Advances in Industrial Control

For further volumes:www.springer.com/series/1412

Igor Boiko

Non-parametricTuning of PIDControllers

A Modified Relay-Feedback-Test Approach

Igor BoikoElectrical Engineering DepartmentThe Petroleum InstituteAbu DhabiUnited Arab Emirates

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ISSN 1430-9491 ISSN 2193-1577 (electronic)Advances in Industrial ControlISBN 978-1-4471-4464-9 ISBN 978-1-4471-4465-6 (eBook)DOI 10.1007/978-1-4471-4465-6Springer London Heidelberg New York Dordrecht

Library of Congress Control Number: 2012947218

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To my family

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technol-ogy transfer in control engineering. The rapid development of control technologyhas 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 inindustrial reports, feasibility study papers and the reports of advanced collaborativeprojects. The series offers an opportunity for researchers to present an extended ex-position of such new work in all aspects of industrial control for wider and rapiddissemination.

The proportional-integral-derivative (PID) controller continues to be an impor-tant controller for industrial applications. It is mainly found at the lower level of thecontrol hierarchy, at the actuator-sensor level, and configured into multiloop struc-tures (for example, cascade loops). Groups of process units may be provided withoptimised set points using advanced techniques like model predictive control, butthe local process loops are likely to be PID controlled. The ease of understandingof PID controller tuning by maintenance and operational staff and the widespreadavailability of PID functionality in distributed-controlsystem/SCADA software andprogrammable logic controllers (PLCs) are important factors that support the con-tinued industrial use of PID control.

Despite this favourable situation for PID control, the large numbers of PID con-trollers in industrial-scale process plant, the economic pressure to contain or reducemaintenance costs and the enabling advances in industrial process computer tech-nology led to interest in creating techniques for the automatic tuning (termed auto-tuning in the 1980s) of PID control loops. In this control-tuning field, the work ofZiegler and Nichols in the 1940s initiated a procedural framework for the rulebasedtuning of PID loops. In the 1980s, the work of Åström and Hägglund involving arelay-test-based procedure, gave the field a new impetus and enabled autonomoustuning to become an online reality.

The methods of PID controller tuning involve two steps:Step 1 A Measurement Experiment—in which some chosen characteristics of the

process are measured. These may be measurements to allow a process model to be

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identified; they may be measurements of a characteristic that can be used directly inthe computation of controller parameters.

Step 2 Controller Tuning Computation—in which control loop performance re-quirements are specified, the structure of the controller is decided and the controllerparameters are computed using the data coming from Step 1.

Automated PID controller tuning seeks theory, algorithms and implementationsthat minimise the use of expert input (from, for example, technical, maintenance,or operational staff) in the tuning process and allows the autonomous tuning ofPID controllers using computer software and hardware installed at the plant. Thesetechniques comprise two categories:

Parametric tuning methods, where the measurements are used to identify a pro-cess model and the controller tuning uses the identified process model data.

Non-parametric tuning methods, where the measurements are of characteristicsor parameters that are used directly in formulas for the computation of the controllerparameters.

The “continuous cycling” procedure and its associated tuning rules devised byZiegler and Nichols, and the relay test procedure and the associated phase margintuning rules are examples of the non-parametric family of tuning methods. Thesetwo contributions have been hugely influential in this particular field of industrialcontrol practice, and they continue to inspire current day research and development.

Monograph contributions to this field in the Advances in Industrial Control seriescover both PID control and the Automatic PID controller tuning field and those inthe PID control field include:

2012 R. Vilanova and A. Visioli (Eds.): PID Control in the Third Millennium:Lessons Learned and New Appoaches, ISBN 978-1-4471-2424-5;

2010 A. Visioli and Q.-C. Zhong, Control of Integral Processes with Dead Time,ISBN 978-0-85729-069-4;

2006 A. Visioli, Practical PID Control, ISBN 978-1-84628-585-1;1999 A. Datta, M.T. Ho and S.P. Bhattacharyya, Structure and Synthesis of PID

Controllers, ISBN 978-1-85233-614-1; and1999 K.K. Tan, Q.-G. Wang and C.C. Hang with T.J. Hägglund, Advances in PID

Control, ISBN 978-1-85233-138-2;

whereas, monograph contributions to the automatic PID control tuning field in-clude:

2011 T. Liu and F. Gao, Industrial Process Identification and Control Design, ISBN978-0-85729-976-5; and

1999 C.C. Yu, Autotuning of PID Controllers: Relay Feedback Approach, ISBN978-3-540-76250-8.

To this activity, it is a pleasure to add this monograph, Non-Parametric Tuningof PID Controllers: Modified Relay Feedback Test Approach, by Igor Boiko. De-spite an extensive literature on the theory and application of the relay test procedureIgor Boiko brings new insights to the field based on his industrial experience byproposing that the tuning and the experimental procedure should be matched to the

Series Editors’ Foreword ix

features of specific categories of control loops found in the process industries. Inthe monograph analyses of the PID controller requirements for flow, level, pressure,and temperature loops are presented. Controller performance is based on gain andphase-margin criteria conjoined with time integral (IAE, ITAE, ISE, ITSE) opti-mality. This novel approach will be especially interesting to the industrial controlpractitioner.

Later chapters in the monograph describe an investigation into the effects of pro-cess nonlinearity on the tuning results for flow loops, and present more theoreticaldetails of the modified relay test procedure reported in earlier chapters. The clos-ing chapter of the monograph examines some of the practical issues involved inperforming online automatic tuning methods and describes software for the tuningtask.

The mix of practical expertise and the theoretical elaboration presented by IgorBoiko in this Advances in Industrial Control monograph will attract a wide reader-ship from the control community and the process industries and continues the se-ries tradition of publishing exemplary research and development in the PID controlparadigm.

M.J. GrimbleM.A. Johnson

Industrial Control CentreGlasgow, Scotland, UK

Preface

The subject of this book is the method of PID controller tuning based on the con-tinuous cycling principle. Relay feedback for tuning was proposed by Åström andHägglund in the 1980s and has been completed since then by numerous modifi-cations aimed at enhancing some features of the original method. The majority ofthese modifications concern parametric methods of tuning that are based on theidentification of certain underlying process models. The method presented hereinis non-parametric. It features a holistic approach to test and tuning, or coordinatedtest and tuning, in which the test parameters are selected not arbitrarily or a prioribut together with the tuning rule to be applied. As a result, this method providesexact values of a specified gain or phase margin and does not require any itera-tive procedure. Another novel feature is the introduction of process-specific optimaltuning rules in the non-parametric setup. This allows an engineer to use the flowloop-optimised tuning rules for flow loop tuning, level loop-optimised tuning rulesfor level loop tuning and so on, and obtain in most cases a better result than generictuning rules would yield.

We hope readers might also find the presented approach to obtaining optimal tun-ing rules an interesting one. It involves a nontraditional solution of the optimisationproblem. We also believe that the recently developed dynamic harmonic balanceprinciple, which is presented in this book, may attract reader interest as well.

A person studying the subject of automatic control might benefit from this bookby learning how linear and nonlinear control theory are brought together to solvea very important practical control problem—optimal tuning of PID controllers.A practising control engineer might gain new insights into PID controller tuning.The presented method is simple in realisation and efficient in terms of practical re-sults. The reader can use the provided MATLAB code to incorporate the variouscomponents of the presented theory and tuning method.

When deriving tuning rules for particular types of industrial processes, focus isgiven to the most common process industry applications: flow, temperature, pressureand level control loops. It is assumed in most situations that the process is manipu-lated through control valves (or dampers), and more specifically through pneumati-cally actuated control valves. In various industries different features of loop tuners

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are important. In this book we assume that tuning could be carried out on a liveprocess (i.e. during plant operation), so that such features as minimum disturbanceto the process and short time required for tuning are of great importance.

Chapter 1 is introductory. It traces the history of tuning—and in particular, non-parametric tuning. Differences between parametric and non-parametric approachesare discussed. The issue of the selection of proper complexity of the process modelis illustrated by an example of curve fitting.

Chapter 2 covers the methods of non-parametric tuning of PID controllers. Theclosed-loop Ziegler–Nichols method and Åström–Hägglund’s relay feedback testare reviewed. It is shown that they allow one to generate test oscillations at the pointcorresponding to –180◦ of the phase characteristic of the process. However, it isalso shown that generation of oscillations in the third quadrant would be beneficial.A number of non-parametric tuning methods based on various modifications of therelay feedback test (RFT), which are capable of producing test oscillations in thethird quadrant, are reviewed.

In Chap. 3, the modified relay feedback test (MRFT) is introduced as a fur-ther logical development of the closed-loop Ziegler–Nichols method and Åström–Hägglund’s relay feedback test. Methods and criteria of tuning are presented in gen-eral and for every considered typical process: flow, level, pressure and temperatureprocesses. With the criterion of optimisation selected and the mechanism of distur-bance generation analysed for each of the considered processes, the optimal tuningrules are obtained by solving the optimisation problem on the domain of parameterscharacterising the situational aspects of the implied process model.

Chapter 4 illustrates possible ways to improve the accuracy of the tuning rules viaan example of the flow loop. It is shown that the precise model of the flow process isnonlinear even if the installed characteristic of the valve is linear. This nonlinearityis revealed as an apparent time constant of the actuator dynamics, which dependson both the amplitude of the relay test and the selection of the operating point. It isshown that we benefit by using the precise nonlinear model of the flow loop whenfinding optimal PID tuning rules.

