Advances in Industrial Control

Series editors

Michael J. Grimble, Glasgow, UKMichael A. Johnson, Oxford, UK

More information about this series at http://www.springer.com/series/1412

M. Chidambaram • Nikita Saxena

Relay Tuning of PIDControllersFor Unstable MIMO Processes

123

M. ChidambaramDepartment of Chemical EngineeringIndian Institute of Technology MadrasChennai, Tamil NaduIndia

Nikita SaxenaDepartment of Chemical EngineeringIndian Institute of Technology MadrasChennai, Tamil NaduIndia

ISSN 1430-9491 ISSN 2193-1577 (electronic)Advances in Industrial ControlISBN 978-981-10-7726-5 ISBN 978-981-10-7727-2 (eBook)https://doi.org/10.1007/978-981-10-7727-2

Library of Congress Control Number: 2017962064

© Springer Nature Singapore Pte Ltd. 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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This work is dedicated to the Almighty,Our family&Our friends

M. ChidambaramNikita Saxena

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technologytransfer in control engineering. The rapid development of control technology has animpact on all areas of the control discipline. New theory, new controllers, actuators,sensors, new industrial processes, computer methods, new applications, newphilosophies..., new challenges. Much of this development work resides in indus-trial reports, feasibility study papers and the reports of advanced collaborativeprojects. The series offers an opportunity for researchers to present an extendedexposition of such new work in all aspects of industrial control for wider and rapiddissemination.

Modern proportional-integral-derivative (PID) controller-tuning rules have theirorigin in the era of implementation using operational amplifiers. It was Ziegler andNichols who initiated procedures for process controller tuning in their landmarkpapers of the early 1940s. Around the mid-1980s, researchers were looking toexploit growing online computing power and to devise methods to automate PIDcontroller tuning. Expert systems was one avenue explored; however, it was thework of Åström and Hägglund, as captured in their US patent of 1985 for example,that introduced control engineers and academics to the elegant technique of therelay experiment for the tuning of PID controllers.

The Editors of the Advances in Industrial Control monograph series are verycommitted to the idea of transferring advanced control techniques into industry.A watching brief of control trends led to the realization that PID control was stillone of the most important controls implemented in the process control field andfurthermore that there was a very active academic and engineering research com-munity for this control method. Consequently over the years, monographs on PIDcontrol were keenly sought for the Advances in Industrial Control series. Thus, theseries now contains a substantial body of published monographs in this fieldincluding the following:

• Autotuning of PID Controllers by Cheng-Ching Yu (e-ISBN 978-1-4471-3636-1[first edition, Advances in Industrial Control], 1999) or (ISBN 978-1-84628-036-8[second edition, stand-alone], 2006);

vii

• Advances in PID Control by Kok K. Tan, Qing-Guo Wang and Chang C. Hangwith Tore J. Hägglund (ISBN 978-1-4471-1219-8, 1999);

• Practical PID Control by Antonio Visioli (ISBN 978-1-84628-585-1, 2006);• Industrial Process Identification and Control Design by Tao Liu and Furong

Gao (ISBN 978-0-85729-976-5, 2012);• Control of Integral Processes with Dead Time by Antonio Visioli and

Qing-Chang Zhong (ISBN 978-0-85729-069-4, 2011);• PID Control in the Third Millennium edited by Ramon Vilanova and Antonio

Visioli (ISBN 978-1-4471-2424-5, 2012);• Non-Parametric Tuning of PID Controllers by Igor Boiko (ISBN

978-1-4471-4464-9, 2012);• Model-Reference Robust Tuning of PID Controllers by Victor M. Alfaro and

Ramon Vilanova (ISBN 978-3-319-28211-4, 2016);

and with Springer as a freestanding text:

• PID Control edited by Michael A. Johnson and Mohammad H. Moradi (ISBN978-1-85233-702-5, 2005)

This field of control is still very active as evidenced by the series of IFACsymposia and conferences devoted to PID control. Indeed at the time of writing thisForeword, preparations are well under way for the 3rd IFAC Conference onAdvances in Proportional–Integral–Derivative Control to be held at Ghent,Belgium, in May 2018.

This new Advances in Industrial Control volume Relay Tuning of PIDControllers: For Unstable MIMO Processes by authors M. Chidambaram and N.Saxena continues the Editors’ search and support for contributions to the PIDcontroller-tuning research. The authors are from the Department of ChemicalEngineering at the Indian Institute of Technology Madras, Chennai. The researchreported investigates the use of the relay experiment for two process themes:unstable processes and multivariable systems. Separately these are difficult systemcharacteristics to work with, but the authors have also put them together to createreally challenging PID control design problems. The monograph contains usefulreview material, new relay tuning results and many comparative simulation studiesof their new procedures.

