Further Studies on $H_{\\infty}$ Filtering Design for Fuzzy System With Known or Unknown Premise Variables

Abstract

In this brief paper, the problem of <inline-formula> <tex-math notation= LaTeX >$H_{\\infty }$ </tex-math></inline-formula> fuzzy filtering design is further studied and two cases (Case <inline-formula> <tex-math notation= LaTeX >$\\mathbf {A}$ </tex-math></inline-formula>: the premise variables are known and Case <inline-formula> <tex-math notation= LaTeX >$\\mathbf {B}$ </tex-math></inline-formula>: the premise variables are unknown) are considered. Different from all the filtering design methods in the literature, in this paper, the time derivatives of the membership functions involved in the Lyapunov function are dealt with by a switching method. Then, for different cases of the premise variables, different switching filters are designed and the membership function dependent Lyapunov function is applied to derive some LMIs (linear matrix inequalities). Furthermore, some discussions on how to further relax the results by applying the polynomial membership functions dependent matrices or by introducing more free variables are given in two remarks. In the end, two numerical examples are given to show the effectiveness of the proposed approach.