Chapter 5 covers the exact model of oscillations in the system arising from themodified relay feedback test. The exact model can be used for parametric tuningthat includes identification of the process model parameters. The development hereis based on the locus of a perturbed relay system (LPRS) method. We present theLPRS for the conventional relay feedback system, the MRFT and the test containingthe two-relay controller.

Chapter 6 provides the model of transient oscillatory motions in a system us-ing the modified relay feedback test. The treatment is based on the concept of thedynamic harmonic balance.

Chapter 7 covers the practical implementation of this book’s tuning method insoftware for distributed control systems. The most critical issues encountered inthe implementation are covered. Industrial software used in the process industry isdescribed.

Some chapters of the book can be read independently of others. For example,practising engineers may be more interested in Chaps. 1, 2, 3 and 7. They can

Preface xiii

omit other chapters without sacrificing an understanding of the main ideas of non-parametric tuning and the modified relay feedback test in particular. At the sametime researchers may find the material of Chaps. 5 and 6 interesting from the per-spective of finding exact models of the oscillations and analysis of transient oscilla-tory modes, respectively.

I express my heartfelt gratitude to Prof. M. Johnson for his careful reading of themanuscript and numerous comments, which allowed me to significantly improve thebook; to Prof. M. Grimble for his initiation of this project and encouragement of mywork on the book; to my co-workers and colleagues at Syncrude Canada: D. Brown,A. Ernyes, W. Oli, and E. Tamayo; to my former MSc student S. Sayedain, who didsimulations presented in Chap. 4; to A. Breslavskaya, who produced a significantshare of the artwork and LaTeX typesetting for the book; and to K. McKenzie forcorrecting and improving the quality of language of the manuscript. I am also grate-ful to my family for their patience and the personal sacrifices which they have givento this my work. Without their support this undertaking would not have been possi-ble.

I also gratefully acknowledge the support of RIFP Project No. 12310 of thePetroleum Institute, Abu Dhabi.

Igor BoikoAbu Dhabi

Acronyms

DCS Distributed control systemDF Describing function (method)DHB Dynamic harmonic balanceDSOPDT Damped second-order plus dead time (model/dynamics)FOPDT First-order plus dead time (model/dynamics)HB Harmonic balanceHMI Human-machine interfaceIAE Integral absolute errorISE Integral square errorITAE Integral time absolute errorITSE Integral time square errorI/P Electric current-to-pressure (transducer)LPRS Locus of a perturbed relay systemMRFT Modified relay feedback testPID Proportional-integral-derivative (controller or algorithm)PLC Programmable logic controllerRFT Relay feedback testSISO Single-input-single-output (system)SM Sliding modeSOPDT Second-order plus dead time (model/dynamics)

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Parametric and Non-parametric Tuning . . . . . . . . . . . . . . . 41.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Non-parametric Tuning of PID Controllers . . . . . . . . . . . . . . 92.1 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Ziegler–Nichols Closed-Loop Test and Tuning . . . . . . . . . . . 112.3 Åström–Hägglund Relay Feedback Test . . . . . . . . . . . . . . 142.4 Generating Test Oscillations in the Third Quadrant . . . . . . . . . 162.5 Tests that Ensure Frequency of Oscillations at Arbitrary Process

Phase Lags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.1 Test Using Additional Time Delay . . . . . . . . . . . . . 192.5.2 Test Using Additional Integrator Term . . . . . . . . . . . 202.5.3 Test Using Additional Derivative Term . . . . . . . . . . . 212.5.4 Test Using Phase-Lock Loop . . . . . . . . . . . . . . . . 22

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Modified Relay Feedback Test (MRFT) and Tuning of PIDControllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 MRFT and Holistic Approach to Test and Tuning . . . . . . . . . . 25

3.1.1 Modified Relay Feedback Test . . . . . . . . . . . . . . . 253.1.2 Homogeneous Tuning Rules . . . . . . . . . . . . . . . . . 283.1.3 Non-parametric Tuning Rules for Specification on Gain

Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.4 Non-parametric Tuning Rules for Specification on Phase

Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Process-Specific Optimal Tuning Rules . . . . . . . . . . . . . . . 333.2.1 General Approach to Producing Process-Specific Optimal

Tuning Rules . . . . . . . . . . . . . . . . . . . . . . . . . 33

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3.2.2 Tuning of Flow Loops . . . . . . . . . . . . . . . . . . . . 393.2.3 Tuning of Level Loops . . . . . . . . . . . . . . . . . . . . 533.2.4 Tuning of Pressure Loops . . . . . . . . . . . . . . . . . . 623.2.5 Tuning of Temperature Loops . . . . . . . . . . . . . . . . 68

