A practical, computationally aware vertex solution for checking robust stability of ´structured´ convex combinations of matrices

This paper presents a practical, computationally aware ´vertex solution´ to the problem of checking the stability of families of matrices described by convex combinations of Hurwitz stable ´vertex´ matrices. The convex combinations resulting from the interval parameter matrix family are labeled as ´structured´ convex combinations to distinguish them from the so called ´unstructured´ convex combinations which are convex combinations of ´user specified´ Hurwitz stable vertex matrices. A previously presented ´vertex algorithm´ by the author for this tough problem is derived under the ´conceptual´ assumption of a ´closed, connected complex plane´. Later, upon realizing that a ´practical´ computational environment assumes an ´open, unconnected complex plane´, a computationally aware vertex solution is offered which accounts for the ´discrepancy´ in the results for some ´ill-conditioned´ problems because the eigenvalues which are supposed to be residing in a closed, connected complex plane are calculated using methods meant for open, unconnected complex plane. This ´discrepancy´ arises for some ill-conditioned problems because in an open, unconnected complex plane, the theoretically present coupling between real eigenvalues and complex (real part) eigenvalues, is destroyed. In this paper, this ´practical, computationally aware´ algorithm is presented in its ´final´ form by carefully highlighting differences between a ´conceptual´ solution and the ´real world, practical´ solution. This in turn clarifies the reason why this problem has been a ´tough nut to crack´ over all these years. Several examples are given which clearly demonstrate effectiveness of the new algorithm, even for ill-conditioned problems. It is hoped that this ´final´ solution paves the way for continued interest in the research related to this important area with many applications