http://simplex.nl/users/rjager/
ref and TOC follow...
Rene'
REFERENCE:
Jager, Rene' (1995). Fuzzy Logic in Control.
Ph.D. thesis Delft University of Technology,
Department of Electrical Engineering,
Control Laboratory, pp. 312.
Delft, The Netherlands. ISBN 90-9008318-9.
TABLE OF CONTENTS:
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Why yet another work on fuzzy control? . . . . . . . . . . . . 1
1.2 Why fuzzy control and where does it fit in? . . . . . . . . . . 3
1.3 Fuzzy control and control systems theory . . . . . . . . . . . 6
1.3.1 Controllers as static functions . . . . . . . . . . . . 6
1.3.2 Stability issues . . . . . . . . . . . . . . . . . . . . 8
1.4 Relation to artificial and computational intelligence . . . . . 9
1.5 What to expect: a road-map for this thesis . . . . . . . . . . 10
2 Fuzzy sets and relations . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Fuzzy sets . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 What are fuzzy sets? . . . . . . . . . . . . . . . . . . 14
2.1.2 Properties of fuzzy sets . . . . . . . . . . . . . . . . 15
2.1.3 Fuzzy numbers and intervals . . . . . . . . . . . . . . 17
2.1.4 The extension principle . . . . . . . . . . . . . . . . 19
2.1.5 Fuzzy set representations . . . . . . . . . . . . . . . 23
2.2 Hedges: linguistic modifiers . . . . . . . . . . . . . . . . . 25
2.2.1 Powered hedges . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Shifted hedges . . . . . . . . . . . . . . . . . . . . . 27
2.2.3 Scaled hedges . . . . . . . . . . . . . . . . . . . . . 27
2.3 Operations on fuzzy sets . . . . . . . . . . . . . . . . . . . 30
2.3.1 Union and intersection . . . . . . . . . . . . . . . . . 30
2.3.2 Complement of fuzzy sets . . . . . . . . . . . . . . . . 33
2.4 Fuzzy relations . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 Projection and cylindrical extension . . . . . . . . . . 36
2.4.2 Composition of fuzzy relations . . . . . . . . . . . . . 37
2.5 Summary and remarks . . . . . . . . . . . . . . . . . . . . . . 42
3 Fuzzy logic and reasoning . . . . . . . . . . . . . . . . . . . . . 43
3.1 Fuzzy propositions . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Logical connectives . . . . . . . . . . . . . . . . . . 44
3.1.2 Negation in fuzzy propositions . . . . . . . . . . . . . 47
3.2 Fuzzy rules . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Representation of a fuzzy rule . . . . . . . . . . . . . 48
3.2.2 Fuzzy implications . . . . . . . . . . . . . . . . . . . 49
3.2.3 Aggregation of fuzzy rules . . . . . . . . . . . . . . . 52
3.2.4 Classification of fuzzy implications . . . . . . . . . . 54
3.2.5 Rule base properties . . . . . . . . . . . . . . . . . . 59
3.2.5.1 Continuity of a rule base . . . . . . . . . . . 60
3.2.5.2 Consistency of a rule base . . . . . . . . . . 61
3.2.5.3 Completeness of a rule base . . . . . . . . . . 62
3.3 Fuzzy reasoning . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Inference of a fuzzy rule . . . . . . . . . . . . . . . 63
3.3.1.1 Compositional rule of inference . . . . . . . . 63
3.3.1.2 Generalized modus ponens and tollens . . . . . 64
3.3.1.3 Criteria for generalized modus ponens . . . . . 65
3.3.1.4 Inference of a rule modeled by
a T-implication . . . . . . . . . . . . . . . . 68
3.3.2 Inference of a fuzzy rule base . . . . . . . . . . . . . 71
3.3.2.1 Local versus global inference . . . . . . . . . 71
3.3.2.2 Rules modeled by
classical-conjunction-based implications . . . 72
3.3.2.3 Rules modeled by
classical-implication-based implications . . . 72
3.4 Summary and remarks . . . . . . . . . . . . . . . . . . . . . . 74
4 Fuzzy control . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Theoretical approach to fuzzy control . . . . . . . . . . . . . 76
4.1.1 Fuzzification of inputs . . . . . . . . . . . . . . . . 77
4.1.2 Defuzzification of output . . . . . . . . . . . . . . . 77
4.1.2.1 Center-of-gravity defuzzification . . . . . . . 78
4.1.2.2 Indexed defuzzification methods . . . . . . . . 79
4.1.2.3 Mean-of-maxima defuzzification . . . . . . . . 80
