Knowledge-Based Control Systems Robert Babuška and Jens Kober Delft Center for Systems and Control Version 19-07-2017 Delft University of Technology Delft, the Netherlands Copyright © 1999–2010 by Robert Babuška. 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 information storage and retrieval system, without permission in writing from the author. Contents 1. Introduction1 1.1. Conventional Control . .1 1.2. Intelligent Control . .1 1.3. Overview of Techniques . .2 1.4. Organization of the Book . .4 1.5. WEB and Matlab Support . .4 1.6. Further Reading . .5 1.7. Acknowledgements . .5 2. Fuzzy Sets and Relations7 2.1. Fuzzy Sets . .7 2.2. Properties of Fuzzy Sets . .8 2.2.1. Normal and Subnormal Fuzzy Sets . .9 2.2.2. Support, Core and α-cut . .9 2.2.3. Convexity and Cardinality . 10 2.3. Representations of Fuzzy Sets . 11 2.3.1. Similarity-based Representation . 11 2.3.2. Parametric Functional Representation . 11 2.3.3. Point-wise Representation . 12 2.3.4. Level Set Representation . 13 2.4. Operations on Fuzzy Sets . 13 2.4.1. Complement, Union and Intersection . 14 2.4.2. T -norms and T -conorms . 15 2.4.3. Projection and Cylindrical Extension . 16 2.4.4. Operations on Cartesian Product Domains . 17 2.4.5. Linguistic Hedges . 18 2.5. Fuzzy Relations . 19 2.6. Relational Composition . 20 2.7. Summary and Concluding Remarks . 22 2.8. Problems . 22 3. Fuzzy Systems 23 3.1. Rule-Based Fuzzy Systems . 24 3.2. Linguistic model . 25 3.2.1. Linguistic Terms and Variables . 25 3.2.2. Inference in the Linguistic Model . 27 3.2.3. Max-min (Mamdani) Inference . 33 3.2.4. Defuzzification . 35 3.2.5. Fuzzy Implication versus Mamdani Inference . 37 iii Contents 3.2.6. Rules with Several Inputs, Logical Connectives . 38 3.2.7. Rule Chaining . 40 3.3. Singleton Model . 42 3.4. Relational Model . 43 3.5. Takagi–Sugeno Model . 48 3.5.1. Inference in the TS Model . 48 3.5.2. TS Model as a Quasi-Linear System . 49 3.6. Dynamic Fuzzy Systems . 50 3.7. Summary and Concluding Remarks . 52 3.8. Problems . 52 4. Fuzzy Clustering 55 4.1. Basic Notions . 55 4.1.1. The Data Set . 55 4.1.2. Clusters and Prototypes . 56 4.1.3. Overview of Clustering Methods . 56 4.2. Hard and Fuzzy Partitions . 57 4.2.1. Hard Partition . 57 4.2.2. Fuzzy Partition . 59 4.2.3. Possibilistic Partition . 60 4.3. Fuzzy c-Means Clustering . 60 4.3.1. The Fuzzy c-Means Functional . 61 4.3.2. The Fuzzy c-Means Algorithm . 61 4.3.3. Parameters of the FCM Algorithm . 63 4.3.4. Extensions of the Fuzzy c-Means Algorithm . 66 4.4. Gustafson–Kessel Algorithm . 66 4.4.1. Parameters of the Gustafson–Kessel Algorithm . 68 4.4.2. Interpretation of the Cluster Covariance Matrices . 69 4.5. Summary and Concluding Remarks . 70 4.6. Problems . 70 5. Construction Techniques for Fuzzy Systems 73 5.1. Structure and Parameters . 74 5.2. Knowledge-Based Design . 74 5.3. Data-Driven Acquisition and Tuning of Fuzzy Models . 75 5.3.1. Least-Squares Estimation of Consequents . 75 5.3.2. Template-Based Modeling . 76 5.3.3. Neuro-Fuzzy Modeling . 78 5.3.4. Construction Through Fuzzy Clustering . 79 5.4. Semi-Mechanistic Modeling . 84 5.5. Summary and Concluding Remarks . 85 5.6. Problems . 85 6. Knowledge-Based Fuzzy Control 89 6.1. Motivation for Fuzzy Control . 89 6.2. Fuzzy Control as a Parameterization of Controller’s Nonlinearities . 90 iv Contents 6.3. Mamdani Controller . 92 6.3.1. Dynamic Pre-Filters . 93 6.3.2. Dynamic Post-Filters . 94 6.3.3. Rule Base . 95 6.3.4. Design of a Fuzzy Controller . 96 6.4. Takagi–Sugeno Controller . 102 6.5. Fuzzy Supervisory Control . 102 6.6. Operator Support . 105 6.7. Software and Hardware Tools . 106 6.7.1. Project Editor . 106 6.7.2. Rule Base and Membership Functions . 106 6.7.3. Analysis and Simulation Tools . 106 6.7.4. Code Generation and Communication Links . 107 6.8. Summary and Concluding Remarks . 108 6.9. Problems . 108 7. Artificial Neural Networks 111 7.1. Introduction . 111 7.2. Biological Neuron . 111 7.3. Artificial Neuron . 112 7.4. Neural Network Architecture . 114 7.5. Learning . 114 7.6. Multi-Layer Neural Network . 115 7.6.1. Feedforward Computation . 116 7.6.2. Approximation Properties . 117 7.6.3. Training, Error Backpropagation . 120 7.7. Radial Basis Function Network . 124 7.8. Summary and Concluding Remarks . 125 7.9. Problems . 125 8. Control Based on Fuzzy and Neural Models 127 8.1. Inverse Control . 127 8.1.1. Open-Loop Feedforward Control . 127 8.1.2. Open-Loop Feedback Control . 128 8.1.3. Computing the Inverse . 128 8.1.4. Inverting Models with Transport Delays . 135 8.1.5. Internal Model Control . 135 8.2. Model-Based Predictive Control . 136 8.2.1. Prediction and Control Horizons . 136 8.2.2. Objective Function . 137 8.2.3. Receding Horizon Principle . 137 8.2.4. Optimization in MBPC . 137 8.3. Adaptive Control . 141 8.4. Summary and Concluding Remarks . 143 8.5. Problems . 143 v Contents 9. Reinforcement Learning 145 9.1. Introduction . ..