Table Of Content

IET Control Engineering Series 56 Symbolic Methods inS Symbolic Methods y c om n in control system analysis and design tb r oo in control system analysis and design l Symbolic computing has made a significant impact in the field of control engineering. Neil Munro (1936–2004) received sli This book, which brings together contributions from leading international experts in his B.Sc. in electrical engineering yc the field, provides an up-to-date treatment of various issues in system modelling, from Strathclyde University in 1962, st analysis, design and synthesis methods. his Ph.D. from UMIST in 1969, and a eM D.Sc. in control engineering from the m C•ovetnhroaeng laeinp iepnalcircl uaedtlieoemsn: eonf tmsultidimensional Laplace transforms to the modelling of Uwconansivt reaorpls pietoyn ignoitnfe eMde aprninrocghf eeastss tUoerrM oinIfS aT1 p9inp8 l3ie. dH e aneth • a survey of customised computer algebra modelling programs for multibody 1979 and was head of the Control a Systems Centre for over 30 years. lo dynamical systems He was chairman of SERC’s Control y • central results for the analysis and robust control of single-input single-output & Instrumentation Subcommittee sid linear systems using a new linear programming approach (1983–1985), Director of Control ss • differential and real algebra for dealing with systems of polynomial equations and and Instrumentation for the SERC and a Edited by Neil Munro inequalities for nonlinear systems, and the computational aspects of algebraic and DTI Joint Framework for Information n geometric invariants for both certain and uncertain systems Technology (1991–1992), and the d • the dynamic sliding mode approach to the robust control of both linear and National Consultant on Control and Instrumentation for EPSRC d nonlinear systems, and robust pole assignment for parametric uncertain linear (1994–1997). Professor Munro was e systems a Fellow of the IEE, IEEE and Institute s • the development and testing of a new branch-and-bound algorithm for global of Measurement and Control, and ig optimisation, using symbolic algebra techniques a recipient of the Sir Harold Hartley n • an overview and assessment of the use of symbolic computation programs in Medal (1982) and the Honeywell the analysis and design of control systems. International Medal (1997) for contributions to control engineering. The book will be of interest to postgraduate students and researchers in control engineering. E d it e d b The Institution of Engineering and Technology y M www.theiet.org u 0 85296 943 0 n r 978-0-85296-943-4 o IET conTrol EngInEErIng sErIEs 56 Series Editors: Professor D.P. Atherton Professor G.W. Irwin Symbolic Methods in control system analysis and design Other volumes in this series: Volume 2 Elevator traffic analysis, design and control, 2nd edition G.C. Barney and S.M. dos Santos Volume 8 A history of control engineering, 1800–1930 S. Bennett Volume 14 Optimal relay and saturating control system synthesis E.P. Ryan Volume 18 Applied control theory, 2nd edition J.R. Leigh Volume 20 Design of modern control systems D.J. Bell, P.A. Cook and N. Munro (Editors) Volume 28 Robots and automated manufacture J. Billingsley (Editor) Volume 30 Electromagnetic suspension: dynamics and control P.K. Sinha Volume 32 Multivariable control for industrial applications J. O’Reilly (Editor) Volume 33 Temperature measurement and control J.R. Leigh Volume 34 Singular perturbation methodology in control systems D.S. Naidu Volume 35 Implementation of self-tuning controllers K. Warwick (Editor) Volume 37 Industrial digital control systems, 2nd edition K. Warwick and D. Rees (Editors) Volume 38 Parallel processing in control P.J. Fleming (Editor) Volume 39 Continuous time controller design R. Balasubramanian Volume 40 Deterministic control of uncertain systems A.S.I. Zinober (Editor) Volume 41 Computer control of real-time processes S. Bennett and G.S. Virk (Editors) Volume 42 Digital signal processing: principles, devices and applications N.B. Jones and J.D.McK. Watson (Editors) Volume 43 Trends in information technology D.A. Linkens and R.I. Nicolson (Editors) Volume 44 Knowledge-based systems for industrial control J. McGhee, M.J. Grimble and A. Mowforth (Editors) Volume 47 A history of control engineering, 1930–1956 S. Bennett Volume 49 Polynomial methods in optimal control and filtering K.J. Hunt (Editor) Volume 50 Programming industrial control systems using IEC 1131-3 R.W. Lewis Volume 51 Advanced robotics and intelligent machines J.O. Gray and D.G. Caldwell (Editors) Volume 52 Adaptive prediction and predictive control P.P. Kanjilal Volume 53 Neural network applications in control G.W. Irwin, K. Warwick and K.J. Hunt (Editors) Volume 54 Control engineering solutions: a practical approach P. Albertos, R. Strietzel