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Guide on set invariance for delay difference equations

Laraba, Mohammed Tahar; Olaru, Sorin; Niculescu, Silviu Iulian; Blanchini, Franco; Giordano, Giulia LU ; Casagrande, Daniele and Miani, Stefano (2016) In Annual Reviews in Control 41. p.13-23
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... (More)

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.

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author
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Discrete time-delay difference equations, Linear time-delay systems, Set invariance
in
Annual Reviews in Control
volume
41
pages
11 pages
publisher
Elsevier
external identifiers
  • Scopus:84974539604
ISSN
1367-5788
DOI
10.1016/j.arcontrol.2016.04.020
language
English
LU publication?
no
id
c8e393a7-8939-4e11-a813-1839a33ffd6c
date added to LUP
2016-07-06 15:32:49
date last changed
2017-01-01 08:30:00
@article{c8e393a7-8939-4e11-a813-1839a33ffd6c,
  abstract     = {<p>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 <sup>D</sup>-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.</p>},
  author       = {Laraba, Mohammed Tahar and Olaru, Sorin and Niculescu, Silviu Iulian and Blanchini, Franco and Giordano, Giulia and Casagrande, Daniele and Miani, Stefano},
  issn         = {1367-5788},
  keyword      = {Discrete time-delay difference equations,Linear time-delay systems,Set invariance},
  language     = {eng},
  pages        = {13--23},
  publisher    = {Elsevier},
  series       = {Annual Reviews in Control},
  title        = {Guide on set invariance for delay difference equations},
  url          = {http://dx.doi.org/10.1016/j.arcontrol.2016.04.020},
  volume       = {41},
  year         = {2016},
}