Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted Nie–Tan algorithms

Abstract

In recent years, researching on general type-2 fuzzy logic systems (GT2 FLSs) has become a hot topic as the development of alpha-planes representation of general type-2 fuzzy sets. The iterative Karnik–Mendel (KM) algorithms are used to perform the key block of type-reduction (TR) of GT2 FLSs. However, the KM algorithms are computationally intensive and time-consuming, which is not adapted to real-time applications. In the enhanced types of algorithms, the noniterative Nie–Tan (NT) algorithms decrease the computational cost greatly. Moreover, the closed-form Nie–Tan algorithms which calculate the outputs by averaging the lower and upper bounds of the membership functions have been proved to be actually an accurate algorithm for performing TR. The paper expands the NT algorithms to three different forms of weighted NT (WNT) algorithms according to the Newton–Cotes quadrature formulas of numerical integration techniques. Four computer simulation examples are adopted to analyze the performances of WNT algorithms when solving the type-reduction of general type-2 fuzzy logic systems. The proposed WNT algorithms have smaller absolute errors and faster convergence speed compared with NT algorithms, which provide the potential value for designers and adopters of GT2 FLSs.