Control Systems Design Vladimir Zakian (Editor) Control Systems Design A New Framework With 89 Figures Vladimir Zakian, PhD 12 Cote Green Road Marple Bridge Stockport SK6 5EH UK British Library Cataloguing in Publication Data Control systems design : a new framework 1. Automatic control I. Zakian, Vladimir 629.8 ISBN 1852339136 Library of Congress Cataloging-in-Publication Data Control systems design : a new framework / [edited by] Vladimir Zakian. p. cm. Includes bibliographical references and index. ISBN 1-85233-913-6 1. Automatic control. I. Zakian, Vladimir. TJ213.C5728 2005 629.8—dc22 2004051252 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. ISBN 1-85233-913-6 Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 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 regula- tions 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. Typesetting: Camera ready by editor Printed in the United States of America 69/3830-543210 Printed on acid-free paper SPIN 11318958 Preface In recent decades, a new framework for the design of control systems has emerged. Its development was prompted in the 1960s by two factors. First, was the arrival of interactive computing facilities, which opened new avenues for design, relying more on numerical methods. In this way, routine compu- tational tasks, which are a significant part of design, are left to the computer, thus allowing the designer to focus on the formulation of the design problem, which requires creative skills. This has led to a major shift in the field of design, with more emphasis placed on general principles for the formulation of design problems. Second, is a perceived disparity, concerning the aims of control, between conventional control theory and a more precise and gener- ally understood meaning of control. Conventional design theory requires the system to have a good margin of stability. In the new framework, control is achieved when the errors and other controlled variables are kept within re- spective specified tolerances, whatever the disturbances to the system. This criterion reflects more accurately the nature of the classical control problem and is the criterion used in some industries. In many situations, involving what are called critical systems, some controlled variables must not exceed their prescribed tolerances and this means that there is no satisfactory alter- native to the explicit use of the criterion that requires certain outputs to be bounded by their tolerances. The conventional framework for the design of control systems was formed during the five-year period ending in 1945. During that period, known ideas from the fields of electrical circuit theory and servomechanisms, with feedback theory as their common ground, were merged to form a practical approach to design that was eventually adopted by the control community. This frame- work has been greatly elaborated and generalised, and now contains many de- sign methods that are essentially equivalent. However, it remains, in essence, a practical approach to design and it gained acceptance because of its wide range of practical applications. An essential characteristic of the new framework is that, like the conven- tional framework, its design methods are useable in practice. This charac- teristic has resulted from the way the new framework has been developed, by an evolutionary process. Starting with a minimum of theory, new design approaches, making use of numerical methods, were evolved and these were tested on challenging design problems. Successful tests led to new design vi Preface theories and methods that were, in turn, tested and so on. Throughout this process, theory has been kept within the level needed to fulfil the aims of design. At present, not only can designs satisfying the conventional crite- rion be achieved but also critical systems, which are beyond the scope of the conventional framework, can be designed with equal facility. The new framework has involved the development of aspects of theory that have no exact counterpart in the conventional framework. These are concerned with general principles of design and comprise the principle of inequalities and the principle of matching. These principles facilitate an ac- curate and realistic formulation of the design problem and form part of the foundation of the new framework, upon which a superstructure of design methods has been built. The time has come to bring together, into a book, the components of the new framework that have been scattered in the research literature. This is accomplished in the chapters of the book, where the authors aim to explain, revise, correct or expand the ideas and results contained in published papers. Some of the material has not been published before. The book is addressed to those who use or develop practical methods for the design of control systems and to those concerned with the theory underlying such methods. Although the new framework is still growing, its conceptual basis is now sufficiently coherent to allow further systematic development and it contains sufficient methods to permit a wide range of practical applications. However, because it is relatively new, it offers the researcher unsolved problems, some of which are highlighted in the book. The principal aim of the book is to present the new framework. A sec- ondary aim is to show how the conventional framework can be made more effective by the use of the method of inequalities. The method involves the use of numerical processes together with two principles of design, one of which is based on the classical concept of stability, and the other is the principle of inequalities. This secondary aim recognises that the conventional framework is likely to have, for the foreseeable future, a continuing role in the field of control, despite the fact that its range of