A sufficient condition on a general class of interval type-2 Takagi-Sugeno fuzzy systems with linear rule consequent as universal approximators

Abstract

Whether a rule-based interval type-2 fuzzy system has the ability to approximate any continuous multivariate function arbitrarily well is a fundamentally important question for fuzzy control and modeling. The only approximation results available are the preliminary ones that we previously obtained. They state that two general classes of the interval T2 fuzzy systems, one for the Mamdani type and the other for the TS type, are universal approximators. We now further our investigation on the same T2 TS fuzzy systems to establish a quantitative sufficient approximation condition. These TS fuzzy systems use the linear rule consequent. Their input fuzzy sets are interval T2 and can be in any shape (e.g., Gaussian or trapezoidal) and the fuzzy AND operators in the rules can be any one type or mixed types. We first proved that the fuzzy systems could uniformly approximate any multivariate polynomials arbitrarily accurately and then utilized the Weierstrass approximation theorem to produce the sufficient approximation condition. The condition is a formula calculating the number of the input fuzzy sets needed to achieve any given approximation accuracy (the number of fuzzy rules needed can be easily computed afterward). A numerical example is provided for illustration.