Relaxed TS Fuzzy model transformation to improve the approximation accuracy / complexity trade-off and relax the computation complexity

Abstract

The primary goal of the paper is to introduce the Relaxed TS Fuzzy model transformation, a method that enhances the original TS Fuzzy model transformation in two ways. First, it focuses on achieving a more efficient reduction of the number of antecedent fuzzy sets– hence, the fuzzy rules of the TS Fuzzy models– while minimizing the approximation error. Second, it aims to reduce the computational load required for the transformation process. With the first enhancement, the proposed transformation strikes a better balance between the number of fuzzy rules and the approximation accuracy of TS Fuzzy models. With the second enhancement, a unique preand post-processing of the TS Fuzzy model transformation is introduced leading to the radical computational improvements. The core part of the original TS Fuzzy model transformation is the Higher Order Singular Value Decomposition (HOSVD) used to balance the approximation quality with the number of fuzzy rules by truncating singular values. The HOSVD itself is a computationally intensive algorithm, the possibilities for advancements in its implementation seem to be limited as much research has focused on its optimization in the past and had reached its pinnacle in terms of computational complexity more than a decade ago. Therefore, the approach presented in this paper does not concentrates directly on enhancing HOSVD further, but instead proposes a unique pre- and post-processing technique for the tensor on which HOSVD is applied, tailored to the special characteristics of the TS Fuzzy model and the system model under consideration. Following a description of the proposed enhancements, the paper presents numerical examples and two examples of real-world engineering models to demonstrate the effectiveness of the Relaxed TS Fuzzy model transformation compared to the original TS Fuzzy model transformation.