Abstract : This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. The notion of D-invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model. The present paper contains a sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D-invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D-invariant sets is not necessarily D-invariant, while the convex hull of a non-convex D-invariant set is D-invariant. Positive invariance is an essential concept in control theory, with applications to constrained dynam-ical systems analysis, uncertainty handling as well as related control design problems [1, 2, 3]. It serves as a basic tool in many topics, such as model predictive control [4, 5, 6], fault tolerant control [7] and reference governor design [8]. Furthermore, there exists a close link between classical stability theory and positive invariant sets. It is worth mentioning that, in Lyapunov theory, invariance is implicitly described ✩ A preliminary version of the paper has been presented at (Silviu-Iulian Niculescu) URL: http://www.supelec.fr/360_p_40423/mohammed-tahar-laraba (Mohammed-Tahar Laraba) by the sub-level sets of a Lyapunov function, which are known to be contractive sets [9]. The response of a dynamical system to external ex-citation is rarely instantaneous, and time-delay models are well suited for describing dynamics related to propagation phenomena and/or communication flows
