Guide on set invariance for delay difference equations. (2016)

• Record Type: Journal Article
• Authors: Laraba, Mohammed-Tahar; Olaru, Sorin; Niculescu, Silviu-Iulian; Blanchini, Franco; Giordano, Giulia; Casagrande, Daniele; Miani, Stefano
• Is Part Of: Annual reviews in control. Volume 41(2016)
• Page Start: 13
• Page End: 23
• Publication Date: 2016
• Subjects: Set invariance -- Linear time-delay systems -- Discrete time-delay difference equations; Automatic control -- Periodicals; Periodicals; 629.805
• DOI: 10.1016/j.arcontrol.2016.04.020
• Languages: English
• ISSNs: 1367-5788

Abstract

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 .