and N. Mort (Editors) Volume 55 Genetic algorithms in engineering systems A.M.S. Zalzala and P.J. Fleming (Editors) Volume 56 Symbolic methods in control system analysis and design N. Munro (Editor) Volume 57 Flight control systems R.W. Pratt (Editor) Volume 58 Power-plant control and instrumentation D. Lindsley Volume 59 Modelling control systems using IEC 61499 R. Lewis Volume 60 People in control: human factors in control room design J. Noyes and M. Bransby (Editors) Volume 61 Nonlinear predictive control: theory and practice B. Kouvaritakis and M. Cannon (Editors) Volume 62 Active sound and vibration control M.O. Tokhi and S.M. Veres Volume 63 Stepping motors: a guide to theory and practice, 4th edition P.P. Acarnley Volume 64 Control theory, 2nd edition J. R. Leigh Volume 65 Modelling and parameter estimation of dynamic systems J.R. Raol, G. Girija and J. Singh Volume 66 Variable structure systems: from principles to implementation A. Sabanovic, L. Fridman and S. Spurgeon (Editors) Volume 67 Motion vision: design of compact motion sensing solution for autonomous systems J. Kolodko and L. Vlacic Volume 69 Unmanned marine vehicles G. Roberts and R. Sutton (Editors) Volume 70 Intelligent control systems using computational intelligence techniques A. Ruano (Editor) Symbolic Methods in control system analysis and design Edited by Neil Munro The Institution of Engineering and Technology Published by The Institution of Engineering and Technology, London, United Kingdom First edition © 1999 The Institution of Electrical Engineers Reprint with new cover © 2007 The Institution of Engineering and Technology First published 1999 Reprinted 2007 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. 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 be reproduced, stored or transmitted, in any form or by any means, only 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. Inquiries concerning reproduction outside those terms should be sent to the publishers at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the author and the publishers believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor the publishers assume any liability to anyone for any loss or damage caused by any error or omission in the work, whether such error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN (10 digit) 0 85296 943 0 ISBN (13 digit) 978-0-85296-943-4 Printed in the UK by Bookcraft, Bath Reprinted in the UK by Lightning Source UK Ltd, Milton Keynes Contents Foreword Contributors I SYSTEM MODELLING Symbolic modelling and analysis of nonlinear systems 3 H. A. Barker 1.1 Introduction 3 1.2 Nonlinear systems and signals 4 1.2.1 System elements and operations 4 1.2.2 Signals and signal components 5 1.3 Nonlinear system relationships 5 1.3.1 Elemental equations 5 1.3.2 Signal transforms 6 1.3.3 Multidimensional transfer functions 6 1.4 Application of symbolic computation 7 1.5 Example 8 1.5.1 Manual procedure 9 1.5.2 Symbolic procedure 11 1.6 Nonlinear system response 11 1.6.1 Time-domain response 12 1.6.2 Frequency-domain response 13 1.7 Example 14 1.7.1 Time-domain response 14 1.7.2 Frequency-domain response 15 1.8 Graphical user interface 16 1.9 Example 19 1.10 References 21 vi Contents 2 Symbolic computation for manipulation of hierarchical bond graphs 23 P. J. Gawthrop and D. J. Ballance 2.1 Introduction 23 2.2 Modelling with bond graphs 24 2.2.1 Variables 25 2.2.2 Bonds 26 2.2.3 Junctions 26 2.2.4 Components 27 2.2.4.1 One-port components 27 2.2.4.2 Two-port components 27 2.2.5 Parameters 28 2.2.6 Constitutive relationships 28 2.2.7 Causality 28 2.3 Hierarchical bond graphs 29 2.4 Representations, languages and software 31 2.5 Example: a motor generator 33 2.5.1 Acausal bond graph 33 2.5.1.1 Subsystems 33 2.5.1.2 System Generator 36 2.5.1.3 System DC 37 2.5.1.4 System Load 38 2.5.1.5 System Motor 39 2.5.1.6 System PSU 39 2.5.1.7 System Shaft 40 2.5.2 System structure 41 2.5.3 System ordinary differential equations 42 2.5.4 System state matrices 44 2.5.5 System frequency response 46 2.5.6 Numerical parameters and initial states 48 49 2.5.7 System response 50 2.6 Conclusion 2.7 References 50 A survey of customised computer algebra programs for multibody dynamic modelling 53 I. C. Brown and P. J. Larcombe 3.1 Introduction 53 3.1.1 General comments 53 3.1.2 Multibody dynamics and computer algebra 54 3.2 Dynamic formulations and modelling approaches 56 3.2.1 Bond graphs 58 3.3 Customised dynamics programs: type and application 59 3.3.1 Numeric methods 60 3.3.2 Symbolic methods 60 3.3.3 Applications 61 3.4 Multibody dynamics: historical development 62 3.4.1 Early dynamics programs 62 3.4.2 Pioneering symbolic dynamics programs 63 3.4.3 Second-generation symbolic programs 64 Contents vn 3.4.4 Continuing use of numeric programs 65 3.4.5 Programs