Table Of ContentControl engineering series 39
Continuous time
controller design
R. Balasubramanian
Peter Peregrinus ltd., on behalf of the institution of electrical engineers
IEE CONTROL ENGINEERING SERIES 39
Series Editors: Prof. D. P. Atherton
Prof. K. Warwick
Continuous time
controller design
Other volumes in this series
Volume 1 Multivariable control theory J. M. Layton
Volume 2 Elevator traffic analysis, design and control G. C. Barney
Volume 3 Transducers in digital systems G. A. Woolver
Volume 4 Supervisory remote control systems R. E. Young
Volume 5 Structure of interconnected systems H. Nicholson
Volume 6 Power system control M. J. H. Sterling
Volume 7 Feedback and multivariable systems D. H. Owens
Volume 8 A history of control engineering, 1800-1930 S. Bennett
Volume 9 Modern approaches to control system design
N. Munro (Editor)
Volume 10 Control of time delay systems J. E. Marshall
Volume 11 Biological systems, modelling and control D. A. Linkens
Volume 12 Modelling of dynamical systems—1 H. Nicholson (Editor)
Volume 13 Modelling of dynamical systems—2 H. Nicholson (Editor)
Volume 14 Optimal relay and saturating control system synthesis
E. P. Ryan
Volume 15 Self-tuning and adaptive control: theory and application
C. J. Harris and S. A. Billings (Editors)
Volume 16 Systems modelling and optimisation P. Nash
Volume 17 Control in hazardous environments R. E. Young
Volume 18 Applied control theory J. R. Leigh
Volume 19 Stepping motors: a guide to modern theory and practice
P. P. Acarnley
Volume 20 Design of modern control systems D. J. Bell, P. A. Cook and
N. Munro (Editors)
Volume 21 Computer control of industrial processes S. Bennett and
D. A. Linkens (Editors)
Volume 22 Digital signal processing N. B. Jones (Editor)
Volume 23 Robotic technology A. Pugh (Editor)
Volume 24 Real-time computer control S. Bennett and D. A. Linkens
(Editors)
Volume 25 Nonlinear system design S. A. Billings, J. O. Gray and
D. H. Owens (Editors)
Volume 26 Measurement and instrumentation for control M. G. Mylroi
and G. Calvert (Editors)
Volume 27 Process dynamics estimation and control A. Johnson
Volume 28 Robots and automated manufacture J. Billingsley (Editor)
Volume 29 Industrial digital control systems K. Warwick and D. Rees
(Editors)
Volume 30 Electromagnetic suspension—Dynamics and control
P. K. Sinha
Volume 31 Modelling and control of fermentation processes J. R. Leigh
(Editor)
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 36 Robot control K.Warwick and A. Pugh (Editors)
Volume 37 Industrial digital control systems—revised edition
K. Warwick and D. Rees (Editors)
Volume 38 Parallel processing in control P. J. Fleming
'an
Peter Peregrinus Ltd. on behalf of the Institution of Electrical Engineers
Published by: Peter Peregrinus Ltd., London, United Kingdom
© 1989: Peter Peregrinus Ltd.
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system or transmitted in any form or by any means-
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the prior written permission of the publisher.
While the author and the publishers believe that the information and
guidance given in this work is correct, all parties must rely upon their own
skill and judgment when making use of it. 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.
Balasubramanian, R.
Continuous time controller design
1. Continuous time control systems
I. Title II. Series
629.8'312
ISBN 0 86341 162 2
Printed in England by Short Run Press Ltd, Exeter
Contents
Preface xi
1 Introduction
1.1 Preliminaries 1
1.2 System models 2
1.3 Single-input/single-output systems (SISO systems) 2
1.4 Classical feedback control design 4
1.5 Control objective 5
1.6 Outline of other Chapters 6
1.7 Selected bibliography 6
2 The state space analysis of systems 7
2.1 The state space representation 7
2.2 Nonuniqueness of a state space representation 10
2.3 Solution of the state equation 13
2.3.1 Properties of eAt 13
2.3.2 The solution 14
2.4 Elementary frequency domain results 15
2.5 Evaluation of eAt 18
2.6 Controllability 20
2.6.1 Definitions 20
2.6.2 The basic result 20
2.6.3 Other conditions for controllability 22
2.6.4 Output controllability 24
2.7 Observability 25
2.7.1 Definition 25
2.7.2 The basic result 25
2.7.3 Other conditions for observability 26
2.8 More on controllability and observability 27
2.8.1 Duality 27
2.8.2 Nonsingular transformations and
controllability-observability 27
vi Contents
2.9 Canonical forms 28
2.9.1 Diagonalisation 29
2.9.2 Controllable companion form 33
2.9.3 Observable companion form 40
2.10 Controllability and observability: revisited 42
2.11 Minimal realisation 46
2.12 Summary 49
2.13 Selected bibliography 51
3 Modal control 53
3.1 The control problem 53
3.2 Modal control 53
3.3 Single-input case 55
3.3.1 Main result 55
3.3.2 Some pole placement algorithms 56
3.3.3 Ackermann's result 59
3.3.4 Assigning only some poles (others left unchanged) 60
3.3.5 What happens to the zeros? 63
3.4 The multi-input problem 65
3.5 Unity rank K 67
3.6 General properties of state and output feedback 73
3.7 More on output and state feedback 77
3.7.1 Eigenvector placement 78
