Stability of polytopes of matrices via affine parameter-dependent Lyapunov functions: Asymptotically exact LMI conditions

This paper investigates necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz (or Schur) stability of a polytope of matrices. A systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function. Necessity is asymptotically attained through a relaxation based on a generalization of Pólya's Theorem to the case of matrix valued functions. Numerical experiments illustrate the results.