reliant on user knowledge 66 3.4.6 Modern programs 66 3.5 Summary 67 3.5.1 Concluding remarks 72 3.6 References 74 II SYSTEM ANALYSIS 79 4 Robust control under parametric uncertainty. Part I: analysis 81 L. H. Keel and S. P. Bhattacharyya 4.1 Introduction 81 4.2 Notation and preliminaries 82 4.2.1 Parametric uncertainty 84 4.2.2 Boundary crossing and zero exclusion 84 4.3 Real parameter stability margin 85 4.3.1 I2 real parametric stability margin 88 4.3.2 l^ stability margin for time-delay systems 92 4.4 Extremal results in RPRCT 94 4.4.1 Kharitonov's theorem 95 4.4.2 The edge theorem 98 4.4.3 The generalised Kharitonov theorem 99 4.4.3.1 Construction of the extremal subset 101 4.5 Frequency-domain results in RPRCT 106 4.5.1 Frequency-domain properties 108 4.5.2 Closed-loop transfer functions 109 4.6 Concluding remarks 111 4.7 Acknowledgments 113 4.8 References 113 5 Using differential and real algebra in control 115 S.T.Glad and M. Jirstrand 5.1 Introduction 115 5.2 Differential algebra: motivating examples 115 5.3 The basic ideas of differential algebra 117 5.3.1 Ranking of polynomials 118 5.3.2 Reducedness 118 5.3.3 Autoreduced sets 119 5.3.4 Remainders 120 5.3.5 Ritt's algorithm 120 5.3.6 Application of Ritt's algorithm to identifiability problems 121 5.4 Real algebra: motivating example 121 5.5 Constructive methods in real algebra 122 5.5.1 Cylindrical algebraic decomposition 123 5.5.2 Quantifier elimination 126 5.5.3 Applications 127 5.5.3.1 Stationarisable points 127 5.5.3.2 Stability 129 5.5.3.3 Curve following 131 viii Contents 5.5.4 Software 133 5.6 Conclusions 133 5.7 References 133 6 Approximate algebraic computations of algebraic invariants 135 N. Karcanias and M. Mitrouli 6.1 Introduction 135 6.2 Models with numerical inaccuracies and classification of algebraic computation problems 137 6.3 Numerical tools for approximate computations 140 6.3.1 Issues related to numerical dependence and independence of vectors 140 6.3.1.1 The SVD criterion 140 6.3.1.2 The Gramian criterion 141 6.3.2 Compound matrices 141 6.3.3 Selection of a best uncorrupted base for a numerically dependent set 142 6.4 Almost zeros of a set of polynomials 146 6.5 GCD method based on ERES 149 6.5.1 The ERES method: theoretical and numerical issues 150 6.6 GCD method based on matrix pencils 153 6.6.1 Connection o£V j with a linear system 153 m 6.6.2 Reduction of the original set 153 6.6.3 Determination of the associated pencil 154 6.7 LCM computation and approximate factorisation of polynomials 156 6.7.1 Fundamental definitions and properties of LCM of polynomials 156 6.7.2 Factorisation and almost factorisation of polynomials 157 6.7.3 Algorithm for computing the LCM of many polynomials 161 6.8 Applications, examples and numerical performance of methods 161 6.8.1 Performance of the ERES and MP methods 165 6.8.2 Performance of the LCM method 166 6.9 Conclusions 167 6.10 References 167 7 Robust stability conditions for MIMO systems with parametric uncertainty 169 E. Kontogiannis and N. Munro 7.1 Introduction 169 7.2 Direct Nyquist array design method 170 7.3 Uncertain parametric systems and polynomial families 174 7.4 From classical to robust control 176 7.5 Robust direct Nyquist array (RDNA) 179 7.5.1 Robust generalised dominance (RGD) stability theorem 183 7.5.2 Robust fundamental dominance (RFD) stability theorem 185 7.6 Illustrative examples 187 7.7 Conclusions 195 7.8 Acknowledgments 197 7.9 References 198 7.10 Appendixes 199 Contents ix 7.10.1 Appendix A 199 7.10.2 Appendix B 200 III DESIGN AND SYNTHESIS METHODS 201 8 Robust control under parametric uncertainty. Part II: design 203 L. H. Keel and S. P. Bhattacharyya 8.1 Introduction 203 8.2 Robust classical controller design using RPRCT 203 8.3 Linear programming approach to design 210 8.3.1 Fixed-order pole assignment and robust stabilisation 213 8.3.2 Robust pole assignment 216 8.4 Conclusions and future directions 225 8.5 Acknowledgments 226 8.6 References 226 9 Dynamic sliding mode control design using symbolic algebra tools 227 S. K. Spurgeon and X. Y. Yu 9.1 Introduction 227 9.2 Dynamic sliding mode control 228 9.3 Design method 231 9.3.1 Direct sliding mode control 232 9.3.2 Indirect sliding mode control 234 9.3.3 Robust design method 235 9.4 Mathematica implementation 237 9.5 Design examples 238 9.6 Conclusion 241 9.7 Acknowledgment 242 9.8 References 242 9.9 Appendixes 243 9.9.1 Appendix A 243 9.9.2 Appendix B 245 9.9.3 Appendix C 247 10 Pole assignment for uncertain systems 251 M. T. Soylemez and N. Munro 10.1 Introduction 251 10.2 General state-feedback pole assignment problem 254 10.2.1 Dyadic methods 254 10.2.2 Full-rank methods 258 10.3 General output-feedback pole assignment problem 262 10.3.1 Static feedback 262 10.3.2 Dynamic feedback 267 10.4 Conclusions 271 10.5 References 271