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Robust H2 and H∞ State Feedback Control for Discrete–time Polytopic Systems Using an Iterative LMI Based Procedure

This paper considers H2 and H∞ state feedback control design for discrete–time linear time–invariant systems subject to polytopic uncertainty. The associated nonlinearities with the computation of feedback gain are resolved through the Elimination Lemma and the use of slack variables. We propose a new result in a form of linear matrix inequality (LMI) to obtain an initial solution that includes quadratic stability results as a special case. Then we propose an algorithm that can iteratively compute non-increasing guaranteed costs for H2 and H∞–norms based on LMIs. The proposed conditions do not contain any scalar parameter and therefore exhaustive searches on scalar parameters can be avoided. The final solution obtained by our algorithm is less conservative than the ones from the literature, as illustrated by a numerical example.