Industrial Control CentreGlasgow, Scotland, UK

M. J. GrimbleM. A. Johnson

viii Series Editors’ Foreword

Preface

This work intends to develop a controller tuning method for the scalar and themultivariable systems, particularly the unstable processes. The book is divided intotwo sections: Chaps. 1–6 comprise Section I, whereas Chaps. 7–9 compriseSection II. The first section discusses the relay auto-tuning method and theimplementation issues. The method is proposed to incorporate the higher-orderharmonics while determining the ultimate points of the systems. The Fourier seriesanalysis of the system output is modified with suitable assumptions. Once theultimate points are determined, the Ziegler–Nichols (ZN) or related tuning rules areused to design the system. It is seen that the settings obtained by ZN method lead toa system performance with high overshoot and oscillations. In Section II, an attempthas been made to improve the system performance by refining the Ziegler–Nicholstuning rules. Douglas (1972) proposed that the system gain changes when theintegral and derivative mode comes into effect. The trial-and-error method is used totake into account the integral and derivative effect. In the present work, an ana-lytical method is proposed to consider the same. In addition, the method is auto-mated with the help of ideal on–off relay. A two-relay test method is suggested incase the process model is unknown.

The organization of the book is in the following manner:Chapter 1 gives a brief overview of the scope of the process control and pro-

portional, integral and derivative control. The different tuning methods based onultimate value and the model parameters are reviewed. Chapter 2 describes in detailthe relay auto-tuning method. The describing function analysis gives the mathe-matical basis behind the method. The different techniques to implement relay toscalar and multivariable systems are explained, along with the pairing criteria andcriteria for limit cycle to occur.

Chapter 3 presents the improved relay auto-tuning method for the unstableSOPTD systems. The Fourier series analysis is modified suitably for the second-order unstable system to account for the effect of the higher-order harmonic terms.The robustness studies and the effect of the measurement noise on the limit cyclesare also conducted.

ix

Chapter 4 extends the improved relay tuning method to design the decentralizedcontrol system for the stable TITO systems. Chapter 5 analyses the implementationof same technique on the unstable TITO systems. The decentralized control systemis designed based on the limit cycles using the ZN method. A simple method todetermine the minimum value of controller gain is also explained.

Chapter 6 introduces a new framework to design the centralized control systemfor the fully unstable systems using the relay auto-tuning method. The method takesinto account the effect of the higher-order harmonics.

Chapter 7 presents the refined Ziegler–Nichols (ZN) tuning obtained using thesystem parameters for unstable FOPTD and SOPTD systems and its implementa-tion on various unstable SISO systems. The need to refine the method arose due tothe oscillatory performance of the system with high overshoot. This chapterhighlights the improved performance of the proposed method over the conventionalZN method. The method is automated using the relay auto-tuning method.

Chapter 8 presents the correlations developed for designing the controller usingthe model parameters and the ultimate value. The expressions are developed basedon the extensive simulation study conducted on first-order plus time delay system.The method is simulated on various systems, and comparison with the differentmethods illustrates the enhanced performance. Chapter 9 describes the relayautomated refined ZN method for unstable TITO systems.

Chennai, India M. ChidambaramNikita Saxena

x Preface

Acknowledgement

Papers in refereed Journals:1. Saxena Nikita & M. Chidambaram (2015): Improved Relay Auto Tuning of

PID Controllers for Unstable SOPTD Systems, Chemical EngineeringCommunications, vol. 203, 769–782, 2016

2. Saxena Nikita & M. Chidambaram (2016): Relay Auto Tuning ofDecentralized PID Controllers for Unstable TITO Systems, Indian ChemicalEngineer (in press) 2017 DOI: 10.1080/00194506.2015.1129293

3. Saxena Nikita & M. Chidambaram (2016): Improved Continuous CyclingMethod of Tuning PID Controllers for Unstable Systems, Indian ChemicalEngineer (in press) DOI:10.1080/00194506.2016.1145557

4. Saxena Nikita & M. Chidambaram (2016): Relay Auto Tuning ofDecentralized PID Controllers for Stable TITO Systems, Indian ChemicalEngineer (in press)

Presentation in Conferences:1. Saxena Nikita & M. Chidambaram (2015): Tuning of PID Controllers for

First Order Plus Time Delay Unstable Systems, International Conference onAdvances in Chemical Engineering (ICACE-2015), 20–22 December, NIT-Surathkal.

