the relay stiction models. The figure contains two plots, one for a FOPTD plant model and the other for a SOPTD model. In both cases, the plant was regulated by a PI-controller that was tuned using the AMIGO method. Figure 9.12 shows that the R values are generally lower for the Choudhury model than for the relay. However, the R values are still significantly greater than one over most of the range of λ for both cases. This result reinforces the earlier claim that the R value being greater than one is a good indicator of stiction-induced oscillations. (a) FOPTD plant

10

(b) SOPTD plant

relay model

10 relay model

R

5

R

5

Choudhury model Choudhury model

1

1 0.2

0.4

0.6

λ

0.8

1

0.3

0.4

0.5 λ

0.6

0.7

Fig. 9.12 Simulation result showing R index values for two stiction models

The reason for the lower R values for the Choudhury model can be explained as follows: when the Choudhury model is used to simulate stiction in a closed loop, the valve output (or the plant input) is not a rectangular wave but a combination of a step and a ramp as shown in Fig. 9.13. The step part of the signal corresponds to the slip jump, while the ramp part is the sliding part of the input/output (I/O) characteristic that was shown in Fig. 9.4. As the delay increases, the sliding part becomes larger compared to the slip jump and the valve output moves with the controller output for a longer period of time and results in a smaller R value. The value of R increases with increasing λ because the controllers become less aggressive (e.g. because MS and MT decrease as was illustrated in Fig. 9.6). The reason for the increasing difference between the two curves with λ is that the difference between the I/O characteristics of a relay and the Choudhury model increases with

200


Valve output

0.12

Choudhury stiction model

0.08 relay

0.04

0 270

280

Time

290

300

Fig. 9.13 Time-domain plot of valve position (plant input) when cycling with two different stiction models

λ , as was shown in Fig. 9.4. For the same reason, the difference in the two curves is also larger for the SOPTD plant compared to the FOPTD plant. Figure 9.14 shows the variation of R for different plant orders with no actual time delay. The plant model transfer functions were: Gp (s) =

1 . (τ s + 1)n

(9.16)

The exponent n defines the order and higher values lead to increasing apparent time delay. The difference between the relay and the Choudhury stiction models is smaller in Fig. 9.14, and the two curves do not diverge as much as they did in Fig. 9.13. Thus, it appears that a pure time delay contributes more to the differences in the two models than higher-order dynamics. This is also an important result because the observed time delay in many practical systems is often due more to higher-order dynamics than physical delays.

9.5 Application to Industrial Loops The stiction-detection method has been tested on a large number of data sets from several different industries and the results of these tests are summarised along with those from all the other methods reported in Chap. 13 of this book. In this section, two test results are shown and explained in more detail.


201

10 L=0 5 R

relay model

1 0 (1)

Choudhury model 0.16 (2)

0.37 (3)

0.55 (4) λ (n)

0.72 (5)

0.88 (6)

1.03 (7)

Fig. 9.14 Comparison between relay and Choudhury model for plant models with different orders, where n denotes model order

9.5.1 Temperature Control Loop with Stiction from a Building Automation System The first result is from a temperature control loop in a building automation system. The loop was identified as oscillating and efforts were first made to detune the controller. Detuning failed to resolve the problem and only caused the oscillations to have a lower frequency. The problem was suspected to be caused by stiction. A snapshot of the data is shown in Fig. 9.15. The measured error signal is the dotted line that is oscillating about zero. The solid line is the median filtered signal and the band around zero was determined using the band-stop filtering method described earlier. It can be seen that the oscillations are outside of the noise band and are thus considered valid for evaluating the R index. One half-period in the oscillations shown in Fig. 9.15 is highlighted to clarify how the R values are calculated. As can be seen from the figure, the area under the response between the zero-crossing point and the peak value is greater than the area after the peak up to the next zero-crossing, i.e. A1 > A2 . This is common for most of the responses that are shown, thus resulting in a set of R values that are on average greater than one. The actual R values calculated for the data set were 3.0961, 1.2044, 4.4751, 2.0674, 6.5869, 1.1853, 4.4675, 1.4356, 1.0219, and 0.6633. Applying the statistical methods outlined earlier, the median value of the log transformed values is 0.5439 and the MAD value is 0.5542. The t-score is then:

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0.5439 √ = 1.9859 . (0.5542 × 1.4826)/ 10 − 1

(9.17)

This translates into there being more than a 95% probability that the R values are indeed greater than one; hence a good indication of stiction.

1 A1 > A2

0.5

A1

peak A2

0 e(t)

Lim

−0.5

−1 raw signal

filtered signal −1.5 0

50

100

150 Time [samples]

200

250

300

Fig. 9.15 Stiction oscillation from building automation system (TC loop BAS 7)

9.5.2 Temperature Control Loop with Stiction from a Pulp and Paper Plant Figure 9.16 shows a data set obtained from the pulp and paper industry, which shows the error signal for a temperature control loop. This loop was known to not have stiction and the source of the oscillations was thought to be an overly aggressive controller. The shape of the oscillations can be seen to be sinusoidal and thus more symmetrical about the peak value in each half-cycle than was observed in the previous case. The resulting R values were therefore all clustered about one. The median of the log of the R values was −0.1522 and the MAD value was 0.2672. The total number of values obtained was 27 and the resulting t-score was calculated to be −1.9586. This led to a probability value of there being stiction of just 0.03.


203

30 20

e(t)

10 0 −10 −20 −30 0

200

400

600

800 1000 1200 Time [samples]

1400

1600

1800

Fig. 9.16 Aggressively tuned controller from pulp and paper control loop (TC loop PAP 9)

9.6 Summary and Conclusions The basic idea behind the method presented in this chapter is to analyse the shape of an oscillating signal and try to match it to a predetermined shape representative of stiction. This idea could have been implemented in many different ways and the method that has been presented represents just one possible realisation. However, the method of realisation is critical in practice. A method needs to be robust to a wide variety of phenomena encountered in real systems that corrupt idealised expectations of behaviour. Often, the mode of implementation of an idea in a practical environment is more important than the core idea itself. Robustness and reliability turn out to be crucial elements in gaining acceptance of any new technology in most industrial application areas. The method reported in this chapter was developed for the building automation industry, which could be considered one of the lowest-cost applications of control technology. Although focusing on this application forced quite severe constraints on the development, the advantage is that this has led to a higher level of robustness and forced the method to remain simple and have relatively low computational requirements. The method reported here also has the advantage that it does not need to be tuned for each loop to which it is applied. Although there are a small number of parameters, these are generic and can be set offline without detailed knowledge of the loops being tested. The chapter has described the method and its component parts and provided some theoretical basis supplemented with simulation tests and two test results from actual control loops. The simplicity of the method means that it will not be able to dis-

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tinguish between behaviours that are manifested in a similar fashion in the features being analysed. Including the control signal as a measurement would definitely open up new possibilities for more reliably differentiating stiction from other cases, but this measurement was assumed not available. Valve-position feedback would also make detection of the stiction problem easier, but each additional measurement usually translates to an additional cost. An important part of evaluating the usefulness of any method is determining the cost part of the equation that includes the number of measurements and parameters and also the engineering or setup costs. These costs have purposely been kept to a minimum in the reported method and this of course has led to some sacrifice in efficacy. However, results so far have shown that the method performs favourably on a number of different data sets from a diverse array of applications.

Chapter 10

Estimation of Valve Stiction Using Separable Least-squares and Global Search Algorithms Mohieddine Jelali

This chapter presents a new procedure for quantifying valve stiction in control loops based on global optimisation. Measurements of the controlled variable (PV) and controller output (OP) are used to estimate the parameters of a Hammerstein system, consisting of a connection of a two-parameter stiction model and a linear low-order process model. As the objective function is non-smooth, gradient-free optimisation algorithms, i.e. pattern search (PS) methods or genetic algorithms (GA), are used for fixing the global minimum of the parameters of the stiction-model, subordinated with a least-squares estimator for identifying the linear model parameters. Some approaches for selecting the model structure of the linear model part are discussed. Results show that this novel optimisation-based technique recovers accurate and reliable estimates of the stiction-model parameters, deadband plus stickband (S) and slip jump (J), from normal (closed-loop) operating data for self-regulating and integrating processes. The robustness of the proposed approach was proven considering a range of test conditions including different process types, controller settings and measurement noise. Several simulation and industrial case studies are described to demonstrate the applicability of the presented techniques for different loops and for different amounts of stiction.

10.1 Introduction Oscillations may be a very drastic form of plant-performance degradation in the process industries. Oscillations in control loops may be caused either by aggressive controller tuning, disturbances, or the presence of non-linearities, such as static friction, deadzone, and hysteresis. Valve stiction, or static friction, is the most severe source of oscillations. According to Ruel [100], stiction of 0.5% is considered too Mohieddine Jelali Department of Plant and System Technology, VDEh-Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected]

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much in many processes, as stiction guarantees cycling and variability, and is thus more harmful than other valve problems. Stiction leads to limit cycles in PV and OP, and thus to increased variability in product quality and disrupted plant operation. Valve stiction is a widespread problem in industrial practice, as reported by many audits [10, 25, 92]. The literature contains several non-invasive methods to detect stiction in control loops by only using OP and PV. Among others, the following approaches are mentioned: the cross-correlation method of [47], the area-peak method of Singhal and Salsbury [102,110], the relay method of Rossi and Scali [99], the curve-fitting technique of He et al. [44], the pattern-recognition technique of Srinivasan et al. [115], the bicoherence and ellipse-fitting method of Choudhury et al. [17]. Some other techniques available are based on additional knowledge about the characteristic curve of the valve or values of MV, i.e. valve position, e.g. Kano et al. [63] and Yamashita [134]. Fairly complicated methods for detecting stiction were proposed in [51, 117]. Some stiction-detection techniques have been recently reviewed and compared by Horch [50]. A more comprehensive comparative study is given in Chap. 13 of this book. This chapter presents a technique for quantification of stiction in control valves. The proposed method, published in [58] the first time, is based on a Hammerstein model for describing the global system and a separated identification of the linear part, i.e. the transfer function between the manipulated variable (MV) and PV, and the non-linear part, i.e. the function between OP and MV, using available industrial data for OP and PV. This reflects industrial practice, where MV is usually not known, except for flow control loops, where PV and MV are considered to be coincident. The proposed procedure is an extension of similar approaches, e.g. [116], having in common the fact that the non-linear part is represented by Hammerstein models, but the stiction model and the identification techniques are different. Global search techniques, i.e. PS methods or GAs, are used here to estimate the non-linear model parameters, subordinated with a least-squares (LS) identification of the linear model parameters. Estimates of both stiction-model parameters, deadband plus stickband (S) and slip jump (J), are provided. These can be used to estimate the inner signal MV. In contrast, the ellipse-fitting method in [17] estimates only the parameter S. The approach in [116] is based on a separable LS identification algorithm proposed by [6] and applicable only to non-linearities with one single unknown parameter. Thus [116] uses a simple relay model for describing the valve stiction. This model is, however, physically unrealistic and fails to capture the behaviour of loops with non-integrating processes. Note also that the slip jump J is very difficult to observe in PV–OP plots because the process dynamics destroys the pattern. This makes the estimation of a two-parameter model much more challenging than those considered so far in the literature. The chapter is organised as follows: Sect. 10.2 presents the technique proposed for stiction quantification. Section 10.4 describes some practical issues when analysing industrial data with the presented method. Simulation and industrial case studies are given in Sects. 10.5 and 10.6, respectively, to demonstrate the practicality and applicability of the proposed technique.

10 Estimation of Valve Stiction Using Separable Least-squares

207

10.2 Basic Approach A novel technique for detection and estimation of valve stiction in control loops from normal closed-loop operating data based on two-stage identification is proposed in this section. The control system is represented by a Hammerstein model including a two-parameter stiction model and a linear model for describing the remaining system part. Only OP and PV data are required for the proposed technique. This not only detects the presence of stiction but also provides estimates of the stiction parameters. Therefore, the method is useful in short-listing a large number of control valves more or less suffering from stiction in chemical or other plants, containing hundreds or thousands of control loops. This helps reduce the plant maintenance cost and increase the overall profitability of the plant. The framework for oscillation diagnosis proposed consists of the following features: • Two-parameter stiction models, which are more accurate and suitable for both self-regulating and integrating processes, particularly when time delay is present, are considered. • The linear dynamics is represented through simple, i.e. low-order, models. Only a partly automated structure selection is suggested. This choice keeps complexity and thus the required computational burden limited. • Global search techniques, i.e. PS methods or GA, are used to estimate the nonlinear model parameters, subordinated with a LS identification of the linear model parameters. • The proposed parameter estimation is recommended as a second diagnosis stage, i.e. for stiction quantification after detecting stiction using other non-invasive methods. • Since both the linear and the non-linear part will be estimated, closed-loop simulations without the stiction model possibly help identify bad controller tuning or external disturbances, affecting simultaneously the loop performance (in addition to stiction). This ability of detecting multiple loop faults occurring simultaneously is a unique feature of this technique. • The method is not limited to sticky loops, but can also be applied when other non-linearities, such as hysteresis or backlash, are present. • This identification-based technique is robust against noise and drifting trends, usually corrupting real-world data. Moreover, since both the linear and the non-linear part will be estimated, closeloop simulations without the stiction model could be carried out to detect possible bad controller tuning or external disturbances. This issue is not presented here, but can be treated in a similar way to the approach of Karra and Karim in Sections 12.3.2 and 12.5.

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M. Jelali

10.2.1 Identification Model Structure: Hammerstein Model A control valve with stiction can be described by a Hammerstein model, as considered in [116] and illustrated in Fig. 10.1. A basic assumption is that the system part without stiction can be approximated by a linear model.

Fig. 10.1 Process-control loop with valve stiction within an identification framework

10.2.2 Linear Model The process dynamics, including the valve dynamics, are represented by an ARMAX model A(q−1 )y(k) = q−τ B(q−1 )uv (k) +C(q−1 )ε (k) ,

(10.1)

where q−i is the backward shift operator q−i u(k) = u(k − i), τ is the discrete time delay, i.e. the number of unit delays, and ε the unmeasured disturbance. A(q−1 ), B(q−1 ) and C(q−1 ) are polynomials in q−1 of specified order n, m and p, respectively: A(q−1 ) = 1 + a1 q−1 + a2 q−2 + . . . + an q−n , B(q−1 ) = b0 + b1 q−1 + . . . + bm q−m , C(q−1 ) = 1 + c1 q−1 + c2 q−2 + . . . + c p q−p .

(10.2)

Throughout the chapter, the model in Eqs. 10.1 and 10.2 is denoted by ARMAX(n, m, p, τ ). We assume throughout the chapter that the time delay and model orders are known a priori or have been determined by applying some suitable technique; see Sect. 10.4.1.


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10.2.3 Stiction Model Figure 1.5 shows the typical input–output behaviour of a valve with stiction, consisting of the components: deadband plus stickband S (AB), slip jump J (BC) and the moving phase (FG). The sequence motion/stop of the valve due to stiction is called stick-slip motion. More details can be found in [15]. In our approach, we use one of the data-driven two-parameter models NLstic (J, S), proposed by Kano et al. [63], Choudhury et al. [15], and He et al. [44]. Such a model can realistically capture the closed-loop behaviour, as shown in [15] and Chaps. 2 and 3 of this book, in contrast to the relay model used in [116]. A comparison of the relay and Choudhury’s models can be found in [110]; see also Chap. 9 of this book. It is pointed out that the relay model is a good approximation of Choudhury’s model only when the ratio time delay to time constant (τ /T ) is small and the system order is low. The discrepancy between the results using both models increases with increasing (τ /T ) and system order. As the processes we mainly consider are those with significant time delays, a two-parameter stiction model is needed. The stiction non-linearity can be written in the general form uv (k) = NLstic (u(k), · · · , u(0), uv (k − 1), · · · , uv (0), J, S) ,

(10.3)

parametrised by the parameter pair J and S, which are assumed to be constant. Note that the internal signal uv (k) is not measurable. Stiction non-linearity is a discontinuous function or “hard” non-linearity, which does not belong to the class of memoryless non-linearities, usually assumed in the estimation of block-oriented models. Stiction non-linearity thus results in non-smooth and non-convex objective maps. These facts have two important implications: • Most of the known techniques, e.g. correlation analysis, prediction-error methods, for the estimation of Hammerstein models cannot be applied for the stiction estimation and diagnosis problem. • Gradient-based algorithms, known as local optimisation methods, would always get stuck in a local minimum near the starting point. In contrast, global search, usually gradient-free, algorithms are not affected by random noise in the objective functions, as they require only function values and not the derivatives. We do not intend to discuss all approaches for the identification of Hammerstein models, but refer the reader to the excellent overview in [116], where the inherent limitations of “classical” methods in the estimation and diagnosis of stiction are pointed out. In this work, the use of a separable least-squares estimator combined with a global search algorithm is proposed.

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10.3 Identification Approach 10.3.1 Separable Least-squares Estimator To simplify the Hammerstein model identification, a decoupling between the linear and non-linear parts is useful, as recommended in [6, 106] in the context of open-loop identification of input non-linearities other than stiction. This technique is extended in this section for the application to the identification of stiction models parametrised by J and S. To simplify the derivation, we first consider an ARX model (with m = n) to describe the linear system part in the time domain as y(k) = Θ T [y(k − 1), · · · , y(k − n), uv (k − τ − 1), · · · , uv (k − τ − n)] + ε

(10.4)

with the unknown parameter vector

Θ = [−a1 , · · · , −an , b1 , · · · , bn ]T

(10.5)

as well as J and S because of Eq. 10.3. Let ˆ , uˆv (k) = NLstic (u(k), · · · , u(0), uˆv (k − 1), · · · , uˆv (0), J,ˆ S)

(10.6)

ˆ which provides an estimate of MV, i.e. the valve position uv (k), using Jˆ and S. Define the prediction error

εΘˆ ,J,ˆ Sˆ (k) = y(k) − y(k) ˆ = Θˆ T [y(k − 1), · · · , y(k − n), uv (k − τ − 1), · · · , uv (k − τ − n)] ,

(10.7)

for k = 1, 2, · · · , N, and the objective function (MSE: mean-squared error) N ˆ Θˆ ) = 1 ∑ ε 2ˆ ˆ ˆ (k) . VN (J,ˆ S, N k=1 Θ ,J,S

The associated estimates are

ˆ Θˆ T T = arg minVN (J,ˆ S, ˆ Θˆ ) , J,ˆ S,

(10.8)

(10.9)

where N is the number of data samples y(k), y(k) ˆ is the estimate of y(k) or Hammerstein-model output, and Θˆ is the estimated parameter vector of the linear model part. With


211

⎤ y(n + τ ) ⎢y(n + τ + 1)⎥ ⎥ ⎢ y=⎢ ⎥ .. ⎣ ⎦ . ⎡

y(N) ⎡ ⎤ y(n + τ − 1) · · · y(τ ) uˆv (n − 1) · · · uˆv (0) ⎢ y(n + τ ) · · · y(τ + 1) ⎥ uˆv (n) ··· uˆv (1) ⎥ ˆ =⎢ Φ (J,ˆ S) ⎢ ⎥ (10.10) .. .. .. .. .. .. ⎣ ⎦ . . . . . . y(N − 1) · · · y(N − n) uˆv (N − τ − 1) · · · uˆv (N − n − τ ) the objective function VN can be rewritten as ( 1( ˆ Θˆ ( . VN = (y − Φ (J,ˆ S) (10.11) 2 N For a given data set y(k), u(k), i.e. PV (k), OP(k), this minimisation involves three ˆ but smooth in Θˆ . Moreover, variables J,ˆ Sˆ and Θˆ . VN is non-smooth in Jˆ and S,

∂ VN ˆ + Φ T (J,ˆ S) ˆ Φ (J,ˆ S) ˆ Θˆ . = −Φ (J,ˆ S)y (10.12) ∂ Θˆ ˆ Φ (J,ˆ S) ˆ is invertible, the necessary and sufficient condition for Θˆ Hence, if Φ T (J,ˆ S) to be optimal is

ˆ Φ (J,ˆ S) ˆ −1 Φ T y , Θˆ = Φ T (J,ˆ S) (10.13) 0=

provided that Jˆ and Sˆ are optimal. Therefore, by substituting Θˆ in terms of Jˆ and Sˆ back into VN , it follows (

1( ( ˆ Φ T (J,ˆ S) ˆ Φ (J,ˆ S) ˆ −1 Φ T y( (10.14) VN = (I − Φ (J,ˆ S) ( . N 2 When the two-parameter stiction model is used to calculate uˆv (k), the dimension of the search space is reduced from 2n + 2 to 2, independent of the linear part. However, note that, in contrast to the non-linearities considered in [6], the stiction non-linearity cannot be expressed in closed form. The identification algorithm proposed for systems with stiction non-linearities parametrised by J and S can now be summarised as follows: Step 0. Determine the time delay and select initial values for J and S. Step 1. Consider the Hammerstein system in Eqs. 10.4 and 10.6, and collect a data set {y(k), u(k)}. Step 2. Calculate an estimate of MV, i.e. uˆv (k), using a stiction model (parametrised ˆ as in Eq. 10.10. by J and S), and construct y and Φ (J,ˆ S), Step 3. Identify the parameter vector Θˆ of the linear model by a least-squares method (Eq. 10.13; inner identification). Step 4. Iterate to find new values of Jˆ and Sˆ (global search). Step 5. Repeat steps 1–4 until the the optimal values of of Jˆ and Sˆ are found, i.e. the minimum of VN is achieved (outer identification).

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Thus, when an ARX is used, iterative optimisation needs only be performed with respect to the non-linear parameters (J and S). To generalise the method, one may use ARMAX instead of ARX, which now implies the need for iterative (non-linear) estimation also for the linear model. Once the optimal values Jˆ and Sˆ are obtained by a global search method, as described below, the stiction model is computed to generate uˆv (k). Based on uˆv (k) and y(k), the parameters Θˆ of the linear model are identified using an LS-IV (instrumental variables) algorithm or a prediction error method (PEM) [73], combined with model-structure selection, i.e. for estimating the model orders and time delay (when not known or specified); see Sect. 10.4.1. This two-stage identification method is illustrated in Fig. 10.2. The key elements of our approach are as follows: • The linear dynamics is modelled using a low-order model, the polynomial parameters of which are estimated using LS or PEM in a subordinated identification task. The number of unit delays τ , when not known, is determined by applying an appropriate time-delay estimation method. • A pattern-search algorithm or a genetic algorithm is employed to estimate the values of the parameters S and J of the stiction model so that the MSE is minimised (Eq. 10.9). The time trend of MV can thus be estimated.

y (k )

Measured OP u (k )

NLstic(J, S)

uˆv (k )

Glp(τ, )

yˆ(k ) −

τ, Θ J, S

Time-delay & Least-squares / Prediction-error method

dˆ ( k )

Pattern search / Genetic algorithms

Fig. 10.2 Two-stage identification of the system parameters

Remark 10.1. Although the proposed stiction-estimation method is principally able to detect stiction, it is more recommended as a second diagnosis stage, i.e. for stiction quantification after valve stiction has been detected by another simpler method. The presence of stiction non-linearity also ensures the identifiability of the model from normal closed-loop operating data. Also, the identifiability is ensured when another non-linearity, such as deadband or deadzone, is present instead of stiction. However, when the loop oscillates due to “linear” causes, such as external disturbances, the closed-loop identification can become difficult when sufficient excitation is not present. In such cases, the stiction diagnosis may yield wrong results.


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10.3.2 Global Search Algorithms Traditional derivative-based optimisation methods, like those found in the Optimiza® tion ToolboxTM of MATLAB , are fast and accurate for many types of similar optimisation problems. These methods are designed to solve “smooth”, i.e. continuous and differentiable, minimisation problems, as they use derivatives to determine the direction of descent. While using derivatives makes these methods fast and accurate, they are not suitable when problems lack smoothness, as is the case in the valve-stiction estimation problem. When faced with solving such non-smooth problems, methods like genetic algorithms or pattern-search methods are the effective alternatives. Both methods are briefly discussed in the following. 10.3.2.1 Genetic Algorithms Genetic algorithms (GAs) are search techniques that imitate the concepts of natural selection and genetics. They were formally introduced by Holland (1975). GAs search the solution space of a function through the use of simulated evolution, i.e. the survival-of-the-fittest strategy. This provides an implicit as well as explicit parallelism that allows for the exploitation of several promising areas of the solution space at the same time. Instead of looking at one point at a time and stepping to a new point for each iteration, a whole population of solutions is iterated towards the optimum at the same time. Using a population allows us to explore multiple “buckets” (local minima) simultaneously, increasing the likelihood of finding the global optimum. GAs are thus well suited for solving difficult optimisation problems with objective functions that possess “bad” properties such as discontinuity and non-differentiability, as is the case for the stiction-estimation problem. In the GA-based optimisation approach (Fig. 10.3), the (unknown) parameters are represented as genes, hence the name “genetic”, on a chromosome, representing an individual. Similar to the simplex search, a GA features a group of candidate solutions, the population or gene pool, on the response surface. Applying natural selection and using the genetic operators, recombination and mutation, chromosomes with better fitness, i.e. degree of “goodness”, are determined. Natural selection guarantees that chromosomes with the best fitness will propagate in future populations. Using the recombination operator, the GA combines genes from two parent chromosomes to form two new chromosomes (children) that have a high probability of having better fitness than their parents. Mutation allows new areas of the response surface to be explored. One of the reasons GAs work so well is that they offer a combination of hill-climbing ability through natural selection and a stochastic method through recombination and mutation. The main drawback of using GAs is the high computational burden. This is due to their prohibitively slow convergence to the optimum, when compared to gradientbased methods, especially for problems with a large number of design variables. Nevertheless, the continual progress of computer technologies has greatly reduced

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the effort required to implement such methods. Therefore, GAs are becoming increasingly popular in the last years. Complete discussions of genetic algorithms can be found in the books by Goldberg [36] and Michalewicz [85].

Create initial population

Evaluate fitness

Selection

Crossover

Mutation

Select next generation

Stopping?

no

yes Best solution found

Fig. 10.3 Basic procedure of genetic algorithms

10.3.2.2 Pattern-search Methods Pattern search is an attractive alternative to GAs, as they are often computationally less expensive. Pattern search operates by searching a set of points called a pattern, which expands or shrinks depending on whether any point within the pattern has a lower objective function value than the current point. The search stops after a minimum pattern size is reached. Like the genetic algorithm, the pattern search algorithm does not use derivatives to determine descent and so works well on nondifferentiable, stochastic and discontinuous objective functions. Also, similar to the genetic algorithm, pattern search is often very effective at finding the global minimum because of the nature of its search. A PS algorithm can be generally described as follows: Step 1. Initialise direction and mesh size. At each iteration k, the mesh is defined by the set Mk = ∪ {x + Δk Dz : z ∈ NnD }, where Sk ∈ Rn is the set of points x∈Sk

where the objective function f had been evaluated by the start of iteration


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k and Δ k > 0 is the mesh size parameter that controls the fineness of the mesh. D is a set of positive spanning directions in Rn . Step 2. Perform the following steps until convergence. • SEARCH Step. Employ some finite strategy seeking an improved mesh point, i.e. where the value of the objective function is lower than that at the current point. The SEARCH step usually includes a few iterations using a heuristic, such as GA, random sampling, or the approximate optimisation on the mesh of surrogate function. • POLL Step. If the SEARCH step was unsuccessful, evaluate the objective function at points in the poll set Pk = {xk + Δk d : d ∈ Dk ⊆ D} ⊂ Mk , i.e. at points neighbouring the current one on the mesh, until an improved mesh point is found. • Parameter Update. – Success, i.e. when SEARCH or POLL finds an improved mesh point: accept new iterate and coarsen the mesh. – Failure: refine the mesh. The mesh size is updated according to the rule Δk+1 = τ wk Δk , where τ > 1 is a fixed rational number, wk ∈ {0, 1, . . . , w+ } for mesh coarsening and wk ∈ {w− , w− + 1, . . . , -1} for mesh refining and w− ≤ −1 and w+ ≥ 0 are two fixed integers. Typically, Δ k + 1 = 2.0Δ k is used for mesh coarsening and Δ k + 1 = 0.5Δ k for mesh refining. Depending on the mesh-forming method, the search heuristic and the polling strategy, different pattern-search algorithms result. For detailed discussions of these algorithms, the reader should consult, for instance, Lewis and Torczon [69, 70]. In fact, our experience with this method soon led to the conclusion that PS is very fast in finding a point somewhere in the region of the global minimum. Therefore, (GA-based) pattern search is the recommended approach for solving the stictionidentification problem.

10.4 Key Issues 10.4.1 Model Structure Selection In control-performance monitoring, it is usually assumed that the time delay is known or can be estimated accurately. At least an interval, where the time delay may lie, should be given. Simulation studies in [116] indicate that a decoupling in the accuracy of the estimate between the non-linear and the linear component of Hammerstein models with stiction non-linearity may exist. In particular, the accuracy of the estimate of the non-linear component may be not affected by the complexity of the model structure used to describe the linear model part. This point also suggests using a simple model for the linear valve and process dynamics. However,

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a good estimate of the process time delay is essential, as stiction itself introduces an additional time delay into the system, i.e. PV will not change up to the time when OP becomes larger than S plus the pure time delay of the process without stiction. Therefore, trying to simultaneously estimate all unknown parameters would always not be successful. This problem is avoided in the two-stage approach proposed in Sect. 10.3, where the time-delay estimation is included in the inner identification process.

10.4.2 Time-delay Estimation Time-delay estimation (TDE) is an integral part of the inner identification stage of the proposed stiction-estimation technique. Many approaches for estimating the time delay exist. A classification, comprehensive surveys and comparative (simulation) studies of TDE methods are given in [11, 87]. In the following, three of the most frequently used approaches for TDE are briefly described. It is very important to realise that time delay cannot be estimated from routine operating data without external excitations or abrupt changes in the control signals. This fact, often ignored by many researchers, has been well proven in the theory of system-identification literature; see [73]. Occasionally, this may be possible due to non-linearity or some natural perturbation present in the process, but this is not always reliable. However, in the case considered here, i.e. since the loop is assumed to be oscillating due to stiction or other reasons, the time delay can be estimated from industrial input/output data OP/PV. 10.4.2.1 Cross-correlation Method The classical method for TDE is based on analysing the cross-correlation between u and y as the two signals of interest. Both signals are put close to each other and then time shifted until they agree the most. This can be formally written as

τˆ = max E{y(k)u(k − τ )} ≈ max ∑ y(k)u(k − τ ) . τ

τ

(10.15)

k

As stated in [57], correlation-based TDE is one of the most reliable methods. MATLAB’s Higher Order Spectral Analysis ToolboxTM offers the function tder for TDE using maximum-likelihood windowed cross-correlation. Other routines are tde (cross-cumulant method), tdeb (conventional bispectrum method) and delayest (comparison of ARX models with different delays).


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10.4.2.2 Relational Approximation Method The delay term in the continuous-time model is approximated by a low-order rational function, typically a Padé approximation. The time delay is then computed from the pole-zero excess. Isaksson [56] proposed to estimate first a Laguerre model, followed by a calculation of the (discrete-time) zeros zi and their conversion into continuous-time zeros si . A comparison with a first-order Padé approximation gives an estimate of the (continuous) time delay, assuming that the plant has no nonminimum phase zeros, except those resulting from the time delay:

τˆ = 1 +

Tˆ ∑r (2/si ) = 1 + i=1 ; Ts Ts

si ≈

1 ln(zi ) , Ts

(10.16)

where r is the number of zeros in the right half-plane. This method has been modified by Horch [48] to avoid approximation error sources (i.e. conversion of discretetime zeros into continuous-time zeros and the Padé approximation itself) by directly estimating the time delay from the discrete-time zeros of the Laguerre model

τˆ = 1 −

ϕ (ω ) |ω 1 . ω Ts

(10.17)

Measures, such as zero guarding, have to be taken to prevent/remove “false zeros” (close to but outside of the unit circle); see [11] for details. Since the intended use here is performance monitoring, it is not at all critical that one actually gets a kind of “apparent time delay” when the system (without time delay) is non-minimum phase. 10.4.2.3 Model Fitting ®

One may use the functions arxstruc/ivstruc and selstruc of MATLAB ’s Optimization ToolboxTM , minimising Akaike’s information criterion (AIC) [73]. This estimation is quick since the ARX model can be written as a linear regression and can be estimated by solving a linear equation system. Our experience reveals that this method often fails to provide a good estimate of time delay, particularly when substantial noise is present. The same approach but using an ARMAX model, as suggested by Srinivasan et al. [116], can be followed. To simplify and thus accelerate the model-identification process, we advise choosing the model order to be equal, i.e. nA = nB = nC = n, so that it is only necessary to search for two parameters. A search is performed over the range of possible orders [nmin :nmax ] and numbers of unit delays [τ min :τ max ] to find the minimum AIC for the ARMAX model. This approach, which requires relatively high computation burden, was not always successful, due to the existence of many local minima. We had good experience and thus recommend using a fixed-order model, e.g. ARMAX(2, 2, 2, τˆ ), for which the optimal time delay is determined (when not known). For this purpose, numerous methods exist; see [11] for an intensive dis-

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cussion and comparison. We investigated many of them, e.g. the prefiltered arxstruc method proposed by Björklund [11] (and called met1struc) and some methods ® included in MATLAB ’s HOSA ToolboxTM , e.g. tder (windowed cross-correlation). None of these techniques was successful in all simulation and practical cases we studied. Therefore, we implemented a simple search over a range [τ min :τ max ] for ARMAX(2, 2, 2, τˆ ) and picked up the one with the minimum AIC. This approach led to good results, at the expense of a higher computation burden; see Sect. 10.5.

10.4.3 Determination of Initial Parameters and Incorporation of Constraints As for every non-linear optimisation technique, pattern search needs some initial values for the parameters at the first iteration. It is obvious that a good initial guess speeds up the convergence of the algorithm and increases the probability of finding the global optimum. A good initialisation for the stiction-estimation approach proposed can be determined as follows: Step 1. Use the ellipse-fitting method [17] to yield an initial guess Sˆ0 of the deadband plus stickband S. Step 2. Taking S = Sˆ0 , use a simple grid search with a rough step size to get a complete picture VN (J), whose minimum represents an initial estimate Jˆ0 for the slip jump J. This procedure provides a region where the global optimum should lie. Therefore, it is useful to constrain the search of the algorithm within this region, i.e. to specify a lower and upper bound for the parameters J and S, say [0.8Jˆ0 , 1.2Jˆ0 ] and [0.8Sˆ0 , 1.2Sˆ0 ]. This helps avoid problems with falling in local minima. The specified regions should be not too narrow, because the curve-fitting method estimates “apparent stiction” which may differ from real stiction. However, it should be stressed that the proposed method can also be applied on its own without using the ellipsefitting method or any other technique. Remark 10.2. It may occur that a loop with a valve, suffering from stiction, has two (or more) distinct behaviours, e.g. stiction undershoot for one part of the data and stiction overshoot for the other part. An example of such behaviour is illustrated in Fig. 10.4, taken from [15]. It is obvious that the stiction-estimation algorithm will have problems in this case. Therefore, the careful inspection of the PV–OP plot should reveal this situation, and the data parts should be separated. This can be done by visual inspection of the data. Remark 10.3. Decisive for successful optimisation is adding constraints to discriminate parameter combinations (J and S) that drive the stiction model output to zero for all time, which is useless as input for the inner identification process. If such a


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PV and OP

Process output PV

parameter combination is found, the estimation error is assigned an extremely large value to make sure that the algorithm does not re-select this candidate solution.

Samples

(a)

Controller output OP

(b)

Fig. 10.4 Data from a flow loop in a refinery: a) time trend of PV and OP (line with circles: PV, thin line: OP; b) PV-OP plot (right) [15]

10.5 Simulation Studies The utility of the proposed stiction-estimation technique is now illustrated in some simulation and industrial studies. In all case studies, the number of cycles taken for the analysis lies in the range 10–15. ® All computations reported in this study were carried out using MATLAB and ® Simulink (Release 14). All open-loop and closed-loop simulations were accom® ® plished using Simulink . To perform the optimisation tasks, MATLAB was employed in conjunction with the Genetic Algorithm and Direct Search ToolboxTM , i.e. the ga function and the patternsearch function (with the option “@searchga”). Most reliable results in the simulation and practical cases we studied were achieved by searching the time-delay value that minimises AIC for an ARMAX model with ® ® fixed orders n = 3, m = p = 2. Computations were performed on an Intel Pentium M Processor. Two of the investigated simulation studies are discussed below in a separate section for each of the loops. The transfer functions and PI-controllers are shown in Table 10.1. The magnitudes of S and J are specified as a percentage (%) of valve input span and process output span, respectively. Kano’s stiction model was used, but the same results can be found by considering Choudhury’s model. Results for the ideal case, where no noise is present and the time delay is specified, are not given, as they are not spectacular: the algorithm yields very accurate parameter estimates. Below, a few simulation results are given; however, a broad range of other test conditions, i.e. low-order/high-order, self-regulating/integrating (τ /T = 0.1–10), and different stiction strengths (J/S = 0–5), have also been successfully tested. (The re-

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sults are not shown here for the scale of brevity.) The proposed algorithm produces good estimates of the stiction-model parameters with deviations less than 10% of the actual values. Table 10.1 Process models and controllers used in the simulation studies Process type Process model Controller * ) 3e−τ s 1 FOPTD Gp (s) = 1+T s , τ = 10, T = 10 Gc (s) = 0.2 1 + 10s * ) e−τ s 1 IPTD Gp (s) = (1+T Gc (s) = 0.4 1 + 0.2s s)s , τ = 1, T = 0.5

Table 10.2 Results for the process simulations FOPTD Test conditions (T s = 1 s) J S 2.0 5.00 5.0 5.00 7.0 5.00

IPTD Estimated stiction [%] Test conditions (T s = 1 s) Jˆ Sˆ J S 2.02 5.00 4.0 6.0 4.97 4.89 3.0 3.0 6.34 4.86 5.0 3.0

Estimated stiction [%] Jˆ 4.30 3.12 4.90

Sˆ 5.66 3.27 3.01

10.5.1 First-order-plus-time-delay Process This example, which models a concentration loop with slow dynamics and a large time delay, is taken from [15]. A zero-mean Gaussian noise signal was added to the process output. The stiction-estimation algorithm with time-delay identification was applied to the “data” obtained for different stiction cases (undershoot, no offset and overshoot). Table 10.2 lists the test conditions and results for each scenario. (Ts denotes the sampling period.) It can be concluded that the presented technique accurately quantifies stiction in all scenarios considered. The estimates of the recovered stiction models are very close to the true values.

10.5.2 Integrating Process with Time Delay The stiction estimation for a closed loop with an integrating plus time delay (IPTD) process is tried next. As before, several scenarios were considered. The conditions tested and the results achieved are given in Table 10.2. They prove the reliability of the presented method for stiction quantification also for integrating processes. The estimates of the recovered stiction models are close to the true values.


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10.6 Industrial Case Studies Over a dozen industrial case studies (not all presented here) from different industrial sites have demonstrated the wide applicability and accuracy of this method as a valuable stiction-quantification technique. The objective of this section is to demonstrate the proposed framework and techniques on different industrial control loops, including flow control (FC), pressure control (PC), level control (LC) and temperature control (TC). For each loop, the setpoint (SP), PV and OP data were available. The procedure suggested in Sect. 10.4.3 was used for finding good initial stiction parameters. Table 10.3 gives the summary of the results achieved by the application of the methodology described in Sect. 10.3. The results are commented on below for each loop. In all examples, the linear part was approximated by an ARMAX(3, 2, 2, τˆ ) model with τˆ determined for minimum AIC. Table 10.3 also contains values of the oscillation regularity factor r and period Tp according to Thornhill et al. [127], and the non-linearity index (NLI) proposed by Choudhury et al. [17] in relation to the ellipse fitting of the PV–OP plot. Table 10.3 Summary of results for the industrial control loops Loop no.

CHEM 25 PAP 2 CHEM 24 POW 2 POW 4 MIN 1 CHEM 70

Oscillation and non-linearity detection results r T p [s] NLI 5.6 192 0.56 11.2 42.4 0.16 2.87 136 0.17 21.4 288 0.55 16.2 237 0.36 4.0 6940 0.15 54.6 3135 -

Initial guess for S S0 1.80 3.00 23.00 11.40 4.80 1.10 -

Estimated stiction [%] Jˆ Sˆ 0.59 1.80 0.84 3.00 0.81 22.90 1.10 11.47 2.49 4.49 0.96 1.02 0.04 0.14

CPU time [min] 21 26 29 24 34 18 20

It is important to see that, when the time delay is known, the computation time reduces significantly, by up to 90%. This means that a major portion of the computation burden results from time-delay estimation. Therefore, an efficient algorithm for this task is worth consideration in future research.

10.6.1 Loop CHEM 25: Pressure Control Loop This is a pressure control loop in a refinery. Data from this loop were analysed by the ellipse-fitting method to give a stiction band S0 = 1.8; see Fig. 10.5. The presence of non-linearity is confirmed by the index value NLI = 0.56. Setting this valve as S0 and varying J yields V (J) shown in Fig. 10.6. It can be seen that the minimum lies in the neighbourhood of J0 = 0.60. Using these initial parameter values, the proposed algorithm was run with the constraints J ∈ [0.25, 1.0] and S ∈ [1.5, 2.2] to give J = 0.59 and S = 1.80.

