TSK Fuzzy Function Approximators: Design and Accuracy Analysis

Fuzzy systems are excellent approximators of known functions or for the dynamic response of a physical system. We propose a new approach to approximate any known function by a Takagi-Sugeno-Kang fuzzy system with a guaranteed upper bound on the approximation error. The new approach is also used to approximately represent the behavior of a dynamic system from its input-output pairs using experimental data with known error bounds. We provide sufficient conditions for this class of fuzzy systems to be universal approximators with specified error bounds. The new conditions require a smaller number of membership functions than all previously published conditions. We illustrate the new results and compare them to published error bounds through numerical examples.

I. Introduction

A class of fuzzy systems is a universal approximator if, for any real continuous function on a compact set, there exists a fuzzy system from this class that can approximate the function to any degree of accuracy [1], [2]. The universal approximation capabilities of fuzzy systems have been discussed extensively in the literature. Many different classes of fuzzy systems have been analyzed and proven to have the universal approximation property [3], [4]. However, the universal approximation property of fuzzy systems is only one aspect of designing a fuzzy function approximator. A critical question is the following: how can one design a fuzzy system to approximate a given real continuous function? Here, the word design means choosing the shape of the membership functions, the number of fuzzy sets needed for each input variable, and the rule base.