Fuzzy Systems: Modeling and Control

531 Pages·1998·14.077 MB·English

by  Didier Dubois, Hung T. Nguyen, Henri Prade, Michio Sugeno B.S. degree (auth.), Hung T. Nguyen, Michio Sugeno (eds.)

FUZZY SYSTEMS
Modeling and Control
THE KLUWER HANDBOOK SERIES
ON FUZZY SETS
Series Editors: Didier Dubois and Henri Prade
FUZZY SETS IN DECISION ANALYSIS, OPERATIONS RESEARCH AND
STATISTICS, edited by Roman Slowmski
ISBN: 0-7923-8112-2
FUZZY SYSTEMS
Modeling and Control
edited by
Hung T. Nguyen
New Mexico State University
and
Michio Sugeno
Tokyo Institute of Techno/ogy
111...
"
SPRINGER-SCIENCE+BUSINESS MEDIA, LLC
ISBN 978-1-4613-7515-9 ISBN 978-1-4615-5505-6 (eBook)
DOI 10.1007/978-1-4615-5505-6
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1998 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1s t edition 1998
All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, mechanical, photocopying,
recording, or otherwise, without the prior written permission ofthe publisher, Springer
Science+Business Media, LLC.
Printed on acid-free paper.
Contents
Series Foreword xiii
Contributing Authors xv
Introduction: The Real Contribution of Fuzzy Systems 1
Didier Dubois, Hung T. Nguyen, Henri Prade, and Michio Sugeno
References 14
1
Methodology of Fuzzy Control 19
Hung T. Nguyen and Vladik Kreinovich
1.1 Introduction: Why Fuzzy Control 19
1.1.1 Traditional control methodology and its limitations 20
1.1.2 What we can use instead of the classical control model 23
1.2 How to Translate Fuzzy Rules into the Actual Control: General Idea 25
1.3 Membership Functions and Where They Come From 28
1.4 Fuzzy Logical Operations 38
1.5 Modeling Fuzzy Rule Bases 43
1.6 Inference From Several Fuzzy Rules 46
1.7 Defuzzification 49
1.8 The Basic Steps of Fuzzy Control: Summary 51
1.9 Tuning 51
1.10 Methodologies of Fuzzy Control: Which Is The Best? 53
References 59
2
Introduction to Fuzzy Modeling 63
Kazuo Tanaka and Michio Sugeno
2.1 Introduction 63
vi FUZZY SYSTEMS: MODELING AND CONTROL
2.2 Takagi-Sugeno Fuzzy Model 64
2.3 Sugeno-Kang Method 65
2.3.1 Structure identification 65
2.3.2 Prediction of water flow rate in the river Dniepr 69
2.4 SOFIA 69
2.4.1 Algorithm 70
2.4.2 Prediction of CO concentration at a traffic intersection 79
2.4.3 Prediction of O2 concentration in a municipal refuse incinerator 81
2.5 Conclusion 84
References 87
3
Fuzzy Rule-Based Models and Approximate Reasoning 91
Ronald R. Yage1' and Dimita1' P. Filev
3.1 Introduction 91
3.2 Linguistic Models 92
3.3 Inference with Fuzzy Models 93
3.4 Mamdani (Constructive) and Logical (Destructive) Models 95
3.5 Linguistic Models With Crisp Outputs 103
3.6 Multiple Variable Linguistic Models 105
3.7 Takagi-Sugeno-Kang (TSK) Models 112
3.8 A General View of Fuzzy Systems Modeling 113
3.9 MICA Operators 114
3.10 Aggregation in Fuzzy Systems Modeling 115
3.11 Dynamic Fuzzy Systems Models 120
3.12 TSK Models of Dynamic Systems 129
3.13 Conclusion 131
References 131
4
Fuzzy Rule Based Modeling as a Universal Approximation Tool 135
Vladik K1'einovich, Geo1'ge C. MOUZOU1'is and Hung T. Nguyen
4.1 Introduction 135
4.1.1 Why universal approximation 135
4.2 Main Universal Approximation Results 137
4.2.1 Universal approximation property for the original Mamdani approach 137
4.2.2 Generalizations of the standard fuzzy rule based modeling methodol-
ogy have a universal approximation property 140
4.2.3 Simplified versions of the standard fuzzy rule based modeling method-
