ISBN: 3790813850
TITLE: Fuzzy Modelling and Control
AUTHOR: Piegat
TOC:
1. Introduction 1
1.1 Essence of fuzzy set theory 1
1.2 Development of fuzzy set theory 6
2. Basic Notions of Fuzzy Set Theory 11
2.1 Fuzzy sets 11
2.2 Characteristic parameters (indices) of a fuzzy set 22
2.3 Linguistic modifiers of fuzzy sets 27
2.4 Types of membership functions of fuzzy sets 34
2.5 Type 2 fuzzy sets 52
2.6 Fuzziness and probability: two kinds of uncertainty 56
3. Arithmetic of Fuzzy Sets 59
3.1 The extension principle 60
3.2 Addition of fuzzy numbers 67
3.3 Subtraction of fuzzy numbers 75
3.4 Multiplication of fuzzy numbers 79
3.5 Division of fuzzy numbers 95
3.6 Peculiarities of fuzzy numbers 100
3.7 Differences between fuzzy numbers and linguistic values 107
4. Mathematics of Fuzzy Sets 111
4.1 Basic operations on fuzzy sets 111
4.1.1 Intersection operation (logical product) of fuzzy sets 112
4.1.2 Union (logical sum) of fuzzy sets 126
4.1.3 Compensatory operators 132
4.2 Fuzzy relations 136
4.3 Implication 149
5. Fuzzy Models 157
5.1 Structure, main elements and operations in fuzzy models 157
5.1.1 Fuzzification 159
5.1.2 Inference 160
5.1.2.1 Premise evaluation 162
5.1.2.2 Determination of activated membership functions of conclusions in particular rules at given input values of a fuzzy model 167
5.1.2.3 Determination of the resulting membership function of the rule-base conclusion 172
5.1.3 Defuzzification of the resulting membership function of the rule-base conclusion 184
5.1.4 Example of fuzzy modeling 199
5.2 Significant features of rules, rule bases and fuzzy models 202
5.2.1 Local character of rules 202
5.2.2 Dependence of the number of rules on the number of inputs and fuzzy sets 204
5.2.3 Completeness of a fuzzy model 208
5.2.4 Consistency of the rule base 216
5.2.5 Continuity of the rule base 219
5.2.6 Redundancy of the rule base 222
5.3 Advice relating to rule base construction 224
5.4 Reduction of the rule base 229
5.5 Normalization (scaling) of the fuzzy model inputs and output 244
5.6 Extrapolation in fuzzy models 250
5.7 Types of fuzzy models 280
5.7.1 Mamdani models 281
5.7.2 Takagi-Sugeno models 301
5.7.3 Relational models 311
5.7.4 Global and local fuzzy models 316
5.7.5 Fuzzy multimodels 323
5.7.6 Neuro-fuzzy models 329
5.7.7 Alternative models 331
5.7.8 Similarity principles of the system and of the system model 338
5.7.9 Fuzzy classification 339
6. Methods of Fuzzy Modeling 363
6.1 Fuzzy modeling based on the system expert's knowledge 366
6.2 Creation of fuzzy, self-tuning models based on input/output measurement data of the system 373
6.2.1 Application of neuro-fuzzy networks for fuzzy model parameter tuning 374
6.2.1.1 Structuring and training of neural networks 374
6.2.1.2 Transformation of a Mamdani fuzzy model into a neuro-fuzzy network 388
6.2.1.3 Transformation of a Takagi-Sugeno fuzzy model into a neuro-fuzzy network 397
6.2.2 Tuning of fuzzy model parameters with the genetic algorithm method 400
6.3 Creation of self-organizing and self-tuning fuzzy models based on input/output measurement data of the system 405
6.3.1 Determination of significant and insignificant inputs of the model 406
6.3.2 Determining of fuzzy curves 410
6.3.3 Self-organization and self-tuning tuning of fuzzy model parameters 417
6.3.3.1 Self-organization and tuning of fuzzy models with the geometric method of the maximum absolute error 420
6.3.3.2 Self-organization and self-tuning of fuzzy models with clustering methods 452
6.3.3.3 Self-organization and self-tuning of fuzzy models with the searching method 487
7. Fuzzy Control 495
7.1 Static fuzzy controllers 495
7.2 Dynamic fuzzy controllers 500
7.3 The determination of structures and parameters for fuzzy controllers (organization and tuning) 511
7.3.1 The design of fuzzy controllers on the basis of expert knowledge concerning plant under control 512
7.3.2 The design of a fuzzy controller on the basis of a model of the expert controlling the plant 516
7.3.3 The design of a fuzzy controller on the basis of the model of controlled plant 522
7.3.3.1 Remarks concerning identification of models of dynamic plants 522
7.3.3.2 Some remarks concerning the identification of inverted models of dynamical plants 525
7.3.3.3 Tuning a fuzzy controller with an a priori chosen structure 554
7.3.3.4 Fuzzy control based on the Internal Model Control Structure (IMC structure) 561
7.3.3.5 Fuzzy control structure with an inverse of a plant model (InvMC structure) 583
7.3.3.6 Adaptive fuzzy control 599
7.3.3.7 Multivariable fuzzy control (MIMO) 601
8. The Stability of Fuzzy Control Systems 609
8.1 The stability of fuzzy control systems with unknown models of plants 614
8.2 The circle stability criterion 618
8.3 The application of hyperstability theory to analysis of fuzzy-system stability 625
8.3.1 The frequency domain representation of hyperstability conditions for control systems with a time invariant non-linear part 627
8.3.2 The time domain conditions for hyperstability of continuous, non-linear control systems containing a time-invariant non-linear part 655
8.3.3 The frequency domain conditions for hyperstability of discrete, non-linear control systems containing a time-invariant non-linear part 657
References 705
Index 725
END