H∞ filter design for linear time-invariant systems with polytopic uncertainties in finite frequency domain

This paper deals with the problem of finite frequency H∞ full‐order filter design for discrete‐time and continuous‐time linear systems, with polytopic uncertainties. Based on the generalized Kalman–Yakubovich–Popov lemma and a parameter‐dependent Lyapunov function, a set of sufficient conditions are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level, from disturbance to the estimation error, is smaller than a given value over a prescribed finite frequency domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we introduce a large number of slack variables by applying Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance. This leads to performance improvement and reduction of conservatism in the solution. It is shown later that the robust filter gains can be obtained by solving a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods.