FLEXIBLE NEURO-FUZZY SYSTEMS Structures, Learning and Performance Evaluation THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE FLEXIBLE NEURO-FUZZY SYSTEMS Structures, Learning and Performance Evaluation by Leszek Rutkowski Technical University of Czestochowa Poland KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 1-4020-8043-3 Print ISBN: 1-4020-8042-5 ©2004 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2004 Kluwer Academic Publishers Boston All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstoreat: http://ebooks.kluweronline.com This book is dedicated to Professor Lotfi Zadeh This page intentionally left blank Contents FOREWORD XI 1. INTRODUCTION 1 2. ELEMENTS OF THE THEORY OF FUZZY SETS 7 2.1. Introduction 7 2.2. Basic Definitions 7 2.3. Triangular Norms and Negations 13 2.4. Operations on Fuzzy Sets 18 2.5. Fuzzy Relations 21 2.6. Fuzzy Reasoning 23 2.7. Problems 25 3. FUZZY INFERENCE SYSTEMS 27 3.1. Introduction 27 3.2. Description of fuzzy inference systems 28 3.3. Mamdani-type inference 32 3.4. Logical-type inference 37 3.5. Generalized neuro-fuzzy system 41 3.6. Data sets used in the book 45 3.7. Summary and discussion 48 3.8. Problems 49 4. FLEXIBILITY IN FUZZY SYSTEMS 51 4.1. Introduction 51 4.2. Weighted triangular norms 51 viii Flexible Neuro-Fuzzy Systems 4.3. Soft fuzzy norms 58 4.4. Parameterized triangular norms 65 4.5. OR-type systems 69 4.6. Compromise systems 70 4.7. Summary and discussion 73 4.8. Problems 74 5. FLEXIBLE OR-TYPE NEURO-FUZZY SYSTEMS 75 5.1. Introduction 75 5.2. Problem description 76 5.3. Adjustable quasi-triangular norms 77 5.4. Adjustable quasi-implications 82 5.5. Basic flexible systems 86 5.6. Soft flexible systems 90 5.7. Weighted flexible systems 99 5.8. Learning algorithms 102 5.9. Simulation results 115 5.10. Summary and discussion 126 5.11. Problems 127 6. FLEXIBLE COMPROMISE AND-TYPE NEURO-FUZZY SYSTEMS 129 6.1. Introduction 129 6.2. Problem description 130 6.3. Basic compromise systems 130 6.4. Soft compromise systems 133 6.5. Weighted compromise systems 140 6.6. Learning algorithms 145 6.7. Simulation results 151 6.8. Summary and discussion 163 6.9. Problems 163 7. FLEXIBLE MAMDANI-TYPE NEURO-FUZZY SYSTEMS 165 7.1. Introduction 165 7.2. Problem description 166 7.3. Neuro-fuzzy structures 166 7.4. Simulation results 174 7.5. Summary and discussion 183 7.6. Problems 183 8. FLEXIBLE LOGICAL-TYPE NEURO-FUZZY SYSTEMS 185 8.1. Introduction 185 8.2. Problem description 185 8.3. Neuro-fuzzy structures 186 Contents ix 8.4. Simulation results 208 8.5. Summary and discussion 233 8.6. Problems 233 9. PERFORMANCECOMPARISON OF NEURO-FUZZY SYSTEMS 235 9.1. Introduction 235 9.2. Comparison charts 236 9.3. Summary and discussion 251 APPENDIX 255 BIBLIOGRAPHY 265 INDEX 277