A Similarity Based Fuzzy System as a Function Approximator

Abstract

In this paper, an attempt is made to prove that some similarity based fuzzy systems can be found to behave as function approximators. A typical similarity based fuzzy system is proposed and its behaviour is shown to have the said property. It elucidates the connection between similarity relation and similarity measure of fuzzy sets to fuzzy inference methodology. The concept of similarity relation is used in fuzzification of crisp input values. Similarity index is used in measuring approximate equality of fuzzy sets over a given universe of discourse of a linguistic variable. The similarity between the observation(s) and the antecedent of a rule is used in selecting rule(s) for possible firing and also in modifying the relation between the antecedent and consequent of the rule based on the specific observation. Inference is drawn through the usual composition and subsequently by projecting the modified fuzzy restriction acting on the variables of interest on the universe of the linguistic variable in the consequent of the rule. A specificity based defuzzification scheme is proposed for multiple-rule firing. It has been proved systematically that such a similarity based fuzzy system can uniformly approximate continuous functions to any desired degree of accuracy on a closed and bounded interval. Simulation results are presented for the well-known dc-motor problem. A comparative study is made to establish the validity and efficiency of the proposed similarity based fuzzy system.