Stability analysis and controller synthesis for discrete-time delayed fuzzy systems via small gain theorem

This paper investigates the problems of stability analysis and controller synthesis of discrete-time Takagi–Sugeno (T–S) fuzzy systems with time-varying delay via an input–output approach. The original system is reformulated as a feedback subsystem and a forward subsystem with two constant time delays and a delay “uncertainty” satisfying the scaled small gain theorem. The lifting method is employed to convert the forward subsystem to a delay-free system, where the main result concerning the stability analysis of the system is obtained based on the scaled small gain theorem. The merit of the new stability condition lies in its reduced conservatism, which is made possible by a more precise estimation of the time-varying delay. Additionally, a state-feedback controller is designed through the Lyapunov–Krasovskii functional approach combined with the utilization of the scaled small gain theorem. Two illustrative examples are given to demonstrate the less conservatism of the stability condition and the effectiveness of the controller design method.