ology also have a universal approximation property 144
4.2.4 Fuzzy rule-based systems are universal approximation tools for dis-
tributed systems 145
4.2.5 Fuzzy rule-based systems are universal approximation tools for dis-
crete systems: Application to expert system design 147
Contents vii
4.3 Can We Guarantee That The Approximation Function Has The Desired Prop
erties
(such as smoothness, simplicity, stability ofthe resulting control, etc.)? 149
4.3.1 Is fuzzy control a universal approximation tool for stable controls? 152
4.3.2 Is fuzzy rule based modeling methodology a universal approximation
tool for smooth systems? 153
4.3.3 Is fuzzy rule based modeling a universal approximation tool for com-
putationally simple systems? 155
4.4 Auxiliary Approximation Results 156
4.4.1 An overview 156
4.4.2 Usin~ additional expert information (fuzzy models and fuzzy control
ru~~ ~7
4.4.3 Expert rules that use unusual logical connectives 158
4.4.4 Taking the learning process into consideration (fuzzy neural networks) 158
4.5 How To Make The Approximation Results More Realistic 161
4.5.1 A general overview 161
4.5.2 Inaccuracies in the input data 161
4.5.3 Can we have fewer rules? 164
4.6 From All Fuzzy Rule Based Modeling Methodologies That Are Universal
Appriximation Tools, Which Methodology Should We Choose? 166
4.6.1 Main results. Part I. Best approximation 167
4.6.2 Main results. Part II. Best model 168
4.7 A Natural Next Question: When Should We Choose Fuzzy Rule Based Mod-
eling In The First Place? And When Is, Say, Neural Modeling Better? 169
4.7.1 Smoothness and stability 169
4.7.2 Computational complexity 170
4.7.3 Conclusions 175
4.7.4 Collaborate, not compete 176
References 177
5
Fuzzy and Linear Controllers 197
Laurent Foulloy and Sylvie Galichet
5.1 Introduction 197
5.2 Modal Equivalence Principle 199
5.2.1 Using the Modal Equivalence Principle 202
5.3 Application to PI Controllers 203
5.3.1 Rule Base 203
5.3.2 Membership functions 204
5.3.3 Outputs for Modal Values 204
5.3.4 Using the Modal Equivalence Principle 205
= =
5.3.5 Example with a f3 1 206
5.3.6 Exact Equivalence 209
5.4 Application to State Feedback Fuzzy Controllers 209
5.4.1 State Feedback Control 209
= =
5.4.2 Single-Input, Single-Output Process (m p 1) 211
5.4.3 Multidimensional Mamdani's Controllers 214
5.4.4 Multi Single-Output Mamdani's Controllers 214
viii FUZZY SYSTEMS: MODELING AND CONTROL
5.4.5 Applications 215
5.5 Equivalence for Sugeno's Controllers 217
5.5.1 Modal Equivalence Principle 217
5.5.2 Control Law between the Modal values 218
5.5.3 Multidimensional Sugeno's Controllers 220
5.6 Conclusion 222
References 223
6
Designs of Fuzzy Controllers 227
Rainer Palm
6.1 Introduction 227
6.2 Fuzzy Control Techniques 229
6.2.1 The Design Goal 230
6.2.2 Fuzzy Regions 233
6.2.3 FC techniques for systems and controllers 234
6.3 The FC as a Nonlinear Transfer Element 237
6.3.1 The Computational Structure of a FC 237
6.3.2 The Transfer Characteristics 246
6.3.3 The Nonlinearity of the FC 248
6.4 Heuristic Control and Model Based Control 251
6.4.1 The Mamdani Controller 252
6.4.2 Sliding Mode FC 253
6.4.3 Cell Mapping 258
6.4.4 TS Model Based Control 261
6.4.5 Model based Control with Lyapunov linearization 263
6.5 Supervisory Control 266
6.6 Adaptive Control 267
References 270
7
Stability of Fuzzy Controllers 273
K azuo Tanaka
7.1 Introduction 273
7.2 Stability Conditions Based on Lyapunov Approach 274
7.2.1 Takagi and Sugeno's fuzzy model 274
7.2.2 Stability conditions 276