applications is more restricted than that of the new framework. It is hoped that the book will provide, students and researchers in uni- versities and practitioners in industry, ready access to this field. It is also intended to be a source book from which other, perhaps more integrated or specifically oriented, books on this subject can be written. Each chapter of the book is almost as self-contained as a paper in a jour- nal. References to the literature mainly indicate the primary sources of the material. However, the chapters were written in accordance with an overall plan for the book. Cross-references to other chapters indicate where certain topics are dealt with more fully or may indicate the sources new material. The chapters are grouped into four parts and their order is intended to give a logical rather than a chronological sequence for the development of the Preface vii subject, with Part I providing some introductions to the material in other parts. Nonetheless, the reader might find it more appropriate to choose an- other sequence. A reader interested in basic principles might start with Part I, while a reader interested in applications might start with the chapters in Part IV, which contain applications and case studies of the basic principles and methods developed in other parts, showing how various challenging practical problems can be formulated and solved. These problems are of two kinds, those that are formulated in terms of the conventional criterion of design and those, involving critical systems, which can be formulated and solved within the new framework. Parts II and III contain the essential computational and numerical methods required to put the framework into practice. V. Zakian February 2005 Acknowledgements The editor acknowledges the cooperation of all the contributors, in the preparation of this book. Thanks are due to James Whid- borne and Toshiyuki Satoh for formatting the book to the publishers speci- fications. Contents List of Contributors xii Part I. Basic Principles 1 Foundation of Control Systems Design Vladimir Zakian . 3 1.1 Need for New Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 ThePrincipleofInequalities 19 1.3 ThePrincipleofMatching 32 1.4 AClassof Linear Couples 43 1.5 Well-constructed Environment-system Models. . . . . . . . . . . . . . . 48 1.6 TheMethodofInequalities . 61 1.7 TheNodeArrayMethodforSolvingInequalities 68 References 92 Part II. Computational Methods (with Numerical Examples) 2 Matching Conditions for Transient Inputs Paul Geoffrey Lane . 97 2.1 Introduction 97 2.2 FinitenessofPeak Output 98 2.3 EvaluationofPeakOutput . 99 2.4 Example . 106 2.5 MiscellaneousResults 111 2.6 Conclusion . 115 References 119 3 Matching to Environment Generating Persistent Distur- bances Toshiyuki Satoh 121 3.1 Introduction 121 3.2 Preliminaries . 123 3.3 Computation of Peak Output via Convex Optimisation . . . . . . 124 3.4 AlgorithmforComputingPeakOutput 135 x Contents 3.5 NumericalExample . 135 3.6 Conclusions 143 References 144 4 LMI-based Design Takahiko Ono . 145 4.1 Introduction 145 4.2 Preliminary 146 4.3 ProblemFormulation 149 4.4 ControllerDesignviaLMI 150 4.5 NumericalExample . 161 4.6 Conclusion . 163 References 164 5 Design of a Sampled-data Control System Takahiko Ono . 165 5.1 Introduction 165 5.2 DesignforSISOSystems . 167 5.3 DesignforMIMOSystems . 184 5.4 DesignExample . 186 5.5 Conclusion . 189 References 189 Part III. Search Methods (with Numerical Tests) 6 A Numerical Evaluation of the Node Array Method Toshiyuki Satoh 193 6.1 Introduction 193 6.2 DetectionofStuckLocalSearch . 194 6.3 SpecialTestProblems . 195 6.4 TestResults 208 6.5 EffectofStoppingRule3 . 210 6.6 Conclusions 211 6.A Appendix — Moving Boundaries Process with the Rosenbrock TrialGenerator . 211 References 215 7 A Simulated Annealing Inequalities Solver James F Whidborne 219 7.1 Introduction 219 7.2 TheMetropolisAlgorithm 220 7.3 ASimulatedAnnealingInequalitiesSolver 221 7.4 NumericalTestProblems . 224 7.5 ControlDesignBenchmarkProblems 225 7.6 Conclusions 228 Contents xi 7.A Appendix–the Objective Functions . 228 References 229 8 Multi-objective Genetic Algorithms for the Method of In- equalities Tung-Kuan Liu and Tadashi Ishihara . 231 8.1 Introduction 231 8.2 Auxiliary Vector Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.3 GeneticInequalitiesSolver . 238 8.4 NumericalTestProblems . 243 8.5 ControlDesignBenchmarkProblems 245 8.6 Conclusions 246 References 247 Part IV. Case Studies 9 Design of Multivariable Industrial Control Systems by the Method of Inequalities Oluwafemi Taiwo 251 9.1 Introduction 251 9.2 Application of the Method of Inequalities to Distillation Columns . 255 9.3 Design of Multivariable Controllers for an Advanced Turbofan EnginebytheMethodof Inequalities 269 9.4 Improvement of Turbo-alternator Response by the Method of Inequalities . 277 References 281 10 Multi-objective Control using the Principle of Inequalities GPLiu 287 10.1 Introduction 287 10.2 Multi-objectiveOptimal-tuningPIDControl 288 10.3 Multi-objective Robust Eigenstructure Assignment . . . . . . . . . . 294 10.4 Multi-objectiveCriticalControl . 302 References 308 11 A MoI Based on H ∞ Theory — with a Case Study James F Whidborne 311 11.1 Introduction 311 11.2 Preliminaries . 313 11.3 A Two Degree-of-freedom H ∞ Method . 314 11.4 A MoI for the Two Degree-of-freedom Formulation . . . . . . . . . . 319 11.5 Example — Distillation Column Controller Design . . . . . . . . . . 320 11.6 Conclusions 325 References 325 . Library Cataloguing in Publication Data Control systems design : a new framework 1. Automatic control I. Zakian, Vladimir 629.8 ISBN 1852339136 Library. suchin .a@ chula.ac.th Tadashi Ishihara Faculty of Science and Technology Fukushima University Kanayagawa 1, Fukushima, 960-1296 Japan e-mail: ishihara@educ.fukushima-u.ac.jp