Set-Invariance for Discrete-Time Linear Systems with Time-Varying Delay: a Polyhedral Approach

In this paper, we address the positive invariance property in linear discrete-time systems in the presence of time-varying delays in the states. We build the study on a recently proposed approach based on an appropriate model transformation which allows the derivation of delay-dependent invariance conditions. For polyhedral sets, we establish necessary and sufficient invariance conditions with respect to the transformed model and prove that such conditions imply the confinement of the state trajectories of the original system in the set for arbitrary realizations of the varying delay, as long as the initial states belong to a set which is positively invariant with respect to an augmented switching system without delay, and can be computed in a finite number of steps known in advance.