The Stabilization of Multi-Dimensional Wave Equation with Boundary Control Matched Disturbance

Guo, Bao‐Zhu; Zhou, Hua‐Cheng; Yao, Cui-Zhen

doi:10.3182/20140824-6-za-1003.00104

Cited by 3 publications

(4 citation statements)

References 5 publications

(7 reference statements)

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“…Theorem 1. Consider the closed-loop system formed of (1), (4), (6), (17), and (18). Assume that Assumptions 1 to 5 are satisfied, then for any x(0) ∈ R n , (0) ∈ R n ,…”

Section: Control Design and Stability Analysismentioning

confidence: 99%

“…In , the convergence of nonlinear ESO (NLESO) and ADRC (NLADRC) for nonlinear time varying uncertain systems was first considered, which show that the ESO estimation errors and the closed‐loop can converge to a neighbourhood of zero in finite time. In a pair of recent papers , ADRC was applied to stabilization for one‐dimensional and multi‐dimensional wave equations. In , a modified ADRC combined with a projected gradient estimator was proposed for a class of uncertain nonlinear systems when there is no prior information about the uncertain dynamics in the input channel.…”

Section: Introductionmentioning

confidence: 99%

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On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

Wang

Ran

Dong

2017

Asian Journal of Control

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This paper designs the active disturbance rejection control (ADRC) to achieve finite-time stabilization for a class of uncertain nonlinear systems. The proposed control incorporates both an extended state observer (ESO) as well as an adaptive sliding mode controller. The ESO is utilized to estimate the full system states and the total uncertainties, and the adaptive strategy is incorporated to deal with the estimation errors. It is proved that, with the application of the proposed control law, semi-global finite-time stabilization can be achieved. Effectiveness of the proposed method is illustrated with a numerical example.

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“…Theorem 1. Consider the closed-loop system formed of (1), (4), (6), (17), and (18). Assume that Assumptions 1 to 5 are satisfied, then for any x(0) ∈ R n , (0) ∈ R n ,…”

Section: Control Design and Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

Wang

Ran

Dong

2017

Asian Journal of Control

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“…ADRC was applied first to stabilization of a one-dimensional anti-stable wave equation subject to boundary disturbance in [17] and then to stabilization of a one-dimensional anti-stable wave equation subject to general control matched disturbance in [18]. The application of this approach has been extended to multi-dimensional systems such as wave equation [19] and Kirchhoff plate [20].…”

Section: Introductionmentioning

confidence: 99%

High-gain adaptive boundary stabilization for an axially moving string subject to unbounded boundary disturbance

Tikialine¹,

Kelleche

Tedjani³

2021

AUCMCS

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In this paper, we are interested to stabilize an axially moving string subject to external disturbances. We assume that the disturbance may increases exponentially. We employ the active disturbance rejection control (ADRC) approach to estimate the disturbance. We design a disturbance observer that has time-varying gain so that the disturbance can be estimated with an exponential way. In order to stabilize the closed loop system, we use a control constructed through a high-gain adaptive velocity feedback. The existence and uniqueness of solution of the closed loop system is dealt with in the framework of the nonlinear semigroup theory by using a theorem due to Crandall-Liggett. It is shown that the formulated control is capable of stabilizing exponentially the closed loop system. The obtained results are also valid for the immobile case ($v=0$) and the present work improves certain previous results.

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“…However, there have not been yet LMI-based results for n-D wave equations, though the exponential stability of the n-D wave equations in bounded spatial domains has been studied in the literature via the direct Lyapunov method (see e.g. [18,9,1,6]).…”

Section: Introductionmentioning

confidence: 99%

New stability and exact observability conditions for semilinear wave equations

Fridman

Terushkin

2016

Automatica

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The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of Linear Matrix Inequalities (LMIs) [5]. In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations, as well as an upper bound on the minimum exact observability time in terms of LMIs. For 1-D wave equations with locally Lipschitz nonlinearities, we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. The efficiency of the results is illustrated by numerical examples.

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The Stabilization of Multi-Dimensional Wave Equation with Boundary Control Matched Disturbance

Cited by 3 publications

References 5 publications

On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

High-gain adaptive boundary stabilization for an axially moving string subject to unbounded boundary disturbance

New stability and exact observability conditions for semilinear wave equations

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