Construction of polynomial polytopes by using segment lemma

A polynomial with all roots lying in the open left half plane of the complex plane is called Hurwitz stable. The convex hull of a finite number of polynomials of the same order is called a polynomial polytope. By the Edge theorem, a polynomial polytope with the invariant degree is Hurwitz stable if and only if all edges are Hurwitz stable. Due to the Edge Theorem, the segment stability criterions are of great importance. In this paper, we consider a construction of stable polytopes by using the Segment Lemma. It is constructed stable polytopes with nonzero volumes. The obtained results can be used, for example, in the stabilization problems of unstable transfer functions by lower-order controllers.