Analysis and Control of Linear Systems This page intentionally left blank Analysis and Control of Linear Systems Edited by Philippe de Larminat First published in France in 2002 by Hermès Science/Lavoisier entitled “Analyse des systèmes linéaires” and “Commande des systèmes linéaires” First published in Great Britain and the United States in 2007 by ISTE Ltd 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 and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd ISTE USA 6 Fitzroy Square 4308 Patrice Road London W1T 5DX Newport Beach, CA 92663 UK USA www.iste.co.uk © ISTE Ltd, 2007 © LAVOISIER, 2002 The rights of Philippe de Larminat to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data [Analyse des systèmes linéaires/Commande des systèmes linéaires. eng] Analysis and control of linear systems analysis and control of linear systems/edited by Philippe de Larminat. p. cm. ISBN-13: 978-1-905209-35-4 ISBN-10: 1-905209-35-5 1. Linear control systems. 2. Automatic control. I. Larminat, Philippe de. TJ220.A5313 2006 629.8'32 dc22 2006033665 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 10: 1-905209-35-5 ISBN 13: 978-1-905209-35-4 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire. Table of Contents Preface xv Part 1. System Analysis 1 Chapter 1. Transfer Functions and Spectral Models 3 Dominique BEAUVOIS and Yves TANGUY 1.1. System representation 3 1.2. Signal models 4 1.2.1. Unit-step function or Heaviside step function U(t) 4 1.2.2. Impulse 4 1.2.3. Sine-wave signal 7 1.3. Characteristics of continuous systems 7 1.4. Modeling of linear time-invariant systems 8 1.4.1. Temporal model, convolution, impulse response and unit-step response 8 1.4.2. Causality 9 1.4.3. Unit-step response 10 1.4.4. Stability 10 1.4.5. Transfer function 12 1.4.6. Causality, stability and transfer function 16 1.4.7. Frequency response and harmonic analysis 17 1.5. Main models 21 1.5.1. Integrator 21 1.5.2. First order system 23 1.5.3. Second order system 27 1.6. A few reminders on Fourier and Laplace transforms 33 1.6.1. Fourier transform 33 1.6.2. Laplace transform 34 1.6.3. Properties 38 vi Analysis and Control of Linear Systems 1.6.4. Laplace transforms of ordinary causal signals 40 1.6.5. Ordinary Fourier transforms 41 1.7. Bibliography 42 Chapter 2. State Space Representation 43 Patrick BOUCHER and Patrick TURELLE 2.1. Reminders on the systems 44 2.1.1. Internal representation of determinist systems: the concept of state 44 2.1.2. Equations of state and equations of measurement for continuous systems 46 2.1.3. Case of linear systems 47 2.1.4. Case of continuous and invariant linear systems 48 2.2. Resolving the equation of state 48 2.2.1. Free state 48 2.2.2. Forced state 49 2.2.3. Particular case of linear and invariant systems 50 2.2.4. Calculation method of the transition matrix 0 A(t -t ) e 51 2.2.5. Application to the modeling of linear discrete systems 55 2.3. Scalar representation of linear and invariant systems 57 2.3.1. State passage → transfer 57 2.3.2. Change of basis in the state space 60 2.3.3. Transfer passage → state 60 2.3.4. Scalar representation of invariant and linear discrete systems 65 2.4. Controllability of systems 66 2.4.1. General definitions 66 2.4.2. Controllability of linear and invariant systems 66 2.4.3. Canonic representation of partially controllable systems 69 2.4.4. Scalar representation of partially controllable systems 73 2.5. Observability of systems 74 2.5.1. General definitions 74 2.5.2. Observability of linear and invariant systems 74 2.5.3. Case of partially observable systems 77 2.5.4. Case of partially controllable and partially observable systems. . . 78 2.6. Bibliography 79 Chapter 3. Discrete-Time Systems 81 Philippe CHEVREL 3.1. Introduction 81 3.2. Discrete signals: analysis and manipulation 83 3.2.1. Representation of a discrete signal 83 3.2.2. Delay and lead operators 84 3.2.3. z-transform 85 Table of Contents vii 3.3. Discrete systems (DLTI) 88 3.3.1. External representation 88 3.3.2. Internal representation 89 3.3.3. Representation in terms of operator 91 3.3.4. Transfer function and frequency response 96 3.3.5. Time response of basic systems 98 3.4. Discretization of continuous-time systems 99 3.4.1. Discretization of analog signals 100 3.4.2. Transfer function of the discretized system 101 3.4.3. State representation of the discretized system 102 3.4.4. Frequency responses of the continuous and discrete system 103 3.4.5. The problem of sub-sampling 104 3.4.6. The problem of over-sampling 105 3.5. Conclusion 107 3.6. Bibliography 107 Chapter 4. Structural Properties of Linear Systems 109 Michel MALABRE 4.1. Introduction: basic tools for a structural analysis of systems 109 4.1.1. Vector spaces, linear applications 110 4.1.2. Invariant sub-spaces 111 