Output-feedback adaptive control of a wave PDE with boundary anti-damping

Bresch‐Pietri, Delphine; Krstić, Miroslav

doi:10.1016/j.automatica.2014.02.040

Cited by 103 publications

(76 citation statements)

References 20 publications

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“…In [12], a novel "backstepping transformation" for boundary control of system (4.1) is proposed for known q > 0. This result is thereafter extended to the case of q being unknown in [8] and [1] where an adaptive control approach is proposed by employing parameter projection. However, the output in these works contains not only the boundary position w(0, t) but also the velocity w t (0, t).…”

Section: Application To Anti-stable Wave Equationmentioning

confidence: 97%

“…In other words, the dynamic feedback (4.2) is equivalent to a direct proportional delayed output feedback. Compared with the exiting controller for anti-stable wave equation in [1], [8], and [12], (4.20) is much simpler. Once again, we allow a time delay for measured output in the feedback.…”

Section: Application To Anti-stable Wave Equationmentioning

confidence: 99%

“…We always suppose that q = 1 because the situation q = 1 is exceptional in the sense that the real parts of eigenvalues approach positive infinity. Similar to [1], we suppose that q is unknown positive number but satisfies the following assumption:…”

Section: Application To Anti-stable Wave Equationmentioning

confidence: 99%

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Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

Feng

Guo

2017

IEEE Trans. Automat. Contr.

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In this paper, we consider finite-time and exponential stabilization for one-dimensional wave equations by boundary displacement measurement only. We limit ourselves in the energy state space where the usual observability inequality is not valid anymore. However, in the optimal state space, the boundary displacement is indeed exactly observable. This motivates us to design observer via displacement output only for these systems. We first discuss a simple case as a motivation. An observer is designed and an output feedback control is then synthesized to make system finite-time stable in energy space. In this same spirit, we consider the same problem for an unstable wave equation and the finite-time stability is also achieved by displacement output feedback. Finally, we consider an antistable wave equation. A direct delayed output feedback control can achieve exponential stability with arbitrary decay rate. Simulation results are presented to validate the theoretical conclusions.

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Section: Application To Anti-stable Wave Equationmentioning

confidence: 97%

Section: Application To Anti-stable Wave Equationmentioning

confidence: 99%

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Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

Feng

Guo

2017

IEEE Trans. Automat. Contr.

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“…Here, we solve these problems within the general setting of Equations (1)- (5). The system is mapped to an exponentially stable target system using a Volterra transformation.…”

Section: Introductionmentioning

confidence: 99%

“…Several contributions have taken advantage of this simplification and designed stabilizing feedback laws, e.g. relying on neutral system approaches [19], flatness approaches [18] or predictor-based approaches [3,5]. However, no existing solution simultaneously allows stabilization…”

Section: Introductionmentioning

confidence: 99%

Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems

et al. 2018

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We solve the problem of stabilizing a linear ODE having a system of a linearly coupled hyperbolic PDEs in the actuating and sensing paths. The system is exponentially stabilized by mapping it to a target system with a cascade structure using a Volterra transformation.Key words: Stabilization, Distributed Parameter Systems IntroductionThe interest for coupled Ordinary Differential EquationsPartial Differential Equations (ODE-PDE) systems has first emerged when considering delays in the actuating and sensing paths of ODE. Delays can be seen as first-order hyperbolic PDEs. There are many approaches to deal with input or measurement delays, usually divided into two categories: memoryless controllers, which extend standard control techniques without explicitly accounting for the delay in the control design [16,25,9]; and prediction-based controllers aiming at explicitly compensating the delay [20,4,2].The use of Lyapunov and backstepping methods enabled dealing with more involved PDEs in the actuating and sensing paths. In [13], an output feedback control law is derived for an ODE having a heat equation in the actuating and sensing paths. The coupled PDE-ODE system is stabilized using an observer-controller structure relying on a backstepping approach. The same approach has been used to deal with ODEs coupled (rather than cascaded) with parabolic PDEs The first application of the backstepping approach to deal with hyperbolic PDE-ODE couplings is [14] where actuator and sensor delays are explicitly compensated. While this problem had already been tackled by, e.g., the Smith predictor [20], the reformulation of the delay as a linear ODE enabled numerous related problem to be tackled, most notably non-constant and uncertain delays [2,4]. In [22], the problem of stabilizing a multi-input ODE with distinct delays is tackled using a backstepping approach. In [6], an observer is designed for an ODE having a homodirectional 1 hyperbolic PDE in the sensing path, relying on a Lyapunov analysis requiring to solve Linear Matrix Inequalities (LMI). As will appear, the systems of [22] are particular cases of the system considered here, although the control design approaches are different.Here, we solve the problem of stabilizing an ODE with a system of first-order linear hyperbolic PDEs in the sensing 1 i.e. where all the states transport in the same direction

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Regulation‐triggered adaptive control of a hyperbolic PDE‐ODE model with boundary interconnections

Wang

Krstić

2021

Adaptive Control & Signal

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Summary We present a certainty equivalence‐based adaptive boundary control scheme with a regulation‐triggered batch least‐squares identifier, for a heterodirectional transport partial differential equation‐ordinary differential equation (PDE‐ODE) system where the transport speeds of both transport PDEs are unknown. We use a nominal controller which is fed piecewise‐constant parameter estimates from an event‐triggered parameter update law that applies a least‐squares estimator to data “batches” collected over time intervals between the triggers. A parameter update is triggered by an observed growth in the norm of the PDE state. The proposed triggering‐based adaptive control guarantees: (1) the absence of a Zeno phenomenon; (2) parameter estimates are convergent to the true values in finite time (from most initial conditions); (3) exponential regulation of the plant states to zero. The effectiveness of the proposed design is verified by a numerical example.

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Output-feedback adaptive control of a wave PDE with boundary anti-damping

Cited by 103 publications

References 20 publications

Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems

Regulation‐triggered adaptive control of a hyperbolic PDE‐ODE model with boundary interconnections

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