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3.9

PV [%]

3.9 3.85 3.8 3.85

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PV [%]

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Fig. 10.5 Data from PC loop CHEM 25: a) PV and OP trends; b) PV–OP plot 0.064 0.063 0.062

VN

0.061 0.06 0.059 0.058 0.057

0

02

04

0.6

0.8

J

1

1.2

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Fig. 10.6 Objective function VN vs. J for loop CHEM 25 setting S = 1.8

Exemplarily for this loop, the estimated inner signal MV is illustrated in Fig. 10.7. This figure clearly indicates both deadband plus stickband and slip-jump effects. The latter is large and visible, especially when the valve is moving in downward and upward directions (J is marked in the figure). Figure 10.8 shows how well the estimated model fits the measured data. The linear model identified has the polynomials (Eq. 10.2, τ = 2) A(q−1 ) = 1 − 1.718q−1 + 0.9234q−2 − 0.1861q−3 , B(q−1 ) = 0.07152 − 0.0682q−1 , C(q−1 ) = 1 − 0.9129q−1 + 0.1184q−2 . If we use a specified linear model structure, i.e. ARMAX(2, 2, 1, 1), and run the algorithm again, we get the estimates J = 0.60 and S = 1.87, which are close to


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the ones obtained above. The linear model has now been estimated to be (Eq. 10.2, τ = 1) A(q−1 ) = 1 − 0.7181q−1 + 0.1256q−2 , B(q−1 ) = 0.00659 + 0.0658q−1 , C(q−1 ) = 1 + 0.2376q−1 .

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Fig. 10.8 Measured vs. predicted PV data for loop CHEM 25

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The total model shows similar prediction quality, but the computation time reduces to 1.7 min. This is to show the price we pay for not a priori knowing the time delay.

10.6.2 Loop PAP 2: Flow Control Loop This loop with valve stiction is taken from [50]/FC525. Figure 10.9 illustrates the data considered for stiction quantification. The application of the approach proposed here led to the estimates J = 0.84 and S = 3.0. 40

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Fig. 10.9 Data from FC loop PAP 2: a) PV and OP trends; b) PV–OP plot

10.6.3 Loop CHEM 24: Flow Control Loop with Setpoint Changes This flow control loop has excessive stiction. It is an inner loop of a cascade control system, and thus is subject to rapid setpoint changes. The plant data are shown in Fig. 10.10. The PV–OP plot shows a shape (parallelogram) very similar to Fig. 1.5. The fact that the oscillations in OP and PV are varying in amplitude and time period makes stiction detection and quantification challenging. The stiction index clearly signals the presence of stiction in the loop. The stiction-parameter estimates were found to be J = 0.81 and S = 22.9, which can be confirmed by a look at the PV–OP plot in the right side of Fig. 10.10.


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Fig. 10.10 Data from FC loop CHEM 24: a) PV and OP trends; b) PV–OP plot

10.6.4 Loop POW 2: Level Control Loop This example represents a level control loop in a power plant; see Fig. 10.11. Data from this loop were already analysed in [15]/Fig. 3 and [17]/Fig. 4, and the ellipsefitting method was applied to give a stiction band S0 = 11.4. The estimated stiction parameters using the technique proposed in this chapter were J = 1.10 and S = 11.47. These results are in good agreement with the data plots given in [15, 17], where the MV trend and MV–OP plot are also shown. 1 0.8

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(a) Fig. 10.11 Data from LC loop POW 2: a) PV and OP trends; b) PV–OP plot

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10.6.5 Loop POW 4: Level Control Loop The data for this level control loop were also obtained from a power plant and are illustrated in Fig. 10.12. The deadband plus stickband was estimated to be S0 = 4.8 by applying the ellipse-fitting method. The stiction parameters in this loop were estimated to be J = 2.49 and S = 4.49. 0.6

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Fig. 10.12 Data from LC loop POW 4: a) PV and OP trends; b) PV–OP plot

10.6.6 Loop MIN 1: Temperature Control Loop This loop on a furnace feed dryer system was also considered in [15]/Fig. 6 and [17]/Fig. 10. Using the data shown in Fig. 10.13, the proposed estimation algorithm leads to the parameter estimates J = 0.96 and S = 1.02 (very small undershoot), which are in good agreement with the data plots given in [15, 17], where the MV trend and MV–OP plot are also shown. The initial value for S was S0 = 1.1, determined using the ellipse-fitting method.

10.6.7 Loop CHEM 70: Flow Control Loop with External Disturbances The purpose of this example is to show that the presented approach can be principally used to detect stiction. A flow control loop is considered, for which the stiction indices indicate no stiction; see Fig. 10.14. It is also known that this loop suffers from external disturbances. This is confirmed by the stiction-estimation algorithm. The latter yields negligible values of J and S. Remember that deadband, i.e. J = 0 and S > 0, cannot induce oscillation for self-regulating processes; see Table 1.3.

10 Estimation of Valve Stiction Using Separable Least-squares 51.5

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Fig. 10.14 Data from FC loop CHEM 70: a) PV and OP trends; b) PV–OP plot

10.7 Summary and Conclusions A novel procedure for quantifying valve stiction in control loops based on two-stage identification has been presented in this chapter. The proposed approach uses PV and OP signals to estimate the parameters of a Hammerstein system, consisting of a connection of a two-parameter stiction model and a linear low-order process model. A pattern search or genetic algorithm subordinated by a least-squares estimator was proposed for the parameter identification. This yields a quantification of the stiction, i.e. estimates of the parameters deadband plus stickband (S) and slip jump (J), thus enabling one to estimate time trends of MV. The method can also be ap-

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plied in the case of one-parameter stiction models. The results on different processes under a range of conditions – low-order/high-order, self-regulating/integrating, different controller settings and measurement noise, different stiction levels – show that the proposed optimisation can provide stiction-model-parameter estimates accurately and reliably. The stiction-quantification technique has been successfully demonstrated on two simulation case studies and on many data sets from different industrial control loops. The relatively high CPU time required for the identification process is not critical, as the analysis is performed offline. Also, this work is inexpensive compared to the savings in experimentation with the process or in downtime costs when invasive methods for stiction quantification would be applied. A faster algorithm for time-delay estimation and a more efficient implementation of the algorithms, e.g. as C-code, should significantly accelerate the computation.

Chapter 11

Stiction Estimation Using Constrained Optimisation and Contour Map Kwan Ho Lee, Zhengyun Ren and Biao Huang

A closed-loop method for valve-stiction detection and quantification based on Hammerstein modelling is discussed. A suitable model structure of valve stiction is chosen prior to conducting valve-stiction detection and quantification. Given the stiction-model structure, a bounded search space of a stiction model is defined and a constrained optimisation problem is performed. The best unknown stiction-model parameters are found by satisfying a mean-squared error criterion within a space of valve stiction model parameters. For this purpose, a multi-start adaptive random search is used. The proposed strategy not only detects but also quantifies valve stiction. The validity of the proposed method is illustrated through industrial examples. Also, the closed-loop identifiability issue is addressed.

11.1 Introduction In a basic control loop with a valve, SP, OP, MV, and PV stand for the setpoint, the controller output, the manipulated variable and the controlled variable, respectively. Note that SP, OP, and PV are usually recorded on the distributed control system in industry, so are readily available, while the manipulated variables such as flow rate are not always available.

Kwan Ho Lee Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected] Zhengyun Ren Department of Automation, Donghua University, Shanghai, China, e-mail: renzhengyun@ 163.com Biao Huang (corresponding author) Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected]

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Valve stiction often causes oscillation in control loops. The presence of oscillation increases the variability of the process variables; thus decreases the quality of product and increases energy consumption. It is known that the undesirable behaviour of control valves is the biggest single contributor to poor control-loop performance and the destabilisation of process operations. Several methods for detection and quantification of valve stiction have been presented in the literature [17, 42, 44, 47, 58, 63, 98, 110]. However, many of them have some practical limitations one way or other, which have to be addressed for real applications in industry. Quantification of control valve stiction is still a challenging issue. Choudhury et al. [17] presented a method to detect and quantify stiction using routine operating data. The non-linearity of the loop is tested using bicoherence. If the non-linearity is detected, stiction is estimated as the maximum width of the cycles of the PV–OP plot in the direction of OP. The PV–OP plot is fitted to an ellipse and the amount of stiction is estimated to be the maximum length of the ellipse in the OP direction, which is called the ellipse-fitting method. The stiction estimated using the method of Choudhury et al. is stated as “apparent stiction” and it provides an indication of the severity of the consequence of the stiction in an oscillatory loop. On the other hand, a simple grid-search method for estimating stiction parameters was presented in [18]. A grid search method with a one-dimensional stiction model was also presented in [115]. Note that the ellipse-fitting method has a clear limitation in the fact that the shape and size of the PV–OP plot depend on several factors: the changes of proportional or integral control gain, the process gain, the process time constant, the time delay of the process, phase lags, etc. will all have effects on the shape and size. Hence, the apparent stiction that the ellipse-fitting method estimates will differ from feedback control and cannot be used for closed-loop stiction quantification. Recently, Jelali [58] independently presented an interesting global optimisationbased method for quantification of valve stiction in control loops. It calculates an initial approximate guess of S, which is the deadband plus stickband, using the ellipse-fitting method [17], and searches for the optimum point near the initial guess using genetic algorithms or pattern search. A good initialisation is needed for the stiction estimation. As noted above, the ellipse-fitting method may not be accurate under closed-loop conditions and hence the optimum solution found near the initial guess may not be a solution that describes the behaviour of the control valve the best. The genetic algorithms adopted require a large number of functional evaluations per iteration and storing of a considerable amount of information in the computer memory. In this chapter, a novel stiction-detection and quantification strategy based on process-model identification is proposed using routine closed-loop operating data. Prior to doing stiction detection and quantification, it is necessary to choose a suitable model structure to describe control-valve stiction. Several data-driven valvestiction-model structures are available in the literature [15, 44, 63]. In this chapter, the one proposed by He et al. [44] will be adopted. Given a valve stiction model structure, a search space of stiction-model parameters is determined by using con-

11 Stiction Estimation Using Constrained Optimisation and Contour Map

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troller output data, which is also called OP. Note that, if a valve-stiction model is exactly known, then a time series of manipulated variable (MV ) can be calculated from given OP data and the valve-stiction model. A process model can be estimated by using system-identification algorithms with MV and OP data. In this chapter, the problem of interest is to find the best unknown stiction-model parameters satisfying a mean-squared error criterion within a space of valve-stiction-model parameters. The proposed strategy not only detects but also quantifies valve stiction. The proposed method has many advantages from the practical implementation point of view. First, it has no requirement to filter original data and can be easily implemented. It can be implemented as an automatic detection tool because it uses only routine operating data. Also, it can naturally deal with open-loop data if OP moves in downward and upward directions several times. The effectiveness of the proposed stiction-detection and quantification method is demonstrated by illustrative industrial examples.

11.2 Stiction Model of Control Valve 11.2.1 General Conception Figure 1.5 shows the schematic operation diagram of a sticky valve, where fd denotes the kinetic friction band, J the slip jump, and fs the static friction band. Some of definitions of stiction can be found in [15, 48] and Chap. 1 of this book.

11.2.2 Physical Model of Valve Sticion Several physical friction models have been presented in [1,66,89]; see Chaps. 2 and 3 of this book. One of the commonly used friction models is the Karnopp model. It includes static and moving friction. The disadvantage when applying the friction model to a generic valve is the need to specify a large set of parameters. In order to overcome this disadvantage, many researchers developed different kinds of empirical data-driven stiction models. The data-driven models have parameters that can be directly related to plant data and they produce the same behaviour as the physical model. The data-driven models need only an input signal and the specification of deadband plus stickband and slip jump. It overcomes the main disadvantages of physical modelling of a control valve.

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11.2.3 Kano’s Valve-stiction Model A valve-stiction model was proposed by Kano et al. [63]; see Sect. 2.3.5. The input and output of this valve-stiction model are the controller output and the valve position, respectively. The controller output is transformed to the range corresponding to the valve position in advance. The first two branches in the model flowchart check if the upper and the lower bounds of the controller output are satisfied. This valve-stiction model has several advantages: • It can cope with the stochastic input as well as the deterministic input. • us (t), which is the controller output at the moment the valve state changes from moving to resting, can be updated at appropriate timings by introducing the valve state. • It can change the degree of stiction according to the direction of the valve movement.

11.2.4 Choudhury’s Valve-stiction Model Choudhury et al. proposed a valve-stiction model in [15], where the control signal is translated to the percentage of valve travel with the help of a linear look-up table; see Sect. 2.3.3. The model consists of two parameters, namely, deadband plus stickband S, which is specified in the input axis, and slip jump J, which is specified in the output axis; see Sect. 2.3.3.

11.2.5 He’s Valve-stiction Model In the valve-stiction model of Kano et al., no matter how small Δ u(t) is, as long as it is greater than zero, the valve will always move and stop, which is not logically correct. The Choudhury et al. stiction model has the same problem. Another issue associated with the Kano et al. and Choudhury et al. models is that the saturation constraints are added to the controller output instead of an actuator (a valve). Based on the typical input–output behaviour of a sticky valve, He et al. proposed a new valve-stiction model [44], which is simpler and more straightforward in logic. If desired, the saturation constraint can be easily added to uv (t) after the model calculation; see Sect. 3.2.


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11.3 Existing Stiction-detection Methods 11.3.1 Open-loop Methods Stiction can easily be detected using invasive methods such as the valve travel or bump test. However, to apply such invasive methods across an entire plant site is neither feasible nor cost effective because of their manpower, cost and time-intensive nature. Several methods have been developed to detect valve stiction in the last decade [37, 51, 121]. However, most methods require either detailed process knowledge or user interaction, which is not desirable for automated monitoring systems.

11.3.2 Closed-loop Methods Horch [47] presented an automatic detection method based on the cross-correlation function (CCF) between the controller output and the process output, which is applicable to self-regulating processes. In a continuing work, Horch [48] proposed another method to address the valve stiction in integrating processes by considering the probability distribution of the second derivative of the controlled variable; see Chap. 6. Singhal and Salsbury [110] presented a valve-stiction-detection method based on comparison of areas before and after the peak of an oscillating control-error signal, i.e. the difference between the setpoint and the process variable being controlled; see Chap. 9. Kano et al. [63] presented two valve-stiction-detection methods: one requires knowing the valve position (VP) and the other is based on the plot of OP and PV with the shape of a parallelogram; see Chap. 5. He and Pottmann [42] presented a valve-stiction-detection technique in which the OP is fitted piecewisely to both triangular and sinusoidal wave using the leastsquares method. A better fit to the triangle indicates valve stiction, while a better fit to the sinusoid indicates non-stiction. Also in the work, a stiction index (SI) was first defined as the ratio of the mean-squared error (MSE) of sinusoidal fitting and the sum of the MSE’s of both sinusoidal and triangular fittings. An SI close to zero would indicate non-stiction, while an SI close to one would indicate stiction. In the meantime, Rossi and Scali [98] independently presented a very similar technique to [42]. In [98], the PV signal instead of OP is fitted using three different signal models: relay, triangular, and sinusoidal wave; see Chap. 8. He et al. [43,44] extended the previous work in [42] to cover both self-regulating and integrating processes based on the following observations: in the case of controlloop oscillation caused by poor controller tuning or external oscillating disturbance, OP and PV typically follow sinusoidal waves for both self-regulating and integrating processes. In the case of stiction, for self-regulating processes, OP will move like a triangular wave, while for integrating processes such as level control, PV will move

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like a triangular wave. The basic idea of this detection method is to fit two different functions, i.e. triangle and sinusoid, to the measured oscillating signal, where OP is for self-regulating processes and PV is for integrating processes. A better fit to the triangle indicates valve stiction, while a better fit to the sinusoid indicates nonstiction. The SI metric is used as a criterion to evaluate the degree of valve stiction; see Chap. 7. Choudhury et al. [17] presented a method to detect and quantify stiction using routine operating data. The non-linearity of the loop is tested using bicoherence. If the non-linearity is detected, stiction is estimated as the maximum width of the cycles of the PV–OP plot in the direction of OP. The PV–OP plot is fitted to an ellipse and the amount of stiction is estimated to be the maximum length of the ellipse in OP direction, which is called the ellipse-fitting method. The stiction estimated using the method of Choudhury et al. is stated as apparent stiction and it provides an indication of the severity of the consequence of the stiction in an oscillatory loop; see Appendix B. On the other hand, a simple grid-search method for estimating stiction parameters was presented in [18]. A grid-search method with a one-dimensional stiction model was also presented in [115]. Recently, Jelali [58] independently presented a global optimisation-based method for quantification of valve stiction in control loops. It calculates an initial approximate guess of S and J, which are the deadband plus stickband and the slip jump, respectively, using the ellipse-fitting method [17], and searches for the optimum point near the initial guess using genetic algorithms or pattern search; see Chap. 10.

11.3.3 Discussion of Existing Methods In Horch [47], one issue is the differentiation of noisy signals. A suitable filter and cut-off frequency have to be carefully chosen in order to filter noise. This can hardly be done automatically since different processes have different system characteristics and different noise levels. It has been observed that even after filtering, the calculation of derivatives amplified a moderate amount of noise and blurred the distinction between the shapes of the two probability distributions [109]. In Singhal and Salsbury [110], there are several practical limitations as mentioned by the authors themselves: • The method can not be applied to integrating processes. • It cannot distinguish other non-linearities from stiction. • The error signal must be sampled many times per oscillation period in order to get accurate peak location and areas calculation. • The noise adds variation to the peak and zero-crossing locations, which could result in problematic diagnosis. In Kano et al. [63], the first method can be used only when flow rate or valve position is measured. The second method is not always reliable even when flow rate or valve position is measured as shown in one of their flow control examples.


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In He et al. [43], there are two issues: • When the controller has bad tuning, the fitting method is not effective. • For processes with large time delay, the opposite result could be obtained. Controllers with poorly tuned parameters and processes with large time delay often exist in industry, which limits the use of the method. The method of Rossi and Scali [98] is very similar to that proposed by He et al. [43]. It is only applicable to self-regulating processes. In Choudhury et al. [17, 18], the ellipse-fitting method has a limitation in the fact that the shape and size of the PV–OP plot is sensitive to several factors: the changes of proportional or integral control gain, the process gain, the process time constant, the time delay of the process, phase lags, etc. Hence, the apparent stiction that the ellipse-fitting method estimates will differ from real stiction. In Jelali [58], a good initialisation is needed for the stiction estimation. The initial point of stiction parameters is obtained by using the ellipse-fitting method and the optimum solution is found near the given initial guess. As noted above, the ellipsefitting method may not be accurate and hence the optimum solution found near the initial guess may not be the solution that describes the behaviour of the control valve the best. The genetic algorithms adopted requires a large number of functional evaluations per iteration and storing a considerable amount of information in the computer memory.

11.4 Closed-loop Stiction Detection and Quantification 11.4.1 Basic Principle and Important Steps The basic idea is to convert the stiction-detection and quantification problem into a low-order Hammerstein-type system-identification problem, followed by a global optimisation search for the stiction parameters. This idea focuses on finding a noninvasive method to determine if stiction exists in a control valve. It approximates process dynamics by a low-order transfer-function model while estimating parameters for the static stiction model to account for the non-linearity induced by the stiction. We consider that most industrial processes can be approximated as the first- or second-order-plus-time-delay process. It is required to choose a suitable stiction-model structure before proceeding. The basic steps to follow in the proposed method are: Step 1. Given a stiction-model structure and OP data, effectively bound a search space of unknown stiction-model parameters. Step 2. Choose stiction-model parameters from the bounded stiction-model space, and a series of manipulated variable (MV ) data is calculated from OP data according to the given valve-stiction model.

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Step 3. With MV and PV data, the process model is identified such that a mean squared error is minimised. By varying stiction-model parameters, different process models are obtained. Step 4. Find the stiction model that describes the characteristics of the control valve behaviour the best. Find the minimum model error and get the corresponding process-model and stiction-model parameters. The key to success of this procedure lies in the efficient optimal search of stiction parameters.

11.4.2 Stiction Detection and Quantification Procedure Step 1. Data Collection and Selection. Tests of the proposed method on simulated data showed that it is necessary to select appropriate low and high limits on the sampling period. Our experience also shows that it is necessary to select the sampling period to make sure there are more than 50 data points in an oscillating period. Step 2. Data Preprocessing. Filtering is not necessary in the proposed method, but de-trending the input and output data is important. De-trending is the process of removing the zero-order trend (the mean) from the original data and is needed not only for input and output data (OP and PV), but also for the generated MV data. Step 3. Selection of Stiction-model Structure. We found that the valve-stiction model structure of He et al. [44] is not only simple in logic but also closer to real stiction behaviour. If we search the expected parameters of the mentioned stiction model according to the data, the model with optimal parameters has been shown to be very close to the real stiction characteristics. Step 4. Bounded Search Space of Stiction Model. A search region of stictionmodel parameters is defined for constrained optimisation. The region of stiction-model parameters is determined using OP data and the given stictionmodel structure. Step 5. Process Model Structure and Identification. Under the assumption that the process is a first- or second-order-plus-time-delay process, the ordinary least-squares method is suitable for identification. The time delay of the process may either be searched in the optimisation or effectively identified by applying time-delay-estimation methods. Step 6. Quantification of Valve Stiction. A cost-effective constrained optimisation technique is adopted for comprehensive stiction-model parameter search over all feasible parameter space. It finds the stiction model that describes the characteristics of the control-valve behaviour the best. The model with a minimum error implies the most possible and realistic stiction model parameters found.


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11.4.3 Search Space of Stiction-model Parameters The control valve is a physical link with movement in a control loop and the characteristic of its behaviour is described by its physical specification. So, a bounded space of stiction parameters to search can be defined and specified by using the OP data and the relationship of stiction parameters. The operation diagram of a sticky valve is shown in Fig. 1.5. Note that fd + fs ≤ Smax ,

(11.1)

where fd ≥ 0, fs ≥ 0, and the upper bound Smax is approximately given by the span of OP. Due to the relation fs = fd + J, it holds that 2 fd + J ≤ Smax .

(11.2)

Figure 11.1a illustrates the constrained search space of stiction-model parameters ( fd , J). Figure 11.1b shows an equivalent search space of stiction parameters in terms of ( fd , fs ). The regions (i)–(iv) indicate typical cases of control-valve stiction: (i) almost weak stiction, (ii) deadband behaviour, (iii) stick-slip behaviour, and (iv) deadband plus stick-slip behaviour. The stiction-model parameter domain in Fig. 11.1b will be used for parameter search in the subsequent sections. Contour maps drawn in the valve-parameter domains are shown in Fig. 11.2 to illustrate typical cases of control-valve behaviours, red contour curves indicate representative solution points. The contour map helps look at the distribution of solution points in the parameter space. Note that the upper bound Smax plays a role in constraining the stiction parameter domain. Restraining the search region of decision parameters efficiently can significantly reduce unnecessary computational cost in quantification of control valve stiction. Given ( fs , fd ), as shown in Fig. 1.5, the stiction parameters (S, J) can be obtained by the following relation: S = fs + fd ,

J = fs − fd .

(11.3)

fd

fd

1 Smax 2

1 Smax 2 ii ii

iv i

iii

i

Smax

0

(a)

J

iv iii

Smax

0

fs

(b)

Fig. 11.1 Search spaces of valve-stiction parameters: a) the search space of stiction parameters ( fd , J); b) the search space of stiction parameters ( fd , fs )

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Fig. 11.2 Stiction-model-parameter domains showing typical cases of control-valve behaviour: a) no-stiction; b) deadband behaviour; c) stick-slip behaviour; d) deadband plus stick-slip behaviour

11.4.4 Constrained Parameter-search Techniques There are two principal goals leading to the design of global optimisation methods: global reliability to ensure that the domain is searched sufficiently to provide a reliable estimate of a global solution and local refinement to produce a fine solution. Most global optimisation algorithms have been developed to achieve these two goals by combining a global strategy and a local strategy [128]. 11.4.4.1 Multi-start Adaptive Random Search Random search algorithms allow in principle to find a global minimum and the solution does not depend on the starting point. Adaptive random search is known as an efficient random search algorithm with systematic reduction of the size of the search region. It adaptively reduces the search region, gradually focusing the global


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search effort on the region that is estimated to contain the global solution point on the basis of the actual sample results. The adaptive random search method may be trapped in a local minimum and hence is recommended for the problems with minimal structure. Note that combining the basic adaptive random search method with a multi-start approach improves the ability to reach the global minimum. The basic procedure of a multi-start adaptive random search is as follows: Step 1. Generate a set of random starting points and iterate an adaptive random search algorithm with rough accuracy on each starting point, which obtains a set of approximate local minimum points. Step 2. Apply an efficient local search algorithm for the optimum search in the vicinity of the minimum points found in Step 1. Step 1 attempts to find promising starting points that are more likely to reach the location of a global optimum. Step 2 is a fine global optimisation within a reduced search domain near the approximate local points given in Step 1. Though this multistart strategy takes more computational effort than the adaptive random search due to the multi-start strategy, it is suitable to address the problems of a large number of local optima and finds a global solution best for complicated models. In reality, it is observed that many local minima are present in the space of stiction-model parameters. Various advanced modifications of the above basic algorithm can be found in the literature [40,72]. Note that in the cases when the model structure of the process is known or determined a priori, the multi-start adaptive search technique can be applied to the constrained optimisation problem to seek a stiction model within the bounded space of stiction-model parameters. The constrained optimisation problem to be minimised is formulated as follows: min J( fs , fd , θ )

(11.4)

subject to fd − fs ≤ 0,

(11.5)

fd + fs − Smax ≤ 0, fs ≥ 0, fd ≥ 0.

(11.6) (11.7) (11.8)

The following form of unconstrained optimisation problem is also useful when applying general global optimisation techniques for stiction-parameter search: 2

min V J( fs , fd , θ ) + ci ∑ (max{0, gi ( fs , fd )})2 ,

(11.9)

i=1

subject to Smax ≥ fs ≥ 0,

(11.10)

1 Smax ≥ fd ≥ 0, 2

(11.11)

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where g1 ( fs , fd ) = fd − fs , g2 ( fs , fd ) = fd + fs − Smax , and ci is a penalty parameter that is assigned to the violated constraints. Note that max{0, gi } = 0 if gi ≤ 0 and max{0, gi } > 0 if gi 0. 11.4.4.2 Contour Map If the process-model structure is unknown, the flowchart in Fig. 11.3 can help determine a suitable model structure and solution of stiction-model parameters. In Fig. 11.3, FOPTD stands for first-order plus time-delay processes and SOPTD stands for second-order plus time-delay processes. It is noted that the process-model structure can affect the local solution structure in the contour map. The basic procedure is as follows: (i) uniform grid search with 1% to 2% resolution, and (ii) followed by the local search. In Step ii, the lowest value of the objective function is taken as a starting point for the fine local search. To reduce the computational load from performing the uniform grid search, a box-bounded search [61] can be utilised to draw the contour map only around local points. It will be shown shortly that the grid search is necessary from the consistency viewpoint. Try FOPTD model structure

Contour shows clear locals?

No

Try SOPTD model structure

Yes Determined by FOPTD

Contour shows clear locals or looks similar to FOPTD model’s contour?

No Not determined

Yes Determined by SOPTD

Fig. 11.3 A simple flowchart for process-model-structure selection and solution determination

11.4.5 Advantages • Simple method and easy implementation. No need to filter the original data. The identification process is to find the most suitable parameters of the valve model and it is effective even under large noises in the output of the process. • Simple process-model structure. A dozen examples, including FOPTD process, SOPTD process, and integrating-plus-time-delay process, are simulated. In these


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cases, if no stiction exists, the stiction parameters of the valve can be identified as zeros undoubtedly by the proposed method. If stiction exists, the stiction parameters of the valve can be identified with satisfactory performance. • Low computational cost. We directly program a least-squares algorithm with an ® analytical structure, and do not use System Identification ToolboxTM of MATLAB to identify the process. This will shorten the run time greatly. Using the original ® MATLAB function, the run time is more than 5 min for a control loop with 1500 data points. Using the proposed method, the average computation time is less than half a minute. • Closed-loop method using routine operating data. This method not only detects stiction but also quantifies it. • It will be shown below that the comprehensive search as we did in the proposed algorithm is necessary for consistency of the estimation.

11.5 Stiction Detection: Identifiability Analysis We have argued in the previous discussion that a comprehensive search of stictionparameter space is necessary. In this section, we shall elaborate this argument theoretically. Many stiction methods are based on closed-loop data. A well-known problem in closed-loop identification is the identifiability. A natural question for the stiction-detection or quantification method proposed in this chapter or methods proposed in other chapters is the consistency issue that has not been addressed in the literature. We shall explore the conditions for the consistency of closed-loop stiction detection and quantification in this section. The stiction-detection and quantification method proposed in this chapter is based on system identification superimposed by non-linear static stiction structure, which is a type of Hammerstein model. Thus, consistency analysis methods used in system identification [53, 73] can be applied. The following derivation follows the basic approach taken by [53].

11.5.1 Heuristic Illustration of Closed-loop Identifiability A general linear model is given by [73, 111]: yt = Gp (q−1 ; θ )ut + Gl (q−1 ; θ )et ,

(11.12)

where Gp (q−1 ; θ ) is the process/plant transfer function, Gl (q−1 ; θ ) is the disturbance transfer function, et is a white-noise sequence with variance of Σe , and ut is the input. The standard assumption in system identification is as follows: −1 −1 −1 −1 G−1 l (q ; θ ) and Gl (q ; θ )Gp (q ; θ ) are asymptotically stable, and Gp (0; θ ) = 0 and Gl (0; θ ) = I. From Fig. 11.4 (by adding appropriate parameters), one can observe that two equations exist in a closed-loop system:

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yt = Gp (q−1 , θ0 )ut + Gl (q−1 , θ0 )et , −1

(11.13)

−1

ut = −Gc (q )yt + Gc (q )rt ,

(11.14)

where rt is the setpoint perturbation (external excitation). These equations can be written as yt = Gp (q−1 , θ0 )ut + Gl (q−1 , θ0 )et , yt =

−1 −G−1 c (q )ut

(11.15)

+ rt .

(11.16)

The input and output data sampled from the closed-loop system should satisfy both these equations. It is clear that et is the disturbance to Eq. 11.15 while rt plays a similar role as a disturbance to Eq. 11.16. An identification procedure is to find a model that fits the input and output data ut and yt the best; say we fit the following model: yt = Gp (q−1 , θ )ut .

(11.17)

Depending on the relative “size” of et and rt , disturbances to the two equations, respectively, an identification result may end up with Eq. 11.15 (i.e. Gp (q−1 , θ ) = −1 Gp (q−1 , θ0 )), Eq. 11.16 (i.e. Gp (q−1 , θ ) = −G−1 c (q )), or somewhere in between. We shall discuss closed-loop identifiability without external excitations, as is often the case in stiction detection. et Gl

rt –

Gc

ut

Gp

yt

Fig. 11.4 Closed-loop system

If rt = 0, i.e. there is no external excitation, Eq. 11.17 fits Eq. 11.16 perfectly. An −1 identification algorithm will usually pick up −G−1 c (q ) as the solution; namely, an inverse of the controller transfer function is identified, and closed-loop identifiability is lost. However, if Gc (q−1 ) is non-linear, such as a non-linear control law, valve stiction, or active constraint on the control actions, then it is possible for an identification algorithm to pick up Eq. 11.15, i.e. Gp (q−1 , θ0 ) may be identified even without external excitation rt . Since the model to be fitted by data is constrained to a linear −1 model structure but G−1 c (q ) in Eq. 11.16 is non-linear, in this case the linear model −1 −1 Gp (q , θ ) of Eq. 11.17 may fit Gp (q−1 , θ0 ) of Eq. 11.15 better than −G−1 c (q ) of Eq. 11.16.


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A more general approach for closed-loop system identification is through the direct method [73,111]. In the next section, we shall elaborate the direct identification method with stiction in the loop.

11.5.2 Identifiability Analysis for Closed-loop Systems with Valve Stiction If there is at least one sample time delay in Gp (q−1 , θ ) and rt is persistent excitation of sufficient order, then as N → ∞, both Gp (q−1 , θ ) and Gl (q−1 , θ ) can converge to their true values by applying the prediction error method directly to input and output data ut , yt . This is known as the direct method for closed-loop identification [73, 111]. However, for a closed-loop system with stiction, the requirement of existence of external excitation may not be necessary. This is shown below. For the general model with stiction, shown in Fig. 11.5 and described by yt = Gp (q−1 ; θ ) f (ut , J, S) + Gl (q−1 ; θ )et ,

(11.18)

the optimal one-step prediction and the prediction error can be derived as −1 −1 −1 −1 y(t|t ˆ − 1) = G−1 l (q ; θ )Gp (q ; θ ) f (ut , J, S) + [I − Gl (q ; θ )]yt ,

ε (t, θ ) =

−1 G−1 l (q ; θ )[yt

−1

− Gp (q ; θ ) f (ut , J, S)],

(11.19) (11.20)

where y(t|t ˆ − 1) is the prediction of yt based on all input and output data up to t − 1. The prediction error is defined as

ε (t, θ ) = yt − y(t|t ˆ − 1).

et Gl

–

Gc

ut

f(ut,J,S)

Gp

yt

Fig. 11.5 Closed-loop system with stiction non-linearity

Let the real system be described by yt = Gp (q−1 ; θ0 ) f (ut , J0 , S0 ) + Gl (q−1 ; θ0 )et .

(11.21)

244


The prediction error method [73, 111] in system identification is to estimate parameters θ0 by minimising the one-step prediction error: J=

1 N ∑ ε (t, θ )ε T (t, θ ) . N t=1

If S = S0 , J = J0 , substituting Eq. 11.21 in Eq. 11.20 gives −1 −1 ε (t, θ ) = G−1 l (q ; θ )[yt − Gp (q ; θ ) f (ut , J0 , S0 )] −1 −1 −1 = G−1 l (q ; θ )[Gp (q ; θ0 ) − Gp (q ; θ )] f (ut , J0 , S0 ) −1 −1 +G−1 l (q ; θ )Gl (q ; θ0 )et

= Φu (q−1 ; θ , θ0 ) f (ut , J0 , S0 ) + Φe (q−1 ; θ , θ0 )et −1

(11.22)

−1

= Φu (q ; θ , θ0 ) f (ut , J0 , S0 ) + (Φe (q ; θ , θ0 ) − I)et + et , where −1 −1 −1 Φu (q−1 ; θ , θ0 ) = G−1 l (q ; θ )[Gp (q ; θ0 ) − Gp (q ; θ )], −1

Φe (q ; θ , θ0 ) =

−1 −1 G−1 l (q ; θ )Gl (q ; θ0 ).

(11.23) (11.24)

Given Gp (0; θ ) = 0 and Gl (0; θ ) = I, which also implies Gp (0; θ0 ) = 0 and Gl (0; θ0 ) = I , it can be verified that

Φu (0; θ , θ0 ) = 0, Φe (0; θ , θ0 ) − I = 0. Namely, both Φu (0; θ , θ0 ) and (Φe (0; θ , θ0 )−I) have at least one sample time delay. Assuming that f (ut , J0 , S0 ) and et are independent, we get cov[ε (t, θ )] = cov[Φu (q−1 ; θ , θ0 ) f (ut , J0 , S0 )] + cov[(Φe (q−1 ; θ , θ0 ) − I)et ] +cov[et ] ≥ cov[et ] , or as a norm expression trace[cov(ε (t, θ ))] ≥ trace[cov(et )]. The minimum is the covariance of the white noise Σe . This minimum is achieved when

Φu (q−1 ; θ , θ0 ) = 0, Φe (q−1 ; θ , θ0 ) = I.

(11.25) (11.26)

Comparing these results with Eqs. 11.23 and 11.24, achieving the minimum implies Gp (q−1 ; θ ) = Gp (q−1 ; θ0 ), −1

−1

Gl (q ; θ ) = Gl (q ; θ0 ), i.e. consistency is achieved.

(11.27) (11.28)


245

In deriving the consistency, we have made two assumptions: S = S0 , J = J0 and f (ut , J0 , S0 ) is independent of et . The first assumption can be satisfied by comprehensive search of S, J over all possible values. Our proposed method is the comprehensive grid-search-based method, which justifies this assumption. Satisfaction of the second assumption depends, however, on any one of the two conditions: (i) extent of the non-linearity, and (ii) “excitation” induced by the nonlinearity. The first condition has been illustrated in the heuristic argument discussed in the previous section. The stronger the non-linearity is, the better the identifiability will be. For example, if S (deadband plus stickband) accounts for most of the OP operating range, then the loop is close to open-loop operation, and the identifiability will be satisfied. The second condition comes as oscillations (limit cycles) induced by the stiction. For loops with limit cycles induced by the stiction, the term f (ut , J0 , S0 ) is dominated by the oscillation. Consequently, the correlation between et and f (ut , J0 , S0 ) will be weak and this second condition is approximately held. However, if the oscillation originates from disturbances, this condition can not be satisfied. In view of the conditions discussed above, it can be concluded that, if stiction does exist, the proposed method will detect and quantify the stiction. The accuracy of quantification is improved if there exists oscillation induced by the stiction. The remaining question is what will be the conclusion if there is no stiction. If there is no stiction, a model estimated without residual would be −Gc (q−1 ), and this only occurs when J = J0 , S = S0 . Thus, it appears that the proposed method is still able to get a correct solution if there is no stiction. The above proof shows consistency of the stiction quantification if the model is a general linear dynamic model, the parameter search is comprehensive, and there is stiction-induced oscillation in the loop. For computational tractability, we have limited our models to first- or second-order in the proposed algorithm. This may introduce some bias in theory. However, since most of the practical processes can be approximated by first- or second-order models, the proposed low-order identification procedure can be justified for most of the practical problems.

11.6 Simulations Consider a level control loop, which was previously studied in [44,63]. The transfer function for the level process is given by the integrating-plus-time-delay (IPTD) type of process model, and the controller is given by a PI controller as follows. 1 −s G(s) = 15s )e ,1 * . Gc (s) = 3 1 + 30s

(11.29)

The level control loop with a valve is studied by applying the proposed method below.

246


We consider typical cases of control-valve-stiction behaviour first for the level control loop, that is, cases 1–3 in Table 11.1. Case 1 is for the simulated study of non-stiction detection, case 2 is for the quantification of weak valve stiction, and case 3 is for the quantification of strong valve stiction. The specific values of valvestiction parameters ( fs , fd ) are given for each case study in Table 11.1. Simulation results are shown in Figs. 11.6 and 11.7, where the left plots are OP and PV timeseries plots and the right plots are OP–PV plots. It is noted that the span of OP data is used to bound the search space, i.e. Smax = span of OP data. The stictionquantification results obtained by using the proposed method are listed in Table 11.2. It is shown that the proposed method quantifies the degree of stiction well to the simple case studies of valve stiction. Contour maps are shown in Figs. 11.8 and 11.9, where the left map is for the FOPTD process model and the right map is for the SOPTD process model. Local points are clearly drawn in the bounded search region. Table 11.1 Case studies of valve stiction in the level control loop Case 1 2 3 4 5 6

Degree of stiction Non-stiction Weak stiction Strong stiction Bad tuning Case 2 coexisting with case 1 Case 3 coexisting with case 1

fd 0 0.65 3 0 0.65 3

fs 0 0.35 2 0 0.35 2

S 0 1 5 0 1 5

Sampling period [s] 0.2 0.2 0.2 0.2 0.2 0.2

Table 11.2 Proposed results and comparisons with existing methods Case 1 2 3 4 5 6

Smax (= Span of OP) 1.8 1.3 5.8 5.1 1.4 7.7

fs 0 0.64 2.79 0 0.66 2.83

fd 0 0.35 1.75 0 0.36 1.76

S 0 0.99 4.54 0 1.02 4.59

S0 0.43 2.59 0.63 3.44

We now consider case 4 in Table 11.1. The capability of distinguishing valve stiction from bad controller tuning is tested in this case study. In Eq. 11.29, the controller gain is increased from 3 to 23.2 to make the closed-loop marginally stable. Simulation results are shown in Fig. 11.7a and the detection and quantification results are given in Table 11.2. It is seen that, given a bounded stiction-parameter domain, it detects non-stiction and distinguishes valve stiction from the bad tuning to this simulated example. We also consider the cases when multiple causes of oscillations are present, that is, cases 5 and 6 in Table 11.1. Case 5 is the mixed case of weak valve stiction and an external oscillatory disturbance and case 6 is the mixed case of strong valve stiction and an external oscillatory disturbance. Simulation results are shown in Figs. 11.7b and c. As shown in Table 11.2, for both cases, where multiple causes of oscillations


247

exist, the proposed method detects and quantifies the degree of stiction well. Table 11.2 also shows the comparison with the ellipse-fitting method. It is seen that the magnitude Sˆ given by the ellipse-fitting method is far from the actual valve stiction to this level-control loop, confirming the problem of the ellipse-fitting method for closed-loop applications.

