Extended Kalman Filter Based Fuzzy Adaptive Filter

Abstract

In classical logic, every statement is either true or false, i.e., it has a truth value of 1 or 0. Classical sets impose rigid membership requirements. Fuzzy logic, which is the principle of imprecise knowledge, was introduced by Lofti A. Zadeh in 1965 (Zadeh, L. A., 1965). It is an extension of classical logic dealing with the partial truth concept. Every statement in fuzzy logic is a matter of degree and exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic, classical/Boolean truth value is replaced with degree of truth. Degree of truth denotes the extent to which a proposition is true. In fuzzy logic, the degree of truth of a proposition may be any real number between 0 and 1, inclusive. This fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Fuzzy logic allows for set membership values between and including 0 and 1, shades of grey as well as black and white, and in its linguistic form, imprecise concepts like “slightly”, “quite” and “very”. Specifically it allows partial membership in a set. It is related to fuzzy sets and possibility theory. Fuzzy sets are an extension of classical set theory and are used in fuzzy logic (Zadeh, L. A., 1975). In classical set theory, the membership of elements in relation to a set is assessed in a crisp condition: either belongs to or not. In contrast, fuzzy set theory allows the gradual assessment of the membership of elements in relation to a set, with the aid of a membership function ┤. A membership function may act as an indicator function, mapping all elements of fuzzy sets to real numbered value in the interval 0 and 1: ┤→[0,1]. In general, there are 6 types of membership functions as depicted in Fig. 1.