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.

Guide on set invariance for delay difference equations / Laraba, Mohammed-Tahar; Olaru, Sorin; Niculescu, Silviu-Iulian; Blanchini, Franco; Giordano, Giulia; Casagrande, Daniele; Miani, Stefano. - In: ANNUAL REVIEWS IN CONTROL. - ISSN 1367-5788. - 41:(2016), pp. 13-23. [10.1016/j.arcontrol.2016.04.020]

Guide on set invariance for delay difference equations

Giordano, Giulia;
2016

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.
Laraba, Mohammed-Tahar; Olaru, Sorin; Niculescu, Silviu-Iulian; Blanchini, Franco; Giordano, Giulia; Casagrande, Daniele; Miani, Stefano
Guide on set invariance for delay difference equations / Laraba, Mohammed-Tahar; Olaru, Sorin; Niculescu, Silviu-Iulian; Blanchini, Franco; Giordano, Giulia; Casagrande, Daniele; Miani, Stefano. - In: ANNUAL REVIEWS IN CONTROL. - ISSN 1367-5788. - 41:(2016), pp. 13-23. [10.1016/j.arcontrol.2016.04.020]
File in questo prodotto:
File Dimensione Formato  
010_2016ARC_LONBGCM.pdf

Solo gestori archivio

Descrizione: Paper
Tipologia: Versione editoriale (Publisher’s layout)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 1.29 MB
Formato Adobe PDF
1.29 MB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/249815
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 13
  • ???jsp.display-item.citation.isi??? 10
social impact