Necessary Stability Conditions for Delay Systems With Multiple Pointwise and Distributed Delays

Cuvas, Carlos; Mondié, Sabine

doi:10.1109/tac.2015.2487478

Cited by 23 publications

(10 citation statements)

References 19 publications

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“…By the presented lemmas, the stability criterion of exponential stability for equation (16) can be deduced as a direct consequence of the similar theorem for equation (17), proven in [17]. This theorem states that the exponential stability of equation (17) is equivalent to the inequality u(0) > | u(τ )|, τ ∈ (0, h].…”

Section: Lemmamentioning

confidence: 86%

“…We present a new exponential stability criterion for equation (16), which is indeed a particular case of conditions (15). Here, the delay Lyapunov matrix is mentioned as the delay Lyapunov function, and denoted by u(τ ).…”

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…The exact stability region for equation (16) in the case |d| < 1 has been presented partially in [27] and [28]. This region is determined by:…”

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…Proof: If equation (16) does not satisfy the Lyapunov condition, by Theorem 1, there exists a complex number s0 such that…”

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…If equation (16) satisfies the Lyapunov condition, the delay Lyapunov functions for equations (16) and (17) satisfy the equality…”

Section: Lemmamentioning

confidence: 99%

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Necessary Stability Conditions for Neutral Type Systems With a Single Delay

Gomez

Egorov

Mondié

2017

IEEE Trans. Automat. Contr.

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Necessary conditions for the exponential stability of timedelay systems of neutral type, and a new stability criterion for the scalar case are presented in this contribution. They are obtained in the framework of Lyapunov-Krasovskii functionals of complete type. The particularity of these conditions is that they depend exclusively on the delay Lyapunov matrix.Index Terms-Neutral type time-delay systems, stability analysis, Lyapunov-Krasovskii functionals, delay Lyapunov matrix. I. INTRODUCTIONNeutral type delay systems are characterized by the fact that the state rate of change depends not only on delayed states, but also on past state rates of change. This class of systems has proven to be suitable for modeling electric transmission lines [1], population dynamics [2], controlled systems with delay in the communication channel [3], drilling operation [4] (see [5]-[7] for more examples). The stability theory of neutral type delay systems via Lyapunov-Krasovskii functionals with prescribed derivative was first approached by Castelan and Infante [8]. Functionals of complete type were later developed for the one delay case by Rodriguez et. al [9]and by Kharitonov [10], and extended to the multiple delays scalar case by Velázquez and Kharitonov [11]. The comprehensive framework of these functionals is exposed in the recent monograph by Kharitonov [12]. This approach has been applied to robust stability analysis [9], exponential estimates [10], determination of critical frequencies and parameters [13], and recently, to the predictor control scheme for neutral type state delay systems with input delay [14].An obvious question is raised by the recent necessary stability conditions for retarded delay systems with pointwise delays and with distributed delays presented by Egorov and Mondié [15] and by Cuvas and Mondié [16], respectively: Is it possible to find stability conditions for neutral time-delay systems that depend only on the delay Lyapunov matrix? An affirmative response to this question is given in this paper. For the multivariable case, necessary stability conditions are provided, while for the scalar neutral type equation a new stability criterion similar to the one presented in [17] is proven.These conditions, provide a new approach for studying the stability of neutral time-delay systems, that complement the frequency domain techniques [13], [18]-[20], as well as the classical LMI sufficient conditions obtained from proposed Lyapunov-Krasovskii functionals (see [21]-[23], among others).The result is achieved by using the methodology introduced in [15] and [16] which relies on an appropriate choice for the class of piecewise initial functions depending on the fundamental matrix, and new properties of the Lyapunov matrix. Because of the jump discontinuities in the fundamental matrix, the result concerning the computation of the functional [12], which was originally proved for differentiable initial functions, had to be relaxed. The new formulation is given in Theorem 2, while the lengthy and technical proof is

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Section: Lemmamentioning

confidence: 86%

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…The exact stability region for equation (16) in the case |d| < 1 has been presented partially in [27] and [28]. This region is determined by:…”

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…Proof: If equation (16) does not satisfy the Lyapunov condition, by Theorem 1, there exists a complex number s0 such that…”

Section: B New Stability Criterion For the Scalar Equationmentioning

confidence: 99%

“…If equation (16) satisfies the Lyapunov condition, the delay Lyapunov functions for equations (16) and (17) satisfy the equality…”

Section: Lemmamentioning

confidence: 99%

See 3 more Smart Citations

Necessary Stability Conditions for Neutral Type Systems With a Single Delay

Gomez

Egorov

Mondié

2017

IEEE Trans. Automat. Contr.

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Necessary exponential stability conditions for linear periodic time‐delay systems

Gomez

Ochoa

Mondié

2016

Intl J Robust & Nonlinear

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Vladimir Kharitonov: Achievements in research

Egorov

2022

Intl J Robust & Nonlinear

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A survey of the main research results of Professor Kharitonov is provided. The following topics are considered: stability analysis of interval polynomials, quasi‐polynomials, and multivariate polynomials; Kalman's hypothesis on the generalization of the Lyapunov matrix equation; critical delay values, complete type functionals and prediction‐based control for time‐delay systems. The fundamentals of Kharitonov's work and the influence of his results on other researchers are discussed.

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Necessary Stability Conditions for Delay Systems With Multiple Pointwise and Distributed Delays

Cited by 23 publications

References 19 publications

Necessary Stability Conditions for Neutral Type Systems With a Single Delay

Necessary Stability Conditions for Neutral Type Systems With a Single Delay

Necessary exponential stability conditions for linear periodic time‐delay systems

Vladimir Kharitonov: Achievements in research

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