2. Saxena Nikita & M. Chidambaram (2015): Improved continuous cyclingmethod for time delay systems with one unstable pole, National Conferenceon Recent Trends in Instrumentation and Control (RTIC-2016), 18–19March, 2016, Madras Institute of Technology.

3. Saxena Nikita & M. Chidambaram (2016): Tuning of pid controllers for timedelay unstable systems with two unstable poles, Advances in Control &Optimization of Dynamical Systems, 1–5 Feb, NIT- Trichy, IFAC-PapersOn-Line 49–1 (2016) 801–806.

4. Saxena Nikita & M. Chidambaram (2016): Improved relay auto-tuningmethod for unstable TITO systems, ETFA 2016 - IEEE InternationalConference on Emerging Technology & Factory Automation, Berlin,Germany, September 6–9, 2016.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Scope of Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Proportional–Derivative–Integral Control . . . . . . . . . . . . . . . . . . . 21.3 Loop Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Manual Loop Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Ziegler–Nichols Method (1942) . . . . . . . . . . . . . . . . . . . . 41.3.3 Refined Ziegler–Nichols Methods . . . . . . . . . . . . . . . . . . . 51.3.4 Process Reaction Curve Method . . . . . . . . . . . . . . . . . . . . 61.3.5 Cohen–Coon Method (1953) . . . . . . . . . . . . . . . . . . . . . . 71.3.6 Synthesis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Relay Feedback Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Real-Time Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Relay Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Relay Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Describing Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Shape of the Relay Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Relay Auto-tuning for Scalar System . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Modified Relay Feedback Method—Sunget al. (1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.2 Modified Fourier Series Analysis of ProcessResponse—Srinivasan and Chidambaram (2004) . . . . . . . . 21

2.4.3 Use of Preload Relay—Tan et al. (2006) . . . . . . . . . . . . . . 222.4.4 Enhanced Process Activation

Method—Je et al. (2009) . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Tuning of Multivariable System . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.1 Pairing Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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2.5.2 Controller Design Using IndependentTuning Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Relay Feedback Test for Multivariable System . . . . . . . . . . . . . . . 342.6.1 Condition for Limit Cycle to Occur . . . . . . . . . . . . . . . . . 352.6.2 Relay Auto-tuning of Decentralized Controllers . . . . . . . . . 362.6.3 Relay Auto-tuning of Centralized Controllers . . . . . . . . . . 40

2.7 Design of PID Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.8 Robust Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Auto-Tuning of Unstable SOPTD Systems . . . . . . . . . . . . . . . . . . . . 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Consideration of Higher Order Harmonics . . . . . . . . . . . . . . . . . . 54

3.2.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Measure of Robust Performance . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 Tuning Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.2 Simple Method to Calculate Kc,Min . . . . . . . . . . . . . . . . . . 61

3.5 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.4 Example 4: Unstable Nonlinear Bioreactor . . . . . . . . . . . . 693.5.5 Effect of Noise on Relay Output . . . . . . . . . . . . . . . . . . . . 73

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Decentralized PID Controllers for Stable Systems . . . . . . . . . . . . . . 754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Simulation Studies on Stable Systems . . . . . . . . . . . . . . . . . . . . . 78

4.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Decentralized PID Controllers for Unstable Systems . . . . . . . . . . . . 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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Contents xv

5.3 Simulation Studies of Unstable Systems . . . . . . . . . . . . . . . . . . . . 955.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Centralized PID Controllers for Unstable System . . . . . . . . . . . . . . . 1136.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 Controllers Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.4 Simulations Studies on Unstable Systems . . . . . . . . . . . . . . . . . . . 117

6.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Refined Ziegler–Nichols Tuning Method for Unstable SISOSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Controller Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.3.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4 Improved Relay Tuning Method . . . . . . . . . . . . . . . . . . . . . . . . . 1447.5 Comparison With the Recent Method . . . . . . . . . . . . . . . . . . . . . 1487.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8 Tuning Rules for PID Controllers for Unstable Systems . . . . . . . . . . 1518.1 Controller Design Method for Unstable FOPTD System Based on

System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.1.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.2 Controller Design for Unstable System Based on UltimateValues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2.2 Comparison with Other Methods . . . . . . . . . . . . . . . . . . . 159

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

9 Auto-tuning of Decentralized Unstable System With Refined ZNMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.2 Controllers Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

xvi Contents

About the Authors

M. Chidambaram is currently a Professor in the Department of ChemicalEngineering, Indian Institute of Technology Madras, Chennai. After completing hisPh.D. at the Indian Institute of Science, Bangalore, he served as a Faculty Memberat the Indian Institute of Technology Bombay, Mumbai, from 1984 to 1991. Sincethen, he has been a Faculty Member at the Indian Institute of Technology Madras.He has also served the institute as Head of the Department of Chemical Engineeringfrom 2000 to 2003, and as the Director, National Institute of Technology (NIT),Tiruchirappalli, from 2005 to 2010. He has 190 journal articles, 7 books and 4 bookchapters to his credit. His primary research interest is in the area of process control.

