Further results on the construction of invariant families of sets for linear systems with delay1

Almost all synthesis methods for the construction of invariant sets for delay difference equations (DDEs) suffer either from computational complexity, i.e., those based on the Krasovskii approach, or come with considerable conservatism, i.e., those based on the Razumikhin approach. Only recently, the notion of an invariant family of sets was introduced in order to allow for a suitable tradeoff between conceptual generality and computational simplicity. In this paper, the construction of such families is further elaborated. In particular, a parametrization of families of sets is proposed that leads to tractable synthesis algorithms while preserving several advantages over the Razumikhin approach. For polyhedral sets, the corresponding synthesis algorithms can be implemented via an algebraic recursion and parametric linear programming. Thus, the invariant family of sets becomes a computationally tractable set invariance notion for DDEs, the advantages of which are illustrated via several examples.