Interval Type-2 Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators

Abstract

Whether a rule-based fuzzy system has the ability to approximate any multivariate continuous function arbitrarily well is an important issue, especially for fuzzy control and fuzzy modeling. The answer to this issue concerning various Mamdani and Takagi-Sugeno (TS) fuzzy systems employing type-1 fuzzy sets is affirmative and well documented in the literature. As for type-2 (T2) fuzzy systems, the only result currently available is ours showing a general class of interval T2 Mamdani fuzzy systems to be universal approximators. In the present paper, we extend our investigation to cover a general class of interval T2 TS fuzzy systems with linear rule consequent. We prove constructively that this class is universal approximator by first proving that the fuzzy systems can uniformly approximate any polynomials arbitrarily accurately and then utilizing the Weierstrass approximation theorem to complete the proof.