4.1.3. Polynomials, polynomial matrices 113 4.1.4. Smith form, companion form, Jordan form 114 4.1.5. Notes and references 115 4.2. Beams, canonical forms and invariants 115 4.2.1. Matrix pencils and geometry 117 4.2.2. Kronecker’s canonical form 118 4.2.3. Controllable, observable canonical form (Brunovsky) 121 4.2.4. Morse’s canonical form 125 4.2.5. Notes and references 128 4.3. Invariant structures under transformation groups 128 4.3.1. Controllability indices 128 4.3.2. Observability indices 129 4.3.3. Infinite zeros 129 4.3.4. Invariants, transmission finite zeros 131 4.3.5. Notes and references 132 4.4. An introduction to a structural approach of the control 132 4.4.1. Disturbance rejection and decoupling: existence of solutions 133 4.4.2. Disturbance rejection and decoupling: existence of stable solutions 135 4.4.3. Disturbance rejection and decoupling: flexibility in the location of poles/fixed poles 135 4.4.4. Notes and references 136 viii Analysis and Control of Linear Systems 4.5. Conclusion 137 4.5.1. Optimal attenuation of disturbance 137 4.6. Bibliography 137 Chapter 5. Signals: Deterministic and Statistical Models 141 Eric LE CARPENTIER 5.1. Introduction 141 5.2. Signals and spectral analysis 141 5.3. Generator processes and ARMA modeling 150 5.4. Modeling of LTI systems and ARMAX modeling 153 5.4.1. ARX modeling 153 5.4.2. ARMAX modeling 154 5.4.3. Output error model 154 5.4.4. Representation of the ARMAX model within the state space 155 5.4.5. Predictor filter associated with the ARMAX model 155 5.5. From the Markovian system to the ARMAX model 156 5.6. Bibliography 157 Chapter 6. Kalman’s Formalism for State Stabilization and Estimation 159 Gilles DUC 6.1. The academic problem of stabilization through state feedback 159 6.2. Stabilization by pole placement 161 6.2.1. Results 161 6.2.2. Example 163 6.3. Reconstruction of state and observers 164 6.3.1. General principles 164 6.3.2. Continuous-time observer 165 6.3.3. Discrete-time observer 166 6.3.4. Calculation of the observer by pole placement 167 6.3.5. Behavior of the observer outside the ideal case 168 6.3.6. Example 169 6.4. Stabilization through quadratic optimization 171 6.4.1. General results for continuous-time 171 6.4.2. General results for discrete-time 173 6.4.3. Interpretation of the results 174 6.4.4. Example 175 6.5. Resolution of the state reconstruction problem by duality of the quadratic optimization 177 6.5.1. Calculation of a continuous-time observer 177 6.5.2. Calculation of a discrete-time observer 178 6.5.3. Interpretation in a stochastic context 179 6.5.4. Example 181 Table of Contents ix 6.6. Control through state feedback and observers 183 6.6.1. Implementation of the control 183 6.6.2. Dynamics of the looped system 184 6.6.3. Interest and limitations of this result 185 6.6.4. Interpretation in the form of equivalent corrector 186 6.6.5. Example 187 6.7. A few words on the resolution of Riccati’s equations 189 6.8. Conclusion 192 6.9. Bibliography 192 Chapter 7. Process Modeling 195 Alain BARRAUD, Suzanne LESECQ and Sylviane GENTIL 7.1. Introduction 195 7.2. Modeling 198 7.3. Graphic identification approached 204 7.3.1. Pseudo-periodic unit-step response 205 7.3.2. Aperiodic unit-step response 207 7.3.3. Partial conclusion 213 7.4. Identification through criterion optimization 214 7.4.1. Algorithms 214 7.4.2. Models 215 7.4.3. Methods 215 7.4.4. Optimization criteria 216 7.4.5. The problem of precision 217 7.4.6. How to optimize 218 7.4.7. Partial conclusion 219 7.4.8. Practical application 220 7.5. Conclusion around an example 222 7.5.1. Simulated procedure 222 7.5.2. In search of a model 223 7.6. Bibliography 226 Chapter 8. Simulation and Implementation of Continuous Time Loops . . 227 Alain BARRAUD and Sylviane GENTIL 8.1. Introduction 227 8.1.1. About linear equations 228 8.1.2. About non-linear equations 228 8.2. Standard linear equations 228 8.2.1. Definition of the problem 228 8.2.2. Solving principle 229 8.2.3. Practical implementation 229 8.3. Specific linear equations 231 8.3.1. Definition of the problem 231 x Analysis and Control of Linear Systems 8.3.2. Solving principle 232 8.3.3. Practical implementation 233 8.4. Stability, stiffness and integration horizon 234 8.5. Non-linear differential systems 235 8.5.1. Preliminary aspects 235 8.5.2. Characterization of an algorithm 236 8.5.3. Explicit algorithms 239 8.5.4. Multi-interval implicit algorithms 240 8.5.5. Solver for stiff systems 242 8.5.6. Partial conclusion 243 8.6. Discretization of control laws 244 8.6.1. Introduction 244 8.6.2. Discretization 244 8.6.3. Application to PID regulators 247 8.7. Bibliography 250 Part 2. System Control 251 Chapter 9. Analysis by Classic Scalar Approach 253 Houria SIGUERDIDJANE and Martial DEMERLÉ 9.1. Configuration of