In the last ten years, a true explosion of investigations into fuzzy modeling and its applications in control, diagnostics, decision making, optimization, pattern recognition, robotics, etc. has been observed. The attraction of fuzzy modeling results from its intelligibility and the high effectiveness of the models obtained. Owing to this the modeling can be applied for the solution of problems which could not be solved till now with any known conventional methods. The book provides the reader with an advanced introduction to the problems of fuzzy modeling and to one of its most important applications: fuzzy control. It is based on the latest and most significant knowledge of the subject and can be used not only by control specialists but also by specialists working in any field requiring plant modeling, process modeling, and systems modeling, e.g. economics, business, medicine, agriculture,and meteorology.

Inhalt

1. Introduction.- 1.1 Essence of fuzzy set theory.- 1.2 Development of fuzzy set theory.- 2. Basic Notions of Fuzzy Set Theory.- 2.1 Fuzzy sets.- 2.2 Characteristic parameters (indices) of a fuzzy set.- 2.3 Linguistic modifiers of fuzzy sets.- 2.4 Types of membership functions of fuzzy sets.- 2.5 Type 2 fuzzy sets.- 2.6 Fuzziness and probability: two kinds of uncertainty.- 3. Arithmetic of Fuzzy Sets.- 3.1 The extension principle.- 3.2 Addition of fuzzy numbers.- 3.3 Subtraction of fuzzy numbers.- 3.4 Multiplication of fuzzy numbers.- 3.5 Division of fuzzy numbers.- 3.6 Peculiarities of fuzzy numbers.- 3.7 Differences between fuzzy numbers and linguistic values.- 4. Mathematics of Fuzzy Sets.- 4.1 Basic operations on fuzzy sets.- 4.1.1 Intersection operation (logical product) of fuzzy sets.- 4.1.2 Union (logical sum) of fuzzy sets.- 4.1.3 Compensatory operators.- 4.2 Fuzzy relations.- 4.3 Implication.- 5. Fuzzy Models.- 5.1 Structure, main elements and operations in fuzzy models.- 5.1.1 Fuzzification.- 5.1.2 Inference.- 5.1.2.1 Premise evaluation.- 5.1.2.2 Determination of activated membership functions of conclusions in particular rules at given input values of a fuzzy model.- 5.1.2.3 Determination of the resulting membership function of the rule-base conclusion.- 5.1.3 Defuzzification of the resulting membership function of the rule-base conclusion.- 5.1.4 Example of fuzzy modeling.- 5.2 Significant features of rules, rule bases and fuzzy models.- 5.2.1 Local character of rules.- 5.2.2 Dependence of the number of rules on the number of inputs and fuzzy sets.- 5.2.3 Completeness of a fuzzy model.- 5.2.4 Consistency of the rule base.- 5.2.5 Continuity of the rule base.- 5.2.6 Redundancy of the rule base.- 5.3 Advice relating to rule base construction.- 5.4 Reduction of the rule base.- 5.5 Normalization (scaling) of the fuzzy model inputs and output.- 5.6 Extrapolation in fuzzy models.- 5.7 Types of fuzzy models.- 5.7.1 Mamdani models.- 5.7.2 Takagi-Sugeno models.- 5.7.3 Relational models.- 5.7.4 Global and local fuzzy models.- 5.7.5 Fuzzy multimodels.- 5.7.6 Neuro-fuzzy models.- 5.7.7 Alternative models.- 5.7.8 Similarity principles of the system and of the system model.- 5.7.9 Fuzzy classification.- 6. Methods of Fuzzy Modeling.- 6.1 Fuzzy modeling based on the system expert's knowledge.- 6.2 Creation of fuzzy, self-tuning models based on input/output measurement data of the system.- 6.2.1 Application of neuro-fuzzy networks for fuzzy model parameter tuning.- 6.2.1.1 Structuring and training of neural networks.- 6.2.1.2 Transformation of a Mamdani fuzzy model into a neuro-fuzzy network.- 6.2.1.3 Transformation of a Takagi-Sugeno fuzzy model into a neuro-fuzzy network.- 6.2.2 Tuning of fuzzy model parameters with the genetic algorithm method.- 6.3 Creation of self-organizing and self-tuning fuzzy models based on input/output measurement data of the system.- 6.3.1 Determination of significant and insignificant inputs of the model.- 6.3.2 Determining of fuzzy curves.- 6.3.3 Self-organization and self-tuning tuning of fuzzy model parameters.- 6.3.3.1 Self-organization and tuning of fuzzy models with the geometric method of the maximum absolute error.- 6.3.3.2 Self-organization and self-tuning of fuzzy models with clustering methods.- 6.3.3.3 Self-organization and self-tuning of fuzzy models with the searching method.- 7. Fuzzy Control.- 7.1 Static fuzzy controllers.- 7.2 Dynamic fuzzy controllers.- 7.3 The determination of structures and parameters for fuzzy controllers (organization and tuning).- 7.3.1 The design of fuzzy controllers on the basis of expert knowledge concerning plant under control.- 7.3.2 The design of a fuzzy controller on the basis of a model of the expert controlling the plant.- 7.3.3 The design of a fuzzy controller on the basis of the model of controlled plant.- 7.3.3.1 Remarks concerning identification of models of dynamic plants.- 7.3.3.2 Some remarks concerning the identification of inverted models of dynamical plants.- 7.3.3.3 Tuning a fuzzy controller with an a priori chosen structure.- 7.3.3.4 Fuzzy control based on the Internal Model Control Structure (IMC structure).- 7.3.3.5 Fuzzy control structure with an inverse of a plant model (InvMC structure).- 7.3.3.6 Adaptive fuzzy control.- 7.3.3.7 Multivariable fuzzy control (MIMO).- 8. The Stability of Fuzzy Control Systems.- 8.1 The stability of fuzzy control systems with unknown models of plants.- 8.2 The circle stability criterion.- 8.3 The application of hyperstability theory to analysis of fuzzysystem stability.- 8.3.1 The frequency domain representation of hyperstability conditions for control systems with a time invariant non-linear part.- 8.3.2 The time domain conditions for hyperstability of continuous, non-linear control systems containing a timeinvariant non-linear part.- 8.3.3 The frequency domain conditions for hyperstability of discrete, non-linear control systems containing a timeinvariant non-linear part.- References.