Identification in Fuzzy Systems

The identification problem which plays a central role in further development of the theory of fuzzy systems and their applications is considered. The systems under consideration are of the type XRY or (X1X2 ••• Xn)RY, where X,X1X2”',Xn denote fuzzy sets of input and Y stands for a fuzzy set of output, while R denotes a fuzzy relation expressing the relationships between input and output. Identification is treated as a three-step procedure which consists of a) the choice of the proper (suitable) structure of fuzzy relational equations, b) the determination of the parameters of this structure, viz., the relation of the equation, and c) validation of the model constructed. Three different types of relational structure fuzzy relational equations are taken into account: sup-min, sup-prod, and inf-min. Appropriate methods of determination of fuzzy relations are presented. The idea of probabilistic sets in the identification problem is discussed in detail.