Notes on the approximation rate of fuzzy KH interpolators

This paper investigates the approximation behaviour of the K1 oczy–Hirota (KH) interpolative fuzzy controllers. First, in accordance with the remarks in (Fuzzy Sets and Systems 125(1) (2002) 105), it is pointed out that it is a fuzzy generalization of the Shepard operator. Shepard operator has thoroughly studied by approximation theorist since the mid-1970s. Exploiting the aforementioned relationship, we establish analog results on the approximation rate of KH controllers. The optimal order and class of approximation (saturation problem) are determined for certain values of the exponent . Corresponding results on the modi8ed alpha-cut based interpolation method, being an improvement of the KH interpolator, are also provided. The results o9er trade-o9 facilities between approximation accuracy and the number of rules. As a consequence, the necessary and su;cient number of rules can be determined for a prescribed accuracy. c © 2002 Elsevier B.V. All rights reserved.