Improved finite-frequency H∞ analysis method and filter synthesis for discrete-time polytopic systems

This paper extends the existing results on the finite-frequency H_∞ filtering problem for discrete-time linear systems with polytopic uncertainty. New analysis conditions for estimating H_∞ gain of transfer functions with known parameters over finite frequency ranges are derived based on the generalized Kalman-Yakubovich-Popov (GKYP) lemma and the projection lemma. The derived conditions are non-conservative in that they become necessary and sufficient when strictly proper transfer functions are considered. The synthesis result is a finite-frequency counterpart of that addressed by Duan et al. Automatica, 42, 1919-1926 (2006) in full-frequency domain, with a distinction that system stability and performance are guaranteed in separate conditions. The proposed approach introduces slack variables which provide extra degree of freedom in the solution space for the finite-frequency H_∞ filtering problem. This leads to performance improvement and reduction of conservativeness in the solution. Two examples are given for demonstration of the efficacy of the new analysis conditions and the proposed synthesis method.