Nikita Saxena completed her Ph.D. under the guidance of Prof. M. Chidambaram.She completed her B.Tech in Chemical Technology at Harcourt Butler TechnicalUniversity (HBTU), Kanpur. She also has a year of experience in the fast-movingconsumer goods (FMCG) industry. She has authored four journal articles andpresented papers at several conferences. Her areas of interest include relay controlsystems and model identification.

xvii

Abbreviations and Notations

Abbreviations

CM Conventional methodHO Higher-order harmonicsIAE Integral of absolute errorIM/PM Improved (proposed) methodIMC Internal model controlISE Integral of square errorITAE Integral of time-weighted absolute errorMIMO Multi-input multi-outputNI Niederlinski IndexPID Proportional–integral–derivative controllerP-PI Proportional/proportional–integral controllerRGA Relative gain arraySISO Single-input single-outputSTR Single test relayTITO Two-input two-outputTRT Two tests relayZN Ziegler–Nichols method

Notations

a Amplitude of oscillations corresponding to principal harmonicsobserved from process output

a1/a* Corrected amplitude of output oscillations using the proposed methodai Amplitude of oscillations observed from ith relay inputAm Gain marginbi ConstantsC(s) Controller transfer functionD Time delayDr Dilution rate

xix

e/e Error (yr – y)Fs Set point filtergc,ij Individual transfer functions of controller matrixgp,ij Individual transfer functions of plant transfer matrixGp(s) Plant transfer matrixG(s) Transfer functionGc(s) Transfer function matrix of the PID controllerh/hi Relay height/relay height of the ith relayKc,des Designed value of the proportional gain of PID controllerKc,max Maximum value/ultimate gain of the controller gainKc,min Minimum value of the controller gainKu Ultimate gain (same as Kc,max)kui Ultimate gain of ith controller corresponding to ith relay inputkp Process gainKp/ G(0) Process steady state gain matrixKI kc/sIKD KcsDn Order of the system/number of inputs or outputsN Number of higher-order harmonic terms to be incorporatedN(a) Describing function matrix of multivariable relay systemP(s) Process transfer functionPu Ultimate period of oscillationsq 4h/pr(t) Reference signals Laplace variableSf Feed substrate concentrationS Substrate concentrationS(s) Sensitivity functiont Timet* 0.5p/xu

T(s) Complementary sensitivity functionu Input variableX Biomass concentrationui Manipulated variableyi Controlled variabley Process outputyr Set point value for y

xx Abbreviations and Notations

List of Figures

Fig. 1.1 Closed-loop scalar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Fig. 1.2 Effect of changing value of proportional gain on system output

for a unit step input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Fig. 1.3 Generating process reaction curve . . . . . . . . . . . . . . . . . . . . . . . . 6Fig. 1.4 Process reaction curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Fig. 2.1 Decomposition of a relay waveform . . . . . . . . . . . . . . . . . . . . . . 14Fig. 2.2 Classification of relay control system based on nature

of the linear part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Fig. 2.3 Block diagram representation of control system

with a nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Fig. 2.4 Relay feedback responses of FOPTD processes with different

D/s values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Fig. 2.5 Relay feedback responses of various processes with different

D/s values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Fig. 2.6 Relay feedback control system . . . . . . . . . . . . . . . . . . . . . . . . . . 19Fig. 2.7 Proposed configuration of the preload relay . . . . . . . . . . . . . . . . 23Fig. 2.8 Decentralized control for 2 � 2 system. . . . . . . . . . . . . . . . . . . . 25Fig. 2.9 Centralized control for 2 � 2 system . . . . . . . . . . . . . . . . . . . . . 26Fig. 2.10 Decoupled control for 2 � 2 system . . . . . . . . . . . . . . . . . . . . . . 27Fig. 2.11 Independent loop tuning technique for 2 � 2 system . . . . . . . . . 27Fig. 2.12 Sequential loop tuning procedure for 2 � 2 system . . . . . . . . . . 28Fig. 2.13 System configuration when a first loop (y11) is closed

and second loop (y22) is open, b first loop (y11) is openand second loop (y22) is closed . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fig. 2.14 System response when first loop (y11) is closed and secondloop (y22) is open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Fig. 2.15 System response when first loop (y11) is open and second loop(y22) is closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Fig. 2.16 Set point response of the system when both the loops are closedwith the controller designed based on the diagonal transferfunction (no detuning) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xxi