4.1.2.4 Center-of-area defuzzification method . . . . . 81
4.1.3 Example of theoretical approach . . . . . . . . . . . . 83
4.2 Practical approach to fuzzy control . . . . . . . . . . . . . . 85
4.2.1 Fuzzy inference in practice . . . . . . . . . . . . . . 85
4.2.1.1 Practical fuzzy inference scheme . . . . . . . 86
4.2.1.2 Inference with T-implications . . . . . . . . . 87
4.2.1.3 Inference with S-implications . . . . . . . . . 88
4.2.1.4 Inference with other implications . . . . . . . 92
4.2.2 Input fuzzification . . . . . . . . . . . . . . . . . . 93
4.2.3 Common ``inference'' methods . . . . . . . . . . . . . . 96
4.2.3.1 Max-min method . . . . . . . . . . . . . . . . 97
4.2.3.2 Max-prod method . . . . . . . . . . . . . . . . 97
4.2.3.3 Sum-prod method . . . . . . . . . . . . . . . . 100
4.2.4 Defuzzification in practice . . . . . . . . . . . . . . 102
4.2.4.1 Averaging defuzzification methods . . . . . . . 103
4.2.4.2 Height-related methods . . . . . . . . . . . . 107
4.2.4.3 Extended defuzzification methods . . . . . . . 107
4.3 Fuzzy control rules . . . . . . . . . . . . . . . . . . . . . . 112
4.3.1 Mamdani fuzzy rules . . . . . . . . . . . . . . . . . . 112
4.3.2 Sugeno fuzzy rules . . . . . . . . . . . . . . . . . . . 113
4.3.3 Differences and similarities . . . . . . . . . . . . . . 117
4.4 Fuzzy linear control . . . . . . . . . . . . . . . . . . . . . 118
4.4.1 Fuzzy linear models . . . . . . . . . . . . . . . . . . 118
4.4.2 Fuzzy linear controllers . . . . . . . . . . . . . . . . 119
4.4.3 Experiments with fuzzy pole-placement controller . . . . 121
4.4.4 Remarks and considerations . . . . . . . . . . . . . . . 122
4.5 Fuzzy controller as input-output mapping . . . . . . . . . . . 124
4.5.1 Fuzzy system as universal approximator . . . . . . . . . 125
4.5.2 Linear controller as subset of fuzzy controller . . . . 126
4.6 Fuzzy controller analysis . . . . . . . . . . . . . . . . . . . 129
4.6.1 Role of fuzzy sets . . . . . . . . . . . . . . . . . . . 129
4.6.1.1 Number of fuzzy sets . . . . . . . . . . . . . 129
4.6.1.2 Overlapping fuzzy sets . . . . . . . . . . . . 131
4.6.1.3 Shape of fuzzy sets . . . . . . . . . . . . . . 132
4.6.1.4 Fuzzy sets for the output . . . . . . . . . . . 135
4.6.2 Role of operators . . . . . . . . . . . . . . . . . . . 136
4.6.2.1 Negation in rule premises . . . . . . . . . . . 137
4.6.2.2 Logical and connective . . . . . . . . . . . . 138
4.6.2.3 Logical or connective . . . . . . . . . . . . . 140
4.6.3 Role of the rule base . . . . . . . . . . . . . . . . . 141
4.6.3.1 Incompleteness and interpolation . . . . . . . 141
4.6.3.2 Exceptions and rule precedence . . . . . . . . 143
4.7 Conclusions and remarks . . . . . . . . . . . . . . . . . . . . 144
5 Adaptive fuzzy control . . . . . . . . . . . . . . . . . . . . . . . 147
5.1 Self-organizing fuzzy control . . . . . . . . . . . . . . . . . 148
5.1.1 Self-organizing controller scheme . . . . . . . . . . . 148
5.1.2 Relation-based approach . . . . . . . . . . . . . . . . 151
5.1.2.1 Numerical example of the
relation-based approach . . . . . . . . . . . . 152
5.1.3 Rule-based approach . . . . . . . . . . . . . . . . . . 155
5.1.4 Simplified rule-based approach . . . . . . . . . . . . . 156
5.2 Fuzzy relations as associative memories . . . . . . . . . . . . 157
5.3 Adaptation by fuzzy supervisors . . . . . . . . . . . . . . . . 163
5.3.1 Fuzzy supervised PID-control . . . . . . . . . . . . . . 163
5.3.2 Adaptive fuzzy expert controller . . . . . . . . . . . . 167
5.4 Gradient-descent adaptation . . . . . . . . . . . . . . . . . . 170
5.4.1 The basic adaptation scheme . . . . . . . . . . . . . . 170
5.4.2 Restrictions on adaptation . . . . . . . . . . . . . . . 172
5.4.3 Maintaining fuzzy partitions . . . . . . . . . . . . . . 174
5.5 Comparison with other ``learning'' systems . . . . . . . . . . 176
5.5.1 Relation to radial-basis function networks . . . . . . . 176
5.5.2 Comparison with generalized CMAC . . . . . . . . . . . . 180
5.6 Conclusions and remarks . . . . . . . . . . . . . . . . . . . . 182
