Infinite polytopes in Hurwitz stability region

Abstract

We consider Hurwitz stability region in the parameter space of monic polynomials. Firstly, a multilinear map from the positive first octant of the cartesian coordinate system to the stability region is defined. Based on this map a necessary and sufficient condition for stability is obtained. Starting from stable points, polytopes with infinite edge lengths are defined. Stability of the edges of these polytopes is equivalent to the stability of the convex combinations of third and fourth order factor polynomials. If all edges of polytope are stable then by the edge theorem the polytope is robust stable. The obtained result is important for understanding the geometry of stability domain, and can be used for generation of convex subsets of stable polynomials and for stabilization by fixed order controllers and other related problems