Practical Stability of Time-Delay Systems: LMI's Approach

Villafuerte, R.; Mondié, Sabine; Poznyak, Alexander S.

doi:10.3166/ejc.17.127-138

Cited by 18 publications

(25 citation statements)

References 8 publications

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“…In fact, in the real world, the problem of practical stability is more appropriate. Then, for practical purpose, practical stability seems desirable ( see [5] ) for systems without delays and [24], [10] and [14] for time-delay systems. In the general case, one can not directly measure the states of a system.…”

Section: Practical Exponential Stabilitymentioning

confidence: 99%

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Observer design and practical stability of nonlinear systems under unknown time‐delay

Echi

2019

Asian Journal of Control

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In the present paper, we study observer design and we establish some sufficient conditions for practical exponential stability for a class of time-delay nonlinear systems written in triangular form. In case of delay, the exponential convergence of the observer was confirmed. Based on the Lyapunov-Krasovskii functionals, the practical stability of the proposed observer is achieved. Finally, a physical model and simulation findings show the feasibility of the suggested strategy.Mathematics Subject Classification. 93C10, 93D15, 93D20.Time-delay systems are one of the basic mathematical models of real phenomena such as nuclear reactors, chemical engineering systems, biological systems [16], and population dynamics models [18]. The analysis of systems without delays is generally simple as compared to nonlinear systems under time delays. However, there are a number of problems relating to nonlinear observer for time delay system. In particular, a problem of theoretical and practical importance is the design of observer-based for time-delay systems. In literature, a separation principle, for nonlinear free-delay systems, using high gain observers is provided in [1] and [2].Much attention has been paid to solve such a problem and many observer designs for time-delay nonlinear systems approaches have been used. In [26], it is shown that observer design for a class of nonlinear time delay systems is solved by using linear matrix inequality. In [11], some sufficient conditions for practical uniform stability of a class of uncertain time-varying systems with a bounded time-varying state delay were provided using the Lyapunov stability theory.[4] and [9] show that state and output feedback controllers of time-delay systems, written in a triangular linear growth condition are reached under delay independent conditions and under delay dependent conditions respectively.The observer design problem for nonlinear systems satisfying a Lipschitz continuity condition has been a topic of numerous papers, such as for nonlinear free-delay systems [1,2,27,25], for nonlinear systems with unknown, time-varying [19,14,12]. A reduced-order observer design method is presented in [27] for a class of Lipschitz nonlinear continuous-time systems without time delays which extend the results in [25].However, in practice, dynamics, measurement, noises or disturbances often prevent the error signals from tending to the origin. Thus, the origin is not a point of equilibrium of the system. An additive term on the right-hand side of the nonlinear system is used to present the uncertainties systems. For this reason, the property is referred to as practical stability which is more suitable for nonlinear free-delay systems ( see [5,7]) and for nonlinear systems with time-delay ( see [10,14,19,24]). Under unknown, bounded timedelay, an observer design for a class of nonlinear system is presented in [19]. [10] concluded that a class of nonlinear time delay systems is conformed due to some assumptions and the time varying delay bounded the practical exponentia...

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Section: Practical Exponential Stabilitymentioning

confidence: 99%

“…The input and the states are bounded, which make assumption H1 holds. It is easy to check that system (30) satisfies Assumption A1 with γ1(ε) = 1 24 (1 + ε), γ2(ε) = 1 12 (1 + ε) and γ3(ε) = (1 + ε). So, λmax(P ) = 1.1101 and λmin(P ) = 0.1099.…”

Section: Numerical Examplementioning

confidence: 99%

Observer design and practical stability of nonlinear systems under unknown time‐delay

Echi

2019

Asian Journal of Control

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“…Some results about practical stability for time-delay systems are derived in [21], but unlike this previous work, now we consider non-linearities that satisfy so called quasi-Lipschitz a Correspondence to: cperez@ctrl.cinvestav.mx.…”

Section: Introductionmentioning

confidence: 98%

On the practical stability for a class of switched system

Pérez¹,

Poznyak²,

Azhmyakov³

2012

2012 9th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)

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This paper addresses a problem of robust control for a class of switched systems with timedelays. We deal with controllable nonlinear models that satisfy a quasi-Lipschitz condition and include bounded perturbations. The stabilization issue is carried out by two linear feedback controls: the static and the switching one. We also apply the extended version of the invariant ellipsoid method in combination with the Lyapunov-Krasovskii stability approach. Sufficient conditions of practical stability are derived in terms of Linear Matrix Inequalities (LMIs). The result is illustrated by a numerical example.

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“…Based on this method, the desired state of a system can be unstable only if it oscillates sufficiently near this state. Recently, there has been a significant development in the theory of practical stability [10][11][12][13][14][15]. Moreover, impulses and time delays exist in many processes of dynamic systems, for example, physics, chemical technology, population dynamics, and neural networks, and they may impact systems seriously [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Practical Stability of Impulsive Discrete Systems with Time Delays

Sun¹,

Liu

2014

Abstract and Applied Analysis

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The purpose of this paper is to investigate the practical stability problem for impulsive discrete systems with time delays. By using Lyapunov functions and the Razumikhin-type technique, some criteria which guarantee the practical stability and uniformly asymptotically practical stability of the addressed systems are provided. Finally, two examples are presented to illustrate the criteria.Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 954121, 10 pages http://dx.

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Practical Stability of Time-Delay Systems: LMI's Approach

Cited by 18 publications

References 8 publications

Observer design and practical stability of nonlinear systems under unknown time‐delay

Observer design and practical stability of nonlinear systems under unknown time‐delay

On the practical stability for a class of switched system

Practical Stability of Impulsive Discrete Systems with Time Delays

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