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(b)

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−1

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−4

−2

0 OP

(c) Fig. 11.7 Time-series OP and PV plots and OP–PV plots: a) case 4; b) case 5; c) case 6

2

4

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25

2

Space of fd

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2.5

1.5 1 0.5 0 0

2 15 1 05

1

2

3 4 Space of fs

0 0

5

(c) Fig. 11.8 Contour maps: a) case 1; b) case 2; c) case 3

1

2

3 4 Space of fs

5


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25

2

2 Space of fd

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250

1.5 1 0.5 0 0

15 1 05

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2 3 Space of fs

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(a)

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1

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1

(b)

3 Space of fd

Space of fd

3 2 1 0 0

2 1

2

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0 0

6

(c) Fig. 11.9 Contour maps: a) case 4; b) case 5; c) case 6

2

4 Space of fs

6

5


251

11.7 Industrial Applications 11.7.1 Illustrative Industrial Examples To demonstrate the validity of the proposed method, industrial control loop data shown in Figs 11.10 and 11.11 are considered. Figures 11.10 and 11.11 show timeseries plots and OP–PV plots of the control-loop data and Figs. 11.12 and 11.13 show contour plots in parameter-search domains. It is noted that CHEM 7 and 8 are open-loop data and others are closed-loop data. A computer system with an ® ® Intel Pentium CPU 3.2GHz and 2GB of RAM was used for computation and the search space of stiction-model parameters was bounded by the span of OP, i.e. Smax = span of OP. From computation results, it is seen that CHEM 7, 10 and 11 show mostly stick-slip behaviour, CHEM 8 and 12 show deadband plus stick-slip behaviour, and CHEM 9 shows mostly no stiction. It is noted that, for CHEM 9, 10 and 12, the results obtained by the proposed method are in agreement with the result of [17], while for oscillating CHEM 8 and 11, the proposed results are not in agreement with those of [17]. Figure 11.12a shows two dominant local solution areas, i.e. S = {0.54, 1.25}, both of which indicate stick-slip behaviour for CHEM 7. Note that one of the dominant solutions, i.e. S = 1.25, is in agreement with those of [17]. Stiction quantification results of [58] would also appear to be similar to the results of [17]. Both [17] and [58] rely on the OP–PV plot method to some extent, and thus show similar results. The proposed method, however, does not depend on an initial guess given by the OP–PV plot but is based on a constrained optimisation with a bounded space of a stiction model. The apparent stiction obtained from the ellipse-fitting method can be different from real stiction and may not be reliable as an initial guess, as shown in the simulation study. It is also noted that the constrained optimisation algorithm adopted in the proposed method is efficient in computation. It takes 19.8 and 33 s on average for the first- and second-order-plus-time-delay model, respectively. The comparison results are summarised in Table 11.3, where the elapsed CPU time is given in seconds for the FOPTD and SOPTD processes. Table 11.3 Stiction quantification results for industrial control loop data (CHEM 7–12) Loop no. CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11 CHEM 12

Loop type Pressure control Pressure control Pressure control Pressure control Flow control Flow control

Data size 4 685 900 2 732 1 000 1 000 2 000

Smax 2 2 2 2 2 2

fs 0.52 0.82 0 1.70 0.10 0.78

fd 0.02 0.18 0 0 0.04 0.64

CPU Time [s] S FOPTD SOPTD 0.54 37.2 49.9 1 14.2 21.1 0 17.4 37.9 1.70 12.9 15.4 0.14 14.0 19.5 1.42 23.4 54.3

S0 1.38 2 0 1.65 1.23 1.84


3 2 500 1000 1500 2000 2500 3000 3500 4000 4500

1 PV

0.5 0 −0.5 −1

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−2

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−3

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(c) Fig. 11.10 Time-series OP and PV plots and OP–PV plots: a) CHEM 7; b) CHEM 8; c) CHEM 9


253

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0.8 Space of fd

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05

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1

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Space of fd

(b)

0.6 0.4 0.2 0 0

06 04 02

05

1 Space of fs

15

0 0

2

(c) Fig. 11.12 Contour maps: a) CHEM 7; b) CHEM 8; c) CHEM 9

2

1

1

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255

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(a)

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(b)

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Space of fd

0.8 0.6 0.4 0.2 0 0

06 04 02

0.5

1 Space of fs

0 0

15

(c) Fig. 11.13 Contour maps: a) CHEM 10; b) CHEM 11; c) CHEM 12

256


11.7.2 Comparative Study A comparative study over 93 industrial loops will be presented in Chap. 13 of this book. In this section, we will elaborate case studies for several selected industrial control loops using the method proposed in this chapter and compare them with the published results in the literature. Figures 11.14, 11.15, 11.18 and 11.19 show time series plots and OP–PV plots of the control loop data, and Figs. 11.16, 11.17, 11.20 and 11.21 show contour plots in parameter search domains. The comparison results are summarised in Table 11.4, where the process model is selected by the flow chart given in Fig. 11.3. Table 11.4 Comparative study of industrial control loop data Loop no. MIN 1 POW 2 POW 4 CHEM 24 CHEM 25 BAS 7 PAP 4 CHEM 4 CHEM 5 CHEM 6

Loop type Temperature Level Level Flow Pressure Temperature Concentration Level Flow Flow

Data [500 1500] [1 700] [1 700] [100 800] [100 350] [1 end] [1 end] [1 end] [1 end] [1 end]

Smax 2.03 17.4 9.73 28.7 2.62 2.18 19.0 30.1 0.74 0.69

Process model FOPTD FOPTD FOPTD FOPTD FOPTD FOPTD FOPTD SOPTD FOPTD FOPTD

fs 1.16 1.04 0.49 10.89 1.70 0.57 2.09 0 0.17 0.014

fd 0 0.17 0.10 9.74 0 0.04 2.09 0 0.09 0

S 1.16 1.21 0.59 20.63 1.70 0.61 4.18 0 0.26 0.014

S0 1.1 11.4 4.8 22.5 1.4 1.24 3.23 0.5 0.14

Sˆ 1.02 11.47 4.49 22.9 1.8

MIN 1, which is a loop from the mining industry, is a temperature control loop, which was considered in [15, 17, 58]. In [58], the stiction parameters were found to be Jˆ = 0.96 and Sˆ = 1.02 using an initial guess S0 = 1.1 obtained from the ellipsefitting method. On the other hand, the result of the proposed method, i.e. fd = 0, fs = S = 1.16, indicates that the loop shows stick-slip behaviour rather than undershoot. However, noting that Jˆ ≈ Sˆ in the result of [58], it can be said that both results are quite in agreement for this industrial example. POW 2 and 4, which are loops from the power industry, are level control loops. POW 2 was analysed in [15, 17, 58] using the ellipse-fitting method. In [58], the ˆ ellipse-fitting method was applied to obtain an initial estimate of the parameter S, ˆ that is, S0 = 11.4. Using the initial guess, the stiction parameters were estimated, which are given Jˆ = 1.10 and Sˆ = 11.47. For this industrial loop, the proposed method produced different estimates of stiction parameters as follows: fs = 1.04, fd = 0.17, in other words, J = 0.88, S = 1.22, which disagree with [17, 58]. For the control loop POW 4, the stiction parameters were given by as Jˆ = 2.49, Sˆ = 4.49 in [58], while the stiction parameters obtained by the proposed method are given by J = 0.39, S = 0.49, which also disagree with [17, 58]. CHEM 24 is a flow control loop with setpoint changes, which is taken from [58]. As mentioned in [58], the loop is an inner loop of a cascade control system and thus it changes rapidly subject to setpoint changes. In [58], the result of the stiction parameters was given as Jˆ = 0.81 and Sˆ = 22.9. For this industrial loop, the proposed


257

method obtained estimates of stiction parameters as follows: fs = 10.89, fd = 9.74, i.e. J = 1.15, S = 20.63. Both are quite in agreement for this industrial example. Noting that fs ≈ fd in both results, it can be said that the flow control loop shows mostly backlash behaviour. CHEM 25 is a pressure control loop, which is taken from [58]. In [58], the result of the stiction parameters were found to be Jˆ = 0.59 and Sˆ = 1.8. For this industrial loop, the proposed method obtained estimates of stiction parameters as follows: ˆ it can be said that both fs = 1.7, fd = 0, i.e. J = 1.7, S = 1.7. Noting that S ≈ S, are partially in agreement.

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(b) Fig. 11.15 Time-series OP and PV plots and OP–PV plots: a) CHEM 24; b) CHEM 25

PAP 4, which is from the paper and pulp industry, is a concentration loop. In [47], it was confirmed by experiments that the loop oscillates because the PI-controller was too tightly tuned and the oscillation could be eliminated by re-tuning the controller. Using the proposed result, we found the stiction parameters, which are given by fs = 2.09, fd = 2.09. It is observed that the loop shows mostly deadband behaviour and the oscillation is not caused by stick-slip of the control valve. The control loops CHEM 4–6, which are from chemical processes, are taken from [44]. CHEM 4 is a level control loop that is over-aggressively tuned. It is known that the level oscillation is not caused by stiction. The proposed method shows no stiction in the control valve and the result is correct. Here, for the level control loop, SOPTD is used as a process model structure. CHEM 5 and 6 are flow control loops, and these loops are known to have valve-stiction problems. The proposed method shows that both loops are sticky but CHEM 6 is less sticky than CHEM 5, which is in agreement with the result in [44].

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(c) Fig. 11.16 Contour maps: a) CHEM 4; b) CHEM 5; c) CHEM 6;

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(b) Fig. 11.17 Contour maps: a) CHEM 24; b) CHEM 25

11.8 Graphical User Interface ®

The proposed method has been programmed using MATLAB [81] and a trial version is available for public testing at http://www.ualberta.ca/˜bhuang/ Research/Research.htm. It needs an Intel Pentium 4 CPU and MATLAB (version 7). In this section, we shall discuss the main features of the software and its graphical user interface (GUI), which is shown in Fig. 11.22. Note that the stiction detection and quantification tool is applied only if there are oscillations in the control loop.

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(c) Fig. 11.18 Time-series OP and PV plots and OP–PV plots: a) MIN 1; b) POW 2; c) POW 4;

The main features of the GUI are described in the following: 1. Main menu • Load data; • Exit; • Figure bar. 2. Control-loop data information • Notations in a basic control loop: SP, OP, MV, and PV; • Comments on the control loop;

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• • • •

Industrial fields; Loop number; OP and PV tag names; Sampling period [s].

3. OP, PV data plotting and data preprocessing • • • • • •

Time series data plot of OP and PV; OP–PV plot; Span of OP; Selection of sample data; Downsampling; Single or double axes in plotting.

4. Process-model-structure selection • FOPTD; • SOPTD; • Time delay [s] if known. Users can select a process-model structure by clicking the radio buttons. Clicking the radio button “FOPTD” selects the first-order plus time-delay process and “SOPTD” the second-order plus time-delay process. If the time delay of the pro-

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cess is known, then check the check box “Time Delay” first and insert the process time delay in the second. Note that if the sampling period is unknown, the check box “Time Delay” is not activated. 5. Optimisation solvers • • • •

Contour map (uniform grid search plus fine local search); Global plus local optimisation; Runtime; Open structure: conopt.m. (Users can edit this file.)


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Note that by default (choosing contour plus local) this GUI does not need an additional optimisation solver. However, for the optional global plus local optimisation (choosing Option 1–3 in the GUI), an optimisation solver is required. This GUI provides open structure in such a way that users can try any other optimisation solvers or techniques in conopt.m. To run the example of global plus local optimisation given in conopt.m, an optimisation solver, called lgo, is needed and can be purchased from TOMLABTM at www.tomopt.com. If it is installed, the options in the popup menu are given by • Multi-start: multi-start adaptive random search; • Adaptive: adaptive random search; • BNB: branch and bound search. 6. Displaying results • Contour map – FOPTD; • Contour map – SOPTD; • Bar graph showing fs and fd . If the GUI completes the calculation, then it shows quantification results for stiction parameters fs and fd . The contour maps show the minimum mean-squared estimation solution in the parameter domain, which is the search space of valve-


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stiction-model parameters given in Fig. 11.1. Note that in the contour maps, minimum regions appear in red colour.

Fig. 11.22 GUI of the stiction detection and quantification tool

11.9 Summary and Conclusions A novel closed-loop stiction-detection and quantification method using routine closed-loop operating data is presented based on a closed-loop model-identification approach. A suitable model structure of valve stiction is chosen prior to conducting valve-stiction detection and quantification. Given the stiction-model structure, a feasible search domain of stiction-model parameters to be found is defined, and a constrained optimisation problem is solved for the search of best stiction-model parameters. A cost effective constrained optimisation technique is adopted to find the best valve-stiction models representing a more realistic valve behaviour in the oscillating control loop. It is proved that the comprehensive search is necessary for

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consistency of the estimation. The validity of the proposed method is demonstrated through simulations, illustrative industrial examples, and comparative studies with existing stiction-quantification methods. A GUI implemented for public testing of the proposed method has been introduced with its main features.

Chapter 12

Oscillation Root-cause Detection and Quantification Under Multiple Faults Srinivas Karra and M. Nazmul Karim

In this chapter a model-based oscillation root-cause detection and quantification approach is presented. The objective of this methodology is to devise a strategy to make the diagnosis applicable to most practical situations encountered in industry by considering the presence of multiple faults occurring simultaneously. Power spectral density (PSD) and auto-correlation function (ACF) based oscillation detection method (described in Chap. 4) followed by a model-based approach for identifying and quantifying the root-cause of the oscillations is presented in this chapter. The application of each component of the algorithm is described with appropriate illustrative examples. Overall methodology is validated on simulation case studies involving various cases of stiction, poor controller tuning, and external disturbances.

12.1 Introduction Occurrence of multiple faults simultaneously, resulting in oscillatory control loops, is a natural consequence of industrial processing units employing hundreds of control loops to regulate the process at desired operating conditions. Hence, in order to devise applicable oscillation diagnostics to troubleshoot industrial control loops, consideration of multiple faults occurring at the same time is inevitable. In this chapter, a model-based fault-detection and isolation technique that can detect and quantify individual faults causing oscillations in control loops under a multi-fault scenario is proposed and validated against various simulation case studies. Its appli-

Srinivas Karra Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected] M. Nazmul Karim (corresponding author) Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA, e-mail: [email protected]

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cability to industrial control-loop troubleshooting is demonstrated by a few industrial case studies. The organisation of the chapter is as follows: In Sect. 12.2, a brief review of existing model-based oscillation diagnosis methodologies is presented. Also discussed are the identified areas of improvement to handle practical issues in industrial control loops. In Sect. 12.3 the overview of the proposed algorithm is presented. In Sect. 12.4, a novel system-identification technique that is used as a tool in the oscillation diagnosis is described. In Sect. 12.5, the oscillation diagnosis algorithm is discussed in detail along with the practical issues involved. In Sect. 12.6, few simulation case studies are presented to validate the capabilities of the proposed methodology under various multiple-fault scenarios. Sect. 12.7 describes the diagnosis of a few industrial control loops.

12.2 Preliminaries and Brief Review of Model-based Oscillation Diagnosis 12.2.1 Root-cause for Oscillations and Compensation Techniques At the stage of controller implementation, the three common causes for oscillatory feedback control loops are: poor controller tuning, oscillatory external disturbances, and control-valve non-linearities, such as stiction, backlash and saturation [10, 26, 79, 84]. If the controller is aggressively tuned, for example high gain and low integral time are used in case of PID-controller, then it results in sustained oscillations in the controlled variable. Proper controller tuning is required to alleviate these “soft” oscillations. Oscillatory disturbances that enter the process may also result in oscillatory behaviour in the control loop. There are many sources of oscillatory disturbances, e.g. the presence of an oscillatory control loop upstream, cyclic events, such as ambienttemperature fluctuations, fluctuations in raw-material quality, and variations in product demand with time, etc. Proper process design and control-system design can eliminate such oscillations caused by external disturbances. Another important factor that may lead to oscillatory control loops is the presence of non-linearities in the control valve, e.g. stiction, backlash, and saturation. Stiction-induced oscillations are well discussed in previous chapters of the book. If the stiction in a control valve is the root-cause for an oscillatory control loop, specialised compensation techniques [39, 68, 113, 114] may be employed to reduce these oscillations. If the stiction is above a threshold (that depends on the process knowledge), it calls for maintenance of the control-valve hardware. In more than 4000 audited control loops in all industries, as reported by Beckman and Jury [9], it was found that performance for more than 50% of the control loops could be significantly improved if the control-loop hardware (control valves, I/P

12 Oscillation Root-cause Detection and Quantification Under Multiple Faults

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converters) were serviced at appropriate times, regularly. Since there are several possible causes for performance degradation of control loops, it is not always easy to improve their performance by taking corrective action, as it is a challenging task to identify the root-cause. Hence, detection diagnosis and quantification of the rootcause for oscillatory behaviour in control loops is of paramount importance in order to take appropriate corrective action.

12.2.2 Oscillation Diagnosis and Root-cause Quantification In the context of performance assessment and troubleshooting, the objectives of oscillation diagnosis of control loops can be stated as: • Identification of control loops with sustained oscillations. • Distinct detection and quantification of root-cause once the loop is confirmed as oscillatory. Various techniques used in identification of oscillatory control loops are described in Chap. 4. The techniques available in the literature address the issue of root-cause detection by distinguishing between stiction-induced and aggressivetuning-induced limit cycles. The non-invasive techniques that are readily adaptable for use in industrial control-loop oscillation diagnosis using the controller output (OP) and process variable (PV) information can be broadly classified into four categories: cross-correlation-function-based, shape-based, non-linearity-detectionbased, and model-based algorithms. Out of these four classes of algorithms, only the model-based algorithms are always associated with quantification of the stiction strength. One exception to this is stiction-quantification methodology using an elliptical fit to the PV–OP signature in conjunction with the non-linearity-detectionbased method for stiction detection [17]. Using the OP–PV data, a Hammerstein model representative of the system comprising of a sticky valve and a linear process has been identified in [20, 58, 116]. It was postulated that if the stiction parameter(s) identified is(are) significant, stiction can be confirmed. A One-parameter empirical stiction model was used in [116] followed by an auto-regressive moving average exogenous (ARMAX) model to map the PV–OP relationship, whereas [20, 58] used a two-parameter empirical stiction model followed by ARMAX model for modelling the PV–OP relationship. Another fundamental difference between these techniques is the optimisation procedure used for identifying the stiction parameter(s). The first two methodologies used a grid-search algorithm, whereas [58] used a genetic-algorithm-based global search method to obtain the stiction parameter(s). As pointed out in [116], the basic hypothesis behind the identifiability of true stiction-model parameters is the decoupling between non-linear and linear components of the Hammerstein model. The necessary condition for this decoupling to exist in stiction diagnosis is the consistency of the model that describes the process. An ARMAX structure for linear process dynamics provides a consistent model, pro-

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vided there are no external non-stationary disturbances (for example an oscillating control loop upstream). The presence of any non-stationary external disturbances introduce bias into the model-parameter estimates, as the constant parametrised noise model used (ARMAX) cannot capture the exact disturbance dynamics in such a case. Hence, there is a possibility that this method may give a wrong diagnosis in the presence of non-stationary external disturbances, which are prevalent in most practical applications. If a linear dynamic model that is consistent with the linear process subjected to non-stationary disturbances is used, this problem can be eliminated. Moreover, if the quantification of external disturbance and degree of aggressiveness of control tuning alongside stiction strength is made possible, this information is used in taking appropriate corrective action, as described in Section 12.2.1.

12.2.3 Challenges to be Addressed It was identified that diagnosis of oscillations in case of multiple faults occurring at the same time and distinct identification and quantification of all types of faults are still challenges that remain unanswered. The objective of this work is to develop a methodology for oscillation detection and diagnosis, which can identify and quantify various faults in control loops.

12.3 Overview of the Root-cause Detection and Quantification Methodology 12.3.1 Revisiting Control-valve Characteristics Under Stiction The subsequent stiction-detection and quantification methodology presented in this chapter is based on the valve-stiction model. An empirical sticky-valve model developed by Kano et al. [63] is used in the present work for stiction diagnosis (Section 2.3.5). However, other empirical models that describe valve stiction [15,44] can also be used (Chap. 2 and 3). As a part of the control-loop oscillation diagnosis the parameters that describe the stiction characteristics are calculated. The characteristics of a control valve suffering from stiction are discussed in detail in previous chapters. The relation between OP and valve position/manipulated variable (MV) under valve stiction is described in Section 1.4 and is shown in Fig. 1.5.


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12.3.2 Oscillation Detection and Diagnosis Methodology Monitoring of control-loop performance is carried out by performing oscillation detection periodically, using the OP–PV data of finite length (moving monitoring window) using the COC methodology outlined in Section 4.6. If significant oscillations are identified in a control loop, the next step to be implemented is the identification of the root-cause of these oscillations. In this work, both root-cause detection and quantification are performed in a single step. An iterative two-tier identification of the Hammerstein model is carried out to estimate parameters of a non-linear stiction model [63] and linear dynamic process model with additive non-stationary disturbances (EARMAX: Extended ARMAX model, [67]) using OP–PV data. The Hammerstein-model structure used in this work is shown in Fig. 12.1. uop (t )

Two-parameter Empirical model Stiction Non-linearity

uˆ(t )

Extended ARMAX model

yˆ(t )

Linear process dynamics and additive non-stationary disturbance

Fig. 12.1 Hammerstein model representing control valve followed by linear process subjected to additive non-stationary disturbance: uop (t): controlled output; u(t): ˆ valve position; y(t): ˆ process variable

Section 12.4 presents the method to identify linear time-invariant systems subjected to non-stationary external disturbances using a novel EARMAX model identification technique. Identification of the Hammerstein model is presented in Section 12.5.1. If the identified stiction-parameter values are close to zero, it can be concluded that there is no stiction present in the control valve. If the bias-term identified, which is a representative of the additive non-stationary external disturbance, is oscillatory, the root-cause for oscillations is oscillatory external disturbances. Alongside the Hammerstein modelling of PV–OP mapping, a PID-like controller transfer function is identified using OP–PV data. Using the identified PID-controller transfer function and the linear process transfer function, a closed-loop simulation is performed to generate OP–PV data for the experimental setpoint (SP) profile. If generated OP or PV is oscillatory then it can be concluded that the controller tuning is underdamped and causing limit cycles in the control loop. The flowchart of the proposed stiction detection and quantification method is given in Fig. 12.2.

12.4 Process-model Identification Under Non-stationary Disturbances The objective of this section is to devise a system-identification technique that can yield unbiased linear dynamic process-model parameter estimates alongside esti-

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mating the additive non-stationary disturbance that is entering the process. An identified linear process model is used in isolating the fault due to aggressive controller tuning, where the identified non-stationary disturbance is used in isolating the fault due to external oscillatory disturbances. Tracking error signal

Oscillatory? yes Stiction parameters

Identify the parameters Of Hammerstein model

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Fig. 12.2 Flow chart of oscillation detection and diagnosis

For this purpose, let us consider the linear time-invariant (LTI) process subjected to additive non-stationary disturbance described by an EARMAX and given by y(t) =

1 C(q−1 ) B(q−1 ) u(t − τ ) + e(t) + η (t) , −1 A(q ) A(q−1 ) A(q−1 )

(12.1)

where, A(q−1 ), B(q−1 ) and C(q−1 ) are polynomials in the time-shift operator q−1 and given as A(q−1 ) = 1 + a1 q−1 + a2 q−2 + ... + ana q−na = 1 + q−1 A∗ (q−1 ) , B(q−1 ) = b1 q−1 + b2 q−2 + ... + bnb q−nb ,

C(q−1 ) = 1 + c1 q−1 + c2 q−2 + ... + cnc q−nc = 1 + q−1C∗ (q−1 ) .

(12.2)

Here, e(t) is a white-noise sequence, τ is the time delay of the process. na , nb and nc are the orders on auto-regressive, exogenous and moving average parts of the process, respectively. The predictor for this process can be given by ˆ y(t) ˆ = −Aˆ ∗ (t)y(t ˆ − 1) + B(t)u(t − τ ) + Hˆ ∗ (t)ε (t − 1) + ηˆ (t) ,

(12.3)

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ˆ and Cˆ ∗ (t) are the estimates of A∗ , B bias term. Hˆ ∗ (t) = Cˆ ∗ (t) − Aˆ ∗ (t) and Aˆ ∗ (t), B(t) ∗ and C , respectively, at time instant t. The vector notation for the predictor is shown in Eq. 12.4, y(t) ˆ = ϕ T (t)θˆ (t) + ηˆ (t) ,

(12.4)

where, the estimated parameter vector at time t

T θˆ (t) = aˆ1 (t) . . . aˆna (t) bˆ 1 (t) . . . bˆ nb (t) hˆ 1 (t) . . . hˆ nh (t)

(12.5)

and the regressor vector

ϕ (t) = [−y(t ˆ − 1) . . . − y(t ˆ − na ) u(t − τ ) . . . u(t − τ − nb + 1) ε (t − 1) . . . ε (t − nh )]T .

(12.6)

12.4.1 Identification of EARMAX Model The EARMAX model structure contains both time-varying and time-invariant parameters. To identify this model, a novel identification technique that is capable of handling the combination of time-invariant and time-varying parameters is used. As there is a time-varying model parameter present in the model structure, the identification is performed in a recursive manner. Covariance updates of parameter estimates of both the LTI part and the time-varying part are carried out separately using different forgetting factors for each of them. The algorithm is listed as follows: Initiation. The recursive estimator is initiated with the following covariance matrices (Pθ and Pη ), parameter estimates (θˆ and ηˆ ), and forgetting factors (λθ and λη ): Pθ (0) = β In×n , Pη (0) = β ; β = 104 , θˆ (0) = 0n×1 , ηˆ (0) = 0 ,

λθ (0) = 1, λη (0) = 0.84 , {ε (t)} = 0 ,

(12.7)

where, n = na + nb + nc . Propagation. The following steps 1 to 6 are to be performed: Step 1.

Formulate the regressor vector

ϕ (t) = [−y(t ˆ − 1) . . . − y(t ˆ − na ) u(t − τ ) . . . u(t − τ − nb + 1) ε (t − 1) . . . ε (t − nh )]T .

(12.8)

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Step 2.

Gain update:

) *−1 Kθ (t) = Pθ (t − 1)ϕ (t) λθ (t − 1) + ϕ T (t)Pθ (t − 1)ϕ (t) , Kη (t) = Pη (t − 1) (λη (t − 1) + Pη (t − 1))−1 . Step 3.

(12.9)

Parameter update:

−1 In−1 Kθ (t) θˆ (t) θˆ (t − 1) + Kθ (t)(y(t) − ϕ T (t)θˆ (t − 1) = . Kη ϕ T (t) 1 ηˆ (t) ηˆ (t − 1) + Kη (t)(y(t) − ηˆ (t − 1)) (12.10) Step 4.

a posteriori prediction error:

ε (t) = y(t) − ϕ T (t)θˆ (t) . Parameter estimate covariance update: ) * 1 I − Kθ (t)ϕ T (t) Pθ (t − 1) , Pθ (t) = λθ (t − 1) 1 (1 − Kη (t)) Pη (t − 1) . Pη (t) = λη (t − 1)

(12.11)

Step 5.

Step 6.

(12.12)

Forgetting factor update:

λθ (t) = 1 ,

) *2 y(t − 1) − ϕ T (t − 1)θˆ (t − 1) − ηˆ (t − 2) λη (t) = 1 − , 1 + Pη (t − 1)σe (12.13) 0.72 ≤ λη (t) ≤ 0.9 . As the objective is to obtain an estimate of the non-stationary additive disturbance along with linear process model parameters, the recursive identification algorithm may be carried out in two or more iterations, where the parameter estimates obtained at the end of ( j − 1)th iteration are chosen as the initial parameter estimates for jth iteration. The development of EARMAX identification algorithm can be found elsewhere [67]. In practical cases, the true model order is unknown and has to be estimated. The order of the model that minimises the test statistic ξÑM (a measure of independence between input sequence and innovation sequence) is chosen as the estimate of true model order. The test statistic ξÑM is given by

ξÑM = rûTζ R−1 N rûζ ,

(12.14)

where rûζ is the CCF between input sequence {u(t)} and whitened residual sequence of model innovations {ε (t)} at different lags [0 M]. RN is the input covariance matrix given by


RN =

1 N ∑ ϕu,ε (t)ϕu,Tε (t) , N t=1

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(12.15)

where ϕu,ε (t) = [u(t) u(t − 1) . . . u(t − M + 1)]T .

12.4.2 Illustrative Example: Identification Under Non-stationary Disturbance The objective of this case study is to validate the efficacy of the EARMAX modelidentification method to obtain an unbiased process model in the presence of nonstationary disturbances. An ARMAX(1,1,2,2) process with non-stationary coloured disturbance subjected to a sinusoidal deterministic disturbance is simulated as a test bed and is given by the following discrete model, y(t) = 0.9429y(t − 1) + 0.6855u(t − 3) + e(t) + c1 e(t − 1) + c2 e(t − 2) + η (t) , (12.16) where

η (t) = 0.1 sin(0.02t) + 0.1 sin(0.05t) and

⎧ t ≤ 300 ⎨ 0.5 300 ≤ t < 700 c1 = 0.7 ⎩ −0.1 700 ≤ t < 1000

⎧ t ≤ 300 ⎨ 0.5 c2 = −1.3 300 ≤ t < 700 . ⎩ 0.2 700 ≤ t < 1000

(12.17)

(12.18) ®

e(t) is the white-noise signal with σe = 0.1, generated by randn in MATLAB . The process is being controlled by a proportional-integral (PI) controller with a proportional gain, KC = 0.04, and an integral gain, K = 0.01. To obtain the input–output data for model identification, the system is excited by introducing a pseudo-random binary signal (PRBS) switching between –0.1 to 0.1 with a switching time of 10 sampling intervals in the SP signal. One hundred Monte-Carlo (MC) simulations are carried out with different realisations of white-noise signal to find the uncertainty in parameter estimates in a stochastic environment. Step-response plots of the identified models (with average model parameters in 100 simulations) using EARMAX, OE, ARMAX and BJ identification techniques along with step responses of the actual process are given in Fig. 12.3. It can be clearly seen from Fig. 12.3 that the model identified using EARMAX is superior in terms of its accuracy in predicting the true process behaviour. The models identified using BJ, ARMAX are equal in terms of their accuracy, both with a significant mismatch from the actual process behaviour. The model identified using OE is inferior and it is evident from the large mismatch in its step response (Fig. 12.3)

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from the actual process behaviour. In EARMAX, as there is an inclusion of a timevarying bias term, it can best explain the dynamics of a time-varying additive nonstationary disturbance. This can be seen from the identified bias term along with the non-stationary additive disturbance plotted in Fig. 12.4 in one of the simulations performed.

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To investigate the effect of controller tuning on the quality of models that are identified using the closed-loop data, ten sets of simulations are carried out with a distinct value of KC in each set ranging from 0.01 to 0.1. Each set consists of 100 MC simulations. The deterministic process-model parameters identified using the EARMAX method are compared with those identified using BJ, ARMAX and OE model-identification methods. Mean-parameter estimates with EARMAX, BJ, ARMAX and OE identification techniques for each set of simulations with distinct KC are plotted in Fig. 12.5. From these plots, it can be seen that all the parameter estimates generated using the EARMAX technique are closer to the real model parameters and the variation of the estimates with controller tuning is negligible as compared to any other identification technique. Next closer estimates are obtained using the BJ identification technique, whereas estimates obtained using ARMAX and OE techniques are biased in almost all sets. EARMAX yields slightly better estimates as compared to BJ. Most importantly, the accuracy of model identification is not affected by controller tuning that qualitatively shows that EARMAX may yield unbiased parameter estimates in closed-loop identification. However, further theoretical analysis is necessary to validate this claim. EARMAX assumes a simple structure for disturbance modelling as compared to BJ and its identification can be recursively carried out by a pseudo-linear estimator.


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0.3

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Fig. 12.5 Performance of EARMAX, BJ, ARMAX and OE identification techniques on closedloop data obtained under different controller tunings: true model parameters a = −0.9429 and b = 0.6855

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12.5 Root-cause Detection and Quantification 12.5.1 OP–PV Model Identification Methodology As the gradient-based optimisation algorithms fail in the stiction Hammerstein model identification [116], a grid-based optimisation approach is adapted for identifying the stiction parameters. The search space in stiction-model parameters (S and J) is divided into a grid. At each node, i.e. for a set of S and J parameters, MV data is generated using the empirical stiction model. For the MV and PV data, the optimal process model parameters are deduced using the identification procedure described in Section 12.4. The stiction parameters that result in the minimum test statistic (ξÑM ) are to be identified. The size of the grid in search space is to be chosen as a balance between computational time and required precision of the estimated stiction parameters. 12.5.1.1 Identifiability of Stiction Parameters It needs to be verified that the predictor for a linear process subjected to additive non-stationary disturbance given in Eq. 12.3, facilitates consistent parameter estimates in the non-linear component of the Hammerstein model in the presence of external non-stationary disturbances. It was reported in [86]: Theorem 1 and Eq. 20) that there always exists a decoupling in the accuracy of the estimate between the non-linear and the linear component of Hammerstein systems. And hence, the accuracy of the estimate of the non-linear component is not affected by the complexity of the model structure used to estimate the linear component under the assumption of consistent linear model structure for characterising the linear dynamics [116]. A time-varying bias term is included in the linear model to represent the additive non-linearity that may enter at the output in the form of external non-stationary disturbances. It was shown in Section 12.4.2 that the inclusion of a bias term with decoupled covariance update yields the consistent linear part of Hammerstein model and unbiased linear process-model parameters. As the linear model part is consistent with the process, the estimation of stiction non-linearity will not be affected by the accuracy of the linear process-model estimation. This ensures the decoupling between the non-linear stiction model and the linear process model leading to the unbiased estimates of stiction-model parameters and hence Jˆ → J and Sˆ → S. Conversely, an ARMAX structure is not consistent with the actual linear process, if subjected to non-stationary disturbances. This results in violating the necessary condition that decouples the estimation of the non-linear component from the linear component in the Hammerstein model and hence introduces bias in the non-linear model-parameter estimates in the presence of non-stationary disturbances. Another issue to be investigated is whether the global optima for parameter estimates can be obtained using the grid-search method proposed in Section 12.5.1 for the Hammerstein-model identification. The optimisation of model parameters in


279

this case can be classified as separable least-squares [6, 58]. References [6] and [58] have shown that when the linear component of the model is identified using a leastsquares estimator, the dimensionality of the optimisation problem reduces to 2, only remaining are the stiction parameters in the non-linear component of Hammerstein model. Hence, an accurate estimate of model parameters can be found from the proposed grid-search method provided the grid size is small. Here, the grid size is chosen as a trade-off between the accuracy of stiction-model-parameter estimates and the computational power required. 12.5.1.2 Persistence of Excitation To be able to obtain good model-parameter estimates, the data has to be rich enough. In other words, the process has to be persistently excited. Normal operating data may not be persistently exciting, especially if the process is near the steady state for lengthy periods of time. If the control loop is under normal operation, the diagnosis is stopped at the stage of oscillation detection, and will not proceed till the Hammerstein-model identification stage. However, the persistence of excitation condition has to be addressed for two scenarios: (i) valve stiction is present in the absence or presence of other faults, and (ii) external oscillatory disturbances and/or underdamped controller tuning are inducing oscillations and there is no stiction in the control valve In the presence of stiction as discussed in Chap. 4, the control loop develops sustained oscillations. For significant stickband and slip jump in the control valve, under feedback the valve output (flow) appears like a continuous square wave with a specific frequency (which is a function of controller tuning and stiction strength). These square wave inputs to the process, in a way, perturb the process similar to open loop tests conducted for identification and aids in identifying the process dynamics accurately, which is evident from the simulation studies, discussed later in Section 12.6. However, for very low quantities of stickband and slip jump, there will not be any realisable effect on the control loop (insignificant oscillations compared to measurement noise) and the diagnosis is stopped at the stage of oscillation detection itself. In the absence of stiction, the excitations that are caused by disturbances or aggressive tuning may not have enough excitations to obtain rich data, using which one cannot obtain a reliable estimate of process model. As stated in Section 12.5.1.1, the estimation of stiction non-linearity will not be affected by the accuracy of the linear process-model estimation. Hence, in the case of oscillations induced by a fault other than stiction in a control valve, using the proposed technique one can obtain an accurate estimate of stiction parameters and non-stationary disturbance even though the process-model estimates are biased. This is proved by simulation studies discussed in Section 12.6. In such cases, the diagnosis will be qualitatively accurate and if needed, one may excite the process using external perturbations to obtain rich data to estimate an accurate process model for further oscillation-isolation tasks.

280


12.5.1.3 Effect of Measurement Noise As the proposed oscillation diagnosis method uses an identification technique, it is no different from any other system-identification method in handling measurement noise. An explicit noise model in addition to the bias term can surpass the problem of bias in process-model-parameter estimates. If the measurement noise is uncorrelated with the stiction, oscillatory disturbance or aggressive tuning introduced excitations in the process, under richness conditions, accurate estimation of Hammerstein model is possible, which leads to accurate oscillation diagnosis.

12.5.2 Identification of Controller Transfer Function As described in Section 12.3.2, the oscillation diagnosis methodology requires the estimation of the PID-controller transfer function from the available OP and PV measurements. The estimation method eliminates the need for excess process knowledge required to carry out the diagnosis. PID-controller parameters are identified from OP–PV data. The discrete PID-controller implementation at nth sampling instant can be given by n

uop (n) = KC εtr (n) + K ∑ εtr (k) + KD {εtr (n) − εtr (n − 1)} ,

(12.19)

k=0

where uop (t) is the OP and εtr (t) is the tracking error (PV − SP) at time t, KC , K, and KD are the proportional gain, integral gain and derivative gains of the controller. Similarly the controller output at (n − 1)th sampling instant n−1

uop (n − 1) = KC εtr (n − 1) + K ∑ εtr (k) + KD {εtr (n − 1) − εtr (n − 2)} . (12.20) k=0

From Eqs. 12.19 and 12.20

uop (n) − uop (n − 1) = KC {εtr (n) − εtr (n − 1)} + Kεtr (n) +KD {εtr (n) − 2εtr (n − 1) + εtr (n − 2)} .

(12.21)

From Eq. 12.21, one can identify the PID-controller tuning parameters using a leastsquares estimation technique.

12.5.3 Oscillation Root-cause Detection and Quantification Methodology The root-cause detection and quantification are based on the identified model parameters of the Hammerstein model of PV–OP mapping. If the estimated stiction


281

parameters, Jˆ and Sˆ are non-zero, there is a quantifiable deadband plus stickband and slip jump in the control valve. In that case one can conclude that the root-cause for the oscillatory behaviour is stiction non-linearity of the control valve. If Jˆ and Sˆ are close to zero, there is no stiction in the control valve. If there are any cyclic external disturbances entering the process causing the control loop to oscillate, it can be captured by the bias-term sequence {ηˆ (t)}. The cyclic nature can be quantified using the comprehensive oscillation characterisation in Chap. 4. If the {ηˆ (t)} is found to be oscillatory, part of the oscillations is attributed to the presence of external oscillatory disturbances. If the closed-loop simulations of the identified linear process transfer function with the identified PID-controller yield an oscillatory control system, then one of the root-causes for oscillations is underdamped tuning.

12.6 Illustrative Example: Oscillation Diagnosis Under Various Faulty Situations The objective of this simulation study is to demonstrate the effectiveness of the proposed oscillation diagnosis method in different fault situations. A discrete linear process bearing the following transfer function with a discrete PI-controller (KC = 1.0, and K = 0.5) with a sampling period of 1 s is considered for the process simulations: y(t) =

1 + 0.5q−1 + 0.5q−2 0.2642q−2 + 0.1353q−3 u(t) + e(t) . 1 − 0.7358q−1 + 0.1353q−2 1 − 0.7358q−1 + 0.1353q−2 (12.22)

Oscillations resulting from stiction (in a control valve), oscillatory external disturbances and/or aggressive controller tuning are simulated, and the proposed diagnosis method is employed. Simulations are carried out for 4000 sampling instants in each case, and the obtained OP–PV data is used for carrying out the oscillation diagnosis. The cases involving stiction in a control valve are simulated using Kano’s [63] valve-stiction model with a slip jump (J) of 0.5 and deadband plus stickband (S) of 1.5. The cases, where an oscillatory external disturbance is one of the root-causes for oscillations in a control loop, a sinusoidal signal, η (t) = 0.2 sin(0.02t) is added to the process output in closed loop. For simulating the cases involving aggressive controller tuning, the controller proportional gain is chosen as 1.32 and the integral gain as 0.66. In the identification stage of a linear process model, the model order is assumed to be known a priori.

