Super-Exponential Convergence of the Karnik–Mendel Algorithms for Computing the Centroid of an Interval Type-2 Fuzzy Set

Computing the centroid of an interval T2 FS is an important operation in a type-2 fuzzy logic system (where it is called type-reduction), but it is also a potentially time-consuming operation. The Karnik-Mendel (KM) iterative algorithms are widely used for doing this. In this paper, we prove that these algorithms converge monotonically and super-exponentially fast. Both properties are highly desirable for iterative algorithms and explain why in practice the KM algorithms have been observed to converge very fast, thereby making them very practical to use

I. Introduction

A N INTERVAL type-2 fuzzy set (IT2 FS) is today the most widely used T2 FS because it is computationally simple to use. When such FSs are used in a rule-based fuzzy logic system (FLS) (e.g., [4], [5], [9]–[14], [17], and [23]–[25]), the result is an interval T2 FLS (IT2 FLS). In such a FLS, fired-rule output sets are also IT2 FSs, and to go from such sets to a number, as is usually required in most engineering applications of a FLS, one must perform two successive operations, type-reduction and defuzzification. Type-reduction maps the output T2 FS into a type-1 (T1) FS, and defuzzification converts that T1 FS into a number.