Delay-Dependent Invariance of Polyhedral Sets for Discrete-Time Linear Systems

Abstract

In this paper we propose a delay-dependent analysis of the positive invariance property with respect to linear discrete-time systems with delayed states. An appropriate model transformation is employed, together with a matrix parametrization, which allow the derivation of delay-dependent invariance conditions of polyhedral sets with respect to the transformed model. We then show that such conditions imply the confinement of the state trajectories of the original system in the set, as long as the initial states satisfy additional constraints related to the system dynamics. The characterization of this set of admissible initial conditions gives rise to the proposition of a less conservative definition of set-invariance. We illustrate through numerical examples the fact that, under the proposed definition, confinement of state trajectories in the set can be achieved even though it is not invariant according to the classical definition.