feedback loops 253 9.1.1. Open loop – closed loops 253 9.1.2. Closed loop harmonic analysis 255 9.2. Stability 258 9.2.1. Nyquist criterion 259 9.2.2. Routh’s algebraic criterion 265 9.2.3. Stability margins 267 9.3. Precision 270 9.3.1. Permanent error 272 9.3.2. Transitional error 277 9.4. Parametric sensitivity 278 9.4.1. Open loop sensitivity 278 9.4.2. Closed loop sensitivity 280 9.5. Bibliography 282 Chapter 10. Synthesis of Closed Loop Control Systems 283 Houria SIGUERDIDJANE and Martial DEMERLÉ 10.1. Role of correctors: precision-stability dilemma 283 10.1.1. Analysis of systems’ behavior 284 10.1.2. Serial correction 288 10.1.3. Parallel correction 289 10.1.4. Correction by anticipation 290 10.1.5. Conclusions 292 [...]... why linear systems are a fundamental theory, and the problems linked to closed-loop control are a big part of it Input-output and the state representations, although closely linked, are explained in separate chapters (1 and 2) Discrete-time systems are, for more clarity, explained in Chapter 3 Chapter 4 explains the structural properties of linear systems Chapter xvi Analysis and Control of Linear Systems. .. using a linear model is generally sufficient to obtain efficient control laws Limits to the automated systems performances generally come from the restricted power of motors, precision of captors and variability of the behavior of the processes, more than from their possible non-linearity It is necessary to know the basics of linear automated systems before learning about the theory of non -linear systems. .. 293 293 300 303 303 306 306 307 307 3 10 3 10 313 315 315 317 3 20 323 325 Chapter 11 Robust Single-Variable Control through Pole Placement Gérard THOMAS 327 11.1 Introduction 11.1.1 Guiding principles and notations 11.1.2 Reminders on polynomial algebra 11.2 The obvious objectives of the correction 11.2.1... 363 3 70 3 70 xii Analysis and Control of Linear Systems Chapter 12 Predictive Control Patrick BOUCHER and Didier DUMUR 12.1 General principles of predictive control 12.1.1 Anticipative aspect 12.1.2 Explicit prediction of future behavior 12.1.3 Optimization by minimization of a quadratic criterion 12.1.4 Principle of the... deterministic and statistical models of signals Chapter 6 introduces us to two fundamental theoretical tools: state stabilization and estimation These two notions are also covered in control- related chapters Chapter 7 defines the elements of modeling and identification All modern control theories rely on the availability of mathematical models of processes to control them Modeling is therefore upstream of the control. .. concept, Chapter 11 addresses pole placement control and Chapter 12 internal control The three following chapters cover modern automation based on state representation They highlight the necessary methodological aspects H2 optimization control is explained in Chapter 13, modal control in Chapter 14 and H∞ control in Chapter 15 Chapter 16 covers linear time-variant systems ... Synthesis of a corrector depending on varying parameters 15.4.1 Problem considered and L2 gain 15.4.2 Polytopic approach 15.4.3 A more general approach 15.4.4 Example 15.5 Conclusion 15.6 Bibliography 500 502 502 504 506 507 507 509 512 515 518 518 Chapter 16 Linear. .. 4 80 4 80 482 488 488 4 90 493 493 495 495 496 xiv Analysis and Control of Linear Systems 15.2.4 Evaluation of structured single value 15.3 The µ-synthesis 15.3.1 A H∞ robust synthesis 15.3.2 Approach by D-K iterations 15.3.3 Example... because the basic systems theory is needed before it can be developed This same theory also constitutes the beginning of Chapter 8, which is about simulation techniques These techniques form the basis of the control laws created by engineers Chapter 9 provides an analysis of the classic invariable techniques while Chapter 10 summarizes them Based on the transfer function concept, Chapter 11 addresses... 10. 6 Proportional integral proportional (PID) correction 10. 6.1 Transfer function 10. 6.2 Experimental adjustment method 10. 7 Parallel correction 10. 7.1 General principle 10. 7.2 Simple tachymetric correction (C(p) = λp) 10. 7.3 Filtered tachymetric correction 10. 7.4 Correction of delay systems: Smith predictor 10. 8 Bibliography . systems analysis and control of linear systems/ edited by Philippe de Larminat. p. cm. ISBN-13: 97 8-1 - 905 20 9-3 5-4 ISBN- 10: 1-9 05 20 9-3 5-5 1. Linear control systems. 2. Automatic control. I. Larminat,. TJ2 20. A5313 200 6 629.8'32 dc22 200 603 3665 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 10: 1-9 05 20 9-3 5-5 ISBN. separate chapters (1 and 2). Discrete-time systems are, for more clarity, explained in Chapter 3. Chapter 4 explains the structural properties of linear systems. Chapter xvi Analysis and Control of