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FUZZY AND NEURAL CONTROL DISC Course Lecture Notes (November 2009)

ˇ ROBERT BABUSKA
Delft Center for Systems and Control

Delft University of Technology Delft, the Netherlands

Copyright c 1999–2009 by Robert Babuˇka. s
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any informationstorage and retrieval system, without permission in writing from the author.

Contents

1. INTRODUCTION 1.1 Conventional Control 1.2 Intelligent Control 1.3 Overview of Techniques 1.4 Organization of the Book 1.5 WEB and Matlab Support 1.6 Further Reading 1.7 Acknowledgements 2. FUZZY SETS AND RELATIONS 2.1 Fuzzy Sets 2.2 Properties of Fuzzy Sets 2.2.1 Normal and Subnormal Fuzzy Sets 2.2.2Support, Core and α-cut 2.2.3 Convexity and Cardinality 2.3 Representations of Fuzzy Sets 2.3.1 Similarity-based Representation 2.3.2 Parametric Functional Representation 2.3.3 Point-wise Representation 2.3.4 Level Set Representation 2.4 Operations on Fuzzy Sets 2.4.1 Complement, Union and Intersection 2.4.2 T -norms and T -conorms 2.4.3 Projection and Cylindrical Extension 2.4.4 Operations onCartesian Product Domains 2.4.5 Linguistic Hedges 2.5 Fuzzy Relations 2.6 Relational Composition 2.7 Summary and Concluding Remarks 2.8 Problems 3. FUZZY SYSTEMS

1 1 1 2 5 5 5 6 7 7 9 9 10 11 12 12 12 13 14 14 15 16 17 19 19 21 21 23 24 25 v

vi

FUZZY AND NEURAL CONTROL

3.1 3.2

3.3 3.4 3.5

3.6 3.7 3.8

Rule-Based Fuzzy Systems Linguistic model 3.2.1 Linguistic Terms and Variables3.2.2 Inference in the Linguistic Model 3.2.3 Max-min (Mamdani) Inference 3.2.4 Defuzzification 3.2.5 Fuzzy Implication versus Mamdani Inference 3.2.6 Rules with Several Inputs, Logical Connectives 3.2.7 Rule Chaining Singleton Model Relational Model Takagi–Sugeno Model 3.5.1 Inference in the TS Model 3.5.2 TS Model as a Quasi-Linear System Dynamic Fuzzy Systems Summary and Concluding Remarks Problems27 27 28 30 35 38 40 42 44 46 47 52 53 53 55 56 57 59 59 59 60 60 61 62 63 64 65 65 66 68 71 71 73 74 75 75 77 78 79 79 80 80 82 83 89

4. FUZZY CLUSTERING 4.1 Basic Notions 4.1.1 The Data Set 4.1.2 Clusters and Prototypes 4.1.3 Overview of Clustering Methods 4.2 Hard and Fuzzy Partitions 4.2.1 Hard Partition 4.2.2 Fuzzy Partition 4.2.3 Possibilistic Partition 4.3 Fuzzy c-Means Clustering4.3.1 The Fuzzy c-Means Functional 4.3.2 The Fuzzy c-Means Algorithm 4.3.3 Parameters of the FCM Algorithm 4.3.4 Extensions of the Fuzzy c-Means Algorithm 4.4 Gustafson–Kessel Algorithm 4.4.1 Parameters of the Gustafson–Kessel Algorithm 4.4.2 Interpretation of the Cluster Covariance Matrices 4.5 Summary and Concluding Remarks 4.6 Problems 5. CONSTRUCTION TECHNIQUES FOR FUZZY SYSTEMS 5.1 Structure andParameters 5.2 Knowledge-Based Design 5.3 Data-Driven Acquisition and Tuning of Fuzzy Models 5.3.1 Least-Squares Estimation of Consequents 5.3.2 Template-Based Modeling 5.3.3 Neuro-Fuzzy Modeling 5.3.4 Construction Through Fuzzy Clustering 5.4 Semi-Mechanistic Modeling

Contents

vii 90 91 93 94 94 97 97 98 99 101 106 107 110 111 111 111 111 112 112 113 115 115 116 117 118 119 119 121 122124 128 130 130 131 131 132 132 133 139 140 140 141 141 142

5.5 5.6

Summary and Concluding Remarks Problems

6. KNOWLEDGE-BASED FUZZY CONTROL 6.1 Motivation for Fuzzy Control 6.2 Fuzzy Control as a Parameterization of Controller’s Nonlinearities 6.3 Mamdani Controller 6.3.1 Dynamic Pre-Filters 6.3.2 Dynamic Post-Filters 6.3.3 Rule Base 6.3.4 Design of a Fuzzy Controller 6.4 Takagi–SugenoController 6.5 Fuzzy Supervisory Control 6.6 Operator Support 6.7 Software and Hardware Tools 6.7.1 Project Editor 6.7.2 Rule Base and Membership Functions 6.7.3 Analysis and Simulation Tools 6.7.4 Code Generation and Communication Links 6.8 Summary and Concluding Remarks 6.9 Problems 7. ARTIFICIAL NEURAL NETWORKS 7.1 Introduction 7.2 Biological Neuron 7.3 Artificial Neuron 7.4 Neural Network...
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