Approximation capability analysis of hierarchical Takagi-Sugeno fuzzy systems

Based on the approach of hierarchical structure analysis of continuous functions, this paper discusses the approximation capacities of hierarchical Takagi-Sugeno fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with the natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and the number of parameters. Then, based on the Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with natural hierarchical structure and can then be approximated by hierarchical Takagi-Sugeno fuzzy systems to achieve the universal approximation property.

I. INTRODUCTION

As a theoretic foundation that fuzzy systems are capable as general system models, the universal approximation property of fuzzy systems is well known today. Since the pioneering works of Backley [1]., Kosko [4], and Wang [9], many results have been published in the fuzzy system literature, for example [5], [12], [15] –[17] for Mamdani's fuzzy systems and [13]–[14] for Takagi-Sugeno (TS) fuzzy systems. For more details, see a recent survey given by [8]. These results show, under very general conditions, that fuzzy systems can approximate any continuous functions to any desired degree of accuracy. However, almost all these fuzzy approximation schemes based on the standard fuzzy systems suffers the curse of dimensionality which can be viewed from the following two perspectives: 1) Rule dimensionality: The total number of rules in the fuzzy rule base increases exponentially with the number of the input variables; 2) Parameter dimensionality: The total number of parameters in the mathematical formulas of fuzzy systems increases exponentially with the number of the input variables.