Lyapunov stability and Lyapunov functions of infinite dimensional systems

Baker, R.; Bergen, A.

doi:10.1109/tac.1969.1099229

Cited by 21 publications

(7 citation statements)

References 3 publications

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“…Although the use of Lyapunov functions in an infinite dimensional setting is not new, see for example [3], it is still an active research topic. Some interesting results for parabolic PDEs can be mentioned: [8], where a Lyapunov function is used to prove the existence of a global solution to the heat equation; [16], where a Lyapunov function is constructed for the heat equation with unknown destabilizing parameters (and subsequent control extensions [28] and [29]).…”

Section: Theoretical Contributionmentioning

confidence: 99%

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

Argomedo

Prieur

Witrant

et al. 2013

IEEE Trans. Automat. Contr.

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In this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as closely as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances.

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Section: Theoretical Contributionmentioning

confidence: 99%

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

Argomedo

Prieur

Witrant

et al. 2013

IEEE Trans. Automat. Contr.

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“…The use of Lyapunov functions to study the solutions or properties of infinite-dimensional systems is not new, see for instance [1], but it is still an active research topic. Other interesting results involving Lyapunov functions applied to parabolic equations can be found in [3], where a Lyapunov approach is used to prove the existence of a global solution to the heat equation; [10], where Lyapunov functions are used to analyze the regularity and well-posedness of Burgers' equation with a backstepping boundary control; [9], where a Lyapunov function is used to analyze the heat equation with unknown destabilizing parameters and its control extensions in [15] and [16].…”

Section: Introductionmentioning

confidence: 99%

D¹-Input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances

Argomedo

Witrant

Prieur

2012

2012 American Control Conference (ACC)

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D 1 -Input-to-state stability (D 1 ISS) of a diffusive equation with Dirichlet boundary conditions is shown, in the L 2 -norm, with respect to boundary disturbances. In particular, the spatially distributed diffusion coefficients are allowed to be time-varying within a given set, without imposing any constraints on their rate of variation. Based on a strict Lyapunov function for the system with homogeneous boundary conditions, D 1 ISS inequalities are derived for the disturbed equation. A heuristic method used to numerically compute weighting functions is discussed. Numerical simulations are presented and discussed to illustrate the implementation of the theoretical results.

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“…This result finds its roots in the dissipativity framework introduced by Willems in his seminal paper [30]. It should be noted that Willems mentioned in [29] that his work extended to dynamical systems a previous result of Baker and Bergen [1] in the context of linear infinite dimensional systems.…”

Section: Lyapunov Stability Analysismentioning

confidence: 77%

Boundary control of hyperbolic conservation laws using a frequency domain approach

Litrico

Fromion

2009

Automatica

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The paper uses a frequency domain method for boundary control of hyperbolic conservation laws. We show that the transfer function of the hyperbolic system belongs to the Callier-Desoer algebra, which opens the way of sound results, and in particular to the existence of necessary and sufficient condition for the closed loop stability and the use of Nyquist type test. We examine the link between input-output stability and exponential stability of the state. Specific results are then derived for the case of proportional boundary controllers. The results are illustrated in the case of boundary control of open channel flow.

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Lyapunov stability and Lyapunov functions of infinite dimensional systems

Cited by 21 publications

References 3 publications

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

D¹-Input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances

Boundary control of hyperbolic conservation laws using a frequency domain approach

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Lyapunov stability and Lyapunov functions of infinite dimensional systems

Cited by 21 publications

References 3 publications

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

D1-Input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances

Boundary control of hyperbolic conservation laws using a frequency domain approach

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D¹-Input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances