Equivalence between Two Results Providing Polyhedral Contractive Sets for Discrete-Time Linear Systems with Arbitrary Switching

Abstract

The paper belongs to the area of qualitative analysis of the dynamics for discrete-time arbitrary switching linear systems. The purpose of our research is to develop a rigorous comparative study between two criteria for testing the existence of λ-contractive polyhedral sets with respect to the switching systems. Despite the common objective of the two criteria, their connections at the level of the mathematical backgrounds have been insufficiently explored. These backgrounds are relatively distanced, namely the operation with convex sets versus solvability of several matrix equalities, with inequality constraints. Our work proposes complete mathematical results with accompanying proofs, which are able to explain all the links that are currently missing from the literature. These theoretical results are illustrated by a numerical example run in Matlab, which also offers visual support for the construction and utilization of λ-contractive polyhedral sets.