7.3 Fuzzy Controller Design 278
7.3.1 Parallel distributed compensation 278
7.3.2 Stability conditions 280
7.3.3 Common B matrix case 286
References 288
8
Learning and Tuning of Fuzzy Rules 291
Contents ix
Hamid R. Berenji
8.1 Introduction 291
8.2 Learning Fuzzy Rules 292
8.2.1 Fuzzy C-Means Clustering 292
8.2.2 Decision tree systems 293
8.2.3 Using Genetic Algorithms to generate Fuzzy Rules 293
8.2.4 Berenji-Khedkar's linear fuzzy rules generation method 293
8.3 Tuning Fuzzy Rules 295
8.3.1 ANFIS 295
8.3.2 GARIC 296
8.3.3 Fuzzy Q-Learning and GARIC-Q 301
8.4 Learning and Tuning Fuzzy Rules 305
8.4.1 CART-ANFIS 305
8.4.2 RNN-FLCS 305
8.4.3 GARIC-RB 306
8.5 Summary and Conclusion 307
References 307
9
Neurofuzzy Systems 311
Witold Pedrycz, Abraham Kandel, and Yan-Qing Zhang
9.1 Introduction 311
9.2 Synergy of Neural Networks and Fuzzy Logic 315
9.3 Fuzzy sets in the technology of neurocomputing 317
9.3.1 Fuzzy sets in the preprocessing and enchancements of training data 317
9.3.2 Metalearning and fuzzy sets 320
9.3.3 Fuzzy clustering in revealing relationships within data 322
9.3.4 A linguistic interpretation of computing with neural networks 323
9.4 Hybrid fuzzy neural computing structures 326
9.5 Fuzzy neurocomputing - a fusion of fuzzy and neural technology 328
9.5.1 Basic types of logic neurons 329
9.5.2 OR / AND neurons 332
9.5.3 Referential logic-based neurons 332
9.5.4 Approximation of logical relationships - development of the logical
processor 337
9.5.5 Learning 340
9.6 Constructing Hybrid Neurofuzzy Systems 343
9.6.1 Architectures 343
9.6.2 Learning algorithms of neural fuzzy networks 345
9.6.3 Learning algorithms ofcompensatory neural fuzzy networks 351
9.6.4 Designing neurofuzzy systems 353
9.7 Summary 360
References 362
10
Neural Networks and Fuzzy Logic 381
Nadipuram (Ram) R. Prasad
x FUZZY SYSTEMS: MODELING AND CONTROL
10.1 Introduction 381
10.2 Liquid Level Control Problem 383
10.3 Fuzzy Rule Development 385
10.3.1 Case 1: No Restrictions on Flow Rates 388
10.3.2 Case 2: Input/Output Flow Rate Control 388
10.3.3 Pseudo-Rules 388
10.4 Integrated System Architectures 390
10.4.1 FNN2 Architecture 391
10.5 FNN3 Training Algorithm 392
10.5.1 FNN2 Simulation Results 396
10.5.2 FNN3 Architecture 396
10.6 Conclusions 400
References 400
11
Fuzzy Genetic Algorithms 403
Andreas Geyer-Schulz
11.1 Introduction 403
11.2 What is a genetic algorithm? 411
11.3 Fuzzy genetic algorithms 423
11.3.1 Table-driven agents: Coding the syntax 423
11.3.2 Table-driven agents: Coding the semantic 427
11.3.3 Table-driven agents: Combinations 431
11.3.4 Simple reflex agents: Variants of messy genetic algorithms 431
11.4 Fuzzy Genetic Programming 433
11.4.1 Context-Free Fuzzy Rule Languages 433
11.4.2 Simple Genetic Algorithms over k-Bounded Context-Free Languages 437
11.4.3 Non-Deterministic Environments 443
11.4.4 Results and Applications 449
References 450
12
Fuzzy Systems. Viability Theory and Toll Sets 461
J. P. Aubin and O. Dordan
12.1 Introduction 461
12.2 Convexification Procedures 462
12.3 Toll Sets 467
12.3.1 Toll Sets and Toll Maps 468
12.3.2 Operations on Toll Sets 469
12.3.3 The Cramer Transform 472
12.3.4 Analogy between Integration and Optimization 476
12.4 Fuzzy or Toll Differential Inclusions 477
12.4.1 Set-Valued Maps 477
12.4.2 The Viability Theorem 478
12.4.3 Toll Differential Inclusion 479
12.4.4 Example: Random Differential Equations 480
12.4.5 Viability Theorem for Toll Differential Inclusions 481