Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics

Abstract

The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes. We analyze the standard concept of stability in the sense of Schur—abbreviated as SS (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölder $p$-norm, $1\le p\le \infty$, abbreviated as ${\text{SDS}}_p$ (resp., ${\text{HDS}}_p$). Our framework incorporates, as the particular case with $p=2$, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e. ${\text{SDS}}_2$ (resp., ${\text{HDS}}_2$) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution. For the considered class of matrix polytopes, we prove the equivalence between SS and ${\text{SDS}}_p$ (resp., HS and ${\text{HDS}}_p$), $1\le p\le \infty$ (fact which is not true for matrix polytopes with arbitrary structures). We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices. From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vector $p$-norms weighted by diagonal positive-definite matrices). The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.