Design of Modern Control Systems - IET Digital Library

Control engineering series 20
Design of
MODERN
CONTROL
SYSTEMS
Edited by
D J Bell PA Cook N Munro
• •
Peter Peregrinus Ltd. on behfalf of the Institution of Electrical Engineers

IEE CONTROL ENGINEERING SERIES 20
Series Editors: Prof. D. P. Atherton
Dr. K. Warwick
CONTROL
SYSTEMS

Other volumes in this series
Volume 1
Volume 2
Multivariable control theory J. M. Layton
Elevator traffic analysis, design and control G. C. Barney and
S. M. Dos Santos
Volume 3 Transducers in digital systems G. A. Woolvet
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: theory and applications K. Warwick and A. Pugh
(Editors)

CONTROL
SYSTEMS
D J Bell • PA Cook*T\J Munro

Published by: Peter Peregrinus Ltd., London, United Kingdom
© 1982 Peter Peregrinus Ltd.
Reprinted 1988
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—electronic,
mechanical, photocopying, recording or otherwise—without the prior written
permission of the publisher.
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 judgment
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.
British Library Cataloguing in Publication Data.
Bell, D. J.
Design of modern control systems.
(IEE control engineering series; v. 20)
1. Control theory
I. Title II. Cook, P. A. III. Munro, N.
ISBN 0 906048 74 5
Printed in England by Antony Rowe Ltd.

Contents
Preface
1 State-space theory 1
1.1 Introduction 1
1.2 Controlability and observability 2
1.3 Standard forms
1.4 Calculation of transfer function
matrix 9
1.5 Realizations 11
1.6 Inverse Systems 14
References 15
Problems 17
2 Complex variable methods in feedback
Systems analysis and design 18
2.1 Introduction 18
2.2 Generalized Nyquist and root-locus
diagrams 19
2.2.1 Characteristic frequencies and
characteristic gains 23
2.2.2 Generalized Nyquist diagrams and
generalized Nyquist stability
criterion 24
2.2.3 Multivariable root loci 25
2.2.4 Conformal nature of mapping between
frequency and gain 25
2.3 Zeros 27
2.3.1 Restriction of operator in domain
and range 29
2.3.2 Spectral characterisation of Zeros 30
2.3.3 Finite zeros when D is non-zero 30
2.4 Bi-linear transformation of
frequency and gain variables 31
2.5 Geometric theory of root locus and
Nyquist diagrams 32
2.6 Angles of arrival at zeros and
angles of departure from poles 35
2.7 Properties of Nyquist and root locus
diagrams for optimal feedback systems 37
2.8 Design techniques 38
2.9 Conclusions 40
References 41

Robustness in variable control systems design 46
3.1 Introduction 46
3.2 Sensitivity of characteristic gain loci 48
3.2.1 Sensitivity indices 49
3.2.2. Analysis 49
3.3 Uncertainty in a feed back system 51
3.3.1 Normality 53
3.4 Relative stability matrices 53
3.5 Multivariable gain and phase margins 55
3.6 Conclusion 60
References 60
Problems 62
A design study using the characteristic locus
method 64
4.1 Introduction 6 4
4.2 Two-bed reactor model 67
4.3 Control system design for Bed II 69
4.4 Compensator design 71
4.5 Design of the 2-bed system 76
4.6 System performance and properties 77
References 82
The inverse Nyquist array design method 83
5.1 Introduction 83
5.2 The multi variable design problem 84
5.3 Stability 88
5.3.1 Diagonal dominance 91
5.3.2 Further stability theorems 92
5.3.3 Graphical criteria for stability 9 3
5.4 Design technique 95
5.5 Conclusions 100
References 101
Appendix 102
Problems 104
Analysis and design of a nuclear boiler control
scheme 106
6.1 Introduction
6.2 Drum boiler plant and its control requirements 107
6.3 Multivariable frequency response methods 109
6.4 Analysis and design approach adopted 111
6.4.1 Transfer function model 112
6.4.2 Analysis of existing control scheme 113
6.4.3 Alternative control scheme design 116
6.5 Performance assessment of improved scheme 120
6.5.1 Three drum version of improved scheme 121
6.6 System integrity 122
6.7 Conclusions 124
References 125
Optimal control 126
7.1 The calculus of variations: classical theory 12 7
7.2 The optimal control problem 129
7.3 Singular control problems 133
7.3.1 The generalised Legendre-Clebsch
condition: a transformation approach 134

7.3.2 Jacobson's necessary condition 136
7.4 Dynamic programming 137
7.5 The Hamilton-Jacobi approach 140
References 142
Problems 14 3
8 Control system design via mathematical programming 14 4
8.1 Introduction 144
8.2 Linear multivariable systems 146
8.3 Non-linear systems
8.4 Semi-infinite programming 152
8.4.1 Feasible-directions algorithms 153
8.4.2 Outer approximations algorithms 155
8.4.3 Non-differentiability 155
8.4.4 Cut-map algorithms 156
8.5 Conclusion 156
References 157
9 Optimisation in multivariable design 159
9.1 Introduction 159
9.1.1 The allocation problem 159
9.1.2 The scaling problem 159
9.1.3 Precompensation design problem 160
9.2 Problem formulation 160
9.2.1 Decomposition 161
9.2.2 Choice of dominance measure 161
9.2.3 General problem 162
9.3 Allocation problem 16 3
9.4 Sealing problem 16 5
9.5 Compensation design 16 7
9.6 Design example 169
References 17 3
Problems 175
10 Pole assignment 177
10.1 Introduction 177
10.2 State-feedback algorithms 179
10.2.1 Dyadic designs 179
10.2.2 Full-rank designs 183
10.3 Output-feedback algorithms 187
10.3.1 Dyadic designs 189
10.3.2 Full-rank designs 192
10.3.3 Example 194
10.4 Concluding remarks 196
References 19 7
Appendix 198
Problems 200
11 Nonlinear systems 202
11.1 Nonlinear behaviour 202
11.2 Fourier series 203
11.3 The describing function 206
11.4 Prediction of limit cycles 210
11.4.1 Frequency response of nonlinear
systems 213
11.5 Nonlinear state-space equations 213
11.6 Lyapunov's method 214

