Necessary Conditions for Some Typical Fuzzy Systems as Universal Approximators

Abstract

We investigate necessary conditions for general single-input single-output fuzzy systems and a class of typical multiple-input single-output fuzzy systems as universal approximators for continuous functions defined on compact domains with arbitrarily small uniform approximation error bounds. Considering the case where the only available data about the continuous function to be approximated are a finite set of its extrema, we have established some necessary conditions for the fuzzy systems to be universal approximators of the function. The conditions can be used practically to determine input fuzzy sets, output fuzzy sets and fuzzy rules of the fuzzy systems. Furthermore, these necessary conditions provide a basis for insightful analysis of the strengths as well as the limitations of the fuzzy systems. The main strength is that only a small number of fuzzy rules may be needed to uniformly approximate continuous functions that have a complicated formulation but a relatively small number of extrema. The limitation is that, in order to approximate highly oscillatory continuous functions, the number of fuzzy rules must be large. © 1997 Elsevier Science Ltd.