Exact solutions for interacting rules using conjunctive logic

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, δ, 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