Fuzzy Rule-Based Systems with Polynomial Membership Functions

Abstract

In order to obtain a richer class of functions to which the fuzzy rule-based system is equivalent, one can use nonlinear membership functions of fuzzy sets, to which polynomials of the second or higher degree belong. Such polynomials are defined by three or more parameters. It would appear that by using nonlinear membership functions, one can get a sufficiently large class of functions, to which the rule-based system is equivalent. However, if we increase the complexity of membership functions of fuzzy sets only, while preserving the number of fuzzy sets assigned for the input variables, our intuition about richness of the class of functions performed by the rule-based system can fail us. The number of fuzzy sets is important, since it determines the number of consequents of the rules; thus, it constrains the class of functions performed by the zero-order TS rule-based systems. This fact will be shown further on.