The relationship between the uniform approximation rates and the shapes of fuzzy sets in fuzzy systems
Purpose - The purpose of this paper is to answer the question that what the best shape of fuzzy sets is in fuzzy systems for function approximation which is essential in many applications of fuzzy systems. Design/methodology/approach - The uniform approximation rates indicate the approximating capabilities of fuzzy systems for function approximation. By Fourier analysis, the uniform approximation rates are estimated for the fuzzy systems with various shapes of if-part fuzzy sets in the case of single-input and single-output. Based on the approximation rates, the relationships between the approximating capabilities and the shapes of fuzzy sets are developed and compared. Findings - The since functions as the input membership functions in fuzzy systems are proved to have the almost best approximation property in a particular class of membership functions. Research limitations/implications - From the viewpoint of function approximation, the input membership functions are not necessarily positive in fuzzy systems. Practical implications - For engineers, the sinc-shaped membership function is a good choice to improve their fuzzy systems in real applications. Originality/value - The uniform approximation rates of fuzzy systems for function approximation are estimated. Mathematically, the relationships between the approximating capabilities and the shapes of fuzzy sets are analyzed for fuzzy systems.
fuzzy control cybernetics systems theory approximation theory