Sufficient conditions on general fuzzy systems as function approximators

Abstract

In a constructive way, we have found sufficient conditions under which general fuzzy systems can uniformly approximate any real continuous function on a compact domain to any degree of accuracy. More importantly, we have revealed the underlying mechanism of such function approximation: the fuzzy systems can uniformly approximate any polynomials which, according to the Weierstrass approximation theorem, can uniformly approximate any continuous function on a compact domain. These findings lead to the following practical results: (1) explicit fuzzy rules of fuzzy systems can now be easily obtained according to functions to be approximated; and (2) formulas are derived which can calculate the number of input fuzzy sets, output fuzzy sets and fuzzy rules needed in order to satisfy any given approximation accuracy. The number increases as required approximation error decreases, and as approximation error approaches zero, the number approaches ∞. These formulas also reveal that the number is minimal when the maximum number of intersection between adjacent input fuzzy sets is one. Practical implications of these results will be discussed, especially in the context of fuzzy control and fuzzy modeling. Two examples are provided to demonstrate how to design fuzzy systems to approximate given functions with required approximation accuracy.