General Takagi-Sugeno fuzzy systems are universal approximators

We prove that a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems are universal approximators. The fuzzy systems studied in this paper are general and typical: they use any types of continuous fuzzy sets, any types of fuzzy logic AND, fuzzy rules with linear TS rule consequent and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. We first constructively prove that the general TS fuzzy systems can uniformly approximate any multivariate polynomial with arbitrarily small error and then prove that the general fuzzy systems can uniformly approximate any multivariate continuous function with arbitrarily high approximation accuracy