Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequent

We have constructively proved a general class of multi-input single-output Takagi-Sugeno fuzzy systems to be universal approximators. The systems use any types of continuous fuzzy sets, any types of fuzzy logic AND, fuzzy rules with linear rule consequent and the generalized defuzzifier. We first prove that the TS fuzzy systems can uniformly approximate any multivariate polynomial arbitrarily well, and then prove they can uniformly approximate ang multivariate continuous function arbitrarily well. We have derived a formula for computing the minimal upper bounds on the number of fuzzy sets and fuzzy rules necessary to achieve prespecified approximation accuracy for any given bivariate function. A numerical example is furnished. Our results provide a solid theoretical basis for fuzzy system applications, particularly as fuzzy controllers and models.