An approach to robust stability of matrix polytopes through copositive homogeneous polynomials

An approach to guaranteeing the stability of a polytope of matrices is proposed. Previous results in the literature have made the obvious connection between various robust stability problems and a test for positivity of a multivariable polynomial. Such results are extended in order to demonstrate that all matrices within a polytope are stable is and only if an associated homogeneous polynomial is strictly copositive. The additional structure obtained by exploiting homogeneity of this multivariable polynomial leads to several computationally tractable sufficiency tests for establishing either robust stability or instability of a polytope of matrices