Fig. 2.17 Set point response of the system when both the loops are closedwith the detuned controller (F = 2) . . . . . . . . . . . . . . . . . . . . . . . 32

Fig. 2.18 System response when first loop (y11) is closed and secondloop (y22) is open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Fig. 2.19 System response when first loop (y11) is open and second loop(y22) is closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Fig. 2.20 Set point response of the system when both the loops are closedwith the controller designed based on the diagonal transferfunction (no detuning) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Fig. 3.1 Servo response for the SISO system . . . . . . . . . . . . . . . . . . . . . . 61Fig. 3.2 Relay feedback oscillations for Example 1, relay

height = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Fig. 3.3 Performance comparisons of the closed-loop system for

Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Fig. 3.4 Relay output with the measurement noise (Example 1) . . . . . . . . 64Fig. 3.5 Relay feedback oscillations for Example 2,

relay height = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Fig. 3.6 Performance comparisons of the closed-loop system for

Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Fig. 3.7 Relay feedback oscillations for Example 3,

relay height = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Fig. 3.8 Performance comparisons of the closed-loop system for

Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Fig. 3.9 Relay feedback oscillations for linearized transfer function

of the bioreactor, relay height = 0.04 . . . . . . . . . . . . . . . . . . . . . 70Fig. 3.10 Nonlinear bioreactor process response . . . . . . . . . . . . . . . . . . . . . 71Fig. 3.11 Relay output for nonlinear bioreactor with the

measurement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Fig. 4.1 TITO Decentralized relay system . . . . . . . . . . . . . . . . . . . . . . . . 76Fig. 4.2 Relay response curve of the process (Example 1) . . . . . . . . . . . . 79Fig. 4.3 Process response and interaction curve for unit step

change in y1r and y2r (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . 80Fig. 4.4 Process response and interaction curve for unit step

change in load entering u1 and u2 (Example 1) . . . . . . . . . . . . . . 80Fig. 4.5 Process response and interaction curve for unit step

change in y1r and y2r (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . 81Fig. 4.6 Process response and interaction curve for unit step

change in load entering u1 and u2 (Example 1) . . . . . . . . . . . . . . 82Fig. 4.7 Limit cycles obtained from the relay feedback test in presence

on random noise of mean 0 and SD = 0.01 (Example 1) . . . . . . 82Fig. 4.8 Limit cycles obtained from the relay feedback test in presence

on random noise of mean 0 and SD = 0.02 (Example 1) . . . . . . 83Fig. 4.9 Process response for unit step change in presence on random

noise of mean 0 and SD = 0.01 (Example 1) . . . . . . . . . . . . . . . 84

xxii List of Figures

Fig. 4.10 Relay response curve of the process (Example 2) . . . . . . . . . . . . 85Fig. 4.11 Process response and interaction curve for unit step change

in y1r and y2r (Example 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Fig. 4.12 Process response and interaction curve for unit step change

in load in y1 and y2 (Example 2). . . . . . . . . . . . . . . . . . . . . . . . . 86Fig. 4.13 Relay response curve of the process (Example 3) . . . . . . . . . . . . 87Fig. 4.14 Process response and interaction curve for unit step change

in y1r and y2r (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Fig. 4.15 Process response and interaction curve for unit step change

in load in y1 and y2 (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . 89Fig. 4.16 Relay responses of the process (Example 4) . . . . . . . . . . . . . . . . 90Fig. 4.17 Process response and interaction for unit step change in inputs

(Example 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Fig. 5.1 Relay response curve of the process (Example 1) . . . . . . . . . . . . 97Fig. 5.2 Closed-loop response of process for step change

in input y1r and y2r for Example 1 with N = 3, 3 . . . . . . . . . . . . 98Fig. 5.3 Closed-loop response of process for step change

in load y1 and y2 for Example 1 with N = 3, 3 . . . . . . . . . . . . . . 99Fig. 5.4 Closed-loop response of process for step change

in input y1r and y2r for Example 1 with N = 5, 5 . . . . . . . . . . . . 100Fig. 5.5 Closed-loop response of process for step change in load

entering u1 and u2 for Example 1 with N = 5, 5 . . . . . . . . . . . . . 100Fig. 5.6 Relay response curve of the process (Example 2) . . . . . . . . . . . . 102Fig. 5.7 Process response and interaction curve for unit step change

in y1r and y2r (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Fig. 5.8 Controller output behaviour of process for step change in input

y1r and y2r for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Fig. 5.9 System response for step change in load entering

in u1 and u2 for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Fig. 5.10 Relay response curve of the process (Example 3) . . . . . . . . . . . . 106Fig. 5.11 Closed-loop responses and interactions of process for step

change in input y1r and y2r for Example 3. . . . . . . . . . . . . . . . . . 107Fig. 5.12 Controller output behaviour to the step change in input

