Approximation theory of fuzzy systems-SISO case

In this paper, the approximation problem of SISO fuzzy systems is discussed. Based on the fact that fuzzy systems can be represented by a linear combination of fuzzy basic functions (FBF´s), we first give a systematic and detailed analysis of FBF´s and present five properties of FBF´s: structure similarity and compatibility between the membership functions and FBF´s, complementarity and less fuzziness of FBF´s, and composition of fuzzy systems. These properties provide a clear picture of the shape and features of FBF´s. Based on these properties of FBF´s, we obtain some basic approximation properties of fuzzy systems: basic approximation property, uniform approximation property, uniform convergent property and universal approximation property. These results reveal the basic approximation mechanism of fuzzy systems and verify a few basic ideas in the design of fuzzy systems