12.6.1 Stiction A sticky valve with a slip jump (J) of 0.5 and stickband (S) of 1.5 is simulated in the control loop. The oscillation detection performed on the tracking error sig-

282


nal confirmed the presence of oscillations in the control loop with a regularity of 1.176 and time period of 33.5 s. As a next step in oscillation diagnosis, the Hammerstein model is identified from OP–PV data. Alongside, a PID-controller transfer function is identified from OP–PV data. The identified stiction parameters (Jˆ = 0.5, and Sˆ = 1.5) are accurately describing the simulated control-valve-stiction phenomena. COC algorithm employed on the bias term and on the PV signal obtained from closed-loop simulations of the identified process model and controller model confirmed the absence of any kind of oscillatory modes in those signals. The regularity of oscillations in these signals is found to be 0.59 and 0.72, respectively. This is indicative of a control loop in which the oscillations are purely caused by stiction in a control valve. Figure 12.6 shows the oscillation diagnosis of this case. Figure 12.6a shows the oscillatory PV signal and the model predictions of PV; Fig. 12.6b the OP signal and predicted MV signal; Fig. 12.6c the identified bias term; and Fig. 12.6d the PV signal obtained in the closed-loop simulation of identified process model and controller models. A measurement white noise with a standard deviation of 0.1 is added in all closed-loop simulations. The process model identified in this case is given by

1.5

1 + 0.4761q−1 + 0.7105q−2 0.264q−2 + 0.153q−3 u(t) + ε (t) . 1 − 0.6947q−1 + 0.1169q−2 1 − 0.6947q−1 + 0.1169q−2 (12.23)

PV PV predicted SP

(a)

1

PV

0.5 0

−0.5 −1 0

50

100

150

Time [s]

200

200

400

600

800

1000

200

400

600

800

1000

Time [s]

(d)

0 −0.5

−1 0

−0.1

0.5 CL

0

External Disturbance Bias Term

0

1

OP MV predicted

1

(b)

0.1

−0.2 0

300

PV

OP and MV

(c)

250

0.2

External Disturbance

y(t) ˆ =

50

100

150

Time [s]

200

250

300

−1 0

Time [s]

Fig. 12.6 Diagnosis of the control loop characterised by stiction-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of identified linear process model and controller model


283

12.6.2 Oscillatory External Disturbance In this case, an external oscillatory disturbance entering the control loop at the output of the process as an additive signal, is simulated using a sinusoidal function η (t) = 0.2 sin(0.02t). The controller is critically tuned (KC = 1.0, and K = 0.5) and there is no stiction in the control valve (J = 0 and S = 0). The diagnosis performed is shown in Fig. 12.7. The OP signal is identified to contain oscillations with a time period of 314 s, and hence the control loop is classified as oscillatory. The identified stiction parameters (Jˆ = 0 and Sˆ = 0) are indicative of the absence of control-valve stiction. The identified bias term is shown in plot c along with the actual sinusoidal disturbance introduced. It can be seen that the time-varying bias term can capture the dynamics of the external oscillatory disturbance accurately. The regularity of oscillations in the bias term is 47.83 with a time period of 314.4 s. The PV signal obtained from closed-loop simulations is found to be non-oscillatory (regularity of oscillations, r = 0.84). This confirms the presence of external oscillatory disturbances as the only root-cause for oscillations in the control loop. The process model identified is given by 1 + 0.6448q−1 + 0.6279q−2 0.1172q−2 + 0.2653q−3 u(t) + ε (t) . 1 − 0.606q−1 + 0.1694q−2 1 − 0.606q−1 + 0.1694q−2 (12.24)

PV PV predicted SP

0.4 (a)

PV

0.2


y(t) ˆ =

0 −0.2 −0.4 0

100 Time [s]

150

0.1 0 −0.1 −0.2

200

0 1

(c)

500

1000 Time [s]

1500

2000

(d)

0.5

0 −0.5 OP MV predicted

−1 −1.5 0


(b)

0.2

PVCL

OP and MV

0.5

50

0.3

50

100 Time [s]

150

0

−0.5 200

−1 0

200

400 600 Time [s]

800

1000

Fig. 12.7 Diagnosis of the control loop characterised by external oscillatory disturbance-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closedloop simulations of the identified linear process model and controller model

284


12.6.3 Aggressive Controller Tuning Proportional gain of the controller is increased by a factor of 1.32 (KC = 1.32 and K = 0.66) to make the controller aggressive, yielding sustained oscillations in the control loop in the absence of external disturbances and stiction in the control valve. The resultant PV and OP signals with distinct oscillations at 7.2 s of time period ˆ are zeare shown in Figs. 12.8a and 12.8c. Identified stiction parameters (Jˆ and S) ros, denoting the absence of stiction in the control valve. The identified bias term is found to be non-oscillatory with a regularity of oscillations of 0.49. The closedloop simulations of the identified controller and process transfer functions yielded an oscillatory trend in OP and PV signals and corresponding regularity is estimated at 2.49 with a time period of 7.23 s. This confirms that the aggressively tuned controller is the root-cause for oscillatory control-loop behaviour. The process model identified is given as 1 + 0.4412q−1 + 0.8129q−2 0.6153q−2 + 0.04297q−3 u(t) + ε (t) . 1 − 0.6185q−1 − 0.4709q−2 1 − 0.6185q−1 − 0.4709q−2 (12.25)

(a)

0.2

PV PV predicted SP

PV

0.5


y(t) ˆ =

0

−0.5 0

40

Time [s]

60

80

100

−0.1

−0.5


−0.2 0 6

OP MV predicted

(c)

0

−1 0

0

500

1000 Time [s]

1500

2000

50

100 Time [s]

150

200

(d)

4 PVCL

OP and MV

0.5

20

(b)

0.1

2 0 −2

20

40

Time [s]

60

80

100

−4 0

Fig. 12.8 Diagnosis of the control loop characterised by aggressive controller-tuning-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closedloop simulations of the identified linear process model and controller model


285

12.6.4 Stiction and Oscillatory External Disturbance Stiction phenomena with J = 0.5 and S = 1.5 is simulated in the control valve along with sinusoidal additive disturbance (η (t) = 0.2 sin(0.02t)) entering the process at output. These two faults introduced sustained oscillations in the control loop. The Hammerstein-model identification performed yielded significant stiction parameters (Jˆ = 0.5 and Sˆ = 1.5) confirming the presence of stiction in the control valve as one of the root-causes for oscillatory behaviour in the control loop. The regularity of oscillations in the identified bias term is ∞ with a time period of 314 s and hence oscillatory external disturbance is found as another root-cause. Closed-loop simulations of the identified process and controller model have shown a non-oscillatory trend in PV and OP signals (regularity of oscillations: r = 0.65) and hence one can confirm that the controller is properly tuned. The identified process model in this case is y(t) ˆ =

1 + 0.4222q−1 + 0.6909q−2 0.2678q−2 + 0.1421q−3 u(t) + ε (t) . −1 −2 1 − 0.7304q + 0.1345q 1 − 0.7304q−1 + 0.1345q−2 (12.26)

1

PV PV predicted SP

(a)

PV

0.5 0

−0.5 −1 0

100

150

Time [s]

200

250

300

0.4

0 −0.2 −0.4 0

1500

2000

(d)

0

−0.5

−1 −2 0

1000

Time [s]

0.5

1 0

500

1

OP MV predicted

(c)


(b)

0.2

PVCL

OP and MV

2

50


The performed diagnosis is plotted in Fig. 12.9.

50

100

150

Time [s]

200

250

300

−1 0

200

400

600

Time [s]

800

1000

Fig. 12.9 Diagnosis of the control loop characterised by stiction and external-disturbance-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closedloop simulations of the identified linear process model and controller model

286


12.6.5 Stiction and Aggressive Controller Tuning In this case, two faults are introduced in the control loop: stiction in the control valve (J = 0.5 and S = 1.5) and aggressive controller tuning (KC = 1.32 and K = 0.66). The diagnostics of this case are plotted in Fig. 12.10. The resultant oscillatory OP– PV data (with distinct oscillations with a time period of 7.4 s) is used for oscillation diagnosis. Significant non-zero values of identified stiction parameters (Jˆ = 0.5 and Sˆ = 1.5) and oscillatory trend in closed-loop simulations of identified process and controller models (regularity of oscillations of 2.47) depicts the presence of both stiction and aggressive controller tuning. The identified process model is given by y(t) ˆ =

1 + 0.4462q−1 + 0.7055q−2 0.2695q−2 + 0.1481q−3 u(t) + ε (t) . 1 − 0.7005q−1 + 0.09789q−2 1 − 0.7005q−1 + 0.09789q−2 (12.27)

The regularity of oscillations of bias term is found to be 0.53, which shows the absence of an oscillatory external disturbance.

0.2

PV PV p edicted SP

(a)

PV

2


4

0 −2 300

400 Time [s]

450

500

−0.1 −0.2 0

500

1000 Time [s]

1500

0.5

OP MV predicted

(c)

0

−5 300

0

PVCL

OP and MV

5

350


(b)

0.1

2000 (d)

0

−0.5 350

400 Time [s]

450

500

0

50

100 Time [s]

150

200

Fig. 12.10 Diagnosis of the control loop characterised by stiction and aggressive controller-tuninginduced oscillations: a) PV signal along with predicted PV; b) identified bias term along with nonstationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model


287

12.6.6 Oscillatory External Disturbance and Aggressive Controller Tuning In this fault scenario, the control loop is subjected to oscillatory disturbances (η (t) = 0.2 sin(0.02t)) and aggressive controller tuning (KC = 1.32 and K = 0.66). The resultant tracking error signal is oscillatory in nature with two distinct time periods of 7.2 s and 308.4 s. Zero values of J and S identified from the OP–PV data show the absence of stiction phenomena in the control valve. Oscillatory behaviour in the identified bias term (r = ∞) and the oscillatory trend in closed-loop simulations of identified process and controller transfer functions (r = 2.46) are indicative of external oscillatory disturbance and aggressive tuning as the root-causes for oscillatory behaviour in the control loop. The process model identified is given by y(t) ˆ =

1 + 0.6057q−1 + 0.702q−2 0.1702q−2 + 0.2211q−3 u(t) + ε (t) . −1 −2 1 − 0.626q + 0.1749q 1 − 0.626q−1 + 0.1749q−2 (12.28)

Figure 12.11 shows the diagnosis results described above.

(a)

PV PV predicted SP

1 PV

0.5 0

−0.5 −1 0

50

Time [s]

100

3

0 −0.2 −0.4 0 1

OP MV predicted

(c)

500

1000 Time [s]

1500

2000

(d)

0.5 CL

0


(b)

0.2

150

1

0

−0.5

−1 −2 0

0.4

PV

OP and MV

2

0.6 External Disturbance

1.5

−1 50

Time [s]

100

150

0

50

Time [s]

100

150

Fig. 12.11 Diagnosis of the control loop characterised by external oscillatory disturbance and aggressive controller-tuning-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model

288


12.6.7 Stiction, Aggressive Controller Tuning and Oscillatory External Disturbance In this case, all three faults are introduced into the control loop: stiction (J = 0.5, S = 1.5), oscillatory disturbance (η (t) = 0.2 sin(0.02t)), and aggressive controller tuning (KC = 1.32 and K = 0.66). Stiction is identified as one of the root-causes, as the identified stiction parameters are Jˆ = 0.5 and Sˆ = 1.5. Part of the oscillations in the control loop can be attributed to the presence of external oscillatory disturbances as the identified bias term is oscillatory (r = 62.63). The third root-cause is identified as aggressive controller tuning, as the closed-loop simulations yielded an oscillatory trend in PV signal (r = 2.46). As an outcome of the identification algorithm, the following process model is estimated:

y(t) ˆ =

1 + 0.4483q−1 + 0.6704q−2 0.265q−2 + 0.1409q−3 u(t) + ε (t) . 1 − 0.7224q−1 + 0.1228q−2 1 − 0.7224q−1 + 0.1228q−2 (12.29)

3

PV

2

PV PV predicted SP

(a)

1 0 −1 −2 0 4

50

Time [s]

100

(c)

0.4

0 −0.2 −0.4 0 5

PVCL

0


(b)

0.2

OP MV p edicted

2

OP and MV

150


The corresponding diagnostics are shown in Fig. 12.12. As seen from the OP and predicted VP plot, there is a clear distinction between the valve position and controller output, confirming the presence of stiction in the control valve. Both estimated disturbance plot and simulated closed-loop PV plot showed oscillatory time trends, depicting the presence of oscillatory external disturbances and aggressive controller tuning.

500

1000 Time [s]

1500

2000

(d)

0

−2 −4 0

50

Time [s]

100

150

−5 0

50

Time [s]

100

150

Fig. 12.12 Diagnosis of the control loop characterised by control-valve stiction, external oscillatory disturbance and aggressive controller-tuning-induced oscillations: a) PV signal along with predicted PV; b) identified bias term along with non-stationary disturbance introduced; c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model


289

In the above-mentioned fault scenarios, the frequency response of the identified linear process models along with the frequency response of the actual process are shown in Fig. 12.13.

Amplitude (dB)

1.5 1 0.5 0

Phase (deg)

0

−1

0

10

10

−50

Actual Process S D A S and D S and A D and T S, D and T

−100 −150 −200 −1

10

0

Frequency (rad/s)

10

Fig. 12.13 Frequency response of actual process and identified process models under various fault scenarios (S: stiction in control valve; D: external oscillatory disturbance; A: aggressive controller tuning)

It can be seen from Fig. 12.13 that in all cases involving stiction in a control valve the process models identified are accurate enough to model the behaviour of the actual process in the entire range of frequencies. In the cases where there is no stiction in the control valve but the process is being excited by external oscillatory disturbances, the identified models are acceptable in their accuracy with small mismatch in the lower-frequency region in the case of amplitude. There is an evident mismatch in the identified model behaviour and actual process at all frequencies in the case where aggressive tuning is the only root-cause for oscillations. This can be attributed to the insufficient excitations caused by aggressive tuning in the process at lower frequencies. Even in such a case, oscillation diagnosis could detect the root-cause accurately, where accurate stiction parameters are estimated even though process-model-parameter estimates are biased. This case offers an additional proof to the arguments that are made in Sects. 12.5.1.1 and 12.5.1.2. The set of seven cases shown above comprises all possible fault scenarios that may lead to oscillatory behaviour in control loops. It is shown from the performed diagnostics using the proposed method that it could identify and quantify the rootcause efficiently.

290


12.7 Industrial Case Studies The objective of this section is to demonstrate the application of the proposed method on a number of selected industrial control-loop data. In the diagnosis, SP, OP and PV data are used. The sampling frequency is assumed to be 1 s for convenience, as this will not affect the diagnosis.

12.7.1 Control Loop 1 The diagnosis is performed on the data obtained from a pressure control loop (CHEM 10) in a chemical process industry. It was known a priori that the control valve in this control loop contained stiction. The diagnosis results of this control loop are plotted in Fig. 12.14.

PV PV pred SP

(a)

PV

2 0 −2 0 3

400 600 Time [s]

800

0 −0.2 200

400 600 Time [s]

800

1000

1

OP MV pred

(c)

(b)

0.2

−0.4 0

1000

(d) 0.5 CL

1 PV

OP and MV

2

200

0.4 Bias Term Identified

4

0

0

−1 −2 0

200

400 600 Time [s]

800

1000

−0.5 0

50

100

150 Time [s]

200

250

300

Fig. 12.14 Oscillation diagnosis performed on industrial control loop 1: a) PV signal along with predicted PV; b) identified non-stationary disturbance (bias term); c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model

The oscillation-detection algorithm applied to tracking error signal found significant oscillations (regularity of 1.028) with a time period of 113.74 s and a decay ratio of 0.8. The identified stiction parameters (Jˆ = 0.05 and Sˆ = 1.85) indicate the presence of quantifiable stiction, which was causing the sustained oscillations in the OP and PV signals. It is found that the identified bias term is also oscillatory with a regularity of 1.25 and a time period of 46.8 s. The identified process and controller are given by


y(t) ˆ =

−0.2667q−1 + 2.688q−2 u(t); 1 − 0.1737q−1 − 0.6511q−2

KC = 0.1903,

291

K = 0.02 .

(12.30)

The tracking error signal obtained from closed-loop simulations of the identified process and controller models are non-oscillatory with regularity of oscillations 0.67. Hence, it can be concluded that stiction in the control valve and external oscillatory disturbance were responsible for the oscillatory trend in this control loop.

12.7.2 Control Loop 2 The data set of the flow-control loop (CHEM 11) was obtained from the chemical industry. The time trends of PV and OP signals for this control loop are plotted in Figs. 12.15a and c. Limit cycles in these time trends were actually caused by stiction in the control valve. The tracking error signal contained sustained oscillations with a regularity of 1.565 at a time period of 111 s and a decay ratio of 0.82. The identified stiction parameters (Jˆ = 0.06 and Sˆ = 0.48) indicate the presence of stiction in the control valve. The identified process and controller are:

6

0.6308q−1 + 0.8412q−2 u(t); 1 − 0.4236q−1

PV PV pred SP

(a)

PV

4 2 0 −2 −4 0 1.5

200

400 600 Time [s]

(c)

(12.31)

(b)

0 −1 200

400 600 Time [s]

800

1000

1.5

(d) CL

1 PV

0

K = 0.016 .

1

−2 0

1000

0.5

0.5 0

−0.5 −1 0

2

OP MV pred

1 OP and MV

800

KC = 0.1225,

Bias Term Identified

y(t) ˆ =

200

400 600 Time [s]

800

1000

−0.5 0

100

200 Time [s]

300

400

Fig. 12.15 Oscillation diagnosis performed on industrial flow control loop 2: a) PV signal along with predicted PV; b) identified non-stationary disturbance (bias term); c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model

292


It is found that the tracking error signal obtained from closed-loop simulations of the identified process and controller models and the identified bias term are nonoscillatory with regularity of oscillations of 0.35 and 0.49. Hence, it can be concluded that only the stiction in the control valve alone is responsible for the oscillatory control loop.

12.7.3 Control Loop 3 Data for a flow-control loop (CHEM 24) are analysed for the presence of oscillations. Figure 12.16 shows the oscillation root-cause analysis results for this control loop. The OP of the controller is found to be oscillatory with oscillations at a time period of 49.6 s. It is identified that there is a significant stiction (Jˆ = 1 and Sˆ = 23) present in the control valve. The identified bias term contains insignificant oscillations at 32 s time period with a regularity of 0.5. This trend in bias term is similar to the trend of set point excitations. The identified process and controller are as follows:

200

4.793q−1 − 4.141q−2 u(t); 1 − 1.239q−1 + 0.358q−2

30

PV PV pred SP

(a)

PV

100 0 −100 0

100

200 300 Time [s]

500

20

K = 0.0278 .

(12.32)

(b)

10 0 −10 −20 0

100

200 300 Time [s]

400

500

10

OP MV pred

10 (c)

(d) 0

0

PVCL

OP and MV

400

KC = 0.0205,

Bias Term Identified

y(t) ˆ =

−10

−10 −20 −20 0

100

200 300 Time [s]

400

500

−30 0

50

100

150 200 Time [s]

250

300

Fig. 12.16 Oscillation diagnosis performed on industrial flow control loop 3: a) PV signal along with predicted PV; b) identified non-stationary disturbance (bias term); c) OP signal along with predicted MV; d) PV signal obtained from closed-loop simulations of the identified linear process model and controller model


293

The closed-loop simulation of the identified process and controller models is found to yield non-oscillatory OP and PV signals (regularity of 0.4). However, as shown in the step response for a change in set point (Fig. 12.16d), there is a large overshoot in the closed-loop response, which may also lead to huge fluctuations under frequent setpoint excitations. From these results, it can be concluded that the root-causes of oscillations in this control loop are stiction in the control valve and frequent SP excitations.

12.8 Summary and Conclusions In this chapter, a model-based diagnosis and root-cause quantification methodology for oscillatory control loops was presented. The Hammerstein model describing the relationship between OP–PV signals, and the PID-controller transfer function obtained from OP–PV were explicitly used in isolating the root-cause. Its effectiveness to identify and quantify the root-cause even in the case of multiple faults occurring simultaneously (poor controller design, control-valve stiction and external disturbances) was proven by simulation examples. A few industrial cases were also discussed to describe the oscillation diagnosis procedure in detail. The proposed methodology can provide means of individual fault detection and isolation in the presence of multiple faults occurring at the same time, which makes it relevant to apply in industrial control loop oscillation diagnosis. Further improvements in the optimisation of Hammerstein-model parameters can strengthen the methodology in obtaining a global solution for stiction-parameter estimates.

Chapter 13

Comparative Study of Valve-stiction-detection Methods Mohieddine Jelali and Claudio Scali

This chapter contains an exhaustive comparative study of methods for detecting stiction in control valves. The study involves 93 different data sets from different process industries, including chemicals, pulp and paper mills, commercial building, and metal processing. The different techniques considered in the study are assessed in terms of efficiency in detecting stiction, possible problems that may occur when applying them to real industrial data, and necessary preprocessing steps and countermeasures to avoid false detections. Achieved results (decision flags and/or index values) are presented and discussed for each method. Statistics for the applicability and indications of the methods are reported. Suggestions are given concerning possible combinations of the methods that lead to reliable stiction-detection results.

13.1 Introduction So far several different techniques for detection of stiction have been presented and discussed throughout the previous chapters of this book. Some of the methods are manual or require strong interaction with the user; other techniques are based on additional knowledge about the characteristic curve of the valve and values of the manipulated variable or valve position. Some techniques only detect the presence of stiction; other approaches also deliver an estimate of the extent of stiction. Moreover, the methods differ in their robustness to some bad effects, such as noise, nonstationary trends and multiple oscillations, that may be present in industrial data.

Mohieddine Jelali Department of Plant and System Technology, VDEh-Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected] Claudio Scali CPCLab, Department of Chemical Engineering (DICCISM), University of Pisa, Via Diotisalvi, n.2 - 56126 Pisa, Italy, e-mail: [email protected]

295

296

M. Jelali and C. Scali

The aim of this chapter is to compare the techniques presented throughout the previous chapters using a large number of data sets from different process industries, including chemicals, pulp and paper, commercial building, mining and metal processing. The rest of the chapter is organised as follows. In Sect. 13.2, the stictiondetection methods considered are briefly reviewed. The industrial control loops involved in the comparative study are described in Sect. 13.3. Some loops are illustrated to show characteristics and differences. The application results are presented and discussed in Sect. 13.4 in terms of efficiency and sensitivity to different effects that may occur when considering industrial data. Weaknesses, pitfalls and counter® measures are reported for the methods. A MATLAB graphical user interface (GUI) created for side-by-side comparison of the methods is described. Some suggestions are given on how to combine different techniques and detect stiction problems reliably.

13.2 Selected Methods The stiction-detection techniques to be compared and their indications (SI: stiction index) are summarised in Table 13.1. Methods A, B and C are based on the shape analysis of MV–OP plots, and thus assume that the valve position is available, making the application of the methods restricted to flow control loops. Methods D and E have been primarily used to detect process non-linearities in control loops. Whereas the bicoherence method has been thoroughly investigated in combination with PV–OP plots (ellipse fitting) to detect stiction, the surrogate analysis was only introduced for non-linearity detection. Methods F, G and H can be utilised to detect stiction by exploiting different features of measured loop signals, i.e. the correlation of OP and PV, control error areas and their distributions, respectively. Methods I and J have in common that certain signal patterns characterising stiction (sinusoid, triangle, relay) are fitted to measured loop signals. Methods belonging to the group K do not only detect stiction, but also quantify the stiction extent, based on the identification of a Hammestein model from closed-loop data. The specific methods differ in the linear model structure selected and the strategy and algorithm used to estimate the model parameters. Methods A–C and F–K have been presented in detail in the previous chapters of this book. Methods D and E are briefly reviewed in Appendix B. For more details about non-linearity detection, the reader should consult the recent book by Choudhury et al. [21]. There are other methods to detect stiction in control loops, such as those proposed by Deibert [24], Ettaleb et al. [28], Hägglund [37], Horch and Isaksson [51], Taha et al. [121] and Wallén [132]. However, all these approaches require either detailed process knowledge, user interaction or rather special process structures and will not be considered in this study.

13 Comparative Study of Valve-stiction-detection Methods

297

Table 13.1 Summary of the indications from different stiction-detection techniques Method A MV–OP-shape-based (sticking-valve positions) (Kano et al.) See Sect. 5.2.1 B MV–OP-shape-based (Parallellogram) (Kano et al.) See Sect. 5.2.2 C MV–OP-shape-based (Qualitative analysis) (Yamashita) See Sect. 5.2.3 D Bicoherence & PV–OP plot (Choudhury et al.) See Sect. B.1. E Surrogates analysis & PV–OP plot (Thornhill) See Sect. B.2. F Cross-correlation (Horch) See Chap. 6. G Histogram (Horch) See Chap. 6. H Area peak (Salsbury and Singhal) See Chap. 9. I Curve fitting (He et al.) See Chap. 7. J Relay (Rossi and Scali) See Chap. 8. K Hammerstein identification (Jelali; Lee et al.; Karra and Karim) See Chaps. 10, 11 and 12.

No stiction ρA ≤ 0.25

Uncertainty –

Stiction ρA > 0.25

Fmax ≤ 0.7

–

Fmax > 0.7 & r ≈ 1

ρC ≤ 0.25

–

ρC > 0.25

NGI < NGIcrit

–

NGI ≥ NGIcrit & NLI ≥ NLIcrit & PV–OP is elliptical

NPI ≤ 1.0

–

NPI ≥ 1.0 & PV–OP is elliptical

0 ≤ Δ ρ ≤ 0.072 0 ≤ Δ τ ≤ 1/3

0.072 < Δ ρ < 1/3 1/3 < Δ τ < 2/3

1/3 ≤ Δ ρ ≤ 1 2/3 ≤ Δ τ ≤ 1

SI ≤ 0.4

0.4 < SI < 0.6

SI ≥ 0.6

R≤1

1 < R ≤ (1 + δ )

R > (1 + δ )

SI ≤ 0.4

0.4 < SI < 0.6

SI ≥ 0.6

SI ≤ −0.21

−0.21 < SI < 0.21

SI ≥ 0.21

J=0&S=0

–

J>0&S>0

Some of these methods will be abbreviated throughout the chapter as follows: • • • • • • •

BIC: bicoherence and ellipse-fitting technique of Choudhury et al. CORR: cross-correlation-based method of Horch. HIST: histogram-based method of Horch. RELAY: relay technique of Scali and Rossi. CURVE: curve-fitting approach of He et al. AREA: area-peak-based method of Salsbury and Singhal. HAMM1: Hammerstein-model-based technique of Jelali.

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• HAMM2: Hammerstein-model-based technique of Lee et al. • HAMM3: Hammerstein-model-based technique of Karra and Karim. Note that the extension of the curve-fitting method for other signal patterns, such as rectangular and trapezoidal waves, is straightforward. For this purpose, Srinivasan and Rengaswamy [115] proposed a technique based on qualitative pattern recognition. The algorithm aims to distinguish square, triangular and saw-tooth-like signal shapes in both OP and PV. The technique used to classify the signals is dynamic time warping (DTW) and this is applied for each oscillation cycle individually rather than a complete data set at once. DTW is a classical technique from speech recognition used to compare signals with stored patterns (Table 1.4).

13.3 Industrial Control Loops Involved in the Study In total, 93 industrial control loops are considered for the comparative study, including flow control (FC), level control (LC), temperature control (TC), pressure control (PC), analyser control (AC), strip thickness or gauge control (GC). Some information about the loops is collected in Table A.1. A few loops will be presented in what follows to show the main characteristics and differences: • The loops can be classified by origin and type; see Table 13.2. The majority of the loops (35%) are flow control loops that come from chemical plants. • Most of the loops show constant setpoint (SP), but some have varying SP, typically those being under cascade or advanced (supervisory) control. For instance, the data for the FC loop CHEM 18 with variable SP are shown in Fig. 13.1. • Some loops have non-linearities other than stiction, such as controller output (OP) saturation (Fig. 13.2) or deadzone (Fig. 11.19b); some others are subject to heavy quantisation; see Fig. 13.3. • Other loops are in open-loop, i.e. manual mode. Data for two examples are shown in Figs. 11.10a and c. • Some loops have too short period of data, which cannot be analysed by many methods. Loop CHEM 5 has the shortest data length, i.e. 200 samples, as can be seen in Fig. 11.14b. • Many loops are affected by noise (e.g. loop POW 3 in Fig. 13.4), non-stationary trends or other disturbances and anomalies (e.g. loop CHEM 33 in Fig. 13.5); others show time windows with different characteristics (e.g. loop MIN 1 in Fig. 13.6) or multiple oscillations (e.g. loop PAP 10 in Fig. 13.7). All this suggests careful preliminary data preprocessing, such as filtering or data windowing. • Some loops have more than one performance problem. For instance, it was known that the TC loop BAS 6 suffers from stiction and aggressive controller tuning; see Fig. 13.8. This represents a challenge for stiction-detection methods.


299

8

SP, PV

5 6 0 4 −5

SP PV PV

2

5

0

OP

−2 −4 0 −6 −8 −5 200

300

400

500

600 700 Samples

800

900

1000

−5

0 OP

(a)

5

(b)

Fig. 13.1 Data for FC loop CHEM 18 (varying SP): a) PV and OP trends; b) PV–OP plot 1.5

1

PV

0.5

1

0 −0.5

0.5

PV

−1 0

0 −2

−0.5

OP

−4 −6 −1

−8 −10 −12

0

500

Samples

(a)

1000

1500

−1.5 −12

−10

−8

−6

OP

−4

−2

0

2

(b)

Fig. 13.2 Data for LC loop POW 5 (OP in saturation): a) PV and OP trends; b) PV–OP plot

Note that also the data are given in very heterogeneous formats. For instance, data for many loops, e.g. loops CHEM 13–17 and CHEM 40–64 are normalised and their ranges are not known (due to confidentiality reasons), so that the real entity/significance of perturbation cannot be stated. This fact has two implications: • It is relevant for the plant operator: for instance, a perturbation can be clearly associated to the presence of stiction, but if the amplitude of oscillation is very small, that loop will be considered as a well-performing loop. • The effect of noise and more generally the quality of data can affect the application of a technique and the final verdict issued; it is evident that by increasing the level of noise the number of uncertain verdicts will increase.

PV

300

M. Jelali and C. Scali 0.2

0.2

0.1

0.15

0

0.1

−0.1

0.05 PV

−0.2 3 2

0 −0.05

1 OP

−0.1

0 −0.15

−1 −2

0

200

400

600

800

1000 1200 Samples

1400

1600

1800

2000

−0.2 −2

−1

0

(a)

OP

1

2

3

(b)

Fig. 13.3 Data for TC loop CHEM 3 (quantisation): a) PV and OP trends; b) PV–OP plot 0.5

0.5 0.4 0.3

PV

0

0.2 0.1 PV

−0.5 4

0 −0.1

2 OP

−0.2 0

−0.3

−2 −4

−0.4 0

200

400

600

800

1000 1200 Samples

(a)

1400

1600

1800

2000

−0.5 −4

−2

0 OP

2

4

(b)

Fig. 13.4 Data for LC loop POW 3 (noisy data): a) PV and OP trends; b) PV–OP plot

It should also be mentioned that the real root-cause of bad performance was not known for all considered loops. However, some indications can be stated/guessed from visual inspection of the data, as given in Table A.1.

SP, PV

13 Comparative Study of Valve-stiction-detection Methods 1

1

0.5

0.8

301

0.6

0

0.4 −0.5

SP PV

0.2 PV

−1 5

0 −0.2

OP

−0.4 0

−0.6 −0.8

−5

0

100

200

300

400 Samples

500

600

−1 −5

700

0 OP

(a)

5

(b)

Fig. 13.5 Data for TC loop CHEM 33 (non-stationary trend and disturbances): a) PV and OP trends; b) PV–OP plot 30

30

PV

20 20

10 0

10

−10

PV

−20 1.5

0

1 −10

OP

0.5 0

−20

−0.5 −1 −1.5

0

500

1000

1500 Samples

2000

2500

(a)

−30 −1.5

−1

−0.5

0 OP

0.5

1

1.5

(b)

Fig. 13.6 Data for TC loop MIN 1 (intermittent stiction): a) PV and OP trends; b) PV–OP plot Table 13.2 Classification of the loops by origin and type Origin BAS CHEM PAP POW MIN MET

Number of loops 3 63 13 5 1 3

Type FC LC TC PC AC GC

Amount 33 22 16 16 3 3

PV

302

M. Jelali and C. Scali 2

2

1

1.5

0

1

−1

0.5 PV

−2 4

−0.5

2 OP

0

0

−1

−2

−1.5

−4

0

200

400

600

800 1000 Samples

1200

1400

1600

1800

−2 −4

−2

(a)

0 OP

2

4

(b)

PV

Fig. 13.7 Data for LC loop PAP 10 (multiple oscillations): a) PV and OP trends; b) PV–OP plot 1

0.8

0.5

0.6

0

0.4

−0.5

0.2 PV

−1 4

−0.2

2 OP

0

0

−0.4

−2

−0.6

−4 250

300

350

Samples

400

450

500

−0.8 −4

(a)

−2

0 OP

2

4

(b)

Fig. 13.8 Data for TC loop BAS 6 (stiction and tight tuning): a) PV and OP trends; b) PV–OP plot

13.4 Application Results and Discussion 13.4.1 Application Results The techniques in Table 13.1 have been applied to the data from the industrial loops in Table A.1. The data assessments were undertaken by the investigators of the techniques using their original codes and provided to the authors of this chapter in the form of (Excel, Word) result sheets. The results are listed in Tables 13.11–13.19. Comments on the data pretreatment and the results for each method are given in the tables and footnotes. Readers are encouraged to particularly look at the valuable


303

comments on the loops given by Thornhill, Horch and Scali in Tables 13.13, 13.14 and 13.16, respectively. The results contained in the individual tables are now grouped and classified to perform a sort of “objective analysis”, without entering into the relative merits of the different techniques. A comparison of verdicts issued on each loop from the different techniques is reported in Table 13.10. The following abbreviations are used: • • • • • •

YES: stiction. NO: no stiction. UNC: uncertain or unknown. NA: not applicable to that data set. XX: not applicable to that kind of loop, e.g. CORR is not applicable to LC loops. NIV: verdict not issued for different causes, depending on the technique, e.g. no suitable feature extracted for the AREA method by Salsbury and Singhal. • MAN: manual (open-loop) valve. • NL: non-linearity. • Q: quantisation. Note that some indications have been compacted, e.g. small stiction to YES; Unknown to UNC.

13.4.2 Synthesis and Discussion The results reported in Table 13.3 are obtained by making the following further assumptions: • Summarising issued verdicts in YES, NO, UNC. • Summing their amounts to give the total amount of applications (Total A). • Computing the total amount of non-applicability (NA) (Total NA) as NA = 93 − A.

Table 13.3 Synthesis of issued verdicts and applicability YES NO UNC Total A Total NA

BIC 35 31 0 48 45

CORR 20 13 26 59 34

HIST 44 21 10 77 16

RELAY 60 14 6 80 13

CURVE 42 14 16 72 21

AREA 21 31 0 52 41

HAMM2 HAMM3 71 75 15 17 0 0 86 92 7 1

It can be clearly seen that the applicability varies largely from one method to another: minimum A = 52 (AREA), maximum A = 92 (HAMM3). Also, the presence of stiction (YES) is indicated for a minimum of 20 loops (21.5% for CORR) to a maximum of 75 loops (80.6% for HAMM3). The absence of stiction is indicated

304


(NO) for a minimum of 13 cases (14% CORR) to a maximum of 32 loops (34.4% for BIC). Obviously, larger applicability (A) yields a larger number of stiction indications, e.g. A(YES): 92(75) for HAMM3, 86(71) for HAMM2, 80(60) for RELAY and 77(44) for HIST. However, larger applicability does not necessarily imply higher power or reliability of any technique. Note that the methods CORR and AREA are applicable to the least number of loops; but one should be aware that CORR can only be applied to self-regulating processes. The common verdicts issued by 8 techniques given in Table 13.10 are now illustrated in Table 13.4, indicated as the number of occurrences of the YES or NO verdict for the loops and their cumulative sum. Table 13.4 Common verdicts issued by different techniques Stiction detected 8 YES 7 YES 6 YES 5 YES 4 YES

Loops 4 6 10 18 13

Cumulative 4 10 20 38 51

No stiction detected 8 NO 7 NO 6 NO 5 NO 4 NO

Loops 0 0 3 3 3

Cumulative 0 0 3 6 9

To exemplify: • Only for four loops do all eight techniques indicate stiction (YES), while there are 51 loops for which at least four techniques indicate stiction. • Only for three loops do six techniques clearly exclude stiction (NO), while there are only nine loops for which at least four techniques exclude stiction. From this analysis, one may conclude that not all stiction-detection techniques considered are reliable enough. However, the previous results analysis is heavily affected by the large number of non-applicability cases, which varies for the different techniques. Attention has been then focused on the 36 loops, to which all techniques are applicable; see Table 13.5. The corresponding verdicts are reported in Table 13.6. It can be seen that indications of different techniques are now more homogeneous: • The presence of stiction (YES) is detected for a minimum of 10 loops (38.4% for AREA), to a maximum 22 loops (84.6% for HAMM2 and HAMM3). • The absence of stiction is indicated (NO) for a minimum of 3 cases (11.5% for RELAY) to a maximum of 10 cases (42.3% for AREA). Limiting the statistic to BIC, RELAY, HAMM2 and HAMM3, the indications would become more homogeneous. Indeed, this confirms that the issue of nonapplicability of certain techniques plays a crucial role. Observe that the highest number of UNC verdicts is given by CORR; the AREA method often delivers no verdict. However, it is not fair to compare the CORR method with other methods, since it is not applicable to integrating processes.


305

Table 13.5 Results for the 36 loops to which all techniques are applicable CHEM 1 CHEM 2 CHEM 3 CHEM 6 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 16 CHEM 18 CHEM 23 CHEM 24 CHEM 28 CHEM 29 CHEM 32 CHEM 33 CHEM 40 CHEM 42 CHEM 43 CHEM 49 CHEM 50 CHEM 51 CHEM 54 CHEM 55 CHEM 60 CHEM 62 PAP 2 PAP 4 PAP 5 PAP 7 PAP 9 MIN 1 MET 1 MET 2 MET 3

BIC YES YES NLQ YES YES YES YES NL-NO YES YES YES YES YES NO YES YES YES NO NL-NO NO YES YES NL-NO YES YES NO NO YES YES YES NO-NO YES YES NO NL-NO NL-NO

CORR YES YES YES UNC YES UNC YES NO YES YES UNC YES YES YES YES UNC NO UNC NO NO UNC NO UNC UNC YES YES UNC YES NO YES NO NO UNC UNC UNC UNC

HIST YES YES NO YES YES NO YES NO YES YES UNC YES YES NO UNC NO YES YES NO NO NO YES YES UNC UNC YES YES NO NO YES YES NO YES NO YES NO

RELAY YES YES YES UNC YES YES YES NO YES YES YES YES YES YES YES YES UNC NIV NO YES NO UNC YES YES YES YES NO YES YES YES YES NO YES NO YES YES

CURVE YES YES YES YES YES NO YES NO YES YES YES YES YES NO NO NO UNC YES NO UNC NO UNC UNC YES YES YES YES UNC UNC YES UNC NO YES UNC UNC UNC

AREA YES NIV NO NO YES YES YES NO NIV NIV NO YES NO YES NO NO NIV NIV NO NIV NO NIV NIV NO YES NO NIV YES YES NO YES NO YES NIV NO NO

HAMM2 YES YES NO YES YES YES YES NO YES YES YES YES YES YES YES YES YES NO YES NO YES NO YES YES YES YES YES YES YES NO YES NO YES YES YES YES

HAMM3 YES YES NO YES YES YES YES YES YES NO YES YES YES YES YES YES YES NO NO YES YES NO YES YES YES YES NO YES YES NO NO YES YES YES YES YES

Table 13.6 Synthesis of issued verdicts for the 36 loops YES NO UNC Others

BIC 23 13 0 0

CORR 15 8 13 0

HIST 19 13 4 0

RELAY 26 6 3 1

CURVE 18 8 10 0

AREA 11 15 0 10

HAMM2 HAMM3 29 28 7 8 0 0 0 0

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13.4.3 Efficiency of the Techniques, Problems and Countermeasures In this section, the different techniques considered in the comparative study will be assessed in terms of efficiency in detecting stiction, possible problems that may occur when applying them to real industrial data, and the necessary preprocessing steps and countermeasures to avoid false detections. All this will be underlined by selected examples from the collected industrial loops (Table A.1). A good stiction-detection method should: • Only use signals usually measured in industrial control loops, i.e. PV, SP and OP trends. • Be robust to bad effects, such as noise, non-stationary trends, disturbances and multiple oscillations. • Run in a completely automatic manner, without user interaction: ideally, the user should only be needed to inspect and check the results. • Not have any uncertainty regions. If this is the case they should be well defined and preferably as small as possible. • Have some index that gives a measure of how much stiction is present in the loop. 13.4.3.1 MV–OP-shape-based Methods (Kano et al.; Yamashita) These techniques are very easy to implement. The main practical problem is that they require measurement of the valve position, which is only available for smart valves. If the valve position is not available, the flow rate in a cascade loop has to be used. The difficulty associated with this is (i) noise, and (ii) the flow loop has dynamics that can distort the shape of the stiction pattern. The methods are therefore straightforward for flow control loops, of which there are a vast number in the chemical process industries. 35% of the analysed loops are of the FC type (Table 13.2). From the results in Table 13.11 it can be concluded that Method C is the most reliable technique in this group. This method is, however, sensitive to noise as the derivative is used for finding the symbolic representations. As an example, take the FC loop CHEM 11: if Method C is applied to the (moderately) noisy data shown in Fig. 11.11b, one gets an index value ρC = 0.18, falsely indicating no stiction. After filtering the data, e.g. with a low-pass filter with cut-off frequency ωc = 0.5, stiction is successfully detected. Note that also the sampling period may affect the performance of the method: lowering the sampling period makes the method inefficient, as the calculation of the differentials will be too dominated by the noise. Setting the sampling period very high is also disadvantageous, so there must be an optimum to be selected. Yamashita’s method has been deeply examined by Manum [76] and Manum et al. [77]. This technique has then been extended by Scali and Ghelardoni [105] to include stiction patterns not considered in the original work.


307

Note that if the valve position is not available, the flow rate in a cascade loop has to be used. The difficulties associated with this is (i) noise, and (ii) the flow loop has dynamics that can distort the shape of the stiction pattern. 13.4.3.2 Bicoherence and Ellipse-fitting Method (Choudhury et al.) The bicoherence technique introduced for non-linearity detection has to be combined with other methods for stiction detection, typically with ellipse fitting. The technique is particularly sensitive to non-stationary slowly varying trends, thus needs frequency filtering. Also, the portion of the signal used for bicoherence calculation should not have any step change or abrupt change, as can be seen in the data of the PC loop CHEM 17 in Fig. 13.9. Moreover, long data sets, i.e. preferably 4096 samples, but not less than 1024, are required. For this reason, many loops could not be analysed, as can be seen in Table 13.12. 2

1

1.5

0

1

−1

0.5

PV

2

−2

0 PV

−3 2

−0.5 −1

1

−1.5

OP

0 −1

−2

−2

−2.5

−3

0

500

Samples

(a)

1000

1500

−3 −3

−2

−1

OP

0

1

2

(b)

Fig. 13.9 Data for PC loop CHEM 17 (step change): a) PV and OP trends; b) PV–OP plot

The necessity of filtering to remove non-stationary trends is illustrated on loop CHEM 30. The row data shown in Fig. 13.10 are from an industrial flow control loop in a refinery. Without filtering, it is clear that no distinct ellipse can be fitted to the PV–OP plot. On the contrary, if a Wiener filter with the boundaries [0.01, 0.25] is applied to the data, the PV–OP mapping has a clear and distinct elliptical pattern, as shown in Fig. 13.11. It is also observed that the filtering has removed the slowly varying mean-shift and high-frequency noise from the PV and OP signal. Equation B.11 yields an estimate of the apparent stiction S = 0.43%. Concerning ellipse fitting, care has to be taken, as the optimisation routine will always try to fit an ellipse to the data, even in situations where human eyes clearly detect no ellipse. An example is shown Fig. 13.12 to which an ellipse would be fit-

308


ted, although no stiction is present in the loop. Thus, it is useful to introduce a measure of fitness for the ellipse. This draws two confidence-limit ellipses around the fitted ellipse and checks how many data points are within the limits; see Fig. 13.13. The percentage of theses data points is defined as the ellipse fitness. If the fitness is below a specified threshold, say 60%, the fitted ellipse should be rejected and thus no stiction is concluded. 400

500

300 400

PV

200 100

300

0 −100

200 PV

−200 1

100

0.5 OP

0

0

−0.5 4000

−100

4100

4200

4300

4400

4500 4600 Samples

4700

4800

4900

5000

−200 −1

−0.5

(a)

0 OP

0.5

1

(b)

Fig. 13.10 Subset of data for FC loop CHEM 30 (non-stationary trend): a) PV and OP trends; b) PV–OP plot

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Fig. 13.11 Subset of data for FC loop CHEM 30, when the data are preprocessed using a Wiener filter [0.01, 0.25]: a) PV and OP trends; b) PV–OP plot

13 Comparative Study of Valve-stiction-detection Methods 3

309

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500 600 Samples

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Fig. 13.12 Data for AC loop CHEM 13 (no stiction): a) PV and OP trends; b) PV–OP plot

Moreover, it is recommended to select an appropriate segment of the data, where the oscillation is as regular as possible, i.e. with a maximum oscillation index for filtered OP. A completely automated procedure of the bicoherence and ellipse-fitting technique has been proposed by Choudhury et al. [17, 21].

