Construction of invariant families of sets for linear systems with delay

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. This paper utilizes the concepts of vector Lyapunov functions and set-dynamics in order to introduce a novel notion of invariance for linear DDEs, termed the invariant family of sets. The proposed concept of an invariant family of sets allows for a suitable trade-off between computational simplicity and conceptual generality. In addition, it reduces the stability analysis for a DDE to the stability analysis for a relatively simple comparison system. The practical implementation for families of ellipsoidal sets proposed in this paper leads to synthesis algorithms that can be realized by solving a sequence of linear matrix inequalities.