Instability conditions for linear time delay systems: a Lyapunov matrix function approach

Mondié, Sabine; Ochoa, Gilberto; Ochoa, B.M.

doi:10.1080/00207179.2011.620632

Cited by 27 publications

(14 citation statements)

References 12 publications

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“…As one can see in Figure 1, the region, obtained with the necessary stability conditions (9) and (10), coincides with the exact stability domain (12). Comparison of the region of instability shows that the instability conditions presented in [23] are outperformed.…”

Section: Examplementioning

confidence: 51%

“…Example Consider the equation

ẋ (t) MathClass-rel= ax (t) MathClass-bin+ bx (t MathClass-bin- 1) MathClass-punc.

It is known that the exact exponential stability domain in the parameter space is

\begin{matrix} leftalign \\ rightalign-odd (\{a + b < 0\} \cap \{a - b ⩽ 0\}) & align-even \cup (\{| a | + b < 0\} \cap & rightalign-label & align-label \\ rightalign-odd & align-even \cap \{a + b cos \sqrt{b^{2} - a^{2}} < 0\} \cap \{\sqrt{b^{2} - a^{2}} < π\}) . & rightalign-label (12) \end{matrix}

As one can see in Figure , the region, obtained with the necessary stability conditions and , coincides with the exact stability domain . Comparison of the region of instability shows that the instability conditions presented in are outperformed. It is worth mentioning that in the scalar case, as shown in , the condition is also sufficient. Example The following example was the subject of investigation by Kolmanovskii and Myshkis :

ẋ (t) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter 0 & arraycenter 1 <...> \end{matrix})

…”

Section: Illustrative Examplesmentioning

confidence: 99%

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Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

Egorov

Mondié

2013

Int. J. Robust Nonlinear Control

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SUMMARYExponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Examplementioning

confidence: 51%

“…Example Consider the equation

ẋ (t) MathClass-rel= ax (t) MathClass-bin+ bx (t MathClass-bin- 1) MathClass-punc.

It is known that the exact exponential stability domain in the parameter space is

\begin{matrix} leftalign \\ rightalign-odd (\{a + b < 0\} \cap \{a - b ⩽ 0\}) & align-even \cup (\{| a | + b < 0\} \cap & rightalign-label & align-label \\ rightalign-odd & align-even \cap \{a + b cos \sqrt{b^{2} - a^{2}} < 0\} \cap \{\sqrt{b^{2} - a^{2}} < π\}) . & rightalign-label (12) \end{matrix}

ẋ (t) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter 0 & arraycenter 1 <...> \end{matrix})

…”

Section: Illustrative Examplesmentioning

confidence: 99%

Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

Egorov

Mondié

2013

Int. J. Robust Nonlinear Control

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“…We now evaluate the Frobenius norm of (18) and apply the triangle inequality and the bounds (19) and (21), which shows that…”

Section: Preconditioningmentioning

confidence: 99%

“…It has been extensively used to characterize stability of delay differential equations, as one can explicitly construct a Lyapunov functional from U (t), where the solution is sometimes referred to as delay Lyapunov matrices. Sufficient conditions for stability are given in [13,21,20] and for neutral systems in [22], and conditions for instability in [19,4]. It has been used to provide bounds on the transient phase of delay-differential equations in the PhD thesis [23] and [14,15].…”

Section: Introductionmentioning

confidence: 99%

Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

Jarlebring

Poloni

2019

Applied Numerical Mathematics

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The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delayWe propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value U (τ /2) ∈ R n×n , i.e., the delay Lyapunov matrix at time τ /2. This linear matrix equation with n 2 unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n 2 × n 2 matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation M X +X T N = C, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n ≈ 1000, i.e., a linear system with n 2 ≈ 10 6 unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.

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“…For more general cases, the pursuit of necessary conditions is based on the substitution of a special initial function and on verifying that the functional satisfies a quadratic lower bound. The first results, obtained with rough constant initial functions, provided LMI type conditions depending on system matrices and the Lyapunov matrix function (Mondié, Ochoa, & Ochoa, 2011). In Egorov and Mondié (2013b), a choice of finite impulses gave a positivity conditions expressed in terms of system matrices and the matrix Lyapunov function; in addition, the extension to the multiple delay case was made possible by a reduction of the problem to the single delay case.…”

Section: Introductionmentioning

confidence: 98%

Necessary stability conditions for linear delay systems

Egorov

Mondié

2014

Automatica

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Instability conditions for linear time delay systems: a Lyapunov matrix function approach

Cited by 27 publications

References 12 publications

Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

Necessary stability conditions for linear delay systems

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