1 0.8 0.6

Outer ellipse

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Inner ellipse

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Fig. 13.13 Check of the validity of fitted ellipse for loop POW 2

10

310


13.4.3.3 Surrogates Analysis (Thornhill) The technique based on surrogates analysis was originally introduced to detect nonlinearities in control loops, but can principally be combined with ellipse fitting to detect stiction. Surrogate analysis requires a minimum number of 12 cycles for reliable assessment of data, and is very sensitive to data non-end-matching. An example showing the extreme need for data end-matching is illustrated for the LC loop PAP 13 (Fig. 13.14) in Fig. 13.15: when the data are not end-matched, then the resulting NPI value of 1.6 signals non-linearity (upper panel), which is, however, a wrong indication. One gets the right decision when end-matching (here based on zero-crossings) is applied (lower panel). 1.5

1.5

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400

500 600 Samples

700

800

900

1000

(a)

−1.5 −5

0 OP

5

(b)

Fig. 13.14 Data for LC loop PAP 13 (no non-linearity): a) PV and OP trends; b) PV–OP plot

13.4.3.4 Cross-correlation Method (Horch) The stiction-detection method based on cross-correlation is simple and thus easy to use. Besides normal operating data, neither detailed process knowledge nor user interaction is needed. The drawbacks/pitfalls are that (i) the method cannot be applied to integrating systems, and (ii) the phase shift depends on controller tuning: the phase lag is π for an aggressive controller when the loop cycles due to controller output saturation; however, when stiction is present and the controller output is not saturated, the phase lag can lie between π/2 and π for a PI controller. Examples, which confirm these problems, have been given by Yamashita [134] and He et al. [44]. It is known that the correlation-based method does not work for loops with Pdominant control, i.e. yields uncertain results [48]. Typical examples for this are the loops CHEM 5 (Fig. 11.14b) and CHEM 25 (Fig. 11.15b). Figure 13.16 shows


311

the cross-correlation function for the latter loop. It can be observed that the crosscorrelation function is neither even nor odd. Non−end−matched data

2

NPI = 1.6

SP−PV

1 0 −1 −2

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100

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250

End−matched data

2

300

NPI = 0.4

SP−PV

1 0 −1 −2

0

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100

150

Samples

200

250

300

Fig. 13.15 Illustration of the effect of data end-matching on the non-linearity index for the industrial control loop (PAP 13), which does not have any non-linearity

0.04

Cross−correlation function

0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 −30

−20

−10

0 Lags

Fig. 13.16 Cross-correlation function for PC loop CHEM 25

10

20

30

312


13.4.3.5 Histogram-based Method (Horch) In contrast to Horch’s first method based on cross-correlation, the histogram-based technique can be applied to both self-regulating and integrating processes. However, it is more complicated than the cross-correlation-based method. Its main weakness is the sensitivity to noise since the first or second derivative of PV is used for detecting stiction. Therefore, filtering is an integral component of the technique (Sect. 6.7.1). This method is based on the assumptions that – for integrating processes – PV is triangular for the stiction case (and sinusoidal when stiction is absent). Obviously, the technique will not work when this assumption is not satisfied. It has been observed that industrial loops do not always show this pattern; see Table 1.4. An example is the LC loop POW4; see Fig. 13.17. 0.6 0.5 0.4 PV

0 0.2

−0.5 0 PV

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4 −0.4

OP

2 0

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300 400 Samples

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600

700

−1 −6

(a)

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−2

0 OP

2

4

6

(b)

Fig. 13.17 Data for LC loop POW 4 (stiction): a) PV and OP trends; b) PV–OP plot

Note that for both loops CHEM 5 and CHEM 25, for which the correlation-based method gives no decision, the histogram-based technique correctly detects stiction. For loop CHEM 25, the MSE for the stiction case (Gaussian distribution) is 2.8 compared to 4.0 the no-stiction case (camel distribution); see Fig. 13.18. Hence, the method correctly detects the presence of stiction. 13.4.3.6 Area-peak Method (Salsbury and Singhal) This method is also simple to implement and computationally undemanding. However, the main drawback of the technique is that it requires data that allow good estimates of the area-peak and zero-crossing locations in the control error trend. These features are usually very sensitive to noise. Indeed, the results for this method in Table 13.15 show that in about the half (46%) of cases considered no suitable features


313

could be extracted from the data for the stiction analysis. For instance, although the signal OP of loop CHEM 5 clearly indicates stiction because it is triangular (Fig. 11.14b), but PV and thus the control error does not show remarkable and regular peaks, thus wrongly giving a very low probability of stiction.

0.15

1 0.5

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d2y(t)/dt2

y = PV

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1

Fig. 13.18 Stiction detection using the histogram-based method for PC loop CHEM 25

13.4.3.7 Curve-fitting Method (He et al.) This method is well suited for consistently detecting stiction for both self-regulating and integrating processes. Although He’s method is claimed to be robust in handling noise owing to the formulation as least-squares problems, one sometimes has to filter the data before analysing stiction. An example situation is when analysing the data from the PC loop CHEM 39, shown in Fig. 13.19. It can be seen that the signals are heavily corrupted by noise. If the method is applied to the raw data, the stiction index is computed to be ηstic = 0.54, so that no decision can be taken. If Wiener filtering is applied to remove excess noise (Fig. 13.20), the index is now ηstic = 0.67, indicating the presence of stiction. Note that He’s technique needs accurate determination of the signal zero-crossings, a non-trivial task when the data are affected by noise. This topic has also been treated by Salsbury and Singhal in Sect. 9.2.3.

314

M. Jelali and C. Scali 0.02

0.01

0.015

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PV

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Fig. 13.19 Data for PC loop CHEM 39 (noisy data): a) PV and OP trends; b) PV–OP plot 0.02

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(b)

Fig. 13.20 Filtered data for PC loop CHEM 39: a) PV and OP trends; b) PV–OP plot

13.4.3.8 Relay Method (Rossi and Scali) The relay method is similar to the curve-fitting technique of He et al., with the difference that it additionally fits the data to a pattern generated by a first-orderplus-time-delay system under relay control. Therefore, all merits and drawbacks of He’s method also apply for Scali’s technique. Moreover, the relay fitting procedure is complex and thus time consuming. However, this option makes the method more powerful than the method of He et al., as it can detect stiction in situations where He’s technique fails. Note that whereas He’s method looks at the OP signal or the PV signal, depending on the process type, Scali’s technique considers the control error signal SP − PV, irrespective of the process type. This means that Scali’s method


315

does not consider the effect of the first integrating component after the valve, which is the key of He’s approach. To exemplify the comparison, take the data from the FC loop CHEM 29 illustrated in Fig. 13.21, which is cascaded within a LC loop in a chemical complex. For this loop, He’s method indicates no stiction because the OP signal is more sinusoidal than triangular. In contrast, Scali’s technique correctly detects stiction since the control error signal has a more triangular wave (not shown in the figure). There are also other loops for which He’s method successfully detects stiction, but Scali’s yields uncertain decision. This situation occurs when analysing the data from FC loop CHEM 6 shown in Fig. 11.14c. 13.4.3.9 Hammerstein-model-based Methods (Jelali, Lee et al., Karra and Karim) Three methods for stiction quantification based on Hammerstein identification have been independently proposed by Jelali, Lee et al., and Karra and Karim. These techniques have the advantage that they not only detect stiction, but also quantify its extent, i.e. deliver estimates of the stiction parameters. This information is very valuable and can be utilised for short listing the control valves in industrial plants typically containing a large number of control loops. Also these methods can be used to diagnose multiple faults occurring simultaneously in control loops, as presented in Sects. 12.3.2 and 12.5 for Karra’s approach. The price to be paid is the higher complexity and thus higher computation burden required.

100 100

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Fig. 13.21 Data for FC loop CHEM 29 (varying SP; stiction): a) PV and OP trends; b) PV–OP plot

316


The three methods have in common: • Identification of a Hammerstein model from closed-loop data (SP, OP, PV). • Use of a two-parameter model for stiction modelling, e.g. Kano’s stiction model. However, the methods differ in • Structure of the Linear Model. Jelali suggests a low-order (ARMAX/BJ) model, the orders and delays (when not known) of which are automatically estimated. Lee et al. approximate the process by a fixed first- or second-order-plustime-delay model, depending on the contour map computed. Karra and Karim introduced an extended ARMAX model for the linear part that contains an additive non-stationary disturbance term. • Parameter-estimation Technique. In Jelali’s method, a global optimisation algorithm, e.g. pattern search or genetic algorithms are suggested to estimate the stiction parameters. Instead, Lee et al. use a multi-start adaptive random search algorithm. Karra and Karim adopted a grid-based optimisation approach. • Strategy of Model Identification. Jelali proposes a two-stage identification, i.e. global identification of the stiction model, subordinated by a LS or PEM identification of the linear model part. Similar strategies, though not the same, are used in Lee’s and Karra’s approaches. A comparison of the methods by Jelali and Lee et al. has already been given in Sect. 11.7.2 on some industrial loops. In general, the methods deliver different values for the stiction-parameter estimates. However, Jelali’s and Lee’s methods produce quite similar results for many loops. Karra’s results mostly disagree with those produced by the other variants. As a first example, consider the data from the LC loop CHEM 4 shown in Fig. 11.14a. It is known that this loop suffers from tight controller tuning, but does not have any stiction problem. Lee’s method correctly detects no stiction (J = 0.03, S = 0.03). On the contrary, Karra’s technique indicates the presence of significant stiction (J = 2.5, S = 3.5), which is a false alert. For the FC loop CHEM 24 (Fig. 11.15a), the techniques used by Jelali and Lee yield similar values for the stiction estimates. In contrast, Karra’s method leads to completely different values. A look at the PV–OP plot (which coincides with the PV–MV plot in this FC case) confirms that the former estimates should be the more accurate ones. For the next example already considered in Sect. 10.6.4 (Fig. 10.11), all three methods produce different estimates of J and S. Since the valve-positioner data were made available for this loop in [18]/Fig. 5, giving S ≈ 11.25, it can be concluded that Jelali’s method should deliver the more accurate values. Note that Karra’s method also yields a good estimate of J. The last example considered in this section is the TC loop MIN 1; its data are illustrated in Sect. 11.7.2 (Fig. 11.18a) and Sect. 13.3 (Fig. 13.6). For this loop, the results of all three methods are in good agreement. From the MV–OP plot which was known for the loop in [15]/Fig. 6, it is observed that the produced parameter estimates are also in good agreement with the real values.


317

13.4.4 Comparison on 20 Loops with Known Problems From Table 13.5, 20 loops are extracted for which: (i) all techniques are applicable, and (ii) the real root-causes for malfunction are known. The results, together with the right stiction indication for these loops, are given in Table 13.7. It can been concluded that BIC and HAMM2 are the best performing methods; see Table 13.8. Table 13.7 Issued verdicts for the 20 loops with known problems CHEM 1 CHEM 2 CHEM 3 CHEM 6 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 16 CHEM 23 CHEM 24 CHEM 29 CHEM 32 PAP 2 PAP 4 PAP 5 PAP 7 PAP 9 MIN 1

Stiction? YES YES NO YES YES YES YES NO NO NO YES YES YES YES YES NO YES NO NO YES

BIC YES YES NO YES YES YES YES NO YES YES YES YES YES YES YES YES YES NO YES YES

CORR YES YES YES UNC YES UNC YES NO YES YES YES YES YES UNC YES NO YES NO NO UNC

HIST YES YES NO YES YES NO YES NO YES YES YES YES UNC NO NO NO YES YES NO YES

RELAY YES YES YES UNC YES YES YES NO YES YES YES YES YES YES YES YES YES YES NO YES

CURVE YES YES YES YES YES NO YES NO YES YES YES YES NO NO UNC UNC YES UNC NO YES

AREA YES NIV NO NO YES YES YES NO NIV NIV YES NO NO NO YES YES NO YES NO YES

HAMM2 YES YES NO YES YES YES YES NO YES YES YES YES YES YES YES YES NO YES NO YES

HAMM3 YES YES NO YES YES YES YES YES YES NO YES YES YES YES YES YES NO NO YES YES

Table 13.8 Synthesis of issued verdicts for the 20 loops (NRI: number of right indications) NRI

BIC 16

CORR 13

HIST 13

RELAY 13

CURVE AREA 12 10

HAMM2 HAMM3 15 14

13.4.5 Selected Examples In this section, some of the considered loops are illustrated in more detail to further exemplify the results obtained for the different stiction-detection techniques: • Evident Presence of Stiction. The FC loop CHEM 10 (Fig. 11.11a) is a typical loop with stiction: the OP signal shows the ideal triangular pattern; the PV trend is of the relay type; the PV–OP plot has a clearly elliptical shape. Indeed, an ellipse can be fitted to the PV–OP plot, giving an apparent stiction S ≈ 1.78

318

•

•

•

•

•

1


(Fig. B.1). Another example with stiction evidently seen in the measured signals is the FC loop CHEM 23 (Fig. 13.22): one should particularly observe the triangular shape of the OP trend and the rectangular shape of the PV signal, to be considered coincident with the MV signal. Look also at the PV–OP(MV) plot, which has a pattern very typical for loops with a sticky valve; see Fig. 1.5. Evident Absence of Stiction. The data from the TC loop PAP 9 in Fig. 13.23 show typical patterns without stiction, i.e. sinusoidal OP trend. Also the LC loop CHEM 4 (Fig. 11.14a) is a typical loop without stiction, i.e. has a sinusoidal PV trend; remember the classification in Table 1.4. Loops with Malfunction Difficult to Be Detected. The FC loop CHEM 33 (Fig. 13.5) is an example for which it is hard to automatically detect oscillation and its possible source(s). Shape-based methods (CURVE, RELAY) are uncertain. Horch’s techniques (CORR, HIST) give contradictory conclusions. When relying on Kano’s methods (A B, C), there should be no stiction in the loop. On the other hand, the non-linearity detection methods signals that there is a kind of non-linearity in the loop. Moreover, the model-based methods indicate the presence of stiction and disturbances simultaneously. This could explain why the shape-based methods give uncertain decisions, i.e. disturbances modify the shape of stiction-induced oscillations and make them more sinusoidal. All in all, it can be believed that this loop suffers from stiction or some other kind of non-linearity (e.g. valve oversizing because of the rapid vertical transitions and the tendency of the OP to be at the extremes), but additional tests are necessary to get a final verdict. Loop with Saturation. For the loop POW 5 (Fig. 13.2), which suffers from saturation, the histogram-based and the relay methods give false alerts. On the contrary, the bicoherence method and the model-based techniques correctly detect that no stiction is present. The bicoherence test gives NGI = 0.28 and NLI = 0.82, indicating non-linearity in the loop. The corresponding bicoherence plot in Fig. 13.24 also shows clear peaks, indicating significant non-linearity. Moreover, observe the specific patterns in the PV–OP plot (Fig. 13.2) that shows a vertical straight line with some random cycles. This is a signature of valve saturation; in this case the valve saturates at one end (fully open condition). This pattern may be due to the use of an anti-wind-up algorithm in the integral action of a PI(D) controller because OP is then kept constant while PV may change [21]. Loop with Sensor Fault. The data sets for loops CHEM 13–17 represent a hard test for the stiction-detection techniques because there is no stiction, but a sensor fault that has shapes very similar to those typically found in loop signals with valve stiction. Table 13.9 contains the results for these loops. In particular, for loop CHEM14, the stiction-detection techniques give false-positive results. Loop with Stiction and Aggressive Tuning. For the loop BAS 6 (Fig. 13.8), all model-free methods1 fail to detect stiction because aggressive tuning modifies the shape of stiction-induced oscillations and makes them more sinusoidal. Model-free methods are then not able to distinguish stiction oscillations from agExcept the bicoherence method that was not applicable due to the short length of data.


319

gressive control. On the contrary, the model-based techniques successfully detect the presence of significant stiction. • Loop with Deadzone and Aggressive Tuning. For the loop PAP 4 (Fig. 11.19b), most of the methods wrongly detect the presence of stiction. Only Horch’s techniques correctly indicate no stiction. Table 13.9 Issued verdicts for the loops CHEM 13–17 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17

BIC NO YES NA YES NA

CORR NO YES YES YES UNC

HIST NO YES YES YES NO

RELAY NO YES NA-MAN YES NO

CURVE NO YES YES YES NO

0.5

AREA NO NIV NIV NIV NIV

HAMM2 NO YES YES YES YES

HAMM3 YES YES YES NO YES

0.5

PV

0.4 0.3

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0 −0.1

10 OP

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1500

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5

10

15

(b)

Fig. 13.22 Data for FC loop CHEM 23 (stiction): a) PV and OP trends; b) PV–OP plot

13.4.6 Graphical User Interface ®

A MATLAB GUI (Fig. 13.25) has been developed by the principal editor of this book, where all techniques for oscillation and stiction detection are implemented, including some preprocessing methods. The code for each method has been provided by the corresponding contributor. The GUI gives users the possibility to compare the method on a data set side-by-side, in a fast and user-friendly way. The upper part of Fig. 13.25 contains functions for data preprocessing (de-trending, filtering, decimation, etc.) and for generating plots (time trends, PV–OP shape, power spectrum, etc.). In the middle part of the figure, methods for oscillation detection can be

320

M. Jelali and C. Scali 30

25

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Fig. 13.23 Data for TC loop PAP 9 (no stiction): a) PV and OP trends; b) PV–OP plot Squared bicoherence

1

Max. bic2(0.015625, 0.023438) = 0.87329

0.8

bic

2

0.6 0.4 0.2 0 0.4 03 0.2 0.1

f2

0

0

0.1

0.2

f1

Fig. 13.24 Squared bicoherence plot for LC loop POW 5 (saturation)

03

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3


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selected and applied to the data. The lower part of the figure provides the user with techniques for stiction detection.

Fig. 13.25 Stiction GUI with the results for loop CHEM 23 (flow control)

13.5 Suggestions A possible “statistic-based” strategy is to apply all methods to each data set and compute the number of YES decisions. If the majority of the methods, say 75%, indicate stiction, then YES is taken as the final verdict for that loop. The application of this strategy to the loops in Table 13.7 yields 12 (60%) right indications. However, such a “blind” strategy does not consider the merits and weaknesses of the techniques. A more systematic procedure to diagnose and solve stiction problems in control loops can be formulated as follows: Phase I.

Apply an appropriate oscillation-detection technique to make sure that the loop is oscillating, and to obtain the (main) oscillation frequency. Methods for oscillation detection have been presented and compared in Chap. 4. If no oscillation is detected, the analysis should be stopped. Phase II. If a control loop is detected to be oscillating, methods that enable one to discriminate between oscillation due to valve non-linearities, aggressive controller tuning or disturbances affecting the loop have to be applied. As stiction is a strong non-linearity, a non-linearity test, i.e. bicoherence technique and/or surrogates analysis, should be performed at a first

322


stage. If the non-linearity test indicates that there is no non-linearity in the loop, it makes no sense to apply any further stiction-detection method. Phase III. Apply one or more signal-based (model-free) techniques, e.g. AREA, CURVE, RELAY, CORR or HIST, to detect the presence of stiction. For flow loops, a method based on MV–OP shape can also be used. Phase IV. Apply a model-based detection method, i.e. based on Hammerstein identification, to confirm the presence of stiction and quantify its extent. This information is very useful in troubleshooting a large plant as loops can be ranked based on their extent of stiction. This phase is also necessary when no stiction is detected in Phase III because the assessed loop may have multiple faults, e.g. stiction and aggressive tuning simultaneously, which may not be correctly diagnosed by model-free methods. Phase V. Once stiction is detected in a control loop, and its extent is considered high, a valve-travel test is needed to confirm the presence and real extent of stiction. Note that confirming stiction by putting the loop in manual is not convenient and cost effective due to the risk of plant upset and production of more “off-spec” products. It should therefore be the last stage of any oscillation-diagnosis procedure. If stiction is confirmed, its negative effects cannot be totally eliminated without repairing the valve at the next plant shut-down. Remember that, for each stiction-detection method, some information about the loop and data preprocessing steps are needed to check the applicability and avoid misleading results. The most important checks and pretreatment steps can be summarised for each method as follows: • MV–OP-shape-based Methods (Kano et al.; Yamashita). Availability of valve position (trivial for flow control loops and loops with smart valves); filtering of excessive noise. • Bicoherence and Ellipse-fitting Method (Choudhury et al.). Availability of at least 1024 data points; steady-state routine operating data; removing of outliers, sharp abrupt changes or step changes, transients and excessive noise. • Surrogates Analysis (Thornhill). Availability of 12 cycles of signal to be assessed; steady-state routine operating data; removal of outliers, sharp abrupt changes or step changes and transients; appropriate end-matching. • Cross-correlation Method (Horch). Applicability only to self-regulating processes; high-pass filtering to remove slow variations (linear trend). • Histogram-based Method (Horch). Filtering and differentiation are integral components of the method. • Area-peak Method (Salsbury and Singhal). Removing of excessive noise; determination of zero-crossings. • Curve-fitting Method (He et al.). Removing of excessive noise; determination of zero-crossings.


323

• Relay Method (Rossi and Scali). Information about range of PV and OP variables, controller parameters and loop configuration; minimum acquisition time; data showing dominant frequency. • Hammerstein-model-based Methods (Jelali, Lee et al., Karra and Karim). Data pretreatment is usually not required.

13.6 Summary and Conclusions An exhaustive comparative study of methods for detecting stiction in control valves has been presented in this chapter. The study involved 93 different data sets from different process industries, including chemicals, pulp and paper mills, commercial building, and metal processing. The different techniques considered in the study have been assessed in terms of efficiency in detecting stiction and possible problems that may occur when applying them to real industrial data. This has been illustrated by various selected data examples from industrial fields. It has been shown that each method has its strengths and weaknesses. Particular attention should be paid to the assumptions to be satisfied when applying any technique. Out of the 93 loops considered, only 36 were found to which all methods are applicable. The quality of data has been revealed to be a very crucial issue: most of the methods require some careful pretreatment of data to avoid false detections. The cross-correlation technique is simple and easy to implement, but may have problems with phase shift induced by controller tuning and is limited to selfregulating processes. The histogram-based method is applicable to both self-regulating and integrating processes, but it is more involved than the correlation-based method and is sensitive to noise in the signals. Curve fitting, which exists in different versions, is a powerful technique to detect stiction. These methods rely on certain signal patterns characterising stiction, which, however, may be modified if other malfunctions, e.g. aggressive tuning, are simultaneously present in the loop. Non-linearity detection (bicoherence or surrogate analysis) followed by ellipse fitting has been proven to be one of the most efficient methods, but the complexity of this technique and the requirement of a large data set (more than 1024 samples at least) free from sharp or abrupt changes are clear weaknesses of the method. Methods for stiction diagnosis based on Hammerstein identification, which exist in different versions, are superior, even though the complexity and thus the computational load required is higher. These techniques have the advantage that they not only detect stiction, but also quantify its extent, i.e. deliver estimates of the stiction parameters. This information is very valuable and can be utilised for short listing the control valves in industrial plants typically containing a large number of control loops. Also, these methods can be used to diagnose multiple faults occurring simultaneously in control loops.

324


It has also been demonstrated that some data sets (sensor fault in CHEM 17) may exist for which all stiction-detection techniques give false-positive results. This is because some signals have shapes very similar to those typically found in loop signals with valve stiction. This reveals the necessity to carry out an invasive test on the valves to confirm the presence of stiction at the last stage. This might seem a strong limitation of the techniques proposed for a completely automatic analysis of plant data making use of data recorded under routine operations. In a real application, the loops indicated as suspected of being affected by stiction are a subset (hopefully small) of the total number of loops under assessment and then the techniques help to focus on this subset. Also, the recommended methods will give a much higher chance that the choice of valve for the invasive test will be right first time. A systematic oscillation-diagnosis procedure has been proposed, combining some oscillation and non-linearity detection techniques, as well as additional tests and methods for checking, elimination or compensation of valve stiction.

13.7 Appendix: Tables of Results of the Comparative Study Table 13.10 Comparison of verdicts from different techniques BAS 1 BAS 2 BAS 3 BAS 4 BAS 5 BAS 6 BAS 7 BAS 8 CHEM 1 CHEM 2 CHEM 3 CHEM 4 CHEM 5 CHEM 6 CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17

BIC NA NLQ NLQ NL-YES NIV NA NA NA YES YES NLQ NA NA YES MAN MAN MAN YES YES YES NL-NO YES NA YES NA

CORR NA NA UNC YES NIV NO UNC NA YES YES YES NIV UNC UNC NA NA UNC YES UNC YES NO YES YES YES UNC

HIST NA NA YES YES YES NO YES NA YES YES NO NO YES YES NA NA YES YES NO YES NO YES YES YES NO

RELAY NA NA YES YES YES NO YES NA YES YES YES NO YES UNC NA NA NA YES YES YES NO YES MAN YES NO

CURVE NA NA YES YES YES NO YES NA YES YES YES NO YES YES YES UNC NA YES NO YES NO YES YES YES NO

AREA NO NO NO NO NIV NO YES NO YES NIV NO NIV NIV NO NIV NIV NIV YES YES YES NO NIV NIV NIV NIV

HAMM2 YES YES NA NA NA YES YES NA YES YES NO NO YES YES YES YES NO YES YES YES NO YES YES YES YES

HAMM3 YES YES YES YES NIV YES YES YES YES YES NO YES YES YES YES YES YES YES YES YES YES YES YES NO YES


325

Table 13.10 (Continued) CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22 CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39 CHEM 40 CHEM 41 CHEM 42 CHEM 43 CHEM 44 CHEM 45 CHEM 46 CHEM 47 CHEM 48 CHEM 49 CHEM 50 CHEM 51 CHEM 52 CHEM 53 CHEM 54 CHEM 55 CHEM 56 CHEM 57 CHEM 58 CHEM 59 CHEM 60 CHEM 61 CHEM 62 CHEM 63 PAP 1 PAP 2 PAP 3

BIC YES NA NA NA NA YES YES NA YES YES NO YES YES YES YES YES NA NA NA NL-NO NA NA NO NO-NL NO-NL NO NA NO NO NO NO-NL YES YES NO-NL NO-NL NO-NL YES YES NO NO NO NO NO NO-NL NO NO NA YES YES

CORR UNC YES YES NO YES YES YES UNC XX XX YES YES UNC NO UNC NO UNC UNC XX XX UNC UNC UNC NO NO NO NA NA NA NA NA UNC NO UNC NA NA UNC YES NA UNC NA NA YES NA UNC NO UNC YES XX

HIST UNC YES UNC UNC NO YES YES YES YES UNC NO UNC YES NO NO YES YES UNC YES YES YES YES YES NO NO NO NA NA NA NA NA NO YES YES NA NA UNC UNC NA YES NA NA YES NA YES UNC YES NO YES

RELAY YES YES YES YES YES YES YES YES YES UNC YES YES YES NO YES UNC YES YES YES YES YES YES NIV YES NO YES NIV UNC YES YES YES NO UNC YES YES YES YES YES YES NA YES YES YES NIV NO NO YES YES YES

CURVE YES YES YES UNC NO YES YES YES YES YES NO NO YES NA NO UNC YES NA YES YES YES YES YES NA NO UNC NA NA NA NA NA NO UNC UNC NA NA YES YES NA NA NA NA YES NA YES NA UNC UNC YES

AREA NO NIV NIV NIV YES YES NO NO YES NO YES NO YES NIV NO NIV NIV YES YES NIV YES NO NIV NO NO NIV YES NIV NIV NIV NIV NO NIV NIV NIV NIV NO YES NIV NIV NIV NIV NO NIV NIV NIV YES YES YES

HAMM2 YES YES NO YES YES YES YES YES YES YES YES YES NA YES YES YES YES NO YES NO YES YES NO NO YES NO YES YES YES YES YES YES NO YES YES YES YES YES NO NA YES NO YES YES YES YES YES YES YES

HAMM3 YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES YES NO NO NO YES YES YES YES YES YES YES NO YES YES YES YES YES NO YES YES NO YES NO NO NO YES YES NO

326


Table 13.10 (Continued) PAP 4 PAP 5 PAP 6 PAP 7 PAP 8 PAP 9 PAP 10 PAP 11 PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

BIC YES YES NA NO-NO NL-NO YES YES YES YES YES YES YES NL-NO YES NL-NO YES NO NL-NO NL-NO

CORR NO YES XX NO XX NO XX XX XX XX XX XX XX XX XX UNC UNC UNC UNC

HIST NO YES NO YES? YES NO YES UNC YES NO YES YES YES NO YES YES NO? YES NO

RELAY YES YES NO YES UNC NO YES NIV NIV NO NO YES YES YES YES YES NO YES YES

CURVE UNC YES NO UNC YES NO YES YES YES NO YES UNC UNC UNC NA YES UNC UNC UNC

AREA YES NO NO YES NIV NO NIV NO NIV NO NO NO NIV NO NIV YES NIV NO NO

HAMM2 YES NO YES YES YES NO YES YES YES YES YES YES YES YES NA YES YES YES YES

HAMM3 YES NO NO NO YES YES YES YES YES YES NO YES YES YES NO YES YES YES YES

Table 13.11 Results of stiction detection using the methods of Kano et al.; only FC loops are analysed Loop name CHEM 1 CHEM 2 CHEM 3 CHEM 4 CHEM 5 CHEM 6 CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17 CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22 CHEM 23

Method A

Method B

Method C

SIA 0.000 0.109

δA 0.00 3.77

Stiction? SIB no 0.426 no 0.938

δB 0.66 3.43

Stiction? SIC no 0.136 yes 0.286

δC 0.00 3.42

Stiction? no yes

0.000 0.000

0.00 0.00

no no

0.003 0.122

0.00 0.13

no no

0.061 0.264

0.00 0.13

no yes

0.000 0.000

0.00 0.00

no no

0.900 0.701

1.13 1.55

yes yes

0.304 0.267

0.59 0.59

yes yes

0.101

0.81

no

0.835

1.89

yes

0.503

1.00

yes

0.050 0.389 0.096 0.212 0.828 0.268

0.82 1.34 0.97 0.86 9.67 1.07

no yes no no yes yes

0.960 0.804 0.163 0.596 0.804 0.596

2.57 3.43 0.17 6.31 7.76 19.71

yes yes no no yes no

0.333 0.391 0.179 0.000 0.628 0.167

1.69 4.90 0.00 0.00 4.00 0.00

yes yes no no yes no


327

Table 13.11 (Continued) Loop name CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39 CHEM 40 CHEM 41 CHEM 42 CHEM 43 CHEM 44 CHEM 45 CHEM 46 CHEM 47 CHEM 48 CHEM 49 CHEM 50 CHEM 51 CHEM 52 CHEM 53 CHEM 54 CHEM 55 CHEM 56 CHEM 57 CHEM 58 CHEM 59 CHEM 60 CHEM 61 CHEM 62 CHEM 63 PAP 1 PAP 2 PAP 3 PAP 4 PAP 5 PAP 6 PAP 7 PAP 8 PAP 9

Method A

Method B

Method C

SIA 0.821

δA Stiction? SIB 13.60 yes 0.867

δB Stiction? SIC 21.01 yes 0.193

δC 0.00

Stiction? no

0.460 0.000 0.007 0.523 0.190 0.000 0.277

5.66 0.00 0.59 2.04 0.87 0.00 0.81

yes no no yes no no yes

0.944 0.414 0.003 0.963 0.004 0.002 0.769

6.83 0.50 0.00 14.62 0.00 0.00 1.36

yes no no yes no no yes

0.432 0.043 0.053 0.486 0.000 0.008 0.121

4.62 0.00 0.00 8.26 0.00 0.00 0.00

yes no no yes no no no

0.000 0.027

0.00 0.69

no no

0.002 0.007

0.00 0.00

no no

0.019 0.081

0.00 0.00

no no

0.000 0.000 0.012 0.035 0.112 0.012 0.155 0.000 0.006 0.000 0.000 0.012 0.049 0.074 0.000 0.063 0.106 0.092 0.112 0.043 0.022 0.015 0.184

0.00 0.00 0.58 1.10 0.81 1.58 0.83 0.00 0.54 0.00 0.00 0.61 0.74 0.96 0.00 0.90 0.95 0.88 1.19 0.73 2.06 0.67 0.92

no no no no no no no no no no no no no no no no no no no no no no no

0.000 0.003 0.004 0.015 0.446 0.443 0.431 0.010 0.003 0.005 0.005 0.006 0.320 0.531 0.040 0.324 0.392 0.409 0.002 0.422 0.002 0.002 0.467

0.00 0.00 0.00 0.02 0.55 4.96 2.11 0.00 0.00 0.00 0.00 0.00 2.45 0.88 1.69 1.44 0.61 2.42 0.00 0.72 0.00 0.00 2.75

no no no no no no no no no no no no no no no no no no no no no no no

0.000 0.000 0.009 0.005 0.000 0.006 0.128 0.000 0.085 0.011 0.005 0.023 0.022 0.121 0.179 0.043 0.202 0.399 0.000 0.195 0.310 0.339 0.404

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 1.00 2.03 1.24

no no no no no no no no no no no no no no no no no yes no no yes yes yes

0.000

0.00

no

0.021

0.06

no

0.048

0.00

no

328


Table 13.11 (Continued) Loop name PAP 10 PAP 11 PAP 12

Method A

Method B

Method C

SIA

δA

Stiction? SIB

δB

Stiction? SIC

δC

Stiction?

0.042 0.026

0.73 0.74

no no

0.00 4.43

no no

1.24 0.00

no no

0.002 0.438

0.093 0.240

Explanations and Comments: 1. The three methods applied assume that flow rate is measured. Therefore, only the results for flow control loops are shown here. 2. To detect stiction, the fixed thresholds 0.25 for Method A, 0.7 for Method B, and 0.25 Method C were used.

Table 13.12 Results of stiction-detection method based on bicoherence and ellipse fitting by Choudhury et al. Loop name BAS 1

Time span

BAS 2

35001–39096

BAS 3

30001–34096

BAS 4

77001–81096

Used Index parameters values

Nfft=128, nsamp=128, overl=50 Nfft=128, nsamp=128, overl=50 Nfft=256, nsamp=256, overl=50;

S

NGI=0.05, NLI=0.04

PV data are heavily quantised, nonlinearity analysis (NLA)∗ not possible, QF=1.0 Small amount of Non-linearity due to quantisation

NGI=0.09, NLI=0.08

Small amount of Non-linearity due to quantisation

NGI=0.28, 0.5 NLI=0.55

Non-linear loop, small amount of stiction

BAS 5

BAS 6 BAS 7 BAS 8 CHEM 1

1–1625

CHEM 2

1–1000

CHEM 3

All

Nfft=256, nsamp=256, overl=50; Nfft=256, nsamp=256, overl=60; Nfft=128, nsamp=128, overl=0

Comments

NGI=0.39, 1.0 NLI=0.60

Data are heavily quantised, something wrong, the OP signal is always zero, probably the valve hits saturation at fully closed position Data length too short for reliable NLA Data length too short for reliable NLA data are heavily quantised Stiction

NGI=0.4, 4.0 NLI=0.38

Stiction

NGI=0.21, NLI=0.19

Non-linearity is due to quantisation, PV data are heavily quantised, QF=1.


329

Table 13.12 (Continued) Loop name CHEM 4 CHEM 5 CHEM 6

Time span

1–1000

CHEM 7

CHEM 8

CHEM 9

CHEM 10 1–1000 CHEM 11 1–1000 CHEM 12 1–2000 CHEM 13 1–1500 CHEM 14 1–1500 CHEM 15 CHEM 16 501–1500 CHEM 17 CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22


S

Nfft=128, NGI=0.15, 0.2 nsamp=128, NLI=0.87 overl=50

Comments Data length too short for NLA Data length too short for NLA Stiction

This data set appears to come from manual control of the loop. The method only works for loops in AUTO mode. This data set appears to come from manual control of the loop. The method only works for loops in AUTO mode. This data set appears to come from manual control of the loop. The method only works for loops in AUTO mode. Nfft=256, NGI=0.57, 1.75 Stiction nsamp=256, NLI=0.5 overl=50 Nfft=256, NGI=0.63, 1.53 Stiction nsamp=256, NLI=0.92 overl=50 Nfft=256, NGI=0.55, 1.93 Stiction nsamp=256, NLI=0.56 overl=50 Nfft=128, NGI=0.31, Non-linear, but no stiction nsamp=128, NLI=0.61 overl=50 Nfft=128, NGI=0.36, 2.68 Stiction nsamp=128, NLI=0.64 overl=50 Sharp/step change in data (bad data for bicoherence). The method requires steady-state data. Nfft=128, NGI=0.31, 2.5 Stiction nsamp=128, NLI=0.42 overl=50 Sharp/step change in data (bad data for bicoherence). The method requires steady-state data Nfft=128, NGI=0.26, 2.0 Stiction nsamp=128, NLI=0.25 overl=50 Data length too short for NLA Data length too short for NLA Data length too short for NLA Data length too short for NLA

330


Table 13.12 (Continued) Loop Time span name CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 1–4096 CHEM 30 1–4096 CHEM 31 1–4096 CHEM 32 All CHEM 33 All CHEM 34 CHEM 35 CHEM 36 CHEM 37 All CHEM 38 CHEM 39 CHEM 40 101–1124 CHEM 41 All CHEM 42 201–1224 CHEM 43 All

Used parameters Nfft=256, nsamp=256, overl=50 Nfft=128, nsamp=128, overl=50

Index S values NGI=0.47, 31 NLI=0.75

Comments

NGI=0.28, 19 NLI=0.90

Stiction, NL valve characteristic may be dominant.

Nfft=256, nsamp=256, overl=50 Nfft=256, nsamp=256, overl=50

NGI=0.66, 1 NLI=0.77

Nfft=128, nsamp=128, overl=50 Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0 Nfft=256, nsamp=256, overl=50 Nfft=256, nsamp=256, overl=50

NGI=0.14, 7.1 NLI=0.27

NGI=0.66, 0.5 NLI=0.77

Data length too short for NLA Stiction Stiction Data length too short for NLA Stiction

NGI=0.31, 0.39 Stiction NLI=0.8 NGI=0.28, 0.7 NLI=0.87

Stiction

NGI=0.28, 18 NLI=0.29

Stiction

NGI=0.31, 1 NLI=0.52

Stiction

Nfft=256, NGI=0.28, nsamp=256, NLI=0.27 overl=50 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=102, overlap=0 Nfft=128, nsamp=92, overlap=0

Stiction, NL valve characteristic may be dominant.

NGI=0 NLI=0

Data length too short for NLA Data length too short for NLA Data length too short for NLA Non-linear, but no stiction Data length too short for NLA Data length too short for NLA Linear loop

NGI=0.21 NLI=0.21

Non-linear loop, no stiction, valve saturation

NGI=0.55 NLI=0.55

Non-linear loop, no stiction

NGI=0 NLI=0

Linear loop


331

Table 13.12 (Continued) Loop Time span name CHEM 44


CHEM 45 All

Nfft=128, nsamp=120, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=102, overlap=0 Nfft=128, nsamp=102, overlap=0 Nfft=128, nsamp=106, overlap=0 Nfft=128, nsamp=74, overlap=0 Nfft=128, nsamp=106, overlap=0 Nfft=128, nsamp=112, overlap=0 Nfft=128, nsamp=102, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=128, overlap=0

CHEM 46 All CHEM 47 All CHEM 48 All CHEM 49 All CHEM 50 All CHEM 51 All CHEM 52 All CHEM 53 All CHEM 54 All CHEM 55 All CHEM 56 All CHEM 57 All CHEM 58 All CHEM 59 All

S

NGI=0 NLI=0

Comments Oscillation period is too large, dowsampled data length too short for NLA. Linear loop

NGI=0 NLI=0

Linear loop

NGI=0 NLI=0

Linear loop

NGI=0.6 NLI=0.8

Non-linear loop, no stiction, ellipse cannot be fitted.

NGI=0.32 2 NLI=0.32

Stiction

NGI=0.63 0.13 Negligible stiction, non-linearity NLI=0.93 may be from somewhere else. NGI=0.2 NLI=0.2

Non-linear, no regular oscillation, no stiction

NGI=0.27 NLI=0.27


NGI=0.46 NLI=0.56

Non-linear, ellipse cannot be fitted, no stiction.

