Exact solutions for interacting rules using conjunctive logic

Abstract

In this paper, I derive an exact solution for the membership function of the output of a fuzzy system consisting of n Mamdani-type rules. The consequents of the rules are triangular fuzzy sets that are evenly spaced on a univariate universe of discourse. All the consequents have the same support width, /spl delta/, which determines the degree to which the consequents overlap. The derivation makes no assumptions about the rule antecedents, since the fuzzy output is expressed as a function of the degree to which each rule is satisfied. Using this function, I derive an exact solution for the defuzzified output of the system using the centroid defuzzification procedure. Finally, a numerical example involving four rules with two input variables illustrates a preliminary investigation into the effect of varying the support width of the consequent fuzzy sets.