Stability and Filtering for Delayed Discrete-Time T-S Fuzzy Systems via Membership-Dependent Approaches

The stability and ${\mathcal {H}}_\infty$ filtering for delayed discrete-time T-S fuzzy systems are studied in this article. The primary objective is to obtain less conservative and more effective analysis and design methods by exploring a combination of the characteristics of T-S fuzzy systems and the delay-dependent methods. First, as the first step of the Lyapunov–Krasovskii functional (LKF) method, a membership-dependent (MD) LKF with delay-product-type term is established to contain more delay and membership function information. Then, to obtain the negative definite condition of the forward difference of the constructed functional, an MD-matrix-separation-based inequality is developed to obtain tighter estimations for the augmented summation terms and an MD-variable-augmented-based free-weighting matrix method is proposed to avoid the generation of delay-dependent nonlinear terms. Based on the abovementioned methods, a less conservative stability criterion and an ${\mathcal {H}}_\infty$ fuzzy filter design method are proposed. Finally, the merits of the proposed methods are verified via two examples.