3.8 Summary 80
3.9 Selected bibliography 80
Quadratic optimal control 82
4.1 Introduction 82
4.2 Mathematical preliminaries 83
4.2.1 Classical minimisation with respect to parameters 83
4.2.2 Euler-Lagrange equations 85
4.2.3 Euler-Lagrange equations with equality constraints 87
4.3 The main problem 87
4.4 A feedback solution 89
4.5 WhatisP(0? 91
4.6 Infinite time problem 92
4.6.1 Limiting behaviour of P(t, t) 94
x
4.7 Design for a prescribed degree of stability 98
4.8 Computation of the steady state P 103
4.8.1 An iterative scheme 103
4.8.2 Method based on eigenvectors 106
4.9 A frequency domain equivalent of the Riccati equation 111
Contents vii
4.10 Scalar case 113
4.10.1 Phase and gain margins of the optimal controller 113
4.10.2 Design via spectral factorisation 114
4.10.3 Sensitivity and tolerance to nonlinearity 117
4.10.4 A root locus interpretation 118
4.11 Summary 120
4.12 Selected bibliography 120
5 Design of observers 122
5.1 Introduction 122
5.2 Full order Luenberger observer 123
5.3 Full order observer/controller combination 124
5.4 Reduced order observers 127
5.5 Reduced order observer/controller combination 130
5.6 General theory of observers 131
5.6.1 Minimal order observers for linear functions of the
states 132
5.6.2 Minimal order observers in the closed loop 135
5.7 Summary 136
5.8 Selected bibliography 136
6 Other selected state space design methods 138
6.1 Introduction 138
6.2 Decoupling control 138
6.2.1 Dynamic decoupling 139
6.2.2 Static decoupling 144
6.3 Model following control 146
6.3.1 Algebraic methods 147
6.3.2 Model following via optimal control 151
6.4 The robust servomechanism problem 154
6.4.1 Motivation 154
6.4.2 The problem 156
6.4.3 The main result 157
6.4.4 Effect of changes in A 161
c
6.4.5 Discussion 163
6.5 Summary 163
6.6 Selected bibliography 164
7 Frequency domain analysis of multivariable systems 165
7.1 Introduction 165
7.2 Basic frequency domain relationships 167
7.2.1 Inverse relationships 167
7.2.2 System sensitivity 168
7.2.3 Other expressions for R(s) 168
viii Contents
7.3 Stability of multivariable systems 169
7.3.1 Some definitions 170
7.4 System matrices 172
7.5 Smith-McMillan form of G(s) 177
7.5.1 Smith form 177
7.5.2 Smith-McMillan form 179
7.6 Poles and zeros of G(s) 181
7.6.1 Poles of G(s) 181
7.6.2 Zeros of G(s) 182
7.7 The design problem 186
7.8 Summary 187
7.9 Selected bibliography 187
8 The inverse Nyquist array method 189
8.1 Introduction 189
8.2 Stability of multivariable feedback systems 189
8.3 The single-input/single-output case 192
8.3.1 Principle of the argument 192
8.3.2 Stability of SISO systems 194
8.3.3 Inverse Nyquist diagram 194
8.4 Multivariable systems 197
8.5 Diagonal dominance and stability 198
8.6 Graphical checking of dominance 200
8.7 Controller design for diagonal dominance 203
8.7.1 Static decoupling 203
8.7.2 Pseudo decoupling 203
8.8 Design examples 204
8.9 Summary 211
8.10 Selected bibliography 211
9 The characteristic locus method 213
9.1 Introduction 213
9.2 Eigenvalues and eigenvectors of G(s) 213
9.2.1 Expansion of G(s) via its eigenvectors 216
9.3 Analysis of feedback systems using characteristic transfer
functions 216
9.3.1 Closed loop stability 217
9.3.2 The quadratic optimal controller and the characteristic
loci 222
9.3.3 Actuator failures and the characteristic loci 224
9.3.4 Performance and the characteristic loci 225
9.3.5 Performance and characteristic directions 225
9.4 Design based on the characteristic loci 227
9.4.1 Commutative controller 228
Contents ix
9.4.2 Controller factorisation 228
9.4.3 The design procedure 228
9.4.4 A design example 229
9.5 Summary 232
9.6 Selected bibliography 233
10 Frequency domain design by factorisation methods 234
10.1 Introduction 234
10.2 Mathematical preliminaries 235
10.3 Matrices over rings 239
10.3.1 Smith-McMillan form 241
10.3.2 Coprime factorisation 242
10.3.3 Relation between RCPF and LCPF 245
10.3.4 Matrices over Euclidean domains 247
10.4 Polynomial matrices 248
10.4.1 Polynomials 248
10.4.2 Polynomial matrices 248
10.4.3 Column and row reduction 249
10.5 Polynomial matrices and design 251
10.6 Matrices over rings of proper stable transfer functions 255
10.6.1 Coprime factorisation of matrices over F(s) 256
10.7 Application of matrices over F (s) in system design 257
x
10.7.1 Feedback and stability 257
10.8 Summary 260
10.9 Selected bibliography 260
11 Selected stochastic problems in control 261
11.1 Introduction 261
11.2 The Wiener filter 261
11.2.1 Linear time invariant systems with stationary inputs 261
11.2.2 The classical Wiener problem 262
11.2.3 Solution of Wiener-Hopf equation 264
11.2.4 Relation to the control problem 265
11.3 A brief review of estimation theory 267
11.3.1 Preliminaries 268
11.3.2 Linear Gaussian problem 271
11.3.3 Least squares linear estimator 272
11.3.4 Weighted least squares linear estimator 273
11.3.5 Recursive estimators 274
11.3.6 Rapproachment with the Gaussian case 275
11.3.7 Solution problems 276
11.3.8 The nonlinear problem 277
11.4 The discrete Kalman filter 281
11.4.1 The extended discrete Kalman filter 284
