Membership-Function-Dependent Stability Analysis for Polynomial-Fuzzy-Model-Based Control Systems via Chebyshev Membership Functions

In this article, we investigate the stability analysis of a polynomial-fuzzy-model-based control system by employing a new form of approximate membership functions called Chebyshev membership functions (CMFs) based on the Lyapunov stability theory. The membership-function-dependent method shows that the approximate functions can introduce useful information to relax the conservativeness of stability analysis. Therefore, in the case of the same order and operating domain, how to design the approximate functions to introduce more information to relax stability is a concern. In this article, given the Chebyshev norm relationship between the approximate membership functions and the original ones, CMFs can be obtained by the Chebyshev approximation theory and the Remez/Remez-type iterative algorithm, which own the minimum of the maximum absolute approximation error. Together with CMFs, the boundary information of premise variables on the overall operating domain is considered to reduce the conservativeness of stability analysis. Furthermore, to further relax the stability analysis without increasing the order of CMFs, the operating domain is divided into subdomains, and piecewise Chebyshev membership functions (PCMFs) are proposed to facilitate the stability analysis. Together with PCMFs, the subdomain boundary information of premise variables is introduced into stability conditions. A simulation example is given to demonstrate the effectiveness of the proposed approach.