11.6.1 Domains of attraction 215
11.6.2 Construction of Lyapunov's
functions 216
11.7 Absolute stability criteria 218
References 220
Problems 222
12 Some DDC system design procedures
12.1 Introduction 223
12.2 Choice of sampling frequency 225
12.3 Frequency domain compensation method 227
12.4 The compensation of Class 0
(Regulator) systems 2 30
12.5 Noisy input or output signals 232
12.6 Structural resonances and digital notch
networks 232
12.7 Coefficient quantisation in a discrete
controller 234
12.8 Arithmetic roundoff-noise in a discrete
controller 237
12.9 Multivate and substrate Controllers 2 38
12.10 Comparison of time domain synthesis and
frequency domain compensation techniques 240
References 244
13 Robust controller design 246
13.1 Introduction 246
13.2 General problem to be considered 24 7
13.3 The servomechanism problem - structural
results 248
13.3.1 Problem description 248
13.3.2 Existence results 251
13.3.3 Robust servomechanism controller
structure 259
13.3.4 Some properties of the robust
controller 261
13.3.5 Various classes of stabilizing
compensators 261
13.4 Extensions of previous results
13.4.1 Control of unknown systems
(Multivariable timing regulators) 263
13.4.2 Multivariable error constants 264
13.5 Robust controller design 267
13.5.1 Design example 268
13.6 Conclusions 270
References 2 71
14 Control of distributed parameter systems 274
14.1 Introduction 274
14.2 Controllability 278
14.3 Observability 281
14.4 Optimal control 281
14.5 Optimal estimation (Kalman filter) 2 82
References 284
Problems 285
15 Decentralised control 286

15.1 Introduction 2 86
15.2 Non-classical information patterns 286
15.2.1 Partially nested information
structures 2 88
15.3 Decentralised stabilisation and pole
placement 2 89
15.4 Design technique of decentralised control 294
15.4.1 The algorithm of Geromel and
Bernussou 294
15.4.2 The model following method 29 8
15.5 Concluding remarks 301
References 302
Solutions
Chapter 1 304
Chapter 3 306
Chapter 5 309
Chapter 7 313
Chapter 10 316
Chapter 11 324
Chapter 14 32 6
Index 331

Preface
Over the last seven years the Science and Engineering
Research Council (previously the Science Research Council)
have been instrumental in the organization and financing of
a series of vacation schools. These have been aimed primarily
at recipients of SFFC postgraduate awards but some
non-SFRC students, together with a few workers from industry
(in the UK and elsewhere) have also taken part. Two of
these vacation schools have been on the subject of analysis
and design of control systems and have been held at the
Control Systems Centre in the University of Manchester
Institute of Science and Technology (UMIST). The chapters
of this present book form the lectures to be given at the
third UMIST vacation school during the week 2 9 March
3 April, 1982. The title of the vacation school is the
same as that for this book.
It is inevitable and indeed desirable that a subject such
as engineering design of multivariable systems will develop
over a period of six years. This becomes apparent if one
compares the contents of the 1976 UMIST vacation school as
given in "Modern approaches to control system design"
(N. Munro, Fd., IFE Control Engineering Series 9, Peter
Peregrinus, 1979) with the contents of the volume before you.
Some of the 1976 topics no longer appear in this 1982 version,
not because they are now obsolete or research is no
longer taking place in those areas but because we have felt
it right and proper to make way for either new subjects
(such as large scale systems) or interesting new applications
(e.g. the nuclear boiler in Chapter 6). Even in the case

where 1976 topics are retained we have arranged for some
different aspects to be presented in this new publication.
It will be noticed that no fundamental background material
on single-input single-output systems and linear algebra is
presented in the present volume, unlike in the reference
given above. This is because we feel that such material is
readily available from many sources and in particular the
quoted reference may be consulted.
The organization of a vacation school and at the same time
the preparation of a book to be published before the school
takes place is a daunting task. We certainly would not have
succeeded without the full cooperation of all the lecturers
involved in the school, the typist who prepared the cameraready
copy and the staff of the publishers. Our grateful
thanks therefore go to all those who honoured deadlines and
delivered the drafts of the chapters which follow this preface,
to Vera Butterworth who typed the whole volume under
exceptionally trying circumstances, and to Pauline Maliphant
of Peter Peregrinus Ltd. who waited so patiently for the
chapters to arrive at Stevenage from Manchester. This book
would not have appeared but for the support given by the
Science and Engineering Research Council and our sincere
thanks go to staff of the SERC and to the other members of
the planning panel which organized the vacation school.
D.J. Bell
P.A. Cook
N. Munro
Manchester, 1982.