NGI=0.27 0.5 NLI=0.29

Little stiction

NGI=0.25 2.08 Stiction NLI=0.27 NGI=0 NLI=0

Linear loop

NGI=0 NLI=0

Linear loop

NGI=0 NLI=0

Linear loop

NGI=0 NLI=0

Linear loop

332


Table 13.12 (Continued) Loop Time span name CHEM 60 All CHEM 61 All CHEM 62 All CHEM 63 All PAP 1 PAP 2

All

PAP 3

All

PAP 4

All

PAP 5

1–4096

PAP 6 PAP 7

1–4096

PAP 8

All

PAP 9

All

PAP 10

All

PAP 11

All

PAP 12

All

PAP 13

All

Used parameters Nfft=128, nsamp=128, overlap=0 Nfft=128, nsamp=166, overlap=0 Nfft=128, nsamp=108, overlap=0 Nfft=64, nsamp=36, overlap=0

Index values NGI=0 NLI=0

S

Comments Linear loop

NGI=0.31 NLI=0.31


NGI=0 NLI=0

Linear loop

NGI=0 NLI=0

Linear loop

Data length too short for reliable NLA Nfft=128, NGI=0.19, 3.24 Stiction nsamp=128, NLI=0.18 overl=50 Stiction Nfft=128, NGI=0.36, 1 nsamp=128, NLI=0.76 overl=50 Nfft=128, NGI=0.58, 2.3 Stiction nsamp=128, NLI=0.81 overl=50 Nfft=128, NGI=0.16, 0.5 Stiction nsamp=128, NLI=0.15 overl=0 Data length too short for reliable NLA Non-linear, but no stiction Nfft=128, NGI=0.15, 0 nsamp=128, NLI=0.15 overl=0 NGI=0.15, Non-linear, but no stiction Nfft=128, nsamp=128, NLI=0.15 overl=50 Nfft=256, NGI=0.35, 0.35 Stiction nsamp=256, NLI=0.66 overl=50 Nfft=128, NGI=0.15, 1 Stiction nsamp=128, NLI=0.15 overl=50 Nfft=128, NGI=0.14, 1.8 Stiction nsamp=128, NLI=0.16 overl=0 NGI=0.22, 5 Stiction Nfft=128, nsamp=128, NLI=0.48 overl=0 Stiction Nfft=128, NGI=0.28, 9 nsamp=128, NLI=0.67 overl=0


333

Table 13.12 (Continued) Loop name POW 1

Time span 1–4096

POW 2

1–4096

POW 3

1–4096

POW 4

1–4096

POW 5

1–4096

MIN 1

All

MET 1

All

MET 2

All

MET 3

All

Used parameters Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0 Nfft=256, nsamp=256, overl=50 Nfft=256, nsamp=256, overl=50 Nfft=128, nsamp=128, overl=0 Nfft=128, nsamp=128, overl=0

Index S values NGI=0.38, 4.5 NLI=0.74

Comments

NGI=0.45, 11 NLI=0.88

Stiction

NGI=0.2, 0 NLI=0.2

Non-linear, but no stiction

NGI=0.3, 4.7 NLI=0.75

Stiction

NGI=0.28, 0 NLI=0.82

Non-linear, but no stiction, valve saturation

NGI=0.22, 1 NLI=0.38

Stiction

NGI=0

Linear loop

0

Stiction

NGI=0.22, 0 NLI=0.53

Non-linear, but no stiction, OP data look quantised.

NGI=0.17, 0 NLI=0.31

Non-linear, but no stiction, OP data look quantised.

Explanations and Comments: 1. For reliable results using the bicoherence method, the data length should be at least 1024 and data segment used for FFT should contain at least one complete oscillation. 2. Data must also be free from compression, quantisation, outliers, sharp abrupt changes or step changes and transients. 3. The loop must be in AUTO mode. No manual experimentation during the collection of the data, i.e. steady-state routine operating data are required. 4. Data of CHEM 12 were downsampled by 2.

334


Table 13.13 Results of non-linearity detection method based on surrogates analysis by Thornhill Loop name

Time span

BAS 1

77 001– 80 000 48 200– 48 700

BAS 2 BAS 3 BAS 4 BAS 5

96 001– 100 000 100 001– 104 000 80 001– 220 000

Preprocessing None

Index values

Non- Comments linear?

PV: 1.29

slightly The quantisation level is high.

None

–

None

–

None

PV: 2.26 OP: 1.76 PV: 1.25

BAS 6

All

BAS 7 BAS 8

All 1–20 000

CHEM 1

All

CHEM 2

All

CHEM 3 CHEM4

200–800 All

Subsampling by 50 None PV: 1.41 OP: 2.22 None – OP: 1.00 Subsampling ERR: 1.48 by 50 Filtered OP: 1.43 [2 400] Filtered – [2 400] None – None –

CHEM 5

All

None

CHEM 6

All

None

PV: 2.14 OP: 3.26 –

CHEM 7

NA

NA

NA

CHEM 8

NA

NA

NA

CHEM 9 625–945 CHEM 10 All

None None

PV: 2.76 PV: 1.13

CHEM 11 All

None

–

CHEM 12 All

None

–

CHEM 13 580–1 081 Filtered – [2 200]

When looking at the whole data set, there are episodes where the OP changes do not have corresponding changes in PV. yes slightly No OP available yes slightly Samples 27 500–32 000 show that a strong oscillation starts whenever there is actuator saturation. PV linear, but OP non-linear Not analysed because there are too few cycles of oscillation. no PV is quantised. no Filtering [2 25] to remove a low-frequency trend did not reveal any non-linearity. yes PV clearly is oscillating. The peaks are of shorter duration than the valleys. no Filtering [2 200] revealed oscillation in PV and OP but no non-linearity. NA Not analysed because there is no oscillation or timescale to use in the analysis. It looks as though the loop is in manual mode. NA Not analysed because there is no oscillation or time scale to use in the analysis. It looks as though the loop is in manual mode. yes yes slightly Are these time trends synthesised or simulated? They seem too regular. no Are these time trends synthesised or simulated? They seem too regular. – Non-linearity not analysed because there are too few cycles of oscillation. no

13 Comparative Study of Valve-stiction-detection Methods Table 13.13 (Continued) Loop Time Prename span processing CHEM 14 580–1 081 Filtered [2 200]

Index values


PV: 2.31 yes OP: 3.01 SP: 1.30 ERR: 2.45 CHEM 15 580–1 081 Filtered – no [2 200] CHEM 16 580–1 081 Filtered – no [2 200] CHEM 17 580–1 081 Filtered OP: 1.26 yes [2 200] (OP) CHEM 18 1–500 Filtered OP: 1.16 yes [2 200] (OP) CHEM 19 All None –

CHEM 20 200–400

Filtered PV: 1.70 [2 25] OP: 1.76

yes

CHEM 21 200–400

no

CHEM 22 All

Filtered – [2 25] None –

CHEM 23 All

None

–

PV: 1.29 OP: 1.51

CHEM 24 501–1 500 Filtered OP: 1.48 [2 100] SP: 3.01

335

yes (OP, SP) yes

CHEM 25 201–600

None

PV: 1.75 OP: 2.58

CHEM 26 All

None

–

no

CHEM 27 All CHEM 28 All

None None

– –

–

The analysis has detected the nonlinearity in the steam flow control loop.

The OP of CHEM17 is the SP to CHEM14 Non-linearity not analysed because there are too few cycles of oscillation. The OP–PV and SP–PV plots show that stiction is present. Analysis is for the high-frequency oscillation at 11 samples per cycle not the lowfrequency oscillation Analysis is for the high-frequency oscillation at 10 samples per cycle Non-linearity not analysed because there are too few cycles of oscillation. The OP–PV and SP–PV plots show that stiction is present. The oscillation index is very high and the data set may be too strongly cyclic for the non-linearity results to be reliable. The OP–PV plot shows stiction is present. The oscillation in PV appears to be a subharmonic of the SP oscillation. The OP–PV plot is an ellipse with sharp corners, which is indicative of stiction in a loop controlling a type-1 process. There are only 8 cycles of oscillation. Parameter k was set to 5 not 8. The OP–PV plot looks like a linear ellipse. The high-frequency content is random noise, not an oscillation The OP–PV plot looks like a linear ellipse Non-linearity not analysed because there are too few cycles of oscillation. The time trends do look as though they were generated by a non-linear system.

336


Table 13.13 (Continued) Loop Time Prename span processing CHEM 29 3 001– None 3 500

Index values

CHEM 31 15 101– 15 300

PV: 2.92 OP: 1.56 SP: 2.17 OP: 2.02 None PV: 2.35 OP: 2.63 ERR: 2.23 Filtered PV: 1.77 [2 25] OP: 1.73

CHEM 32 1–1 400

None

CHEM 30 6 901– 7 900

CHEM 33 100–600

Non- Comments linear? yes (OP)

Data are compressed. The non-linearity analysis may be unreliable.

yes yes

PV: 3.02 yes OP: 1.14 ERR: 1.08 Filtered PV: 1.05 yes [2 50] OP: 1.23 ERR: 1.23

CHEM 34 350–650

Filtered – [2 50]

–

CHEM 35 1–400

CHEM 36 All

Filtered PV: 1.24 yes OP: 1.52 [2 50] SP: 1.04 ERR: 1.39 None – –

CHEM 37 All

None

–

–

CHEM 38 All

None

–

–

The oscillation period is at the lower permitted limit for non-linearity analysis. There may be a subharmonic because every second cycle is smaller. The sampling period is too low to be sure. It may just be an aliasing effect, i.e. missing the peaks because of too few samples per cycle. The OP–PV and SP–PV plots show stiction is present. SP oscillation is an artifact at the edge of the filter. The rapid vertical transitions and the tendency of the OP to be at the extremes suggests an oversized actuator that jumps when it moves. There is no oscillation and therefore no automated way of choosing an embedding (E) and prediction horizon (H). The reason for using a prediction horizon of 10 is that there is a spectral peak at 0.1 on the normalised frequency axis corresponding to features in the time trend with a characteristic duration of 10 samples. Non-linearity not analysed because with no oscillation there is no way of choosing an embedding and prediction horizon (E and H). Analysis is for the high-frequency oscillation at 17 samples per cycle not the lowfrequency disturbance. Non-linearity not analysed because there are too few cycles of oscillation. Non-linearity not analysed because there are too few cycles of oscillation. Non-linearity not analysed because there are too few cycles of oscillation.

13 Comparative Study of Valve-stiction-detection Methods Table 13.13 (Continued) Loop Time Prename span processing CHEM 39 All None CHEM 40 30–512

None

CHEM 41 All

None

CHEM 42 All CHEM 43 All CHEM 44 All

Index values


–

no

–

no

PV: 1.19 OP: 1.57 Filtered – [2 40] Filtered – [2 40] None –

yes

Non-linearity analysed using E = H = 30. because there are features in the data with a characteristic duration of 30 samples. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. The OP is saturating (33%), assessed using the method in [82].

no no –

CHEM 45 All

Filtered – [2 40]

–

CHEM 46 All

None

–

no

CHEM 47 All

None

–

no

CHEM 48 1–500

None

–

no


None None

CHEM 52 All

PV: 1.83 PV: 3.21 OP: 3.21 Filtered – [2 80] None –

CHEM 53 All

None

–

no

CHEM 54 All

None

–

no

CHEM 51 All

337

yes yes no no

Non-linearity not analysed because there are too few cycles and no clear oscillation. OP is saturating. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Filtering with [2 40] filter did not reveal any non-linearity. Use samples 1–500 to avoid a large deviation. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Using samples 1 to 420 duplicates exactly the result reported in [122]. Oscillation period is 51 samples per cycle. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation.

338


Table 13.13 (Continued) Loop Time Prename span processing CHEM 55 All None

Index values


–

no

Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. no Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. no OP not analysed because it has no dynamic features. no Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Filtering with [2 40] filter did not detect any non-linearity. no Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. Filtering with [2 40] filter did not detect any non-linearity. slightly Embedding and prediction horizon of 17 used in non-linearity analysis because this is the period of the plant-wide oscillation. yes PV is oscillating but the peaks and valleys are of unequal duration. Embedding and prediction horizon of 5 used in non-linearity analysis because this is the period of the features visible in the time trend. The feature period is below the lower permitted limit for non-linearity analysis. no

CHEM 56 All

None

–

CHEM 57 All

None

–

CHEM 58 All

None

–

CHEM 59 All

None

–

CHEM 60 All

None

OP: 1.05

CHEM 61 151–251

None

PV: 1.98 OP: 1.11


Filtered – [2 40] None SP: 3.27

PAP 1

All

None

–

–

PAP 2

All

None

yes

PAP 3

All

None

PAP 4

All

None

PV: 1.80 OP: 2.01 PV: 2.39 OP: 2.79 PV: 4.16 OP: 3.94

yes

yes yes

Period of oscillation in PV and OP is 3.6 samples per cycle. Period in SP is 16.4 samples per cycle. The oscillation period for PV and OP is below the lower permitted limit for nonlinearity analysis. Non-linearity not analysed because there are too few cycles of oscillation.

13 Comparative Study of Valve-stiction-detection Methods Table 13.13 (Continued) Loop Time Prename span processing PAP 5 6 000– None 9 500


–

no

–

–

PAP5 looks like an oscillatory loop with poor damping, i.e. resonant and oscillatory tuning, when it is driven by a random disturbance such as instrument noise. The variable oscillation amplitude is characteristic of this tuning fault. Non-linearity not analysed because there are too few cycles of oscillation

PAP 6

All

PAP 7 PAP 8

401–1 400 None 601–1 000 Filtered [2 250] All None 601–1 000 Filtered [2 250] 3 001– None 3 500 1 501– None 3 000 501–1 500 None

– PV: 1.1

– slightly Filter first, then use samples 601–1 000.

– –

no no

–

no

PV: 1.06 OP: 1.45 –

slightly

PV: 1.62 OP: 1.08 PV: 1.28 OP: 1.20 –

yes

MIN 1

4 501– 5 000 4 001– 5 000 3 001– 5 000 4 001– 5 000 2 001– 3 000 1–2 000

MET 1

All

MET 2

2 501– 3 000 2 501– 3 000

PAP 9 PAP 10 PAP 11 PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5

MET 3

None

Index values

None None None None

no

yes no

PV: 1.47 OP: 1.41 PV: 1.47

yes

PV: 1.26 OP: 1.73 Filtered – [2 500]

yes

None None

yes

no

339

Filter first, then use samples 601–1000.

Could be a tuning problem or a limit cycle oscillation propagating from elsewhere in the plant. The data are compressed and non-linearity results are unreliable The data are compressed and non-linearity results are unreliable The data are compressed and non-linearity results are unreliable The data are compressed and non-linearity results are unreliable The data are not compressed. the OP is saturated.. The vertical jumps in PV suggest a sensor problem OP was oscillating but not detected because there is a second oscillation in OP at 6.67 samples per cycle.

None

–

no

None

PV: 1.08

slightly PV has a series of spectral peaks that are not harmonics

Explanations and Comments: 1. For instance, “Filtered [2 500]” means that a Wiener filter with frequency range [1/500, 1/2] = [0.005, 0.2] was applied to the data to remove the low-frequency trend. 2. “–” means that the loop is linear, i.e. NPI < 1.

340


Table 13.14 Results of stiction detection using both methods of Horch Loop no.

Stiction? Comments (Histogram) NA No clear oscillation NA No clear oscillation yes

BAS 6 BAS 7 BAS 8 CHEM 1 CHEM 2 CHEM 3 CHEM 4 CHEM 5 CHEM 6 CHEM 7 CHEM 8 CHEM 9

Time span Stiction? (Correlation) – NA – NA 211 000– uncertain 220 000 214 000– yes 218 000 150 000– uncertain 200 000 All no All uncertain – NA All yes All yes All yes All −99 All uncertain All uncertain – NA – NA 600–900 uncertain

CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17 CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22 CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36

All All All All All All All All All All All All All All All All All All All 2 000–3 000 2 000–4 000 2 000–3 000 All All All All All

yes no yes no yes yes yes no uncertain yes uncertain uncertain no yes yes yes yes uncertain no uncertain yes no no yes yes uncertain yes

BAS 1 BAS 2 BAS 3 BAS 4 BAS 5

yes uncertain yes no yes yes yes uncertain uncertain yes yes no yes yes yes uncertain −99 −99 yes yes uncertain no uncertain no uncertain uncertain −99

yes yes no yes NA yes yes no no yes yes NA NA yes

CORR not possible since OP = const.

Oscillation not clear enough

Very little data Very little data Very little data Experiment data, methods not suitable Experiment data, methods not suitable “Chose oscillatory data; OP not moving: only HIST possible” Very clear data Analyser control loop SP oscillatory, faulty sensor Oscillation hard to find automatically Oscillation hard to find automatically SP oscillatory SP oscillatory

SP oscillatory

Too little data per cycle for HIST CORR cannot be used since u = const. 180 degrees. Little data per cycle. Hard to detect oscillation automatically Hard to detect oscillation automatically Gives result (HIST = 1 for 200–400)


341

Table 13.14 (Continued) Loop no. CHEM 37 CHEM 38 CHEM 39 CHEM 40 CHEM 41 CHEM 42 CHEM 43 CHEM 44 CHEM 45

Time span Stiction? (Correlation) All −99 All uncertain All uncertain All uncertain All no All no All no – NA – NA

Stiction? (Histogram) yes no yes yes no no no NA NA

CHEM 46

–

NA

NA

CHEM 47

–

NA

NA

CHEM 48

–

NA

NA

CHEM 49 CHEM 50 CHEM 51 CHEM 52 CHEM 53 CHEM 54 CHEM 55 CHEM 56 CHEM 57

All All All – – All All – All

uncertain no uncertain NA NA uncertain yes NA uncertain

no yes yes NA NA uncertain uncertain NA yes

CHEM 58

–

NA

NA

CHEM 59

–

NA

NA

CHEM 60 CHEM 61 CHEM 62 CHEM 63 PAP 1 PAP 2 PAP 3 PAP 4 PAP 5

All – All All All All All All 10 000– 12 000 All 10 000– 12 000 All All All 500–2 000

yes NA uncertain no uncertain yes −99 no yes

yes NA yes uncertain yes no yes no yes

−99 no

no yes?

−99 no −99 uncertain

yes no yes uncertain

PAP 6 PAP 7 PAP 8 PAP 9 PAP 10 PAP 11

Comments

Different frequencies Outlier removed Data not valid for methods “Too irregular to be found oscillatory; methods not designed to work here” “Too irregular to be found oscillatory; methods not designed to work here” “Too irregular to be found oscillatory; methods not designed to work here” “Too irregular to be found oscillatory; methods not designed to work here”

No clear enough oscillation No clear enough oscillation Just noise, no clear enough oscillation Strange control (OP hardly moving) ⇒ CORR not applicable “All three signals oscillate irregularly; very noisy” “All three signals oscillate irregularly; very noisy” Too few samples per cycle SP oscillatory ”SP oscillatory; slow sampling”

No clear oscillation

342


Table 13.14 (Continued) Loop no. PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

Time span Stiction? Stiction? Comments (Correlation) (Histogram) 500–2000 yes yes Automatic oscillation detection difficult 500–2000 −99 no 2000–4000 −99 yes 2000–4000 −99 yes 2000–4000 −99 yes 2000–4000 −99 no 2000–4000 −99 yes Strange OP 500–1500 uncertain yes CCF too unsymmetric 1500–2500 uncertain no? Different frequencies 1500–2500 uncertain yes CCF too unsymmetric 1500–2500 uncertain no

Explanations and Comments: 1. Analysis used all data when data sets were below 2000 samples. 2. For CORR, the data is high-pass filtrered in order to remove slow variations such that the data keeps a “constant” mean value, based on the detected oscillation frequency. 3. The actual HIST algorithm includes filtering and differentiation as described in Chap. 6; so both steps belong to the algorithm itself, and are not considered as pretreatment. 4. Methods are in general tuned to be careful, i.e. return −1 in cases where it is uncertain (especially CORR). 5. For integrating loops (level control) and steam flow control loops, method CORR cannot be used (return −99). 6. For self-regulating processes, both methods can be used. In general, it is recommended to rely on results when both methods give the same conclusion. 7. Some data contained only very few samples per oscillation cycle. Such data is difficult to use. 8. CORR and HIST are designed for clearly oscillatory loops. Results on loops that are not oscillatory may be misleading

Table 13.15 Results of stiction detection using the area-peak method by Salsbury and Singhal Loop no. BAS 1 BAS 2 BAS 3 BAS 4 BAS 5 BAS 6 BAS 7

Index value 0.07 0.00 0.00 0.00 −1 0.00 1.00

Stiction? no no no no ? no yes

Comments

No suitable features could be extracted.

13 Comparative Study of Valve-stiction-detection Methods Table 13.15 (Continued) Loop no. BAS 8 CHEM 1 CHEM 2 CHEM 3 CHEM4 CHEM 5 CHEM 6 CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17 CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22 CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39 CHEM 40 CHEM 41 CHEM 42 CHEM 43 CHEM 44 CHEM 45 CHEM 46 CHEM 47 CHEM 48 CHEM 49 CHEM 50

Index value 0.00 0.97 −1 0.00 −1 −1 0.03 −1 −1 −1 1.00 0.19 0.45 0.00 −1 −1 −1 −1 0.00 −1 −1 −1 1.00 0.27 0.00 0.00 0.21 0.03 0.77 0.00 1.00 −1 0.00 −1 −1 0.13 0.88 −1 1.00 0.00 −1 0.00 0.00 −1 0.16 −1 −1 −1 −1 0.00 −1

Stcition? no yes ? no ? ? no ? ? ? yes little evidence possible no ? ? ? ? no ? ? ? yes some evidence no no some evidence no strong evidence no yes ? no ? ? some evidence yes ? yes no ? no no ? little evidence ? ? ? ? no ?

Comments No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

343

344


Table 13.15 (Continued) Loop no. CHEM 51 CHEM 52 CHEM 53 CHEM 54 CHEM 55 CHEM 56 CHEM 57 CHEM 58 CHEM 59 CHEM 60 CHEM 61 CHEM 62 CHEM 63 PAP 1 PAP 2 PAP 3 PAP 4 PAP 5 PAP 6 PAP 7 PAP 8 PAP 9 PAP 10 PAP 11 PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

Index value −1 −1 −1 0.00 0.12 −1 −1 −1 −1 0.01 −1 −1 −1 0.99 1.00 0.39 0.96 0.00 0.07 0.35 −1 0.03 −1 0.00 −1 0.00 0.00 0.00 −1 0.00 −1 0.97 −1 0.0 0.0

Stcition? ? ? ? no little evidence ? ? ? ? no ? ? ? yes yes some evidence yes no not likely some evidence ? no ? no ? no no no ? no ? yes ? no no

Comments No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

No suitable features could be extracted. No suitable features could be extracted. No suitable features could be extracted.

Explanations and Comments: 1. The whole span was used for all data sets. 2. The data has been preprocessed as follows: Step 1. Use second-order band-stop filter with frequency determined from zero crossing frequency in data to determine a noise band. Step 2. Apply 5-point median filter to raw data. Step 3. Extract features for each classifiable half-period. 3. Only the following parameters were used: band-stop filter window size and order – we used second-order with a “quality factor” of 1/10; number of samples in median filter – we used 5; and noise band σ limit we used ±2σ .


345

4. The reported value of the index is the probability of stiction, between zero and one, where 1 is strong belief in stiction. A value of −1 indicates that no suitable features could be extracted from the data for analysis. 5. The existence of stiction is somewhat subjective and relies on picking a threshold for the probability value.

Table 13.16 Results of stiction-detection method based on relay by Scali and Rossi Loop no.

BAS 6

Time span [min] 4 627 4 627 13 319 [244:247] xe03 13 319 [244:247] xe03 4 258 [130:230] xe03 8 [300:501]

BAS 7

9 [1:560]

De-trend

BAS 8 CHEM 1 CHEM 2 CHEM 3

None De-trend De-trend None

CHEM 4 CHEM 5 CHEM 6 CHEM 7

4 2512 27 [1:1 625] 17 [1:1 000] 972 [0:20]xe03 3 [1:200] 3 [1:201] 17 [1:1 000] 2 342

0.26 (0.14) – 0.25 0.28 0.27

De-trend De-trend De-trend None

−0.25 0.53 0.08 –

No stiction Stiction Uncertain –

CHEM 8

450

None

CHEM 9

1 366

None

CHEM 10 CHEM 11

500 500

None None

CHEM 12 CHEM 13 CHEM 14

1 000 500 500 [15:20]xe03 500

None None None

NA: oscillating with offset (?manual) – – NA: few oscillations (with quantization?) – – NA: anomalous oscillating with offset (?) 0.331 Stiction Certainly stiction 0.24 Asymmetric Probably stiction (0.18) stiction 0.264 Stiction Certainly stiction −0.472 No stiction No stiction 0.30 Stiction SP oscillating

None

–

–

BAS 1 BAS 2 BAS 3 BAS 4 BAS 5

CHEM 15

Preprocessing None None None

Index value(s) – – 0.29

Stiction?

Comments

– – Stiction

NA: Anomalous NA: Anomalous Index value may change with window

None

0.29

Stiction

Index value may change with window

None

0.59

Stiction

OP not available

De-trend

−0.41

No stiction

Index value may change with window Asymmetric Index value may change with stiction window – NA: Anomalous Stiction Stiction Stiction

NA: SP step change; PV does not change: loop in manual?

346


Table 13.16 (Continued) Loop no. CHEM 16

Time span PreIndex Stiction? [min] processing value(s) 500 None 0.22 Stiction [20:30]xe03

Comments Close to threshold; changing window may become uncertain. Value may change with window Stiction not evident SP Oscillating; stiction very likely Stiction not evident Likely stiction Certainly stiction Certainly stiction Certainly stction Certainly stiction Likely stiction Stiction not evident

CHEM 17 CHEM 18 CHEM 19

500 [0:9 500] None 204 [0:6 000] None 141 None

−0.26 0.51 0.408

No stiction Stiction Stiction

CHEM 20 CHEM 21 CHEM 22 CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27

140 141 140 250 250 141 214 251

None None None None None None None None

0.234 0.311 0.594 0.253 0.474 0.302 0.314 0.17 (0.30)

CHEM 28

142

None

0.322

CHEM 29

7 200 None [50:60]xe03 4 320 None

0.33

−0.74

CHEM 32

4 320 None [10-15]xe03 333 None

Stiction Stiction Stiction Stiction Stiction Stiction Stiction Uncertain / Asymmetric stiction Asymmetric Stiction not evident Stiction Stiction SP and PV oscillating, both high frequency Stiction SP and PV oscillating, SP lower frequency No stiction Almost sinusoidal oscillation

0.597

Stiction

CHEM 33 CHEM 34

141 120

None None

0.10 0.278

CHEM 35 CHEM 36

333 158

None None

0.369 0.274

CHEM 37

334

None

0.216

CHEM 38

155

None

0.379


120 511 511 511 [5:20]xe03 511 511

None None None None

0.276 – 0.294 -0.786

SP and PV oscillating; same frequency; stiction evident Uncertain Varying SP; Stiction Not significant oscillation (Hägglund’s test [37]) Stiction Likely stiction Stiction Not significant oscillation (Hägglund’s test [37]) Asymmetric Stiction not evident Values close to threshold stiction / Uncertain Asymmetric Likely stiction stiction Stiction Stiction not evident – No verdict (?) Stiction Likely stiction No Stiction Almost sinusoidal disturbance

None None

0.291 –

Stiction –

CHEM 30 CHEM 31

CHEM 43 CHEM 44

0.38

Stiction not evident NA: anomalous only one oscillation; saturation (?)


347

Table 13.16 (Continued) Loop no.

CHEM 47

Time span [min] 511 [0:10]xe03 511 [15:20]xe03 511

CHEM 48 CHEM 49 CHEM 50 CHEM 51 CHEM 52

511 511 511 511 511

None None None None None


511 511 511 511

None None None None

Asymmetric stiction 0.245 Stiction −0.620 No stiction 0.117 Uncertain 0.265 Stiction 0.230 Asymmetric stiction / Uncertain 0.303 Stiction 0.418 Stiction 0.372 Stiction 0.30 Stiction

CHEM 57

511

None

–

–

CHEM 58 CHEM 59

None None

0.38 0.53

Stiction Stiction

CHEM 60 CHEM 61

511 511 [10:30]xe03 511 511

None None

0.355 –

Stiction –

CHEM 62

511

None

−0.21

No stiction

CHEM 63

511 [0:20]xe03 14 [1:849] 20 [1:500] 19 [1:500] 20 [1:500] 300

None

−0.21

No stiction

De-trend De-trend De-trend De-trend None

0.45 0.67 0.27 0.50 0.250

CHEM 45 CHEM 46

PAP 1 PAP 2 PAP 3 PAP 4 PAP 5

PreIndex Stiction? processing value(s) None 0.19 Uncertain

Comments

None

0.33

Irregular oscillations

None

0.259

Stiction

Irregular oscillations

Stiction not evident Stiction not evident Probably disturbance Stiction not evident Stiction not evident Stiction not evident values close to threshold Stiction not evident Stiction not evident Stiction not evident SP and PV oscillating, SP higher frequency NA: Anomalous PV not affected by OP (?) Irregular oscillations SP and PV oscillating, both high frequency Stiction not evident Anomalous; values change with window SP and PV oscillating, both high frequency SP and PV oscillating, both high frequency

PAP 6 PAP 7 PAP 8

14 [1:846] De-trend 20 [200:700] De-trend 30 De-trend [960:1 250]

−0.76 0.37 0.22 (0.20)

PAP 9

−0.04

PAP 10 PAP 11

30 De-trend [200:1 500] 30 [200:700] De-trend 1045 None

Stiction Stiction Stiction Stiction Asymmetric stiction No stiction Stiction Asymmetric Stiction not evident stiction values close to threshold / Uncertain No stiction

0.41 –

Stiction –

PAP 12

1 115

None

–

–

PAP 13

1 059

None

−0.287 No Stiction

Irregular oscillations: verdict changes with window Irregular oscillations: verdict changes with window Almost sinusoidal disturbance

348


Table 13.16 (Continued) Loop no. POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

Time span [min] 720 [5 000:5 300] 720 [1:500] 720 [100:1 500] 720 [1:500] 720 [2 100:2 700] 2 640 [400:1 500] 1 [200:1 500] 4 [300:600] 5 [3 000:3 400]

PreIndex Stiction? processing value(s) De-trend −0.30 No stiction De-trend De-trend

0.40 0.45

Stiction Stiction

De-trend De-trend

0.41 0.44

Stiction Stiction

None

0.36

Stiction

De-trend De-trend De-trend

0.09 0.35 0.55

No stiction Stiction Stiction

Comments

Likely stiction

Explanations and Comments: 1. The relay technique is adopted inside a closed-loop-performance monitoring system, which performs data analysis according to a default path, to avoid inappropriate applications and then limit the number of false alerts and wrong verdicts. The analysis is limited to loops: • Oscillating with a significant amplitude, i.e. positive to the Hägglund’s test [37], • Assimilated to constant SP (according to a test based on a differential index, a global index and a maximum number of changes), • Not decaying and showing a dominant frequency, • With a minimum acquisition time. 2. The correct application of the analysis requires the following information for each loop: • Range of PV and OP variables, • Controller parameters, • Loop configuration, i.e. primary, cascade, under supervisory/advanced control. 3. In this application, a preliminary visual analysis of loop data was performed to “force” the application of the technique to all loops, which seemed to be not affected by evident anomalies. The cause of anomaly for 15 (out of 93) loops, for which the technique has not been applied, is indicated in the last column of the table (on the right).


349

Table 13.17 Results of stiction-detection method based on curve fitting by He and Qin Loop no.

Time span NA NA All All

Preprocessing NA NA None Savitzky– Golay smoothing

Index value(s) NA NA 0.71 0.74

BAS 1 BAS 2 BAS 3 BAS 4

Stiction? Comments NA NA yes yes

BAS 5

All

None

0.77

yes

BAS 6

All

None

0.40

no

BAS 7 BAS 8 CHEM 1 CHEM 2

All NA All All

None NA None None

0.86 NA 0.83 0.63

yes NA yes yes

CHEM 3

All

De-trend

0.68

yes

CHEM4

All

None

0.34

no

Assuming self-regulating process. De-trend was applied first. Integrating, fit PV

CHEM 5 CHEM 6

All All

None None

0.77 0.74

yes yes

Self-regulating, fit OP Self-regulating, fit OP

CHEM 7

All

None

0.88

yes

Assuming self-regulating process.

CHEM 8

All

None

0.56

CHEM 9

All

NA

NA

unknown It looks like the values were held for a period of time for each sample. NA No oscillation

CHEM 10

All

None

0.91

yes


CHEM 11

All

None

0.37

no


CHEM 12

All

None

0.86

yes


CHEM 13

All

De-trend

0.33

no

CHEM 14

All

None

0.66

yes

Assuming self-regulating process. De-trend was applied first. Self-regulating, fit OP

CHEM 15

All

De-trend

0.61

yes

CHEM 16

All

None

0.63

yes

CHEM 17

[1:460]

None

0.28

no

No oscillation No oscillation Assuming self-regulating process. Assuming integrating process. Savitzky–Golay smoothing was applied to remove excess noise. Otherwise, SI = 0.40 Assuming integrating process. OP is not available. Too few points. It looks like aggressive tuning. Assuming self-regulating process. No oscillation Self-regulating, fit OP Self-regulating, fit OP

Assuming self-regulating process. De-trend was applied first. Assuming self-regulating process. Assuming self-regulating process. Only the first 460 samples were used.

350



Time span

CHEM 18

All

PreIndex Stiction? Comments processing value(s) None 0.64 yes Self-regulating, fit OP

CHEM 19

All

None

0.67

yes

Self-regulating, fit OP

CHEM 20

All

None

0.68

yes


CHEM 21

All

None

0.52

unknown Self-regulating, fit OP

CHEM 22

All

None

0.36

no


CHEM 23

All

None

0.95

yes


CHEM 24

All

None

0.62

yes


CHEM 25

All

None

0.87

yes


CHEM 26

All

Smoothing 0.66

yes

CHEM 27

All

None

0.63

yes

Assuming integrating process. Savitzky–Golay smoothing was applied to remove excess noise. Integrating, fit PV

CHEM 28

All

None

0.07

no

CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39

[2 000:5 000] [1:2 000] NA All All [51:400] NA All All All All

0.38 0.86 NA 0.22 0.53 0.60 NA 0.72 0.69 0.90 0.65

CHEM 40

[51:end] [51:end] NA [101:end] [201:end] [201:end] NA NA NA NA NA

None None NA None None None NA None None None Savitzky– Golay smoothing None None NA De-trend De-trend De-trend NA NA NA NA NA

CHEM 41 CHEM 42 CHEM 43 CHEM 44 CHEM 45 CHEM 46 CHEM 47 CHEM 48

0.61 0.63 NA 0.20 0.31 0.46 NA NA NA NA NA

Assuming self-regulating process. Otherwise, SI = 0.67. no Self-regulating, fit OP yes Self-regulating, fit OP NA Too few points (6–7) per cycle no Self-regulating, fit OP unknown Self-regulating, fit OP yes Self-regulating, fit OP NA No oscillation yes Integrating, fit PV yes Integrating, fit PV yes Self-regulating, fit OP yes Savitzky–Golay smoothing was applied to remove excess noise yes yes NA no no unknown NA NA NA NA NA

Assuming self-regulating, fit OP Assuming integrating, fit PV Assuming self-regulating, fit OP Self-regulating, fit OP Assuming self-regulating, fit OP Assuming integrating, fit PV No oscillation No oscillation No oscillation No oscillation No oscillation


351


Time span

CHEM 63 PAP 1

All All All [1:200] [1:200] NA NA All All All All NA NA NA NA All NA All All NA All

Preprocessing None None None None Smoothing NA NA None None None None NA NA NA NA None NA None None NA None

Index value(s) 0.14 0.18 0.56 0.52 0.56 NA NA 0.67 0.69 0.72 0.75 NA NA NA NA 0.67 NA 0.66 0.62 NA 0.47

CHEM 49

no no unknown unknown unknown NA NA yes yes yes yes NA NA NA NA yes NA yes yes NA unknown

PAP 2

All

None

0.53


PAP 3

All

None

0.70

yes

PAP 4

All

None

0.59


PAP 5

All

None

0.68

yes


PAP 6

All

None

0.07

no

Integrating, fit PV

PAP 7

All

None

0.54


PAP 8

All

De-trend

0.60

yes

PAP 9

All

None

0.34

no

PAP 10

All

De-trend

0.62

yes

PAP 11 PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5

All All All All All All All NA

None None None None None None None NA

0.65 0.78 0.34 0.69 0.50 0.54 0.53 NA

CHEM 50 CHEM 51 CHEM 52 CHEM 53 CHEM 54 CHEM 55 CHEM 56 CHEM 57 CHEM 58 CHEM 59 CHEM 60 CHEM 61 CHEM 62

Stiction? Comments Assuming self-regulating, fit OP Assuming integrating, fit PV Integrating, fit PV Assuming self-regulating, fit OP Assuming integrating, fit PV No oscillation No oscillation Assuming self-regulating, fit OP Assuming integrating, fit PV Assuming self-regulating, fit OP Assuming integrating, fit PV No oscillation No oscillation No oscillation No oscillation Assuming self-regulating, fit OP No oscillation Assuming self-regulating, fit OP Assuming integrating, fit PV No oscillation Self-regulating, fit OP

Integrating, fit PV

Integrating, fit PV De-trend was applied first Self-regulating, fit OP

Integrating, fit PV De-trend was applied first yes Self-regulating, fit OP yes Self-regulating, fitted OP no Integrating, fitted PV yes Integrating, fitted PV unknown Integrating, fitted PV unknown Integrating, fitted PV unknown Integrating, fitted PV NA Saturation in OP

352


Table 13.17 (Continued) Loop no. MIN 1 MET 1 MET 2 MET 3

Time span

Preprocessing All None All None All None [2 001:4 000] None

Index value(s) 0.67 0.54 0.57 0.59

Stiction? Comments yes unknown unknown unknown

Self-regulating, fit OP Self-regulating, fit OP Self-regulating, fit OP Self-regulating, fit OP

Table 13.18 Results of stiction-detection and quantification method by Lee et al. Loop name

Time span

Stiction estimates

BAS 1 BAS 2 BAS 3 BAS 4 BAS5 BAS 6 BAS 7 BAS8 CHEM 1 CHEM 2 CHEM 3 CHEM 4 CHEM 5 CHEM 6

All All

All All

0.33 0.53

0.49 0.61

yes yes

All All 100–900 All All All

0.04 0.65 0.00 0.02 0.09 0.01

0.39 2.52 0.00 0.03 0.26 0.01

CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11 CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17 CHEM 18 CHEM 19 CHEM 20 CHEM 21

All All All All All All 100–700 100–700 800–1 200 800–1 200 100–400 All All All 100–300

0.54 0.67 0.00 1.73 0.06 0.13 0.04 0.28 0.11 0.10 0.11 0.31 1.61 0.00 2.11

0.54 1.00 0.00 1.77 0.26 1.42 0.04 0.76 0.18 0.10 0.18 1.93 1.61 0.00 2.26

yes yes no no yes yes (minor) yes yes no yes yes yes no yes yes yes yes yes yes no yes

0.69 0.28 1.07

8.69 21.57 20.64

yes yes yes

J 26.31 13.68

S 26.31 13.68

Stiction? Comments yes yes Not sure if OP is moving enough. Not sure if OP is moving enough. No OP change Not sure if OP is moving enough.

CHEM 22 All CHEM 23 All CHEM 24 100–800

Open-loop data Open-loop data

OP, PV are like disturbances and look symmetrical to each other.

13 Comparative Study of Valve-stiction-detection Methods Table 13.18 (Continued) Loop name

Time span

CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29

100–350 All All All 2 000– 2 500

Stiction estimates J 1.62 1.59 1.93 0.35 0.51

S 1.62 4.11 2.01 1.63 5.35

Stiction? Comments yes yes yes yes yes

CHEM30 CHEM 31 4 000– 4 300 CHEM 32 All CHEM 33 All

0.04

0.04

0.08 0.42

12.28 6.13

yes (minor) yes yes

Not sure if downsampling needed.

CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39

All All All All All All

0.54 0.00 1.24 0.00 1.95 1.47

0.68 0.00 1.25 0.00 4.25 1.47

yes no yes no yes yes

CHEM 40 All

0.00

0.00

no

CHEM 41 300–500 CHEM 42 All

0.00 0.00

0.00 0.12


0.00 0.15 1.61

0.00 1.15 3.45

no yes (minor) no yes yes

CHEM 46 All

1.14

3.74

yes

CHEM 47 All

0.09

0.09

CHEM 48 All

0.17

5.08

yes (minor) yes


0.13 0.05

0.13 0.05

yes no


1.06 1.19 2.12

1.13 3.09 5.29

yes yes yes

CHEM 54 CHEM 55 CHEM 56 CHEM57 CHEM 58 CHEM 59 CHEM 60 CHEM 61

All All All

0.66 0.66 0.00

0.89 1.29 0.00

yes yes no

All All All All

0.22 0.00 0.14 0.29

0.79 0.00 0.14 0.29

yes no yes yes

OP, PV are like disturbances and look symmetrical to each other.

OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other.

OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other. OP, PV are like disturbances and look symmetrical to each other

Not sure if OP is moving enough.

353

354


Table 13.18 (Continued) Loop name

Time span

CHEM 62 CHEM 63 PAP 1 PAP 2 PAP 3 PAP 4 PAP 5 PAP 6 PAP 7 PAP 8 PAP 9 PAP 10 PAP 11 PAP 12

All 1–200 All 1–450 All All 1–3 000 All All 1–500 All All 1–700 1 000– 2 000 1–1 000 100–500 1–700 1–1 000 1–700

PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

Stiction estimates

Stiction? Comments

J 0.15 0.02 1.38 2.52 0.96 0.12 0.01 0.44 0.07 0.05 0.00 1.42 0.21 2.17

S 0.15 0.79 1.63 2.52 1.05 4.27 0.01 0.44 0.07 3.89 0.00 3.84 1.39 3.84

yes yes yes yes yes yes no yes yes yes no yes yes yes

0.19 0.49 0.87 1.02 0.39

0.38 0.69 1.15 4.27 0.58

yes yes yes yes yes

1.16 0.018 0.0098 0.0049

1.16 0.018 0.0100 0.0076

yes yes yes yes

Not sure if OP is moving enough. 500–1 500 All 1–800 100–500

Explanations and Comments: 1. Preprocessing for some loops: BAS 1,2; PAP 7 2. For loops BAS 1, 2, 7, CHEM 4, 7, 8, 22, 26, 28, 36, 42–44, 48, 50, PAP 10, a first-order model; for other loops, a second-order model was used. 3. Stiction decison: within 1% of the span of OP: no stiction; within [1, 2]% of the span of OP: yes (minor).

