Linear autonomous neutral functional differential equations

Henry, Daniel

doi:10.1016/0022-0396(74)90089-8

Cited by 164 publications

(97 citation statements)

References 8 publications

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“…As mentioned in the introductory section, it has been observed already by [14,2], see also [9,10,21], that the stability can be destroyed by even infinitesimally small changes in the delays. This was the main motivation for defining the concept of strong stability with respect to time delays.…”

Section: Problem Statementmentioning

confidence: 85%

“…For retarded systems, the spectral abscissa is nonsmooth but continuous in all parameters of the system, including time delays, see [27]. However, it results from [14,2,9,10], that, in general, it is not the case for neutral systems and kernel operators -the so-called associated difference equations, see also [20,21,22]. It is well-known that the spectral abscissa of the difference equation is not continuous in delays.…”

Section: Introductionmentioning

confidence: 99%

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Positive trigonometric polynomials for strong stability of difference equations

Henrion¹,

Vyhlídal²

2011

IFAC Proceedings Volumes

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We follow a polynomial approach to analyse strong stability of continuous-time linear difference equations with several delays Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

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Section: Problem Statementmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Positive trigonometric polynomials for strong stability of difference equations

Henrion¹,

Vyhlídal²

2011

IFAC Proceedings Volumes

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show abstract

“…[11,13,16,24], which carries over to (2.6). As a consequence, we are from a practical point of view led to the smallest upper bound on the real parts of the characteristic roots, which is 'insensitive' to small delay changes:…”

Section: Difference Equationmentioning

confidence: 99%

Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

Michiels¹,

Vyhlídal²,

Zı́tek³

et al. 2009

SIAM J. Control Optim.

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Abstract. The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.Key words. neutral system, strong stability, spectral theory Notations.C set of complex numbers [27] in mechanical engineering. Equations of neutral type also arise in boundary controlled hyperbolic partial differential equations subjected to small feedback delays [22,6] and in implementation schemes of predictive controllers for time-delay systems [7,25]. In this paper we discuss stability properties of the linear neutral equatioṅ

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“…For retarded systems, the spectral abscissa is nonsmooth but continuous in all parameters of the system, including time delays, see [24]. However, it results from [12,2,7,8], that, in general, it is not the case for neutral systems and kernel operators -the so-called associated difference equation, see also [17,18,19]. It is well-known that the spectral abscissa of the difference equation is not continuous in delays.…”

Section: Introductionmentioning

confidence: 99%

Positive trigonometric polynomials for strong stability of difference equations

Henrion

Vyhlídal

2012

Automatica

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We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

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Linear autonomous neutral functional differential equations

Cited by 164 publications

References 8 publications

Positive trigonometric polynomials for strong stability of difference equations

Positive trigonometric polynomials for strong stability of difference equations

Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

Positive trigonometric polynomials for strong stability of difference equations

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