New LMI Approach to Fuzzy $H_{\infty}$ Filter Designs

This brief is concerned with the $H_{\infty}$ filtering problem for a class of discrete-time fuzzy systems. The objective is to find a new way to design a stable filter guaranteeing the asymptotic stability and a prescribed $H_{\infty}$ performance of the filtering error system. By using the fuzzy Lyapunov function approach and adding slack matrix variables, a new condition for $H_{\infty}$ performance analysis is proposed. Then, a linear matrix inequality (LMI)-based approach is developed for designing the fuzzy $H_{\infty}$ filter, which can be obtained from the solution of convex optimization problems. When these LMIs are feasible, an explicit expression of a desired $H_{\infty}$ fuzzy filter is given. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach.