Optimal necessary conditions for general SISO Mamdani fuzzy systems as function approximators within a given accuracy

In this paper, necessary conditions are investigated for a single input/single output (SISO) Mamdani fuzzy systems as function approximators of continuous functions within a given accuracy. Since general SISO Mamdani fuzzy systems are monotonic on subintervals, the optimal configuration of fuzzy systems is that the number of division points is at least the times of its monotonicity changes. Thus with the extreme of the desired continuous function, necessary conditions are obtained through generating intervals that contain division points and pruning redundant intervals. Furthermore, a dynamically constructive method is proposed to show the conditions are optimal. It has been shown that existing results concerning necessary conditions are only special cases of our results. Finally, simulation examples are given to illustrate the conclusions, the strength of the fuzzy systems as function approximators are analyzed.

I. Introduction

In the past decades, fuzzy systems were widely used and extensively studied for a variety of applications, especially in control applications because of their excellent performance in representing nonlinear functions in an intuitive and informative manner. In many applications of fuzzy theory, fuzzy systems involved can be viewed as rule-based function approximators. Many scholars have proved that fuzzy systems are universal approximators in different aspects [1]–[5]. To our knowledge, the smaller the approximation accuracy, the more fuzzy rules are required in order to approximate a certain continuous function, especially in high dimensions. Large numbers of rules render difficulty in designing fuzzy systems, such as high storage consumption, increased computational complexity, etc.