Function approximation by fuzzy systems

Abstract

This paper provides an overview of our recent work on function approximation by fuzzy systems. Some scalar definitions in fuzzy logic control (FLC) are extended to the n-dimensional case, including the vector fuzzy number and membership vector. A mathematical expression is given for the function g(x) manufactured by a fuzzy system. It is shown that, under suitable assumptions, the fuzzy associative memory function g(x) is Lipschitz and approximates arbitrarily closely on compact set any specified continuous function. Relations are given between the accuracy of the approximation and the number of membership functions selected in each dimension. A major role is played in the analysis by the notion of the 'convex combination', which considerably simplifies the analysis compared to other approaches in the literature.