Multiobjective Optimisation and Control

330 Pages·2003·1.727 MB·English

by  G. P. Liu, Jian-Bo Yang, J. F. Whidborne

Multiobjective
Optimisation and
Control
ENGINEERING SYSTEMS MODELLING AND CONTROL
SERIES

Series Editor: Professor D H Owens
University of Sheffield, UK

3. Controllability Analysis and Control Structure Selection
Y Cao, D Rossiter and D H Owens *
4. Multiobjective Optimisation and Control
G P Liu, J B Yang and J F Whidborne

* forthcoming
Multiobjective
Optimisation and
Control

G P Liu, University of Nottingham, UK
J B Yang, UMIST, UK
J F Whidborne, King’s College London, UK

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Library of Congress Cataloguing-in-Publication Data

Liu, G.P.(Guo Ping), 1962 –
Multiobjective optimisation and control / G.P. Liu, J.B. Yang, J.F. Whidborne.
p. cm. – (Engineering systems modelling and control series ; 4)
Includes bibliographical references and index.
ISBN: 0-86380-264-8
1. Mathematical optimisation. 2. Nonlinear programming. 3. Multiple criteria decision
making. I. Yang, Jian-Bo, 1961 – II Whidborne, J.F. (James Ferris), 1960- III. Title. IV.
Series.

QA402.5 .L57 2001
519.3--dc21 2001019195

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library.

