Further remarks on asymptotic stability and set invariance for linear delay-difference equations

Stanković, Nikola; Olaru, Sorin; Niculescu, Silviu–Iulian

doi:10.1016/j.automatica.2014.05.019

Cited by 12 publications

(10 citation statements)

References 8 publications

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“…One can verify that the spectral radius of A 1 is not subunitary, ρ(A 1 ) = 1.2837 > 1. More than that, the necessary condition proposed in [21] is not verified. We can easily verify that the spectral radius of the sum ρ(A 1 + A 2 ) = 1.1422 > 1, and the set of generalized eigenvalues possesses four elements on the unit circle:…”

Section: Illustrative Examplementioning

confidence: 91%

On the Structure of Polyhedral Positive Invariant Sets with Respect to Delay Difference Equations

Laraba

Olaru

Niculescu

2017

Springer Proceedings in Mathematics &Amp; Statistics

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This chapter is dedicated to the study of the positive invariance of polyhedral sets with respect to dynamical systems described by discrete-time delay difference equations DDE. Set invariance in the original state space, also referred to as D-invariance, leads to conservative definitions due to its delay independent property. This limitation makes the D-invariant sets only applicable to a limited class of systems. However, there exists a degree of freedom in the state-space transformations which can enable the positive invariant set-characterizations. In this work we revisit the set factorizations and extend their use in order to establish flexible set-theoretic analysis tools. With linear algebra structural results, it is shown that similarity transformations are a key element in the characterization of low complexity invariant sets within the class of convex polyhedral candidates. In short, it is shown that we can construct, in a low dimensional state-space, an invariant set for a dynamical system governed by a delay difference equation. The basic idea which enables the construction is a simple change of coordinates for the DDE. The obtained D-invariant set exists in the new coordinates even if its existence necessary conditions are not fulfilled in the original state space. This proves that the D-invariance notion is dependent on the state-space representation of the dynamics. It is worth to recall as a term of comparison that the positive invariance for delay-free dynamics is independent of the state-space realization.

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Section: Illustrative Examplementioning

confidence: 91%

On the Structure of Polyhedral Positive Invariant Sets with Respect to Delay Difference Equations

Laraba

Olaru

Niculescu

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…3.1.3. Specific algebraic conditions for 2 delay dDDEs For dDDEs with two delay parameters, in order to decrease the conservativeness of the time-domain methods, [46] has used the frequency-domain framework. The D-invariance concept was studied, along with its relation to robust asymptotic stability, considered as a strong stability of dDDEs.…”

Section: Alternative Algebraic Conditionsmentioning

confidence: 99%

“…This condition covers, for the two delay case, the existing necessary conditions in the literature and proves to reduce considerably the gap with respect to sufficient conditions. In the present work, we provide an interesting example for which the condition in [46] is verified but the existing algorithms fail to construct a D-invariant set. As discussed in [47], from the stability point of view a pertinent analysis of D-invariance can be made in relationship with delay-independent stability.…”

mentioning

confidence: 96%

“…Particularly, as far as the construction of D-invariant sets is concerned, we can find a series of results in [44,45], which will be appropriately recalled in the present paper. Recently, [46] has proposed a computationally efficient numerical routine which is necessary to guarantee the existence of D-invariant sets for the delay difference equations with two delay parameters. This condition covers, for the two delay case, the existing necessary conditions in the literature and proves to reduce considerably the gap with respect to sufficient conditions.…”

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confidence: 99%

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Guide on set invariance for delay difference equations

Laraba

Olaru

Niculescu

et al. 2016

Annual Reviews in Control

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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 convex- ity (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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“…Therefore, the study of delay-difference equations has attracted considerable attention in recent years (see e.g. [11], [40].…”

Section: Introductionmentioning

confidence: 99%

Strong Stability Analysis of Linear Delay-Difference Equations With Multiple Time Delays

Song

Michiels

Zhou

et al. 2021

IEEE Trans. Automat. Contr.

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This paper is concerned with the strong stability problem of linear continuous-time delay-difference equations with multiple timedelays. A family of linear matrix inequalities (LMIs), indexed by a positive integer k, is derived to assess strong stability. A time-domain interpretation of the proposed LMI based condition is given in terms of a quadratic integral Lyapunov functional, which allows to reveal relations with an existing result. The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference equation is affine, which is instrumental to establishing a sufficient LMIcondition for robust strong stability of delay-difference equations with norm bounded uncertainty. A necessary and sufficient condition for robust strong stability is also given, in the form of a structured singular value characterization. Examples are given to illustrate the effectiveness of the presented results.

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Further remarks on asymptotic stability and set invariance for linear delay-difference equations

Cited by 12 publications

References 8 publications

On the Structure of Polyhedral Positive Invariant Sets with Respect to Delay Difference Equations

On the Structure of Polyhedral Positive Invariant Sets with Respect to Delay Difference Equations

Guide on set invariance for delay difference equations

Strong Stability Analysis of Linear Delay-Difference Equations With Multiple Time Delays

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