Table 13.19 Results of stiction detection and quantification method by Karra and Karim Loop name BAS 1

Time span 1–1 000

Stiction estimates Stiction? Comments J = 0.2, S = 0.2

yes

BAS 2

28 000– 29 000 100 000– 101 000

J = 0, S = 0.25

yes

J = 1.8, S = 3.0

yes

BAS 3

Disturbance is the major cause, alongside stiction. Insignificant oscillations caused by disturbance. Stiction – major cause, disturbance is also present.


355

Table 13.19 (Continued) Loop name BAS 4

J = 2.8, S = 2.8

yes

BAS 5 BAS 6

Time span 100 000– 101 000 – All

Stiction estimates Stiction? Comments

– J = 0.0, S = 1.0

– yes

BAS 7

All

J = 0.1, S = 1.6

yes

BAS 8

1–1 000

J = 0.022, S = 0.022 yes

CHEM 1 CHEM 2

1–1 000 All

J = 0.5, S = 0.5 J = 0, S = 4.0

yes yes

CHEM 3 CHEM4

1–1 000 All

J = 0, S = 0 J = 2.5, S = 3.5

no yes

CHEM 5

All

J = 0, S = 0.4

yes

CHEM 6

All

J = 0.2, S = 0.2

yes

CHEM 7

All

J = 0.8, S = 0.8

yes

CHEM 8

All

J = 1.5, S = 1.5

yes

CHEM 9

All

J = 0.6, S = 0.6

yes

CHEM 10 All CHEM 11 1–500

J = 1.3, S = 1.3 J = 0.2, S = 0.44

yes yes

CHEM 12 All

J = 0.5, S = 0.5

yes

CHEM 13 All

J = 2.0, S = 2.0

yes

CHEM 14 All

J = 0, S = 1.6

yes

CHEM 15 All

J = 0.2, S = 0.5

yes

CHEM 16 All

J = 0, S = 0

no

CHEM 17 All

J = 1.2, S = 1.2

yes

CHEM 18 All

J = 1.5, S = 2.5

yes

CHEM 19 All

J = 0, S = 4.0

yes

CHEM 20 All

J = 0.1, S = 0.1

yes

Stiction and disturbance are both significant. Diagnosis cannot be performed (no OP). Stiction is the major cause, disturbance is causing drift in OP. Stiction is the major cause, disturbance – 10% in PV. Disturbance is the major cause, alongside stiction. Stiction is the major cause. Stiction is the major cause, aggressive controller tuning is also present. Disturbance is the major cause. Stiction and disturbance are both significant. Stiction is the major cause, disturbance is causing drift in OP. Stiction and disturbance are both significant. Stiction and disturbance are both significant. Stiction and disturbance are both significant Stiction and disturbance are both significant, poor controller tuning. Stiction is the major cause. Disturbance is the major cause, presence of stiction. Stiction and disturbance are both significant. Stiction is the major cause, disturbance is causing drift in OP. Stiction is the major cause, significant presence of disturbance is causing drifting OP and PV. Stiction is the major cause, disturbance is also significant. Disturbance is the major cause, possibility of transients in case of sudden setpoint changes. Stiction is the major cause for oscillations, poor controller tuning is causing the upward drift in OP and PV. Stiction is the major cause, significant presence of disturbance. Stiction, disturbance and oscillatory trend in SP. Aggressive controller tuning and oscillatory trend in SP.

356


Table 13.19 (Continued) Loop Time name span CHEM 21 All


yes


J = 0, S = 7.0 J = 9.0, S = 9.0

yes yes

CHEM 24 All

J = 0, S = 17.0

yes

CHEM 25 All

J = 0.3, S = 1.8

yes

CHEM 26 All

J = 0.6, S = 0.6

yes

CHEM 27 1–1 000

J = 1.5, S = 1.5

yes

CHEM 28 All

J = 0.4, S = 1.4

yes

CHEM 29 1–1 000

J = 0.2, S = 3.2

yes

CHEM 30 1–600

J = 0.0, S = 0.4

yes

CHEM 31 1–600

J = 0.2, S = 0.3

yes

CHEM 32 1–600

J = 4.0, S = 15.0

yes

CHEM 33 1–500

J = 1.5, S = 5.5

yes

CHEM 34 All

J = 0.3, S = 0.5

yes

CHEM 35 1–500

J = 0.6, S = 0.6

yes

CHEM 36 All CHEM 37 1–600

J = 1.2, S = 1.4 J = 1.5, S = 1.5

yes yes

CHEM 38 All

J = 0, S = 0.5

yes

CHEM 39 All

J = 0.1, S = 0.3

yes

CHEM 40 50th to J = 0, S = 0 end to avoid transient CHEM 41 All J = 0, S = 0 CHEM 42 60th to J = 0, S = 0 end to avoid transient

no

no no

Stiction is the major cause, disturbance is also significant. Stiction is the major cause. Stiction and disturbance are both significant. Stiction is the major cause, presence of disturbance and aggressive controller tuning. Aggressive controller tuning and stiction are both significant. Disturbance is the major cause, stiction is contributing towards high-frequency oscillations. Stiction is the major cause, significant disturbance. Stiction is the major cause, aggressive controller tuning is also significant. All stiction, disturbance, aggressive controller tuning and oscillatory SP. Stiction is the major cause, significant disturbance. Stiction is the major cause, low-frequency disturbance. Stiction is the major cause, small contributions from disturbance and aggressive controller tuning. Stiction and disturbance are both significant. Disturbance and stiction are both significant. Stiction is causing oscillations, SP and disturbance are causing the slow drift in PV and OP. Stiction and disturbance are both present. Stiction is the major cause, about 50% variability is because of disturbance. Aggressive controller tuning is the major cause, insignificant stiction. Disturbance is the major cause, small stiction. Aggressive controller tuning is the major cause, significant disturbance. Aggressive controller tuning is the major cause, presence of disturbance. Aggressive controller tuning is the major cause, presence of disturbance.


357

Table 13.19 (Continued) Loop Time name span CHEM 43 All


yes


J = 2.9, S = 2.9 J = 0.5, S = 1.7

yes yes

CHEM 46 All

J = 1.6, S = 1.6

yes

CHEM 47 All

J = 0.6, S = 0.6

yes

CHEM 48 All

J = 1.9, S = 1.9

yes

CHEM 49 All

J = 1.3, S = 1.9

yes


J = 0, S = 0 J = 2.1, S = 2.1

no yes

CHEM 52 All

J = 1.6, S = 2.8

yes

CHEM 53 All

J = 0.6, S = 1.4

yes


J = 0.6, S = 2.6 J = 0, S = 1.5 J = 0, S = 0

yes yes no

CHEM 57 All

J = 0, S = 3.0

yes

CHEM 58 All

J = 0.6, S = 1.1

yes

CHEM 59 All

J = 0.1, S = 0.1

no

CHEM 60 All

J = 0, S = 2.5

yes

CHEM 61 All

J = 0.1, S = 0.1

no

CHEM 62 All

J = 0, S = 0.1

no

CHEM 63 All

J = 0, S = 0

no

PAP 1 PAP 2

All All

J = 3.0, S = 4.5 J = 1.8, S = 2.6

yes yes

PAP 3

All

J = 0, S = 0

no

Stiction is the major cause, drift in OP is caused by disturbance. Stiction is the major cause. Stiction is the major cause, 25% variability in PV accounted for the presence of disturbance. Stiction is the major cause, 25% variability in PV accounted to the presence of disturbance. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes. Aggressive controller tuning and stiction are the major causes. Disturbance is the major cause. Stiction is the major cause, and 25% variability can be attributed to the presence of disturbance. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes, and poor controller tuning. Stiction is the major cause. Stiction is the major cause Disturbance and oscillating SP are the major causes. Stiction and disturbance are the major causes. Stiction, disturbance and oscillating SP are the major causes. Disturbance and oscillating SP are the major causes; aggressive controller tuning is also contributing as there are significant initial transients in CL for a change in SP. Stiction and disturbance are the major causes. Disturbance is the major cause, insignificant stiction. Disturbance and oscillating SP are the major causes; aggressive controller tuning is also contributing as there are significant initial transients in CL for a change in SP. Disturbance and oscillating SP are the major causes. Stiction is the major cause. Stiction and disturbance are the major causes. Disturbance is the major cause.

358


Table 13.19 (Continued) Loop name PAP 4

Time span All


yes

PAP 5

1–1 000

J = 0, S = 0

no

PAP 6

All

J = 0.1, S = 0.1

no

PAP 7 PAP 8

1–2 000 1–1 000

J = 0, S = 0 J = 0.05, S = 0.8

no yes

PAP 9 PAP 10

1–600 All

J = 2.0, S = 2.0 J = 1.0, S = 1.1

yes yes

PAP 11

1–2 000

J = 0, S = 1.5

yes

PAP 12

1–2 000

J = 0, S = 3.8

yes

PAP 13

1–1 000

J = 0, S = 1.0

yes

POW 1

1–2 000

J = 0, S = 0.6

no

POW 2 POW 3

1–1 000 1–1 000

J = 12.0, S = 12.0 J = 0.7, S = 0.8

yes yes

POW 4

1–1 000

J = 1.2, S = 3.6

yes

POW 5 MIN 1

1–2 000 1–2 000

J = 0, S = 0 J = 1.2, S = 1.2

no yes

MET 1

1–1 000

J = 0.006, S = 0.006 yes

MET 2

1–1 000

J = 0.004, S = 0.016 yes

MET 3

1–1 000

J = 0.005, S = 0.005 yes

Disturbance is the major cause, presence of quantifiable stiction. Disturbance and aggressive controller tuning are the major causes. Aggressive controller tuning is the major cause, minor stiction. Disturbance is the major cause. Stiction and disturbance are the major causes. Stiction is the major cause. Disturbance and stiction are the major causes. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes. Stiction, disturbance and aggressive controller tuning are the causes. Disturbance and aggressive controller tuning are the major causes. Stiction is the major cause. Disturbance is the major cause, presence of stiction. Stiction and disturbance are the major causes. Disturbance is the cause. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes. Stiction and disturbance are the major causes.

Explanations and Comments: 1. Data were de-trended (MATLAB’s function detrend). 2. Data have been mean-centred and scaled to unit standard deviation prior to the analysis.

Chapter 14

Conclusions and Future Research Challenges Biao Huang, Mohieddine Jelali and Alexander Horch

14.1 Summary of the Book To close this book, let us reiterate the goal of the book and ponder what we have achieved. As stated in the introduction, the aim of control loops is to maintain processes at the desired operating conditions, safely and efficiently. A poorly performing control loop can result in disrupted process operations, degraded product quality, higher material or energy consumption, and thus decreased plant profitability. Therefore, control loops have been increasingly recognised as important capital assets that should be routinely monitored and maintained. The performance of the controllers, as well as of the other loop components, should thus be improved continuously, ensuring products of consistently high quality. To achieve this goal, each component of the control loops should be monitored. The components of control loops include the actuator, sensor, controller and process. Each component is subject to possible faults or less-than-expected performance. Process and control monitoring has been a topic of active research over the last two decades. Actuator-fault detection along with sensor-fault detection has also been studied extensively in the faultdetection research community. The most important actuator in process industries is the control valve, and it has surprisingly received little attention until recently. The main problem of a control valve is the stiction, which is not only common in process control loops but has also caused the most trouble. Many surveys [10, 26, 92] have indicated that about 20–30% of all control loops oscillate due to valve problems. Biao Huang Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada, e-mail: [email protected] Mohieddine Jelali Department of Plant and System Technology, VDEh Betriebsforschungsinstitut GmbH (BFI), Sohnstraße 65, 40237 Düsseldorf, Germany, e-mail: [email protected] Alexander Horch Group Process and Production Optimization, ABB Corporate Research Germany, Wallstadter Str. 59, 68526 Ladenburg, Germany, e-mail: [email protected]

359

360

B. Huang, M. Jelali and A. Horch

This astonishing number motivates an active research in valve-stiction detection, which is the key step in diagnosing the root-cause of oscillations. This book provides a comprehensive overview of the state-of-the-art stiction-detection and quantification methods developed over the last 10 years. Several key researchers who have been actively engaged in seeking solutions for valve-stiction detection have contributed to this book. The topics covered are widely ranged including stiction modelling, oscillation detection, stiction detection and quantification, and diagnosis of the root-cause of oscillations. The book starts from an introduction to the physical mechanism behind the phenomena of stiction. Mathematical models are the best description of the physical phenomena. There are two types of models to describe the stiction: models that are derived from physical principles and models that are derived from process data. A detailed physical model has a number of unknown parameters, and it has been found difficult to use such a model for the purpose of stiction detection. It, however, provides useful insight into the phenomena that are observed in practice and indicates which effects must be captured in a simplified model. Data-based models are commonly used in monitoring and diagnosis. They are simple and easy to use, but they also have limited ability in extrapolation, particularly when a model is completely determined from data. The data-based models presented in this book follow certain physical principles of the stiction and thus they capture the phenomena better than the completely black-box models, while keeping simplicity in the structure. These models are represented by only two or three parameters. Due to their effectiveness in practical use proven by numerous simulations and applications, this book has devoted its attention to the data-based stiction models only. Chapters 2 and 3 present three data-based stiction models, which are adopted by several subsequent chapters. The three models presented here are complementary but also competing to some extent. This comes as no surprise as the physical phenomena of stiction are complicated and require the knowledge of many physical parameters; the data-based models attempt to reproduce the physical phenomena with only two or three parameters. Different models therefore capture different aspects of the stiction phenomena. Valve stiction in control loops very often causes oscillations in the form of limit cycles. However, oscillations can be introduced by other root-causes, such as poor controller tuning and/or external oscillating disturbances. Therefore, appropriate oscillation detection methods have to be developed. Oscillation detection is certainly not new and has been studied extensively, particularly in the signal-processing community. The most classical methods are based on the power spectrum density (PSD) and the auto-correlation function (ACF). Oscillations appear as marked peaks in PSD. Their magnitude and frequencies can be determined directly from it. However, the PSD can be easily blurred by noise, non-sinusoidal oscillations, and timevarying frequencies. The ACF is a smoothed version of the original time-domain signal; thus it has similar limitations as the original one. Both PSD and ACF of oscillation signals may be visualised by human eyes. However, to be able to visualise the oscillations is different from being able to detect them automatically by a computer program. Computer-based detection is important since a real process consists of hundreds of interacted variables, and routinely visualising PSD or ACF for

14 Conclusions and Future Research Challenges

361

each variable is not possible. The best-known approach to automate the procedure is through monitoring the zero-crossing of the control error signals. Oscillation is a regulated phenomenon (e.g. periodic) and regularity is an indication of oscillation versus randomness of the signal. The regularity can be determined through the measures of the oscillation signals, such as the integral absolute error (IAE), which is the integration of the control error signal between the zero crossings. If such a calculated IAE is regularly repeating itself in terms of frequency and magnitude and its magnitude has a considerable size, an oscillation is detected. Chapter 4 has provided a detailed overview of various oscillation-detection methods. Once oscillations are confirmed, root-cause detection proceeds. It is, however, important to distinguish the internal oscillations from the external ones. The internal oscillations are introduced by stiction or poor tuning of the controller. The external ones can be introduced by disturbances or interaction from other loops. Chapter 12 provides a model-based approach to the isolation of the oscillations among stiction, external disturbance, and poor control tuning. The main challenge is, however, still the stiction detection. Stiction detection can be broadly classified into three categories: descriptive statistics-based, pattern-recognition-based, and model-based. Descriptive statistics methods discussed in this book include: cross-correlation function and histogram plots (Chap. 6), and these methods are simple and easy to apply. The pattern-recognition-based methods include: shape comparison of the OP versus MV plot with reference to the stiction pattern (Chap. 5), curve fitting to distinguish between the sine wave and the triangular wave (Chap. 7); curve fitting to distinguish among the sine wave, the triangular wave and the relay-generated wave (Chap. 8); quantifying the asymmetric shape of the oscillation before and after the oscillation peak (Chap. 9). The model-based methods are based on the system-identification approach for the Hammerstein models. The main difference between one modelbased method and another is in the optimisation and stiction models used. Chapters 10–12 are devoted to model-based stiction detection. Since these methods are model-based, they are all capable of quantifying the stiction and providing more accurate results. Stiction quantification is valuable and can be utilised for short listing the control valves in industrial plants typically containing a large number of control loops. Also these methods can be used to diagnose multiple faults occurring simultaneously in control loops, as presented in Chap. 12. They, however, belong to a class of closed-loop identification methods. The key to the success of model-based methods is therefore the identifiability assurance and appropriate use of the optimisation methods. The stiction provides a favourable condition for closed-loop identification and the identifiability of the closed loops with sticky valves is analysed in detail in Chap. 11. Stiction exhibits complex physical phenomena and the data-based models have tried to capture them with simplified structures. Stiction-detection methods are then developed with adoption of one model or another. Therefore, it will come as no surprise that not all models provide the same prediction of the stiction and not all methods of stiction detection give the same conclusion. It is important to know the strength and weakness of each model and each method. This book provides a comparative study for the three stiction models (Chaps. 2 and 3) using both simulation

362


data and real industrial data that exhibit stiction. More importantly, with almost 100 industrial loops, Chap. 13 provides a comprehensive comparative study for all stiction methods introduced in this book. The study revealed the importance of careful pretreatment of data and check of applicability assumptions for any method. Also, it has been shown that model-based stiction-detection techniques have superior performance, even though they are more complicated and thus computationally demanding.

14.2 Future Research Challenges 14.2.1 Stiction Modelling Data-based models have been the choice for stiction detection and they will remain so. However, the physical phenomena of stiction are complicated and described by many physical parameters. The data-based models attempt to predict the stiction with only a handful of parameters. It has been shown in this book that the data-based models do not always provide consistent results. There is a need to devote some of the future research efforts to a better understanding of stiction physics through theoretical analysis as well as experimental investigation. Better understanding of the stiction mechanism can provide a more precise description of the stiction process. Questions such as, whether stiction exhibits the same behaviour from one type of valve to another, whether there exists a general model to describe the stiction, and whether two or three parameters are sufficient to predict the stiction, may be answered. A general first-principles model should be developed. This model should not be constrained by the number of the parameters used, but it should be verified by a wide range of actual stiction data. This model can then be used as a reference model to guide the developments of data-based models. With a reference model, the structure of data-based models along with their strength and weakness can be determined more easily, and the stiction-detection research community can benefit from a common standard to verify the models developed. Noise introduces difficulty in programming stiction models. For example, the two-parameter stiction model developed by Choudhury et al. (Chap. 2) uses the sign function to determine the change of valve direction. In the case of a noisy signal, this will cause numerical problems. Thus, a filter has been used to reduce the randomness. On the other hand, Kano et al. (Chap. 2) use a first-order transfer function to describe the dynamics of the air chamber of the valve, which is equivalent to a filter. The concept of filtering has been used frequently in modelling and detection throughout the book. The choice of the filtering parameters has largely been heuristic, however, and it is worth further research to provide a better guideline.


363

14.2.2 Oscillation Detection From PSD and ACF to IAE and the spectral-envelope method, oscillation detection has been considered largely as a solved problem (Chap. 4). However, distinguishing the shapes of the oscillations is still a challenging problem. In addition to the square wave, triangular wave, sine wave, and relay wave, there could be other shapes of waves associated with stiction or other loop problems. To detect oscillations or to distinguish among a set of candidate shapes of oscillations, a threshold is needed. Most of the existing threshold values are heuristically determined. More rigorous statistical analysis should be considered. In addition to the difference in the shapes, oscillations could be time varying in terms of frequency and magnitude. Time–frequency methods such as wavelet analysis should also be considered. When dealing with multiple oscillations, bandpass filters have to be used in order to apply time-domain methods such as IAE and zero-crossing to detect the oscillation. The determination of the cut-off frequency for the filter is heuristic and PSD has been used for this purpose. However, the choice of the cut-off frequency can have a great impact on the filtered signal. Once again, further research is needed to provide a more accurate guideline for the filter design. When dealing with intermittent oscillations, data segmentation has to be used. At this point, it should be emphasised that the decision of dividing data into different segments is to be done heuristically by visual inspection and no systematic methods have been used, as pointed out in Chap. 4. Further research in this direction is necessary. Having detected an oscillation, it is important to distinguish between internally and externally generated oscillations. At times, it becomes a challenging task to distinguish between oscillatory signals resulting from external disturbances and aggressive controller tuning as they possess identical properties. Is it possible at all to distinguish in a reliable way? Chap. 12 suggests a model-based solution to this problem. This is an interesting topic. Other more robust statistics-based methods should also be considered. From the applications point of view, detecting oscillations from a few loops may not be a problem, the challenge is to develop an automated solution to monitor hundreds of loops. It is desired that the process-specific knowledge demands should be minimal or null. To be applied on hundreds of control loops in a process plant routinely, the method should be based on just the time series of process variables. If it requires more process knowledge, implementation of such a method may not be feasible. A good oscillation-detection methodology for industrial applications should have the following features (Chap. 4): • Usage of only time-series information of process variables with limited or no additional process knowledge. • Robustness to the high-frequency measurement noise and disturbances. • Ability to handle the presence of multiple and intermittent oscillations. • Amenablity to complete automation without human intervention. Certainly, the techniques recently proposed by Thornhill and co-workers, e.g. [7, 123], for plant-wide disturbance detection and diagnosis should be mentioned,

364


but they can only be seen as first approaches and further research is needed in this direction. An overview of the topic was written by Thornhill and Horch [124].

14.2.3 Stiction Detection and Estimation Stiction detection methods developed so far have all targetted one valve at a time by matching or fitting data to the known stiction patterns. It has not been investigated, however, that, if two or more control valves have the stiction and the loops are interacted, what can be the combined effect of the stiction? What will be the shape of the oscillation then? While the methods described so far work well for single loops, it is of interest to extend the methods to interacting processes with multiloop controllers where oscillations may be caused by one or more sticky valves and propagated to other loops through interaction. This is related to another question, whether we should solve the problem through the bottom-up approach or the topdown approach. Direct analysis of the MV and PV data for a specific valve can lead to a conclusion of stiction or not. Plant-wide disturbance analysis may also lead to the same conclusion. Both aim at finding similar things and have not been really combined. It would be interesting to see if the two approaches can be conducted together to improve the reliability of the analysis. Most of the stiction-detection methods assume that there is an oscillation in the first place. It is, however, well known that stiction does not always introduce oscillations. Chapter 5 has a discussion on detection of stiction without oscillations, but the method proposed there relies on MV–OP data. If there is no oscillation, how to detect the stiction without MV is an open problem. From the practical applications point of view, the stiction-detection method has to be robust, such as insensitivity to noise, different data sampling rate, irregular sampling of data, etc. A common scenario is that a method is proven theoretically and rigorously tested in simulation, but then fails to get adopted in practice because of robustness issues. The main problem is that it is never possible to anticipate all of the uncertainties and unusual phenomena encountered in practical systems. Robustness and reliability turn out to be crucial elements in gaining acceptance of any new technology in most industrial application areas (Chap. 9). Thus improving robustness or developing robust detection and quantification methods is certainly a direction of future research. In this context, the performance of Hammerstein-model-based methods for stiction quantification may be improved when more effective strategies for time-delay estimation can be introduced in the identification schemes. Other additional open problems are also worth investigating. For example, most stiction models and detection methods have been introduced and largely tested on pneumatic valves that are the standard choice in process industry. However, in several industrial sites, hydraulic servo-valves are widely used. Moreover, such valves are employed in combination with hydraulic cylinders. In these systems, the valve, the cylinder or both may suffer from too high a static friction. Therefore, tailored techniques are needed to detect and diagnose stiction in hydraulic servo-systems.


365

As another example of the open problems, it is sometimes difficult to distinguish between stiction problems and some other sensor or actuator faults, just by using SP, PV and OP data. An example in terms of loops CHEM 13–17 has been shown in Chap. 13. Hence, stiction-detection techniques have to be combined with other fault-detection methods to solve this problem. There exist many good stiction-detection methods and new methods continue to appear. As shown in this book, there is no single method that can solve all problems and different methods may not even provide consistent results. An increasing number of the methods would not help practitioners but may introduce more confusion. Therefore, future research should be directed more towards analysis of the existing methods and a clever synthesis of them. As final words for stiction detection, over time, there will be more and more smart valves with positioners and the need for stiction-detection methods will reduce in the long term. For a foreseeable future, however, stiction detection remains as one of the most important research as well as engineering practising activities in process control and instrument engineering.

14.2.4 Stiction Control Once stiction of a valve is confirmed, what should be done next? Leaving it alone, compensating it by control, or even requiring valve repair or replacement? Certainly, the negative effects of stiction may be eliminated by repairing or replacing the valve. In many cases, however, this is not possible because of reasons such as that stated in [35]: (i) it is economically not feasible to stop process operation in order to service the valves, (ii) the type of valve/actuator used may be the problem but it is necessary to use this type for safety considerations, and (iii) replacing the valve/actuator could be too expensive. In these cases, the negative effects of stiction cannot be totally eliminated, but methods for combating the stiction to reduce these effects are beneficial. Although this topic has not been treated in this book, it seems useful to give some hints about techniques for stiction control, which may be improved or extended in future research. Simple and practical measures for combating stiction have been proposed by Gerry and Ruel [35]. These techniques essentially detune the integral effect in the controller; thus reduce the stiction-induced oscillations at the cost of steady-state control errors. Another way to reduce the effect of friction is to compensate for it. Several techniques have been reported for stiction compensation. Perhaps the most prominent method is the “knocker” approach proposed by Hägglund [39]: a dither signal is added to OP to compensate stiction. The effect of the dither is that it introduces extra forces that make the system move before the stiction level is reached. The effect is thus similar to removing the stiction. The effects of dither in systems with dynamic friction were studied by Panteley et al. [91]. Frameworks that integrate stiction-estimation procedures for effective compensation were proposed by Srini-

366


vasan and Rengaswamy [112, 113]. These techniques can be improved by using the data-driven models presented in Chaps. 2 and 5 instead of the (physically unrealistic) one-parameter model used up to now. Systems for motion control typically have a cascade structure with a current loop, a velocity loop and a position loop. Since friction appears in the inner loop it would be advantageous to introduce friction compensation in that loop. The friction force is estimated using some model, and a signal that compensates this force is added to the control signal. Some friction-compensation techniques for servo-hydraulic control systems have also been discussed by Jelali and Kroll [59]. However, comprehensive approaches combining stiction detection, quantification and compensation for these systems are still missing.

Appendix A

Evaluated Industrial Control Loops

The methods presented throughout the book are demonstrated on different data sets from different process industries, including chemicals, pulp and paper mills, commercial building, and metal processing. Some information about these loops are collected in Table A.1.

Table A.1 Information about the industrial control loops analysed throughout the book, particularly in Chap. 13 Loop name BAS 1 BAS 2 BAS 3 BAS 4 BAS 5 BAS 6 BAS 7 BAS 8 CHEM 1 CHEM 2 CHEM 3 CHEM 4 CHEM 5 CHEM 6 CHEM 7 CHEM 8 CHEM 9 CHEM 10 CHEM 11

Industrial field Buildings Buildings Buildings Buildings Buildings Buildings Buildings Buildings Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals

Control loop Temperature control Temperature control Temperature control Pressure control Pressure control Temperature control Temperature control Temperature control Flow control Flow control Temperature control Level control Flow control Flow control Pressure control Pressure control Pressure control Pressure control Flow control

Ts [s] Comments, known problems or visual observations 1 No oscillation 1 No oscillation 3 Intermittent oscillation 3 Intermittent oscillation 1 OP not available 1 Stiction and tight tuning 1 Stiction 60 No oscillation 1 Stiction 1 Stiction 30 Quantisation 1 Tuning problem 1 Stiction 1 Stiction 1 Open-loop data; stiction 1 Open-loop data; stiction 1 Stiction 1 Stiction 1 Stiction

367

368

A Evaluated Industrial Control Loops

Table A.1 (Continued) Loop name CHEM 12 CHEM 13 CHEM 14 CHEM 15 CHEM 16 CHEM 17

Industrial field Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals

Control loop Flow control Analyser control Flow control Pressure control Pressure control Temperature control

Ts [s]

CHEM 18 CHEM 19 CHEM 20 CHEM 21 CHEM 22 CHEM 23 CHEM 24 CHEM 25 CHEM 26 CHEM 27 CHEM 28 CHEM 29 CHEM 30 CHEM 31 CHEM 32 CHEM 33 CHEM 34 CHEM 35 CHEM 36 CHEM 37 CHEM 38 CHEM 39 CHEM 40

Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals Chemicals

Flow control Flow control Flow control Flow control Flow control Flow control Flow control Pressure control Level control Level control Temperature control Flow control Flow control Flow control Flow control Flow control Flow control Flow control Level control Level control Pressure control Pressure control Temperature control

12 12 12 12 12 12 12 12 12 12 12 60 15 15 10 12 10 10 12 12 10 60 60


Chemicals Chemicals Chemicals Chemicals

Temperature control Temperature control Temperature control Temperature control

60 60 60 60

CHEM 45

Chemicals

Pressure control

60

CHEM 46

Chemicals

Pressure control

60

CHEM 47

Chemicals

Pressure control

60

CHEM 48

Chemicals

Pressure control

60

CHEM 49 CHEM 50

Chemicals Chemicals

Pressure control Level control

60 60

1 20 20 20 20 20

Comments, known problems or visual observations Stiction Faulty steam sensor; no stiction Faulty steam sensor; no stiction Interaction (likely); no stiction Interaction (likely); no stiction Faulty steam sensor; no stiction. The OP of CHEM17 is the setpoint to CHEM14. Stiction (likely) Stiction (likely) Stiction (likely) Disturbance (likely) Stiction (likely) Stiction (likely) Stiction (likely) Possible margin stability Stiction (likely) Disturbance (likely) Stiction (likely) Stiction Stiction Stiction (likely) Disturbance (likely) Disturbance (likely) Stiction (likely) Disturbance (likely) Disturbance (likely) Disturbance (likely) Disturbance (likely) No clear oscillation (according to power spectrum) OP saturation, as assessed in [82] Too few cycles; no clear oscillation; OP saturation No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum)

A Evaluated Industrial Control Loops

369

Table A.1 (Continued) Loop name CHEM 51 CHEM 52

Industrial field Chemicals Chemicals

Control loop Level control Level control

Ts [s]


Chemicals Chemicals Chemicals Chemicals

Level control Level control Level control Flow control

60 60 60 60

CHEM 57 CHEM 58

Chemicals Chemicals

Flow control Flow control

60 60

CHEM 59

Chemicals

Flow control

60

CHEM 60 CHEM 61

Chemicals Chemicals

Flow control Flow control

60 60

CHEM 62

Chemicals

Flow control

60

CHEM 63 CHEM 64 PAP 1 PAP 2 PAP 3 PAP 4

Chemicals Chemicals Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers

Flow control Gas flow control Flow control Flow control Level control Concentration control

60 60 1 1 1 1

PAP 5 PAP 6 PAP 7 PAP 8 PAP 9 PAP 10 PAP 11 PAP 12 PAP 13 POW 1 POW 2 POW 3 POW 4 POW 5 MIN 1 MET 1 MET 2 MET 3

Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Pulp and Papers Power Plants Power Plants Power Plants Power Plants Power Plants Mining Metals Metals Metals

Concentration control Level control Flow control Level control Temperature control Level control Level control Level control Level control Level control Level control Level control Level control Level control Temperature control Gauge control Gauge control Gauge control

0.2 1 0.2 5 5 5 15 15 15 5 5 5 5 5 60 0.05 0.05 0.05

60 60

Comments, known problems or visual observations No clear oscillation (according to power spectrum) No clear oscillation No clear oscillation No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) No clear oscillation (according to power spectrum) Stiction Stiction Stiction Deadzone and tight tuning; no stiction Stiction No stiction External disturbance No stiction No stiction Stiction Stiction Stiction Stiction Stiction No stiction Stiction No stiction Stiction External disturbance (likely) External disturbance (likely) No oscillation

Appendix B

Review of Some Non-linearity and Stiction-detection Techniques

The detection and quantification of process non-linearities is useful in diagnosing the root-cause of (plant-wide) oscillations in control loops. The extent of nonlinearity of a process depends on the operating region of the process and the input excitation signals to the process. This section briefly reviews two prominent measures of non-linearity and the corresponding detection methods. The first is based on higher-order statistics, i.e. bicoherence; the second uses surrogate data sets to detect process non-linearity. The descriptions in this chapter are largely adopted from the references [14, 19, 122].

B.1 Bicoherence Method Bispectrum and bicoherence have been used to detect the presence of non-linearity in process data. Two indices for testing the “non-Gaussianity” and “non-linearity” of a time series have been defined by Choudhury et al. [14, 19]. The basic principle is to test the zero-squared bicoherence bic2 ( f1 , f2 ) :=

|B( f1 , f2 )|2 , E{|X( f1 )X( f2 )|2 }E{|X( f1 + f2 )|2 }

(B.1)

which implies that the signal is Gaussian and thereby the signal-generating process is linear. The other is to test for a non-zero constant (squared) bicoherence, which shows that the signal is non-Gaussian but the signal-generating process is linear. Remember that the bispectrum is defined as B( f1 , f2 ) = E{X( f1 )X( f2 )X ∗ ( f1 + f2 )} ,

(B.2)

where X(f 1 ) is the discrete Fourier transform of the test data x(k) at the frequency f 1 , X ∗ (f 1 ) the complex conjugate and E the expectation operator. All frequencies are normalised such that the sampling frequency is 1.

371

372

B Non-linearity and Stiction-detection Techniques

If Welch’s periodogram method is used to estimate the bicoherence, the expectation operator can be replaced with a summation operator over the number M of data samples assuming the process is ergodic: bic2 ( f1 , f2 ) :=

1 M

∗ 2 | M1 ∑M i=1 Xi ( f 1 )Xi ( f 2 )Xi ( f 1 + f 2 )|

2 1 M 2 ∑M i=1 |Xi ( f 1 )Xi ( f 2 )| · M ∑i=1 |Xi ( f 1 + f 2 )|

.

(B.3)

In the estimation of the bicoherence, many spurious peaks arise due to the occurrence of small magnitudes of the denominator used to normalise the bispectrum (Eq. B.1). Therefore, Choudhury et al. [19] suggested to add a small and dynamically adapted constant ε to the denominator to remove these spurious peaks: bic2 ( f1 , f2 ) :=

|B( f1 , f2 )|2 . E{|X( f1 )X( f2 )|2 }E{|X( f1 + f2 )|2 } + ε

(B.4)

The selection of ε depends on the noise level. To obtain the value of ε automatically, it can be chosen as the maximum of the Pth percentiles of the columns of D (the denominator of Eq. B.1). If it is assumed that there will be a maximum of 25% peaks in each column of D, the value of P can be chosen as 75 [19]. A simple way to check the constancy of the squared bicoherence (i.e. the linearity) is to have a look at the 3D squared bicoherence plot and observe the flatness of the plot. This method is, however, tedious and cumbersome when a large number of signals have to be analysed. Based on these approaches, Choudhury et al. [14] have proposed practical automated tests, which are described in the following.

B.1.1 Non-Gaussianity Index To check for the significance of bicoherence magnitude at each individual bifrequency, the following statistical test is suggested [19]: χ2

P(2Kbic2 ( f1 , f2 ) > cα ) = α

(B.5)

or P(bic2 ( f1 , f2 ) >

χ2

cα )=α, 2K

(B.6) χ2

where K is the number of data segments used in the bicoherence estimation and cα the critical value calculated from the central χ 2 distribution table for a significance level of α with two degrees of freedom. Typically, one selects

α = 0.05,

giving

χ2

cα = 5.99 .

The non-Gaussianity index (NGI) is defined as

(B.7)

B.1 Bicoherence Method

NGI :=

373

∑ bic2significant czα − , L 2K

(B.8)

where bic2significant are those bicoherence values that fail the hypothesis test in z

cα Eq. B.5, i.e. bic2 ( f1 , f2 ) > 2K and L the number of bic2significant . Using NGI, the following rule-based decision can be taken:

• If NGI ≤ α , the signal is Gaussian. • If NGI > α , the signal is non-Gaussian.

B.1.2 Non-linearity Index If the squared bicoherence is of a constant magnitude at all bifrequencies in the principal domain, the variance of the estimated bicoherence should be zero. To practically check the flatness of the plot or the constancy of the squared bicoherence , the maximum squared bicoherence can be compared with the average squared bicoherence plus two or three times the standard deviation of the estimated squared bicoherence (depending on the confidence level desired). The automatic detection of this can be performed using the following non-linearity index (NLI): NLI := bic2max − (bic2robust + 2σbic2 ,robust ) ,

(B.9)

where bic2robust and σbic2 ,robust are the robust mean and the robust standard deviation of the estimated squared bicoherence, respectively. They are calculated by excluding the largest and smallest Q% of the bicoherence. A good value of Q may be chosen as 10. Ideally, the NLI should be 0 for a linear process, i.e. the magnitudes of squared bicoherence are assumed to be constant or the surface is flat. This is because if the squared bicoherence is a constant at all frequencies, the variance will be zero and both the maximum and the mean will be same. Therefore, it can be concluded that • If NLI ≤ 0, the signal-generating process is linear. • If NLI > 0, the signal-generating process is non-linear. Since the squared bicoherence is bounded between 0 and 1, the NLI is also bounded between 0 and 1. Whenever possible, a large number of data points (e.g. 4096 samples) have to be used for the non-linearity-detection algorithm. Standard choices of the parameters are: data length (N) of 4096, a segment length of 64, a 50% overlap, Hanning window with a length of 64 and a discrete Fourier transform (DFT) length of 128. However, these values should be adapted from case to case depending on the data available for the analysis; see Table 13.12.

374


B.1.3 Total Non-linearity Index A total non-linearity index (TNLI) has been introduced by Choudhury et al. [19] as a metric or measure for quantifying non-linearities: TNLI := ∑ bic2significant .

(B.10)

TNLI is bounded between 0 and L. TNLI quantifies the total non-linearity present in a time series if it is detected as non-linear by the NGI and NLI defined above. This is particularily important when comparing the extent of non-linearities in different time series to detect the source of non-linearity.

B.1.4 Ellipse Fitting In this method proposed by Choudhury et al. [17], the detection of valve or process non-linearity is first carried out using the aforementioned indices NGI and NLI. Once a non-linearity is detected, then the data are treated by a Wiener filter (given PVf and OPf ) and the PVf –OPf plot, generated from a segment of the data that has regular oscillations, is used to isolate its cause. A signature of valve stiction is when PVf –OPf plot shows cyclic or elliptic patterns; see Fig. B.1. If no such patterns are observed, it is concluded that there are valve problems but these are not due to the stiction. The complete procedure can be found in [17]. As a by-product of this method, apparent stiction can be quantified from the maximum width of the ellipse fitted in the PV–OP plot measured in the direction of OP quantifies stiction, i.e.: Stiction plus deadband S [%] ≈

2ab a2 sin2 α

+ b2 cos2 α

,

(B.11)

where a and b are the length of the minor and major axes of the fitted ellipse, respectively and α is the angle of rotation of the ellipse from the positive x-axis. Fig. B.1 illustrates the ellipse fitting for the PC loop CHEM 10. The quantified stiction is termed as “apparent stiction”, because it may be equal to or different from the actual amount of stiction due to the influence of loop dynamics on PV and OP, in particular due to the effect of the controller to regulate PV and thus smooth the stiction effect [17].

B.2 Surrogates Analysis Surrogate data are synthetic data sets having some of the same properties as the time series under test, for instance, the same power spectrum, but with the phase

B.2 Surrogates Analysis

375

Fig. B.1 Ellipse fitted to the PVf –OPf plot of PC loop CHEM 10

coupling removed by randomisation. Non-linearity-detection methods based on surrogate data consider a key statistic of the time series under test compared to that of a sufficiently large number of surrogates: non-linearity is diagnosed if the statistic significantly differs in the test data; otherwise, the null hypothesis, that a linear model fully explains the data, is expected.

B.2.1 Surrogate Data Generation Typically, surrogate data are generated using the following three-step procedure: Step 1. Calculate the FFT of the test data, which gives amplitude and phase at each frequency: z = FFT(test series) .

(B.12)

Step 2. Randomise the phase at each frequency to be uniformly distributed in [0, 2π], but preserve the asymmetry around frequency 0:

376


⎧ ⎪ ⎪ z(i) jφ ⎨ z(i)e i−1 zsurr = ⎪ z(i) ⎪ ⎩ z(i)e− jφN−i+1

i=1 i = 2, . . . , N/2 . i = N/2 + 1 i = (N/2 + 2), . . . , N

(B.13)

Step 3. Take the inverse Fourier transformation: surrogate data = IFFT(zsurr ) .

