Abstract
In this paper, we present an algorithm for representing polynomials of two variables by a fuzzy system. The fuzzy system is based on the cubic spline interpolation of polynomials of the form ∑ ijc ijB i(x)B j(y) where B i(x) and B j(y) are the cubic B-splines. These B-spline functions are used as the fuzzy sets for input fuzzification while the spike functions are used for the output fuzzy sets, with c ij 's as support boundaries after sorted in an increasing order. The ordinal number of c ij in the sorted list is taken to be the output fuzzy set number in the (i,j)th entry of the fuzzy rule table. We prove that this fuzzy system is an exact representation of the cubic spline interpolation function and hence the evaluation error for the fuzzy system is the same as that of spline interpolation which is of order O (h 4) . An algorithm to compute the spline coefficients explicitly without solving the matrix equation involved is also included along with evaluation results of the fuzzy system for various examples with different sizes of the rule table.
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