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Improving the Accuracy of Tuning of PID Controllers . . . . . . . . 814.1 Improving the Accuracy of Tuning Through Nonlinear Model of

Control Valve in Flow Loop . . . . . . . . . . . . . . . . . . . . . 814.1.1 Model of Flow Process . . . . . . . . . . . . . . . . . . . 824.1.2 Lyapunov Linearisation of Flow Process Dynamics . . . . 854.1.3 Local Probing of Incremental Nonlinear Dynamics

Through MRFT . . . . . . . . . . . . . . . . . . . . . . . 874.1.4 Example of Analysis . . . . . . . . . . . . . . . . . . . . . 88

4.2 Optimal Tuning Rules for Flow Loop, Based on Nonlinear Model . 914.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Exact Model of MRFT and Parametric Tuning . . . . . . . . . . . . 975.1 Locus of a Perturbed Relay System (LPRS) as Frequency-Domain

Characteristic of Process . . . . . . . . . . . . . . . . . . . . . . . 975.1.1 From Describing Function Analysis to LPRS Analysis . . . 975.1.2 Symmetric Oscillations in Relay Feedback Systems . . . . 1005.1.3 Asymmetric Oscillations in Relay Feedback Systems and

Propagation of External Constant Inputs . . . . . . . . . . 1015.2 Introduction to the LPRS . . . . . . . . . . . . . . . . . . . . . . 103

5.2.1 Computation of LPRS . . . . . . . . . . . . . . . . . . . . 1035.2.2 Computation of the LPRS from Differential Equations . . . 1055.2.3 Computation of the LPRS from Process Transfer Function . 1115.2.4 LPRS of Low Order Dynamics . . . . . . . . . . . . . . . 1135.2.5 Some Properties of the LPRS . . . . . . . . . . . . . . . . 117

5.3 LPRS Model of Oscillations in MRFT . . . . . . . . . . . . . . . 1195.3.1 Exact Frequency-Domain Analysis of Oscillations . . . . . 1195.3.2 Describing Function Analysis of External Signal

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3.3 Exact Frequency-Domain Analysis of External Signal

Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 1255.4 Exact Model of Oscillations in Two-Relay Controller . . . . . . . 131

5.4.1 LPRS-Based Analysis . . . . . . . . . . . . . . . . . . . . 1335.4.2 Poincaré Map-Based Analysis of Orbital Stability . . . . . 137

5.5 Example of Identification . . . . . . . . . . . . . . . . . . . . . . 1385.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 Analysis of Transient Oscillations in Systems with MRFT . . . . . . 1416.1 Dynamic Harmonic Balance . . . . . . . . . . . . . . . . . . . . . 141

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1416.1.2 Harmonic Balance for Transient Oscillations . . . . . . . . 142

Contents xix

6.2 Analysis of Motions in the Vicinity of a Periodic Solution . . . . . 1456.3 Dynamic Harmonic Balance Including Frequency Rate of Change

(Full Dynamic Harmonic Balance) . . . . . . . . . . . . . . . . . 1476.4 Model of Transient Oscillations in the Presence of Delay . . . . . . 1506.5 Describing Function of MRFT for Sinusoidal Input of

Exponentially Changing Amplitude . . . . . . . . . . . . . . . . . 1516.6 Dynamic Harmonic Balance in System with MRFT Algorithm . . 1536.7 Example of Analysis of Transient Motions Through Dynamic

Harmonic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Software for Loop Tuning in Distributed Control Systems (DCS) . . 1597.1 Specifics of Loop Tuning in DCS . . . . . . . . . . . . . . . . . . 1597.2 Methods of Mitigating Effects Existing in Real Processes . . . . . 161

7.2.1 Invasive Character of Tuning . . . . . . . . . . . . . . . . 1617.2.2 Mitigation of Noise Effects . . . . . . . . . . . . . . . . . 1627.2.3 Mitigation of Effect of External Disturbances . . . . . . . . 1657.2.4 Accounting for Process Nonlinearities . . . . . . . . . . . 166

7.3 DCS Loop Tuning Software Description . . . . . . . . . . . . . . 1667.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.1 Sample Simulink Models Used in Optimisation . . . . . . . . . . . 1718.2 Matlab Code Used in Book . . . . . . . . . . . . . . . . . . . . . 172

8.2.1 ISE Optimisation of Tuning Rules for PI Flow Controller(Response to Set Point) . . . . . . . . . . . . . . . . . . . 172

8.2.2 Library of Functions for LPRS Computing . . . . . . . . . 175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185