6 Fuzzy logic in knowledge-based systems . . . . . . . . . . . . . . . 185
6.1 Knowledge-based systems for control . . . . . . . . . . . . . . 186
6.1.1 Knowledge representation . . . . . . . . . . . . . . . . 186
6.1.2 Real-time control requirements . . . . . . . . . . . . . 188
6.2 Possibility theory . . . . . . . . . . . . . . . . . . . . . . 189
6.2.1 Possibility distributions . . . . . . . . . . . . . . . 190
6.2.1.1 The concept of a possibility distribution . . . 190
6.2.1.2 Fuzzy sets and possibility distributions . . . 191
6.2.1.3 Different interpretations of propositions . . . 192
6.2.2 Possibility and necessity measures . . . . . . . . . . . 194
6.2.3 Principles of minimum and maximum specificity . . . . . 196
6.2.3.1 Principle of minimum specificity . . . . . . . 196
6.2.3.2 Principle of maximum specificity . . . . . . . 196
6.2.4 Rules and conditional possibility distribution . . . . . 197
6.3 Approximate reasoning . . . . . . . . . . . . . . . . . . . . . 198
6.3.1 Reasoning modes . . . . . . . . . . . . . . . . . . . . 198
6.3.2 Translation rules . . . . . . . . . . . . . . . . . . . 200
6.3.2.1 Quantification rules . . . . . . . . . . . . . 200
6.3.2.2 Qualification rules . . . . . . . . . . . . . . 202
6.3.3 Practical considerations . . . . . . . . . . . . . . . . 205
6.4 Reasoning with possibility distributions . . . . . . . . . . . 206
6.4.1 Interpretation of rules . . . . . . . . . . . . . . . . 206
6.4.1.1 Possibility-qualifying rules . . . . . . . . . 207
6.4.1.2 Certainty-qualifying rules . . . . . . . . . . 208
6.4.1.3 Truth-qualifying rules . . . . . . . . . . . . 211
6.4.1.4 Fuzzy control rules in terms of rule types . . 213
6.4.2 An inference break-up method . . . . . . . . . . . . . . 215
6.4.2.1 Breaking up the inference . . . . . . . . . . . 215
6.4.2.2 Reduction of inference break-up . . . . . . . . 216
6.4.2.3 Summary of inference break-up . . . . . . . . . 219
6.5 Other and derived approaches to fuzzy reasoning . . . . . . . . 220
6.5.1 Reasoning with fuzzy truth values . . . . . . . . . . . 220
6.5.1.1 Baldwin's method . . . . . . . . . . . . . . . 220
6.5.1.2 Tsukamoto's method . . . . . . . . . . . . . . 224
6.5.1.3 Mizumoto's method . . . . . . . . . . . . . . . 225
6.5.2 Fuzzy reasoning based on similarity measures . . . . . . 226
6.5.2.1 Yager's method . . . . . . . . . . . . . . . . 227
6.5.2.2 Turksen and Zhong's method . . . . . . . . . . 228
6.5.2.3 Reasoning with domain scaling . . . . . . . . . 229
6.5.3 Reasoning with linguistic qualifiers . . . . . . . . . . 230
6.5.4 Remarks and considerations . . . . . . . . . . . . . . . 233
6.6 Conclusions and remarks . . . . . . . . . . . . . . . . . . . . 234
7 Conclusions and suggestions . . . . . . . . . . . . . . . . . . . . 237
A Fuzzy logic operators . . . . . . . . . . . . . . . . . . . . . . . 243
B Linear controller as subset of fuzzy controller proof . . . . . . . 247
C Derivation of restricted learning rule . . . . . . . . . . . . . . . 251
D GCMAC: Generalized Cerebellar Model Articulation Controller . . . . 255
E RICE: Routines for Implementing C Expert systems . . . . . . . . . . 259
E.1 The inference engine and supporting tools . . . . . . . . . . . 259
E.2 Examples using RICE in simulation and control . . . . . . . . . 262
F Proofs for inference break-up method . . . . . . . . . . . . . . . . 267
F.1 Rule break-up . . . . . . . . . . . . . . . . . . . . . . . . . 267
F.2 Rule inference break-up . . . . . . . . . . . . . . . . . . . . 268
F.3 Rule base inference break-up . . . . . . . . . . . . . . . . . 269
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . 289
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Vage logica in de regeltechniek . . . . . . . . . . . . . . . . . . . . 295
Curriculum vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
-- * Rene' Jager * * * * * * * * * * * * * * * * * * * * Wevershof 19, 1483 XJ De Rijp, The Netherlands * * e-mail: rjager@simplex.nl * voice: +31-29974297 * * World Wide Web: http://simplex.nl/users/rjager/ * * * * * * * * * * * * * * * * * * * * * * * * * * *
------------------------------