FUZZY BASIS FUNCTIONS, UNIVERSAL APPROXIMATION, AND ORTHOGONAL LEAST-SQUARES LEARNING

In this paper, fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, we prove that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs. Specifically, an initial fuzzy system is first constructed which has as many fuzzy basis functions as input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The most important advantage of using fuzzy basis functions is that a linguistic fuzzy IF-THEN rule from human experts is directly related to a fuzzy basis function. Therefore, a fuzzy basis function expansion provides a natural framework for combining both numerical and linguistic information in a uniform fashion. Finally, the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some commonsense fuzzy control rules.