ISBN 0 86380 264 8

Printed in Great Britain by SRP Ltd., Exeter
v
Dedication
To Weihong and Louise
(G.P. Liu)
To Dong-Ling and Lin
(J.B. Yang)
To C(cid:19)ecile
(J.F. Whidborne)
Contents
Preface xiii
Symbols and Abbreviations xv
1 Introduction 1
1.1Multiobjective Optimisation 1
1.1.1 Constrained Optimisation 1
1.1.2 Conventional Multiobjective Optimisation 2
1.1.3 Method of Inequalities 7
1.1.4 Multiobjective Genetic Algorithms 9
1.2Multiobjective Control 10
1.2.1 Con(cid:13)icts and Trade-o(cid:11)s in Control Systems 11
1.2.2 Multiobjective Robust Control 14
1.2.3 Multiobjective Critical Control 15
1.2.4 Multiobjective Eigenstructure Assignment 16
1.2.5 Multiobjective PID Control 16
1.2.6 Multiobjective Optimisation of Controller Implementations 17
1.2.7 Multiobjective Nonlinear Identi(cid:12)cation 18
1.2.8 Multiobjective Fault Detection 19
1.3Outline of the Book 20
2 Nonlinear Optimisation 23
2.1One-Dimensional Optimisation 23
2.1.1 The Dichotomy Method with Derivatives 23
2.1.2 The Dichotomy Method without Derivatives 25
2.1.3 The Fibonacci Method 26
2.1.4 The Golden Section Search Method 31
2.2Optimisation Conditions 32
2.2.1 Necessary Conditions for Local Optimality 32
2.2.2 SuÆcient Conditions for Local Optimality 34
vii
viii CONTENTS
2.3Unconstrained Optimisation Methods 34
2.3.1 Steepest Decent Method 34
2.3.2 Newton’s Method 38
2.3.3 Quasi-Newton’s Methods 41
2.4Summary 44
3 Constrained Optimisation 45
3.1Introduction 45
3.2Optimality Conditions 46
3.2.1 Basic Concepts 46
3.2.2 Kuhn-Tucker Necessary Condition 47
3.2.3 Second Order SuÆcient Conditions 48
3.3Primal Methods 52
3.3.1 Sequential Linear Programming 52
3.3.2 Sequential Quadratic Programming 55
3.4Dual Methods 60
3.4.1 LagrangeanMethods 61
3.4.2 Method of ExteriorPenalties 64
3.4.3 Method of Interior Penalties 68
3.5Summary 71
4 Multiple Objective Optimisation 73
4.1Introduction 73
4.2Basic Concepts and Methods 74
4.2.1 Concepts and De(cid:12)nitions 74
4.2.2 Method Classi(cid:12)cation 77
4.2.3 Simple Weighting Method 78
4.3p-Norm Methods 82
4.3.1 Minimax (Ideal Point) Method 82
4.3.2 Goal Attainment Method 89
4.3.3 Goal Programming 91
4.3.4 The Minimax Reference Point Method 95
4.4Interactive Methods 103
4.4.1 Geo(cid:11)rion’s Method 103
4.4.2 The STEM Method 108
4.4.3 The ISTM Method 112
4.4.4 The Gradient Projection Method 116
4.5Summary 123
5 Genetic Algorithms and Optimisation 125
5.1Introduction 125
CONTENTS ix
5.2What are Genetic Algorithms 125
5.3Basic Structure of Genetic Algorithms 127
5.4Population Representation and Initialisation 129
5.4.1 Binary Representation 129
5.4.2 Real-Valued Representation 129
5.4.3 Initialisation 130
5.5Fitness Functions 130
5.6Selection 132
5.6.1 Roulette Wheel Selection Methods 133
5.6.2 Stochastic Universal Sampling 134
5.7Crossover 135
5.7.1 Single-Point Crossover 135
5.7.2 Multi-Point Crossover 135
5.7.3 Uniform Crossover 136
5.7.4 Other CrossoverOperators 137
5.7.5 Intermediate Recombination 137
5.7.6 Line Recombination 138
5.8Mutation 138
5.9Reinsertion and Termination 140
5.9.1 Reinsertion 140
5.9.2 Termination 141
5.10 Multiobjective Optimisation with GAs 141
5.10.1Constrained Optimisation 141
5.10.2Non-ParetoOptimisation 142
5.10.3Pareto-BasedOptimisation 143
5.11 An Example 143
5.12 Summary 146
6 Robust Control System Design by Mixed Optimisation 147
6.1Introduction 147
6.2An H Loop Shaping Design Procedure 148
1
6.2.1 Overview 148
6.2.2 Preliminaries 149
6.2.3 Normalised Left Coprime Factorisation 150
6.2.4 Coprime Factor Robust H Stability Problem 151
1
6.2.5 A Loop-Shaping Design Procedure (LSDP) 153
6.2.6 Example { The Inverted Pendulum 156
6.3Mixed-Optimisation for the LSDP 160
6.3.1 MATLAB Implementation - The MODCONS Toolbox 162
6.4Example { The Distillation Column 163
x CONTENTS
6.5Example { High Speed EMS Maglev Vehicle 167
6.6Summary 175
7 Multiobjective Control of Critical Systems 177
7.1Introduction 177
7.2Critical Control Systems 178
7.3Critical System Descriptions 180
7.4Input Spaces of Systems 183
7.5Multiobjective Critical Control 184
7.6Control Design of SISO Critical Systems 186
7.7Control Design of MIMO Critical Systems 191
7.8An Example 197
7.9Summary 198
8 Multiobjective Control Using Eigenstructure Assignment 199
8.1Introduction 199
8.2What is Eigenstructure Assignment 200
8.3Allowable EigenvectorSubspaces 203
8.4ParametricEigenstructure Assignment 206
8.5Multiobjective Eigenstructure Assignment 210
8.6Controller Design Using the Method of Inequalities 214
8.7Controller Design Using Genetic Algorithms 217
8.8Summary 221
9 Multiobjective PI Controller Design for a Gasi(cid:12)er 223
9.1Introduction 223
9.2Modelling of the Gasi(cid:12)er 224
9.3System Speci(cid:12)cations of the Gasi(cid:12)er 226
9.4Multiobjective PI Control Formulation 228
9.5Multiobjective Optimal-Tuning PI Control 230
9.6Simulation Results and Discussions 231
9.7Summary 238
10 Multiobjective PID Controller Implementation Design 239
10.1 Introduction 239
10.2 FWL Fixed-Point Representation 241
10.2.1A Linear System Equivalence Completion Problem 242
10.3 MOGA for Optimal FWL Controller Structures 245
10.3.1Multiobjective Genetic Algorithm 245
10.3.2Procedure Outline 248
10.3.3Encoding of Solution Space 249