(B.14)

B.2.2 Non-linear Predictability Index Predictability of a time trend relative to its surrogate gives the basis for a nonlinearity measure. Prediction errors for the surrogates are considered as a reference probability distribution for the null hypothesis. A non-linear time series is more predictable than its surrogates. A statistic based on a three-sigma test can be formulated using non-linear predictability as [122]: NPI =

Γ¯surr − Γtest , 3σsurr

(B.15)

where Γ test is the mean-squared prediction error of the test data, Γ¯surr the mean or the reference distribution and σ surr its standard deviation. This non-linearity test can be interpreted as follows: • If NPI > 1, then non-linearity is inferred in the data. The higher NPI the more non-linear is the underlying process. • If NPI ≤ 1, the process is considered to be linear. • Negative values in the range −1 ≤ NPI < 0 are not statistically significant and arise from the stochastic nature of the test. • Results giving NPI < −1 do not arise at all because the surrogate sequences, which have no phase coherence are always less predictable than non-linear the time series with phase coherence. The basis of this test proposed by Thornhill [122] is to generate predictions from near neighbours under exclusion of near-in-time neighbours so that the neighbours are only selected from other cycles in the oscillation. When nn nearest neighbours have been identified, then those near neighbours are used to make a H-step-ahead prediction. A sequence of prediction errors can thus be created by subtracting the average of the predictions of the nn nearest neighbours from the observed values. The root mean square (RMS) value of the prediction error sequence is built to give the overall prediction error. The complete algorithm for surrogate analysis can be found in [21, 122].

References

1. Armstrong-Helouvry B, DuPont P, Canudas de Wit C (1994) A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines With Friction. Automatica 30:1083–1138. ˚ om KJ (1991) Assessment of achievable performance of simple feedback loops. Int J 2. Astr¨ Adapt Control Signal Process 5:3–19. ˚ 3. Astrom KJ, Hägglund T (1995) PID Controllers: Theory, Design, and Tuning. ISA, Resarch Triangle, NC. ˚ om K, Hägglund T (2005) Advanced PID Control. ISA, Resarch Triangle, NC. 4. Astr¨ 5. Aubrun C, Robert M, Cecchin T (1995) Fault detection in control loops. Contr Eng Pract 3:1441–1446. 6. Bai EW (2002) Identification of linear systems with hard input nonlinearities of known structure. Automatica 38:853–860. 7. Bauer M, Thornhill NF (2008) A practical method for identifying the propagation path of plant-wide disturbances. J Proc Control 18:707–719. 8. Bendat P, Piersol AG (1971) Random Data: Analysis and Measurement Procedures. Wiley, New York. 9. Beckman D, Jury F (1997) Reducing Process Variability With Control Valves. Internal Report (November), Fisher Controls International Inc. 10. Bialkowski WL (1993) Dreams versus reality: a view from both sides of the gap. Pulp and Paper Canada 94:19–27. 11. Björklund S (2003) A Survey and Comparison of Time Delay Estimation Methods in Linear Systems. PhD Thesis, Lund Institute of Technology, Sweden. 12. Box GEP, Jenkins GM, Reinsel GC (1994) Time Series Analysis: Forecasting and Control. Prentice-Hall, Englewood Clifs, NJ. 13. Choudhury MAAS, Thornhill NF, Shah SL (2004) A data-driven model for valve stiction. Proc IFAC Symposium on Advanced Control of Chemical Processes (ADCHEM), Hong Kong. 14. Choudhury MAAS, Thornhill NF, Shah SL (2004) Diagnosis of poor control loop performance using higher order statistics. Automatica 40:1719–1728. 15. Choudhury MAAS, Thornhill NF, Shah SL (2005) Modelling valve stiction. Contr Eng Pract 13:641–658. 16. Choudhury MAAS, Kariwala V, Shah SL, Douke H, Tadata H, Thornhill NF (2005) A simple test to confirm control valve stiction. Proc IFAC World Congress, Praha, Czech R. 17. Choudhury MAAS, Shah SL, Thornhill NF, Shook D (2006) Automatic detection and quantification of stiction in control valves. Contr Eng Pract 14:1395–1412. 18. Choudhury MAAS, Jain M, Shah SL (2006) Detection and quantification of valve stiction. Proc American Control Conference, Minneapolis, Minnesota, pp 2097-2106.

377

378

References

19. Choudhury MAAS, Shook D, Shah SL (2006). Linear or nonlinear? a bicoherence based metric of nonlinearity measure. Proc IFAC symposium SAFEPROCESS, Beijing, China. 20. Choudhury MAAS, Jain M, Shah SL (2008) Stiction - definition, modelling, detection and quantification. J Proc Control 18:232–243. 21. Choudhury MAAS, Shah SL, Thornhill NF (2008) Diagnosis of Process Nonlinearities and Valve Stiction – Data Driven Approaches. Springer, London. 22. Clarke DW (1995) Sensor, actuator, and loop validation. IEEE Control Syst Mag 15:39–45. 23. Crow EL, Shimizu K (1988) Lognormal Distributions: Theory and Applications. Dekker, New York. 24. Deibert R (1994) Model based fault detection of valves in flow control loops. Proc IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS), Espoo, Finland, pp 445–450. 25. Desborough L, Nordh P, Miller R (2001) Control system reliability: process out of control. Ind Comput 8:52–55. 26. Ender DB (1993) Process control performance: Not as good as you think. Control Eng 9:180– 190. 27. EnTech (1998) EnTech Control Valve Dynamic Specification. Version 3.0. 28. Ettaleb L, Davies MS, Dumont GA, Kwok E (1996) Monitoring oscillations in a multiloop system. In Proc IEEE Int Conf Control Applications, Dearborn, USA, pp 859–863. 29. Fisher-Rosemount (1999) Control Valve Handbook. Fisher Controls International Inc., USA. 30. Fitzgerald B (1995) Control Valve for the Chemical Process Industries. McGraw-Hill, New York. 31. Floyd TL, Buchla DM (1998) Basic Operational Amplifiers and Linear Integrated Circuits. Prentice-Hall. 32. Forsman K (2000) On detection and classification of valve stiction. Proc TAPPI Conf Process Control, Williamsburg, USA. 33. Forsman K, Stattin A (1999) A new criterion for detecting oscillations in control loops. In Proc European Control Conference, Karlsruhe, Germany. 34. Geddes KO, Czapor SR, Labahn G (1992) Algorithms for Computer Algebra. Kluwer, Boston. 35. Gerry J, Ruel M (2001) How to measure and combat valve stiction online. Proc ISA International Fall Conference, Houston, TX. http://www.expertune.com/articles/ isa2001/StictionMR.htm. 36. Goldberg DE (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA. 37. Hägglund T (1995) A control-loop performance monitor. Contr Eng Pract 3:1543–1551. 38. Hägglund T (1999) Automatic detection of sluggish control loops. Contr Eng Pract 7:1505– 1511. 39. Hägglund T (2002) A friction compensator for pneumatic control valves. J Proc Control 12:897–904. 40. Hagen LW, Kahng AB (1997) Combining problem reduction and adaptive multistart: a new technique for superior iterative partitioning. IEEE Trans Computer-Aided Des Integ Circuits Systems 16:709-717. 41. Harris TJ (1989) Assessment of closed loop performance. Can J Chem Eng 67:856–861. 42. He QP, Pottmann M (2003) Detection of Valve Stiction Using Curve Fitting. Internal Report, Process Dynamics and Control, DuPont Engineering. 43. He QP, Wang J, Pottmann M, Qin SJ (2005) A curve fitting method for detecting valve stiction in oscillating control loops. Texas-Wisconsin Modeling and Control Consortium Spring Meeting, Austin, TX. 44. He QP, Wang J, Pottmann M, Qin SJ (2007) A curve fitting method for detecting valve stiction in oscillating control loops. Ind Eng Chem Res 46:4549–4560. 45. Hines M, Montgomery DC (1990) Probability and Statistics in Engineering and Management Science. Wiley, New York. 46. Hoo K, Piovoso M, Schnelle P, Rowan D (2003) Process and controller performance monitoring: overview with industrial applications. Int J Adapt Control Signal Process 17:635–662.

References

379

47. Horch A (1999) A simple method for detection of stiction in control valves. Contr Eng Pract 7:1221–1231. 48. Horch A (2000) Condition Monitoring of Control Loops. PhD thesis, Royal Institute of Technology, Stockholm, Sweden. 49. Horch A (2006) Method and a System for Evaluation of Static Friction. US Patent 7047152, http://www.freepatentsonline.com/7047152.html. 50. Horch A (2007) Benchmarking control loops with oscillations and stiction. In: Ordys A, Uduehi D, Johnson MA (eds) Process Control Performance Assessment, Springer, pp 227– 257. 51. Horch A, Isaksson AJ (1998) A method for detection of stiction in control valves. Proc IFAC Workshop on On-Line Fault Detection and Supervision in The Chemical Process Industry, Lyon, France. 52. Horch A, Isaksson AJ, Forsman K (2000) Diagnosis and characterization of oscillations in process control loops. Proc Control Systems, Victoria, Canada, pp 161–165. 53. Huang B, Kadali R (2008) Dynamic Modeling, Predictive Control, and Performance Monitoring. Springer, London. 54. Huang B, Shah SL (1999) Performance Assessment of Control Loops: Theory and Applications. Springer, London. 55. ISA Subcommittee SP75.05 (1979) Process Instrumentation Terminology. Technical Report ANSI/ISA-S51.1-1979, Instrument Society of America. 56. Isaksson AJ (1997) A Comparison of Some Approaches to Time-delay Estimation. PhD Thesis, Royal Institute of Technology, Stockholm, Sweden. 57. Jelali M (2006) An overview of control performance assessment technology and industrial applications. Contr Eng Pract 14:441–466. 58. Jelali M (2008) Estimation of valve stiction in control loops using separable least-squares and global search algorithms, J Proc Control 18:632–642. 59. Jelali M, Kroll A (2003) Hydraulic Servo-Systems: Modelling, Identification and Control. Springer, London. 60. Jiang H, Choudhury MAAS, Shah SL (2007) Detection and diagnosis of plant-wide oscillations from industrial data using the spectral envelope method. J Proc Control 17:143–155. 61. Jones DR, Perttunen CD, Stuckman BE (1993) Lipschitzian optimization without the Lipschitz constant. J Optimiz Theory Appl 79:157-181. 62. Kaiser G (1994) A Friendly Guide to Wavelets. Birkhäuser, Boston. 63. Kano, M., Maruta, H., Kugemoto, H., Shimizu, K. (2004). Practical model and detection algorithm for valve stiction. Proc IFAC DYCOPS, Cambridge, USA. 64. Kedem B (1994) Time Series Analysis by Higher Order Crossings. IEEE Press, New York. 65. Khalil HK (2001) Non-Linear Systems. Prentice-Hall, Upper Saddle River, NJ. 66. Karnopp D (1985) Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems. Trans ASME 107:100–103. 67. Karra S, Karim MN (2008) Alternative model structure with simplistic noise model to identify linear time invariant systems subjected to non-stationary disturbances. J Proc Control, in press, doi:10.1016/j.jpcont.2008.12.007. 68. Kayihan A, Doyle III FJ (2000) Friction compensation for a process control valve. Contr Eng Pract 8:799–812. 69. Lewis RB, Torczon V (1999) Pattern search algorithms for bound constrained minimization. SIAM J Optimiz 9:1082–1099. 70. Lewis RB, Torczon V (2000) Pattern search methods for linearly constrained minimization. SIAM J Optimiz 10:917–941. 71. Li Y, Ang KH, Chong GCY (2006) Patents, software, and hardware for PID control: an overview and analysis of the current Art. IEEE Control Syst Mag 26:42–54. 72. Litinetski VV, Abramzon BM (1998): MARS – a multistart adaptive random search method for global constrained optimization in engineering applications. Eng Optimiz 30:125-154. 73. Ljung L (1999) System Identification: Theory for the User. Prentice-Hall, Upper Saddle River, NJ.

380

References

74. Lunze J (2007) Automatisierungstechnik. Oldenbourg, München. 75. Luyben WL (2001) Getting more information from relay feedback tests. Ind Eng Chem Res 40:4391–4402. 76. Manum H (2006) Analysis of techniques for automatic detection and quantification of stiction in control loops. Diploma Thesis, Norwegian University of Science and Technology. 77. Manum H, Ghelardoni C, Scali C (2009) Automatic diagnosis of valve stiction by means of a qualitative shape analysis technique. In: Ferrarini L, Veber C (eds) Modeling, Control, Simulation and Diagnosis of Complex Industrial and Energy Systems, ISA, pp 115–136. 78. Marchetti G, Scali C (2000) Use of modified relay techniques for the design of model based controllers for chemical processes. Ind Eng Chem Res 39:3325–3334. 79. Martin GD, Turpin LE, Cline RP (1991) Estimating control function benefits. Hydrocarbon Process June:68–73. 80. Maruta H, Kano M, Kugemoto H, Shimizu K (2005) Modeling and detection of stiction in pneumatic control valves. Trans Soc Instrum Control Engineer E-4:18–26. 81. MATLAB 7 (2008) Getting Started Guide. The MathWorks, Natick, MA. 82. Matsuo T, Tadakuma I, Thornhill NF (2004) Diagnosis of a unit-wide disturbance caused by saturation in manipulated variable. Proc IEEE Advanced Process Control Applications for Industry Workshop, Vancouver, Canada. 83. McMillan GK (1995) Improve control valve response. Chem Eng Prog 91:77–84. 84. Miao T, Seborg DE (1999) Automatic detection of excessively oscillatory feedback control loops. Proc IEEE Confer Control Applications, Hawaii, pp 359–364. 85. Michalewicz Z (1994) Genetic Algorithms + Data Structures = Evolution Programs. Springer, Berlin. 86. Ninness B, Gibson S (2002) Quantifying the accuracy of Hammerstein model estimation. Automatica 38:2037–2051. 87. O’Dwyer A (1996) The Estimation and Compensation of Processes with Time Delays. PhD Thesis, Dublin City University, Ireland. 88. O’Dwyer A (2003) Handbook of PI and PID Tuning Rules. Imperial College Press, London. 89. Olsson H (1996) Friction in Control Valves. PhD Thesis, Lund Institute of Technology, Sweden. 90. Ordys AW, Uduehi D, Johnson MA (2007) Process Performance Control Assessment: From Theory to Practice. Springer, London. 91. Panteley E, Ortega R, Gäfvert M (1998) An adaptive friction compensator for global tracking in robot manipulators. Syst Contr Lett 33:307–313. 92. Paulonis MA, Cox JW (2003) A practical approach for large-scale controller performance assessment, diagnosis, and improvement. J Proc Control 13:155–168. 93. Pipponen J (1996) Controlling processes with nonideal valves: tuning of loops and selection of valves. Proc Control Systems, Chateau, Halifax, Nova Scotia, Canada, pp 179–186. 94. Pitas I, Venetsanopoulos AN (1990) Nonlinear Digital Filters: Principles and Applications. Springer, Berlin. 95. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) Numerical Recipes. Cambridge University Press, New York. 96. Qin SJ (1998) Control performance monitoring: a review and assessment. Comp Chem Eng 23:173–186. 97. Rossi M (2007) Development of Techniques for Closed Loop Performance Monitoring of Chemical Processes. PhD Thesis, University of Pisa, Italy. 98. Rossi M, Scali C (2004) Automatic detection of stiction in actuators: a technique to reduce the number of uncertain cases. Proc IFAC DYCOPS, Cambridge, USA. 99. Rossi M, Scali C (2005) A comparison of techniques for automatic detection of stiction: simulation and application to industrial data. J Proc Control 1:505–514. 100. Ruel M (2000) Stiction: the hidden menace. Control Magazine 13:69–75. http://www. expertune.com/articles/RuelNov2000/stiction.html. 101. Salsbury TI (2005) A practical method for assessing the performance of control loops subject to random load changes. J Proc Control 15:393–405.

References

381

102. Salsbury TI (2006) Control performance assessment for building automation systems. Proc IFAC Workshop on Energy Saving Control in Plants and Buildings, Bansko, Bulgaria. 103. Salsbury TI, Singhal A (2005) A new approach for ARMA pole estimation using higher-order crossings. Proc Am Control Conf, Portland, USA. 104. Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least squares procedures. Analytical Chemistry 36:1627–1639. 105. Scali C, Ghelardoni C (2008) An improved qualitative shape analysis technique for automatic detection of valve stiction in flow control loops. Contr Eng Pract 16:1501–1508. 106. Seborg J, Viberg M (1997) Separable non-linear least-squares minimization – possible improvements for neural fitting. Proc IEEE Workshop on Neural Nets for Sig Pro 7, pp 345–354. 107. Seborg DE, Edgar TF, Mellichamp DA (2004) Process Dynamics and Control. Wiley. 108. Sharif MA, Grosvenor RI (1998) Process plant condition monitoring and fault diagnosis. Proc Instn Mec Engrs 212(Part E):13–30. 109. Siegel R, John RH (1992) Thermal Radiation Heat Transfer. Taylor & Francis, New York. 110. Singhal A, Salsbury TI (2005) A simple method for detecting valve stiction in oscillating control loops. J Proc Control 15:371–382. 111. Söderström T, Stoica P (1988) System Identification. Prentice-Hall, New York. 112. Srinivasan R, Rengaswamy R (2005) Stiction compensation in control loops: a framewok for integrating stiction measure and compensation. Industrial and Engineering Chemistry Research 44:9164–9174. 113. Srinivasan R, Rengaswamy R (2008) Approaches for efficient stiction compensation in process control valves. Comput Chem Eng 32:218–229. 114. Srinivasan R, Rengaswamy R, Miller R (2005) Stiction compensation in process control loops: A framework for integrating stiction measure and compensation. Ind Eng Chem Res 44:9164–9174. 115. Srinivasan R, Rengaswamy R, Miller R (2005) Control loop performance assessment. 1. A qualitative approach for stiction diagnosis. Ind Eng Chem Res 44:6708–6718. 116. Srinivasan R, Rengaswamy R, Narasimhan S, Miller R (2005) Control loop performance assessment. 2. Hammerstein model approach for stiction diagnosis. Ind Eng Chem Res 44:6719–6728. 117. Stenman A, Gustafsson F, Forsman K (2003) A segmentation-based method for detection of stiction in control valves. Int J Adap Control Signal Proc 17:625–634. 118. Stoffer DS, Tyler DE, McDougall AJ (1993) Spectral analysis for categorical time series: scaling and spectral envelope. Biometrika 80:611–622. 119. Stoffer DS, Tyler DE, Wendt DA (2000) The spectral envelope and its applications. Statist Sci 15:224–253. 120. Stribeck R (1902) Die wesentlichen Eigenschaften der Gleit- und Rollenlager. Z Ver Dtsch Ing XXXXVI:1341–1348. 121. Taha O, Dumont GA, Davies MS (1996) Detection and diagnosis of oscillations in control loops. Proc IEEE Conference on Decision and Control, Kobe, Japan, pp 2432–2437. 122. Thornhill NF (2005) Finding the source of nonlinearity in a process with plant-wide oscillation. IEEE Trans Contr Syst Technol 13:434–443. 123. Thornhill NF (2007) Locating the source of a disturbance. In: Ordys A, Uduehi D, Johnson MA (eds) Process Control Performance Assessment, Springer, pp 199–225. 124. Thornhill NF, Horch A (2007) Advances and new directions in plant-wide disturbance detection and diagnosis. Contr Eng Pract 15:1196–1206. 125. Thornhill NF, Hägglund T (1997) Detection and diagnosis of oscillation in control loops. Contr Eng Pract 5:1343–1354. 126. Thornhill N, Cox J, Paulonis M (2003) Diagnosis of plant-wide oscillation through datadriven analysis and process understanding. Contr Eng Pract 11:1481–1490. 127. Thornhill NF, Huang B, Zhang H (2003) Detection of multiple oscillations in control loops. J Proc Control 13:91–100. 128. Torn A, Zilinskas A (1988) Global Optimization. Springer, Berlin. 129. Thyagarajan T, Yu CC (2003) Improved autotuning using the shape factor from relay feedback. Ind Eng Chem Res 40:4425–4440.

382

References

130. Townsend WT, Salisbury JK (1987) The Effect of Coulomb Friction and Sticktion on Force Control. Proc IEEE Internat Confer Robotics & Automation, Raleigh, NC, pp 883–889. 131. Whalen BR (1983) Basic Instrumentation. Petroleum Extension Service (PETEX), Austin, TX. 132. Wallén A (1997) Valve diagnostics and automatic tuning. Proc American Control Conference, Albuquerque, New Mexico, pp 2930–2934. 133. Williams C (1950) On the choice of the number and width of classes for chi-square test of goodness of fit. J Am Statist Assoc 45:77–86. 134. Yamashita Y (2006) An automatic method for detection of valve stiction in process control loops. Contr Eng Pract 14:503–510. 135. Yu CC (1999) Autotuning of PID Controllers. Springer, London. 136. Zang X, Howell J (2004) Comparison of methods to identify the root cause of plant-wide oscillations. Proc IEEE Conf Control Applications, Taipei, Taiwan, pp 695–700. 137. Zhu Y (2001) Multivariable System Identification for Process Control. Elsevier, Oxford.

Contributor Biographies

M.A.A. Shoukat Choudhury received his Ph.D. in Process Control from the University of Alberta, Canada. He obtained both his B.Sc. Engineering and M.Sc. Engineering degrees from Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh. He has recently published a book entitled Diagnosis of Process Nonlinearities and Valve Stiction – data driven approaches. He possesses an international patent on Methods for Detection and Quantification of Control Valve Stiction. He has also co-authored more than 25 refereed papers in international journals and conference proceedings. His main research interests include modelling and diagnosis of stiction in control valves, data-quality assessment, control-loop performance assessment, condition-based monitoring, and plant-wide oscillations diagnosis. Q. Peter He received his B.Sc. degree in chemical engineering from Tsinghua University, Beijing, China, in 1996 and M.Sc. and Ph.D. degrees in chemical engineering in 2002 and 2005 from the University of Texas, Austin. He is currently an assistant professor at Tuskegee University. His research interests are in the general areas of process modelling, monitoring, optimization and control, with special interests in the modelling and optimisation, fault detection and classification of batch processes such as semiconductor manufacturing and pharmaceutical processes. He is also interested in biological system modelling and disease diagnosis. He has had over 3 years of experience in the semiconductor and chemical industries. Alexander Horch received the M.Sc. degree in Engineering Cybernetics from the University of Stuttgart, Germany, and the Ph.D. degree in Automatic Control from the Royal Institute of Technology, Stockholm, Sweden. He has been with ABB Corporate Research Germany since 2001. He is working in the area of process and production optimization in the process industries, and is one of the leading experts in control performance monitoring. Since 2006, he has been group leader at ABB Corporate Research Germany.

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384


Biao Huang obtained his Ph.D. degree in Process Control from the University of Alberta, Canada, in 1997, M.Sc. degree in 1986 and B.Sc. degree in 1983 in Automatic Control from the Beijing University of Aeronautics and Astronautics. He joined the University of Alberta in 1997 as an Assistant Professor in the Department of Chemical and Materials Engineering, and is currently a full Professor. He is a recipient of Germany’s Alexander von Humboldt Research Fellowship, Canadian Chemical Engineer Society’s Syncrude Canada Innovation Award, University of Alberta’s Killam Professorship and McCalla Professorship, Petro-Canada Young Innovator Award, and a best paper award from Journal of Process Control. His research interests include: process control, system identification, control-performance assessment, Bayesian methods, fuel-cell modelling and control, with publications of two books in control-loop performance monitoring and subspace methods, respectively. He is currently an Associate Editor for Control Engineering Practice and Canadian Journal of Chemical Engineering. Mohieddine Jelali obtained his Dipl.-Ing. and Dr.-Ing. degrees in Mechanical Engineering from the University of Duisburg, Germany, in 1993 and 1997, respectively. Before joining VDEh-Betriebsforschungsinstitut (BFI) in 1999, he worked for three years with Mannesmann Demag Metallurgy as an R&D Engineer. He is currently working as Project Group Leader and Vice-Head of the Department of Plant and System Technology at BFI. His has submitted a “Habilitation” (Post-doc) thesis on control performance monitoring and optimisation to the University of DuisburgEssen, in January 2009. He has been working for more than a decade in the fields of advanced control and control-performance monitoring as initiator and coordinator of multinational research projects in the field of metal processing. He has several patents and has published more than 40 articles. He is the principal author of the book Hydraulic Servo-systems: Modelling, Identification and Control, Springer, 2002. Manabu Kano received the B.Sc. degree in 1992, M.Sc. degree in 1994, and Dr.Eng. degree in 1999, all in chemical engineering from Kyoto University. He is an Associate Professor of Kyoto University, Japan. He was an Instructor at Kyoto University from 1994 to 2004, and visiting scholar at Ohio State University, US, from 1999 to 2000. Dr. Kano is the recipient of the Research Award for Young Investigators of “Naito Prize” from SCEJ in 2000, the Best Paper Award of “Takeda Prize” from SICE in 2005, the Instrumentation, Control and System Engineering Research Award from ISIJ in 2007, and others. He is a member of SCEJ, SICE, ISIJ, ISCIE, and AIChE. M. Nazmul Karim is a Professor, and the Department Chair of Chemical Engineering at Texas Tech University, Lubbock, Texas. He received his B.Sc. (Honors) degree in Chemical Engineering from the Bangladesh University of Engineering and Technology, Dhaka. He earned his MSc. degree in Control Engineering and Ph.D. degree in Chemical Engineering from UMIST, UK. Before joining Texas Tech University as the Chair of Chemical Engineering in 2004, he taught at Colorado State


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University for more than twenty years. Currently, he is the co-director of the Texas Tech University Process Control and Optimization Consortium (PCOC), which has ten industrial members. He has co-authored over eighty refereed journal papers and published over two hundred conference papers, and has given numerous invited and keynote talks in professional meetings and universities. He has co-authored a textbook Chemical and Bioprocess Control, in 2006. He has also co-edited a book Modeling and Control of Biotechnical Processes, in 1992, with Professor Gregory Stephanopoulos (MIT). He is a Fellow of the American Institute of Chemical Engineers. Srinivas Karra recieved the B.Sc. and M.Sc. in chemical engineering from the National Institute of Technology Warangal and the Indian Institute of Technology Bombay in 2003 and 2005, respectively. Now he is working for a doctoral degree in chemical engineering at Texas Tech University specialising in advanced process control and optimisation. His research interests lie in the areas of first-principles modelling, system identification, model-based fault detection and isolation, and advanced process control of complex processes. He is a member of the American Institute of Chemical Engineers, and the Institute of Electrical and Electronics Engineers Control Systems Society. Hidekazu Kugemoto received the B.Sc. degree in 1990 and the M.Sc. degree in 1992 from Tokyo Metropolitan Institute of Technology. He is a senior research associate of process systems engineering in Sumitomo Chemical Co., Ltd. He has researched the advanced control and the process monitoring for chemical plants. He is the recipient of the Best Paper Award of “Takeda Prize” from SICE in 2005. Kwan H. Lee received his B.Sc. and M.Sc. degree in control and instrumentation engineering from University of Seoul, Seoul, Korea, in 1997 and 1999, respectively. He received his Ph.D. from the School of Electrical Engineering and Computer Science of Seoul National University, Seoul, Korea, in 2004. From 2004 to 2005, he was a visiting scholar at Purdue University, West Lafayette, IN. He is currently a post-doctoral fellow at University of Alberta. His main research interests include economic performance analysis of advanced process control, control-loop performance monitoring, and optimisation. S. Joe Qin obtained his B.Sc. and M.Sc. degrees in Automatic Control from Tsinghua University in Beijing, China, in 1984 and 1987, respectively, and his Ph.D. degree in Chemical Engineering from University of Maryland in 1992. Dr. Qin is a professor at the Mork Family Department of Chemical Engineering and Materials Science, University of Southern California (USC). Prior to joining the USC, he worked as a professor in Chemical Engineering at the University of Texas at Austin and as Principal Engineer at Fisher-Rosemount. His research interests include system identification, process monitoring and fault diagnosis, model predictive control, semiconductor process control, and control performance monitoring. He is currently an Associate Editor for Journal of Process Control and a Member of the Editorial

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Board for Journal of Chemometrics. He served as an Editor for Control Engineering Practice and an Associate Editor for IEEE Transactions on Control Systems Technology. Zhengyun Ren received both the B.Sc. and M. Sc. degrees from Liaoning Shihua University, Fushun, Liaoning, China, and the Ph.D. degree from Shanghai Jiaotong University, Shanghai, China. From 1990 to 1998, he was a Process Engineer in PetroChina Company Limited, China. He was a post-doctoral fellow with the University of Alberta from 2006 to 2007. He is now an Associate Professor in the Department of Automation, Donghua University, Shanghai, China. His research interests are in the areas of advanced process control, process modelling, and process analysis. He has finished almost 20 practical control projects successfully in the area of chemical, refinery, steel, tobacco, and biology industries. Maurizio Rossi received the Laurea Degree in Chemical Engineering at the University of Pisa in 2002. He received the Philiosophy Degree in Chemical Engineering at the University of Pisa in 2006, starting a collaboration with the university of Alberta in Edmonton (Canada). He worked from 2005 to 2006 in Huntsman Tioxide as process Engineer and control Engineer. Currently, he is working for AspenTech as a control Engineer in the field of Advanced Process Control implementation in refineries, ranging from CDU, to FCC, HDS, splitters, hydrocraking etc. He covers the position of Project Manager and Lead Engineer for some of the projects under development. He is the author of several papers published in international journals and conferences proceedings. Timothy I. Salsbury is a Principal Member of the Research Staff at Johnson Controls in Milwaukee, Wisconsin, USA. His current research interests are controlperformance assessment, fault detection, adaptive control, optimisation, and statistical data analysis. He has a Ph.D. degree from Loughborough University of Technology in the UK, with parts of his graduate research carried out at the University of Oxford and also at the Massachusetts Institute of Technology. Previous positions include Research Scientist at the National Research Center of Finland and Research Fellow at Lawrence Berkeley National Laboratory in California. He has several patents and has authored more than 50 articles. Claudio Scali is Full Professor of Chemical Engineering at the University of Pisa (I), with teaching activity in Process Control and Chemical Plants Design. His present research interests are in the areas of performance assessment of industrial controllers, operability analysis of integrated plants. In the past, he was Visiting Professor at the California Institute of Technology, Pasadena (CA, USA) and at the Laboratoire des Systems Automatiques, University of Picardie, Amiens (F). Presently, Dr. Scali is Associate Editor of Journal of Process Control and a member of the CPC scientific committee of IFAC. He is responsible for organization of Ph.D. schools, permanent courses for professional advancement and for research contracts with leading Italian companies in the petrochemical and energy fields.


387

Sirish L. Shah has been with the University of Alberta since 1978, where he currently holds the NSERC-Matrikon-Suncor-iCORE Senior Industrial Research Chair in Computer Process Control. He was the recipient of the Albright & Wilson Americas Award of the Canadian Society for Chemical Engineering (CSChE) in recognition of distinguished contributions to chemical engineering in 1989, the Killam Professorship in 2003 and the D.G. Fisher Award of the CSChE for significant contributions in the field of systems and control. He has held visiting appointments at Oxford University and Balliol College as a SERC fellow, Kumamoto University (Japan) as a senior research fellow of the Japan Society for the Promotion of Science (JSPS) , the University of Newcastle, Australia, IIT-Madras India and the National University of Singapore. The main area of his current research is process and performance monitoring, system identification and design and implementation of soft sensors. He has co-authored two books, the first entitled Performance Assessment of Control Loops: Theory and Applications, and a more recent book titled Diagnosis of Process Nonlinearities and Valve Stiction: Data Driven Approaches. Ashish Singhal is currently a member of the Advanced Controls and Operations Research group at Praxair R&D in Tonawanda, NY. His primary research interests include multivariable control system design, model predictive control, process monitoring and statistical analysis. He has published several research papers in international journals and conferences and has five patents issued or pending based on his work. Prior to working for Praxair, Dr. Singhal was with the Building Efficiency Research Group at Johnson Controls, Inc. He received a Ph.D. degree in Chemical Engineering from the University of California, Santa Barbara, in 2002. Nina F. Thornhill was educated as an undergraduate at Oxford University (1976) and received M.Sc. and Ph.D. from Imperial College London and University College London. She worked for ICI from 1976 to 1982 and joined University College London as a lecturer in 1984. She is currently a professor in the Department of Chemical Engineering at Imperial College London where she has held the ABB/RAEng Chair of Process Automation since 2007. Her research interests are process data analysis and performance assessment with applications in oil and gas, refining and chemicals and electrical transmission systems. Jin Wang received the B.Sc. degree in chemical engineering from Tsinghua University, Beijing, China, in 1994, and the Ph.D. degree in chemical engineering from the University of Texas at Austin in 2004. From 2002 to 2006, she was a Process Development Engineer and later a Senior Process Development Engineer at Advanced Micro Devices, Inc. Since 2006, she has been an Assistant Professor in the Department of Chemical Engineering at Auburn University. Her research interest consists of two major areas: systems biology with focus on modelling of metabolic networks, and manufacturing process monitoring, control and optimisation, with special interest in system identification, control-performance monitoring, fault detection and classification. She holds twelve US patents.

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Yoshiyuki Yamashita received the B.Sc., M.Sc., and Dr.Eng. degrees, all in Chemical Engineering from Tohoku University, in 1982, 1984, and 1987, respectively. He is a Professor at Tokyo University of Agriculture and Technology (TUAT), Japan. Before joining TUAT, he was a Research Associate at Tohoku University from 1987 to 1993, and Associate Professor at Tohoku University from 1993 to 2007. He is a recipient of the SCEJ Young Investigator Researcher Award (1993) and Outstanding Paper Awards of JCEJ (2000 and 2008). He is currently an Editor in Chief for Journal of Chemical Engineering of Japan (JCEJ).

Index

Area-based stiction detection, 185 hypothesis test, 190 index, 186 industrial study, 200 noise band, 193 procedure, 188 simulation study, 198 theoretical basis, 187 zero-crossing, 195 Bicoherence definition, 371 estimation, 372 Bispectrum definition, 371 Cross-correlation function (CCF), 117 Cross-correlation method assumptions, 117 industrial study, 120 measures, 118 theoretical explanation, 126 Curve fitting, 151 industrial study, 159 simulation study, 154, 155 sinusoidal, 152 stiction index, 153 triangular, 153 Ellipse fitting, 374 Friction model, 8, 23, 41 Histogram-based stiction detection camel distribution, 139 differentiation and filtering, 133 Gaussian distribution, 136

industrial study, 142 procedure, 141 sample histogram, 135 Non-Gaussianity index (NGI), 372 Non-linear predictability index (NPI), 376 Non-linearity index (NLI), 373 Oscillation fractional power, 77 index, 72, 74, 77 period, 65 power, 66 regularity, 66, 77 root-causes, 62, 268 Oscillation characterisation auto-covariance function (ACF), 64 period, 65 power, 66 power spectrum, 65 regularity, 66 strength, 65 Oscillation detection auto-correlation function (ACF), 74, 76 dampened oscillation, 85 decay ratio, 74 describing function, 68 frequency, 68 graphical user interface (GUI), 319 industrial study, 94 integral of absolute error (IAE), 69, 72 intermittent oscillations, 90 multiple faults, 271 multiple oscillations, 87, 96, 97 one predominant oscillation, 84 power spectrum, 67 procedure, 93 389

390 SEA refinery case, 97 signal with coloured noise, 83 simulation study, 81 spectral envelope, 78 zero-crossing, 72, 76 Oscillation-diagnosis procedure, 321 Pneumatic control valve, 4, 23 Power spectrum, 65 Preprocessing bandpass filter, 88 digital FIR filter, 83 low-pass filter, 73, 133 median filter, 192 noise variance, 194, 195 second-order filter, 193 Wiener filter, 87 Regularity factor, 77 Relay-based stiction detection, 170 fitting procedure, 172 index, 171 industrial study, 178 simulation study, 175 Spectral envelope, 78 statistical procedure, 80 statistical test, 79 Stiction apparent, 374 asymmetric, 172 definition, 5 describing function, 9 index, 105, 106, 153, 171, 175, 186 limit cycles, 9, 11 Nyquist plot, 11 Stiction compensation, 365 Stiction detection graphical user interface (GUI), 260 area-based, 185 comparative study, 302 cross-correlation function (CCF), 117 curve fitting, 151 graphical user interface (GUI), 319 histogram-based, 133 industrial study, 111 parallelogram, 105 qualitative shape analysis, 106 relay-based, 170 simulation study, 108 Stiction identification ARMAX, 208, 210, 270 EARMAX, 272, 273

Index non-stationary disturbances, 271 simulation study, 275 Stiction index (SI), 105, 106, 153, 171, 175, 186 Stiction model Choudhury, 25, 187 comparison, 32, 40, 43, 56 data-driven, 25, 29, 32, 38, 49 general, 209 He, 38, 49 Kano, 29 one-parameter, 24 physics-based, 23, 41 relay, 187 simulation, 28, 40, 51 Stenman, 24 three-parameter, 49 two-parameter, 25, 29, 38, 237 Stiction pattern, 13 Stiction quantification comparative study, 256 contour map, 237, 240 ellipse fitting, 374 genetic algorithms, 213 global search, 213, 238 grid-based optimisation, 278 Hammerstein model, 208, 211, 235, 269, 271 identifiability, 241, 243, 278 identification, 210, 211, 236, 271 industrial study, 221, 251, 290 model structure selection, 215 multi-start adaptive random search, 238 pattern search, 214 persistence of excitation, 279 procedure, 211, 218, 236, 271 simulation study, 219, 245, 281 Surrogate analysis, 374 data, 374 data generation, 375 Total non-linearity index (TNLI), 374 Valve backlash, 4 deadband, 5 deadzone, 5 hysteresis, 4 MV–OP plot, 8 parameters, 4 stiction, 6

Other titles published in this series (continued): Soft Sensors for Monitoring and Control of Industrial Processes Luigi Fortuna, Salvatore Graziani, Alessandro Rizzo and Maria G. Xibilia Adaptive Voltage Control in Power Systems Giuseppe Fusco and Mario Russo

Magnetic Control of Tokamak Plasmas Marco Ariola and Alfredo Pironti Real-time Iterative Learning Control Jian-Xin Xu, Sanjib K. Panda and Tong H. Lee

Advanced Control of Industrial Processes Piotr Tatjewski

Deadlock Resolution in Automated Manufacturing Systems ZhiWu Li and MengChu Zhou

Process Control Performance Assessment Andrzej W. Ordys, Damien Uduehi and Michael A. Johnson (Eds.)

Model Predictive Control Design and Implementation Using MATLAB® Liuping Wang

Modelling and Analysis of Hybrid Supervisory Systems Emilia Villani, Paulo E. Miyagi and Robert Valette

Fault-tolerant Flight Control and Guidance Systems Guillaume Ducard

Process Control Jie Bao and Peter L. Lee

Predictive Functional Control Jacques Richalet and Donal O’Donovan

Distributed Embedded Control Systems Matjaž Colnarič, Domen Verber and Wolfgang A. Halang

Fault-tolerant Control Systems Hassan Noura, Didier Theilliol, Jean-Christophe Ponsart and Abbas Chamseddine

Precision Motion Control (2nd Ed.) Tan Kok Kiong, Lee Tong Heng and Huang Sunan

Control of Ships and Underwater Vehicles Khac Duc Do and Jie Pan

Optimal Control of Wind Energy Systems Iulian Munteanu, Antoneta Iuliana Bratcu, Nicolaos-Antonio Cutululis and Emil Ceangǎ Identification of Continuous-time Models from Sampled Data Hugues Garnier and Liuping Wang (Eds.) Model-based Process Supervision Arun K. Samantaray and Belkacem Bouamama Diagnosis of Process Nonlinearities and Valve Stiction M.A.A. Shoukat Choudhury, Sirish L. Shah, and Nina F. Thornhill

Stochastic Distribution Control System Design Lei Guo and Hong Wang Publication due March 2010 Advanced Control and Supervision of Mineral Processing Plants Daniel Sbárbaro and René del Villar (Eds.) Publication due March 2010 Active Braking Control Design for Road Vehicles Sergio M. Savaresi and Mara Tanelli Publication due April 2010 Nonlinear and Adaptive Control Design for Induction Motors Riccardo Marino, Patrizio Tomei and Cristiano M. Verrelli Publication due April 2010

Detection and Diagnosis of Stiction in Control Loops: State of the Art and Advanced Methods (Advances in Industrial Control)

Detection and Diagnosis of Stiction in Control Loops: State of the Art and Advanced Methods (Advances in Industrial Control)

Diagnosis of Process Nonlinearities and Valve Stiction: Data Driven Approaches (Advances in Industrial Control)

Nonlinear Control of Vehicles and Robots (Advances in Industrial Control)

Drives and Control for Industrial Automation (Advances in Industrial Control)

Control of Traffic Systems in Buildings (Advances in Industrial Control)

Advanced Control of Industrial Processes: Structures and Algorithms (Advances in Industrial Control)

Precision Motion Control: Design and Implementation (Advances in Industrial Control)

Fuzzy Logic, Identification and Predictive Control (Advances in Industrial Control)

Model Predictive Control of Wastewater Systems (Advances in Industrial Control)

Fuzzy Logic, Identification and Predictive Control (Advances in Industrial Control)

Advanced Control and Supervision of Mineral Processing Plants (Advances in Industrial Control)

Advanced Industrial Control Technology

Soft Sensors for Monitoring and Control of Industrial Processes (Advances in Industrial Control)

Advances in Dynamics and Control

Advances in Control and Communication

Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)

Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)

Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)

Dry Clutch Control for Automotive Applications (Advances in Industrial Control)

Advanced Methods in Adaptive Control for Industrial Applications

Characterizers for control loops

Mycotoxins in Food: Detection and Control

Mycotoxins in Food: Detection and Control

Mycotoxins in Food: Detection and Control

State Variable Methods in Automatic Control

Stochastic Processes, Estimation, and Control (Advances in Design and Control)

Receding Horizon Control: Model Predictive Control for State Models (Advanced Textbooks in Control and Signal Processing)

Predictive Functional Control: Principles and Industrial Applications (Advances in Industrial Control)

Modelling and Control of Mini-Flying Machines (Advances in Industrial Control)

Control and Monitoring of Chemical Batch Reactors (Advances in Industrial Control)

Detection and Diagnosis of Stiction in Control Loops: State of the Art and Advanced Methods (Advances in Industrial Control)