On the Feedback Control of the Wave Equation

Alli, Hasan; Singh, Tarunraj

doi:10.1006/jsvi.1999.2893

Cited by 21 publications

(8 citation statements)

References 6 publications

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“…Now we consider the question: Given a constant state ū = λ ∈ (0, ∞), is there a solution (q, ρ) of ( 1), (2) that corresponds to the constant velocity ū? For λ = 0 we obtain the constant solution of (1), (2) where q = 0. For λ > 0 there is a corresponding solution of travelling wave type (in particular the corresponding solution of (1), ( 2) is not stationary), namely…”

Section: Stationary States Of the Systemmentioning

confidence: 99%

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

Gugat¹,

Leugering²,

Wang³

2017

Mathematical Control &Amp; Related Fields

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For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H 2 -Lyapunov function and show that the boundary feedback constant can be chosen such that the H 2 -Lyapunov function and hence also the H 2 -norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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Section: Stationary States Of the Systemmentioning

confidence: 99%

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

Gugat¹,

Leugering²,

Wang³

2017

Mathematical Control &Amp; Related Fields

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“…As can be seen from Eqs (11), (12) or (13), the control law has eliminated the infinite number of vibratory poles, hence the name Absolute Vibration Suppression (AVS). The special structure of the AVS controller leads to several observations:…”

Section: The Absolute Vibration Suppression (Avs) Controllermentioning

confidence: 99%

“…The stability of the nominal closed loop system is evident from the denominator of the closed loop transfer function (12). A Root Locus analysis (see e.g.…”

Section: Stability and Robustnessmentioning

confidence: 99%

“…The delays in the transfer function correspond to the wave motion and from that point of view, the AVS performance is achieved by making the actuating end a sink for the returning wave. In [10] it is shown that the AVS controller bears similarities to wave based control methods [11][12][13]. The current paper is devoted to several stability and robustness issues associated with the AVS.…”

Section: Introductionmentioning

confidence: 99%

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Absolute Vibration Suppression (AVS) Control – Modeling, Implementation and Robustness

Halevi

Peled

2010

Shock and Vibration

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Absolute Vibration Suppression (AVS) is a control method for flexible structures governed by the wave equation. Such system may be a rotating shaft, a rod in tension or a crane for which the cable mass is not negligible. First an accurate, infinite dimension, transfer function, relating arbitrary actuation and measurement points, with general boundary conditions, is derived. The transfer function consists of time delays, due to the wave motion, and low order rational terms, which correspond to the reflection from the boundary. Thus the compact mathematical representation has a clear physical interpretation. Furthermore, it leads to the special structure of the AVS controller which is a collocated, rate to force feedback that completely eliminates the vibration. The paper investigates the properties of the AVS controller and its robustness to modeling errors, in particular small non-collocation.

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“…The objective is to determine a sound-speed function which damps the vibration of the system, and by considering the case where the sound speed takes only two values, a simple control law is proposed. Alli and Singh (2004) addressed the problem of design of collocated and non collocated controllers for a uniform bar whose dynamics are described by the wave equation, without structural damping. The root-locus technique is used to control the non-collocated system by means of a time delay controller.…”

Section: Introductionmentioning

confidence: 99%