<inline-formula> <tex-math notation="LaTeX">$H_{\infty }$</tex-math> </inline-formula> Stabilization for Polynomial Fuzzy Time-Delay System: A Sum-of-Squares Approach

Abstract

The $H_{\infty }$ stabilization problem for polynomial fuzzy time-delay system (PFTDS) is investigated in this paper. First, a polynomial fuzzy controller is proposed to stabilize the PFTDS. Based on Euler's homogeneity relation, homogeneous polynomial Lyapunov–Krasovskii functional, and the developed polynomial slack variable matrices, a novel stabilization condition without the assumption in common use is presented in terms of sum-of-squares (SOS). In order to reduce the conservative and implementation cost, the property of membership function, and the right-hand side technique are adopted. Also, two relaxed $H_{\infty }$ stabilization conditions for PFTDS, which may or may not share the same fuzzy rules, are proposed. To illustrate the feasibility and merit of the proposed result, three examples are provided. In the first example, a well-known Takagi–Sugeno fuzzy time-delay system is demonstrated to show that the proposed SOS-based methods can provide a longer allowable delay time than do some existing linear matrix inequality based methods. The second and third examples show the validity and feasibility of the proposed results for PFTDS with the same or fewer rule numbers of the polynomial fuzzy controller.