LMI RELAXATIONS FOR HOMOGENEOUS POLYNOMIAL SOLUTIONS OF PARAMETER-DEPENDENT LMIS

This paper presents a systematic procedure to generate families of linear matrix inequality (LMI) conditions of finite dimension which provide a homogeneous polynomial solution of arbitrary degree for parameter-dependent LMIs defined in the unit simplex. Differently from other approaches in the literature, the LMI conditions are written directly in terms of the entries of the homogeneous polynomial matrix solution. The proposed procedure can be applied to a wide variety of robust control and analysis problems for linear systems with uncertain parameters belonging to a polytope. As shown in the paper, the solutions of parameter-dependent LMIs defined in the unit simplex can be constrained to the class of homogeneous polynomial without loss of generality. For a given degree, sufficient LMI conditions assuring the existence of a solution are provided. As the degree increases, a feasible homogeneous polynomial solution is obtained whenever it exists. The relaxation method is presented in terms of general parameter-dependent LMIs defined in the unit simplex, with two special cases analyzed in details.