Toward a Fuzzy Logic System Based on General Forms of Interval Type-2 Fuzzy Sets

Abstract

Recent years have witnessed a widespread in the use of interval type-2 fuzzy logic systems (IT2 FLSs) in real-world applications. It has been shown recently that interval type-2 fuzzy sets (IT2 FSs) are more general than interval-valued fuzzy sets (IV FSs) [1]. Hence, there is a need to explore the capabilities of the more general forms of IT2 FSs (beyond IV FSs) and the applications areas they will be more suitable for. In addition, there is a need to develop the theory of the general forms of IT2 FLSs (gfIT2 FLSs), which employ IT2 FSs that are not equivalent to IV FSs and can have nonconvex secondary membership functions (MFs). Although these systems could be considered within the scope of general type-2 FLSs (GT2 FLSs), the practical implementation of GT2 FLSs has traditionally required the secondary MFs to be convex and normal type-1 fuzzy sets (T1 FSs). In addition, the type-reduction operation still presents a challenge for GT2 FLSs because of its computational complexity. In this paper, we present a complete framework for a type-2 FLS that uses the most recent perception of IT2 FSs (the so called general forms of interval type-2 fuzzy sets, gfIT2 FSs), whose secondary grades can be nonconvex T1 FSs. This framework includes new equations for the meet and join operations of gfIT2 FSs, as well as a new type reduction procedure for the type-2 FLS involving gfIT2 FSs. In addition, we present the type-2 FLS operation for singleton and nonsingleton fuzzification. We will introduce the various operations employed within a gfIT2 FLSs, from fuzzification (including singleton and nonsingleton) to inference, type-reduction, and defuzzification. We will also present two examples in which these gfIT2 FSs arise naturally when modeling sonar sensors input noise and the antecedents/consequents from a survey including different users.