LMI RELAXATIONS FOR H∞ CONTROL OF TIME-VARYING POLYTOPIC SYSTEMS BY MEANS OF PARAMETER-DEPENDENT QUADRATICALLY STABILIZING GAINS

This paper provides necessary and sufficient conditions in terms of LMI relaxations to compute H∞ parameter-dependent state feedback control gains that ensure closed-loop quadratic stability for linear systems affected by arbitrarily fast time-varying parameters inside a polytope. The proposed conditions, based on an extension of Pólya's Theorem and on the systematic construction of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities, are written as a sequence of progressively less and less conservative linear matrix inequality conditions. Necessity is attained as the level of relaxation increases, providing an H∞ parameter-dependent state feedback gain that quadratically stabilizes the system whenever such a gain exists. An extension to compute decentralized control gains is also presented and numerical results illustrate the efficiency of the proposed conditions.