Enhancing the role of column representatives in testing the invariance properties of switching systems

Abstract

The role played by column representatives is expanded in the study of the properties exhibited by the trajectories of arbitrary switching linear systems. Previous contributions on the employment of column representatives focused on positive dynamics and linear co positive Lyapunov functions associated with exponentially contractive positive sets that are invariant with respect to such dynamics. Our approach refers to arbitrary dynamics and invariant sets with general form for time-dependence. We address both discrete and continuous-time cases. Our key finding is that the existence of such invariant sets is fully characterized (if and only if) by the Schur (Hurwitz respectively) stability of the column representatives corresponding to a matrix set adequately built from the original system matrices. Our mathematical developments are illustrated by a numerical example. These developments incorporate the previous contributions mentioned above as particular cases.