(Example 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Fig. 5.13 Closed-loop response of process for step change

in load y1 and y2 for Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . 108Fig. 5.14 Controller output behaviour to step change in disturbance

for Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Fig. 5.15 Maximum singular value curve for robustness analysis

for Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Fig. 6.1 Centralized control structure and corresponding proposed relay

scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Fig. 6.2 Limit cycles obtained to relay feedback for Example 1 . . . . . . . . 118

List of Figures xxiii

Fig. 6.3 Closed-loop performances of centralized control systemfor step change in set point. Example 1 . . . . . . . . . . . . . . . . . . . 119

Fig. 6.4 Closed-loop performances of the centralized control systemfor step change in load Example 1 . . . . . . . . . . . . . . . . . . . . . . . 120

Fig. 6.5 Inverse maximum singular value plot for robustness analysisfor Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Fig. 6.6 Limit cycles obtained to relay feedback (Example 2) . . . . . . . . . 122Fig. 6.7 Process response to the designed centralized control structure

for step change in input (Example 2) . . . . . . . . . . . . . . . . . . . . . 123Fig. 6.8 Process response to the designed centralized control structure

for step change in load (Example 2) . . . . . . . . . . . . . . . . . . . . . . 124Fig. 6.9 Inverse maximum singular value plot for robustness analysis

(Example 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Fig. 7.1 Feedback controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Fig. 7.2 Servo and regulatory process response to step change in input

(Example 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Fig. 7.3 Process response of the servo and the regulatory problems

(Example 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Fig. 7.4 Robustness analysis (Example 2). . . . . . . . . . . . . . . . . . . . . . . . . 134Fig. 7.5 Comparison of proposed method with other methods present

in the literature (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Fig. 7.6 Servo and regulatory responses for a step

change (Example 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Fig. 7.7 Process servo response for step change (10% decrease

in set point) (Example 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Fig. 7.8 Servo response and the regulatory response of the system

(Example 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Fig. 7.9 Servo response and regulatory response of nonlinear CSTR

(Example 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Fig. 7.10 Step 1—relay feedback loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Fig. 7.11 Sustained oscillations obtained from relay feedback

loop test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Fig. 7.12 Step 2—improved relay feedback loop test . . . . . . . . . . . . . . . . . 146Fig. 7.13 Process response for step change in input . . . . . . . . . . . . . . . . . . 147Fig. 7.14 Servo response and the regulatory response of the system . . . . . 149Fig. 8.1 Process response for step change. (i) e = 0.05; (ii) e = 0.5;

(iii) e = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Fig. 8.2 Process response for step change in X from

0.9951 to 1.294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Fig. 8.3 Process response when Sf decreased by 10%. . . . . . . . . . . . . . . . 156Fig. 8.4 Process response when Sf decreased by 10%. . . . . . . . . . . . . . . . 157Fig. 8.5 Servo and regulatory responses for Case 1 . . . . . . . . . . . . . . . . . 158Fig. 8.6 Process response of FOPTD integrating

system—Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xxiv List of Figures

Fig. 8.7 Process response of unstable FOPTD system—Example 2 . . . . . 162Fig. 8.8 Process response of SOPTD system with one unstable

pole—Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Fig. 8.9 Process response of SOPTD nonminimum phase system

without time delay—Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . 163Fig. 8.10 Process response of SOPTD system with two unstable

poles—Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Fig. 8.11 Process response of SOPTD system with complex

poles—Example 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Fig. 8.12 Process response of nonlinear bioreactor—Example 7. . . . . . . . . 164Fig. 9.1 TITO decentralized control system . . . . . . . . . . . . . . . . . . . . . . . 168Fig. 9.2 Updated TITO decentralized control system for the

proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Fig. 9.3 Limit cycles of relay test I (Example 1) . . . . . . . . . . . . . . . . . . . 171Fig. 9.4 Limit cycles of relay test II (Example 1) . . . . . . . . . . . . . . . . . . . 172Fig. 9.5 Process response to unit step change in set point

for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Fig. 9.6 Robustness study on the designed controller for Example 1

in the presence of uncertainty in process time constant(solid line: original system/no mismatch; dashed line:process time constant increased by 10%; dotted line:process time constant decreased by 10%) . . . . . . . . . . . . . . . . . . 174

