Delay-dependent set-invariance for linear difference equations with multiple delays: a polyhedral approach

Abstract

This paper revisits the set-invariance for linear delay-difference equations and proposes novel delay-dependent notions in contrast with the existing delay-independent constructions available in the literature. The main tool in this endeavour is the model transformation by means of a matrix parametrization, which opens the way to the elaboration of delay-dependent conditions for set invariance. For the class of polyhedral sets, it will be shown how these transformed models enable particularly structured invariance conditions. Aside from the mathematical developments, a discussion of the delay-dependent conditions with respect to the confinement of the state trajectories of the original system will be presented. The relationship will be established by means of constraints involving the initial states for the time-delay systems. The characterization of this set of admissible initial conditions can be analysed in terms of complexity and conservativeness with respect to the classical delay-independent set-invariance. An illustrative example is provided to underline that confinement of state trajectories in a set can be achieved even though this is not invariant according to the classical definition.