Study on Approaching Precision to Standard Fuzzy Systems with Two Different Basic Functions

The standard fuzzy systems are established with partition of normal quadratic polynomial membership functions and normal trigonometric membership functions. Universal approximation error bounds of these fuzzy systems from SISO to MISO are given and their relations are founded. The error remainder term and auxiliary function are employed for the first time in proving process. Moreover, advantage and shortcoming of the two fuzzy systems are compared and correlative conclusions are obtained. Finally, computing examples are given and the validity of the conclusions is confirmed.

I. Introduction

The approximation properties of fuzzy systems have been attracting a lot of attention in recent years. Owing to the foundation work of BackLey [1], Kosko [2], Wang [3], [4], Zeng [5]–[7] and Hao [9]–[11], many results have been obtained in literature for the fuzzy system, especially on the universal approximation property. For the given class of fuzzy systems, we can state that this class of universal approximation property is a universal approximator if any real continuous function on a compact set can be approximated to any degree of accuracy by this class of fuzzy systems. Since standard fuzzy systems suffer the curse of rule dimensionality and parameter dimensionality, many scholars change their directions to hierarchical fuzzy systems. However, most important fuzzy systems are standard fuzzy systems.