Fig. 9.7 Maximum singular value plot (Example 1) . . . . . . . . . . . . . . . . . 174Fig. 9.8 Comparison of methods for a unit step change in the set point

for Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Fig. 9.9 Comparison of methods for a unit step change in load for

Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Fig. 9.10 Limit cycles of relay test I (Example 2) . . . . . . . . . . . . . . . . . . . 176Fig. 9.11 Limit cycles of relay test II (Example 2) . . . . . . . . . . . . . . . . . . . 177Fig. 9.12 Process response to step change set point for Example 2 . . . . . . 178Fig. 9.13 Process response to unit step change in the load

for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Fig. 9.14 Controller output behaviour to the step change

in the set point for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 179Fig. 9.15 Controller output behaviour to step change in load

for Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Fig. 9.16 Maximum singular value curve (Example 2) . . . . . . . . . . . . . . . . 181

List of Figures xxv

List of Tables

Table 1.1 Tuning effects of PID controller terms . . . . . . . . . . . . . . . . . . . 2Table 1.2 Tuning rules proposed by Ziegler–Nichols . . . . . . . . . . . . . . . . 5Table 1.3 Modified Ziegler–Nichols tuning rules . . . . . . . . . . . . . . . . . . . 5Table 1.4 Controller settings by Ziegler–Nichols

for FOPTD system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Table 1.5 Cohen–Coon method tuning rules. . . . . . . . . . . . . . . . . . . . . . . 8Table 1.6 Transfer function, controlled variable and manipulated

variable of different real-time systems. . . . . . . . . . . . . . . . . . . . 10Table 2.1 Classification and the corresponding equations of the

symmetric relay element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Table 2.2 Comparison of harmonic effect by new proposed signal

with the conventional relay method . . . . . . . . . . . . . . . . . . . . . 20Table 2.3 Simulation results of the proposed methods and the

conventional method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Table 2.4 Controller tuning rules proposed by Zhuang

and Atherton (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Table 2.5 Various robust stability analysis methods reported

in the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Table 3.1 Calculations for improved ultimate gain incorporating

higher-order harmonics (Example 1) . . . . . . . . . . . . . . . . . . . . . 62Table 3.2 Effect of measurement noise on the relay feedback system

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Table 3.3 Calculations for improved ultimate gain incorporating

higher-order harmonics (Example 2) . . . . . . . . . . . . . . . . . . . . . 66Table 3.4 Calculations for improved ultimate gain incorporating

higher-order harmonics (Example 3) . . . . . . . . . . . . . . . . . . . . . 67Table 3.5 Comparisons of the closed-loop time integral

performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Table 3.6 Calculations for improved ultimate gain incorporating

higher-order harmonics in linearized model of bioreactor(Example 4: bioreactor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xxvii

Table 3.7 Calculations for improved ultimate gain incorporatinghigher-order harmonics in the nonlinear bioreactor(Example 4: bioreactor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Table 3.8 Closed-loop time-domain performance comparison(bioreactor problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Table 3.9 Effect of measurement noise on the relay feedback system(bioreactor problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Table 3.10 Maximum sensitivity, phase margin, gain margincomparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Table 4.1 Results using the limit cycles to the relay feedback test(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Table 4.2 Ultimate gain and PID controller parameters using theconventional method and improved method(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Table 4.3 Time integral performance analysis (Example 1) . . . . . . . . . . . 81Table 4.4 Data obtained on the analyzing the limit cycles in presence

of noise (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 4.5 Ultimate gain of the process incorporating higher-order

harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 4.6 Results using the limit cycles to the relay feedback test

(Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Table 4.7 Ultimate gain and PID controller parameters using the

conventional and improved methods (Example 2). . . . . . . . . . . 85Table 4.8 Time-domain performance analysis (Example 2) . . . . . . . . . . . 87Table 4.9 Results using the limit cycles to the relay feedback test

(Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Table 4.10 Ultimate gain and PID controller parameters using the

conventional method and improved method(Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Table 4.11 Time-domain performance analysis (Example 3) . . . . . . . . . . . 90Table 4.12 Results using the limit cycles to the relay feedback test

(Example 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Table 4.13 Ultimate gain and PID controller parameters using the

conventional method and improved method (Example 4) . . . . . 91Table 4.14 Time-domain performance analysis (Example 4) . . . . . . . . . . . 92Table 5.1 Results using the limit cycles to the relay feedback test

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Table 5.2 Ultimate gain and PID controller parameters using the

conventional method and improved method (Example 1) . . . . . 97Table 5.3 Time integral performance analysis (Example 1) . . . . . . . . . . . 99Table 5.4 Comparison of ultimate values using conventional method

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Table 5.5 Results using the limit cycles to the relay feedback test

(Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xxviii List of Tables

Table 5.6 Ultimate gain and PID controller parameters using theconventional method and improved method (Example 2) . . . . . 103

Table 5.7 Time integral performance analysis for step change in inputin y1r and y2r (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Table 5.8 Results using the limit cycles to the relay feedback test(Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table 5.9 Implementation of the correction term to incorporatehigher-order harmonics to the data obtained from the relaytest (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table 5.10 PID controller parameters using the conventional methodand improved method (Example 3). . . . . . . . . . . . . . . . . . . . . . 106

Table 5.11 Total variation in the controller output (U)for (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Table 5.12 Time-domain integral performance (Example 3) . . . . . . . . . . . . 109Table 6.1 Data obtained by analysing the limit cycle

of the relay test (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 118Table 6.2 Correction in Kc,max to include higher-order harmonics

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Table 6.3 Controller parameters of the centralized control structure

based on the conventional and the improved relay auto-tuningmethod (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Table 6.4 Time integral performance (Example 1) . . . . . . . . . . . . . . . . . . 120Table 6.5 Data obtained by analysing the limit cycle

of the relay test (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 122Table 6.6 Incorporation of higher-order harmonics

(Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Table 6.7 Controller settings for the designed centralized control

structure based on the conventional and the improved relayauto-tuning method (Example 2). . . . . . . . . . . . . . . . . . . . . . . . 123

Table 6.8 Time integral performance analysis (Example 2) . . . . . . . . . . . 124Table 7.1 Details of calculation for ZN method (Example 1) . . . . . . . . . . 130Table 7.2 Details of calculation for the updated method (Example 1) . . . 130Table 7.3 Comparison study of conventional and proposed methods

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Table 7.4 Controller settings (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . 132Table 7.5 Time integral performance (Example 2) . . . . . . . . . . . . . . . . . . 133Table 7.6 Maximum sensitivity, gain margin and phase margin

comparison (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Table 7.7 Controller settings (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . 136Table 7.8 Time integral performance for servo response (Example 3) . . . 137Table 7.9 Time integral performance for regulatory

response (Example 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Table 7.10 Controller settings (Example 4) . . . . . . . . . . . . . . . . . . . . . . . . 140Table 7.11 Time integral performance (Example 4) . . . . . . . . . . . . . . . . . . 140

List of Tables xxix

Table 7.12 Maximum sensitivity, gain margin and phase margincomparison (Example 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Table 7.13 Controller settings (Example 5) . . . . . . . . . . . . . . . . . . . . . . . . 143Table 7.14 Time integral performance (Example 5) . . . . . . . . . . . . . . . . . . 144Table 7.15 Data obtained from sustained oscillations from relay

feedback loop for first test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Table 7.16 Data obtained from sustained oscillations from relay

feedback second test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Table 7.17 Controller settings from the relay test . . . . . . . . . . . . . . . . . . . . 147Table 7.18 Time integral performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Table 7.19 PID controller settings and performances indices . . . . . . . . . . . 148Table 8.1 Controller parameters for FOPTD system

for different values of e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Table 8.2 ITAE values for different cases. . . . . . . . . . . . . . . . . . . . . . . . . 154Table 8.3 Maximum sensitivity, gain margin and phase margin

for different values of e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Table 8.4 Controller parameters for different methods and

corresponding ITAE values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Table 8.5 Time integral performance criteria (Example 1) . . . . . . . . . . . . 159Table 8.6 Tuning rules for comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Table 8.7 Example considered for the comparison purpose . . . . . . . . . . . 159Table 8.8 Ultimate values and the controller parameters using different

methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Table 8.9 Time-domain integral performance comparison for a unit

change in set point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Table 8.10 Time-domain integral performance comparison for a unit

change in load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Table 9.1 Data obtained by analysing limit cycle to relay test

(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Table 9.2 Implementation of the correction term to incorporate

higher-order harmonics to the data obtained from relay test(Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Table 9.3 Controller settings (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . 172Table 9.4 Total variations in the controller output (Example 1) . . . . . . . . 173Table 9.5 Time-domain integral criteria (Example 1) . . . . . . . . . . . . . . . . 173Table 9.6 Data obtained by analysing limit cycle to relay tests

(Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Table 9.7 Implementation of the correction term to incorporate

higher-order harmonics to the data obtained from relay tests(Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Table 9.8 PID controller settings (Example 2) . . . . . . . . . . . . . . . . . . . . . 178Table 9.9 Total variation in the controller output (Example 2) . . . . . . . . . 180Table 9.10 Time integral performance analysis (Example 2) . . . . . . . . . . . 181

xxx List of Tables