Some Characterizations of Schur Matrices and Their Application to the Stability of a Polytope of Matrices

Abstract

Motivated by a general Hurwitz matrix expression involving a triplet of matrices, its Schur counterparts are first derived. This time, they are expressed by a pair of matrices satisfying a certain nonlinear or norm condition. It is then shown that the results can find applications in Schur stability analysis of a polytope of matrices. Using the obtained expessions, two kinds of quadratic Lyapunov functions are proved to work for the polytope:a fixed quadratic function and a parameter-dependent quadratic function. The first kind gives a well known extreme point result on quadratic stability of polytopes of matrices, while the second yields a new result for the stability test of the polytope.