A necessary and sufficient extreme point solution for checking robust stability of polytopes of matrices

Addresses the issue of developing a finitely computable necessary and sufficient test for checking the robust stability of a polytope of matrices and provides a complete solution to the problem in the form of an extreme point result. The result uses the fact that the robust stability problem can be converted to a robust nonsingularity problem involving the original matrix and the associated Kronecker Lyapunov matrix. The special nature of the Kronecker Lyapunov matrix is exploited with the introduction of a new concept labeled real axis nonsingularity. Another important concept introduced is that of virtual matrix family. The virtual matrix concept indirectly captures the interior of the uncertain matrix family. Using measures labeled weighted real axis determinant and real axis nonsingularity scalar which are positive for an asymptotically stable matrix, the proposed necessary and sufficient condition involves checking if a set of real axis nonsingularity matrices (formed in terms of the vertex matrices in the Kronecker Lyapunov space) are asymptotically stable or not. This condition thus involves the eigenvalue information of the higher dimensional matrices in the Kronecker Lyapunov space. The proposed methodology is illustrated with a variety of examples. The importance of this result and possible extensions of this result are discussed.