Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

Ramírez, Héctor; Zwart, Hans; Gorrec, Yann Le

doi:10.1016/j.automatica.2017.07.045

Cited by 45 publications

(45 citation statements)

References 19 publications

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“…Linear static and dynamic output feedback has been studied by Villegas. Related results for nonlinear dynamic output feedback were obtained in Ramirez et al [27] Here we only mention the following results for impedance passive port-Hamiltonian systems. Augner [23] extended these results for nonlinear static and dynamic output feedback.…”

Section: Stabilizability By Output Feedbackmentioning

confidence: 56%

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An operator theoretic approach to infinite‐dimensional control systems

Jacob

Zwart

2018

GAMM-Mitteilungen

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In this survey we use an operator theoretic approach to infinite‐dimensional systems theory. As this research field is quite rich, we restrict ourselves to the class of infinite‐dimensional linear port‐Hamiltonian systems and we will focus on topics such as well‐posedness, stability and stabilizability. We combine the abstract operator theoretic approach with the more physical approach based on Hamiltonians. This enables us to derive easy verifiable conditions for well‐posedness and stability.

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Section: Stabilizability By Output Feedbackmentioning

confidence: 56%

“…Boundary control and observation systems satisfying (27) are called scattering passive. We now return to the port-Hamiltonian system (4).…”

Section: Port-hamiltonian Systems With Boundary Control and Observationmentioning

confidence: 99%

An operator theoretic approach to infinite‐dimensional control systems

Jacob

Zwart

2018

GAMM-Mitteilungen

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“…In [11] it is shown that the port-Hamiltonian system (16), (18) In this paper we have studied the notion of exact controllability for a class of linear port-Hamiltonian system on a one dimensional spacial domain with full boundary control and no internal damping. We showed that for this class wellposedness implies exact controllability.…”

Section: Example Of An Exactly Controllable Port-hamiltonian Systemmentioning

confidence: 99%

“…Here we follow the functional analytic point of view. This approach has been successfully used to derive simple verifiable conditions for well-posedness [13,10,9,1,11,14], stability [1,15] and stabilization [16,17,15,18] and robust regulation [19]. For example, the port-Hamiltonian system (1) Provided the port-Hamiltonian system (1) is well-posed, we aim to characterize exact controllability.…”

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confidence: 99%

On Exact Controllability of Infinite-Dimensional Linear Port-Hamiltonian Systems

Jacob

Kaiser

2019

IEEE Control Syst. Lett.

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Infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domain with full boundary control and without internal damping are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result shows that well-posed port-Hamiltonian systems, with state space L 2 ((0, 1); C n ) and input space C n , are exactly controllable.invertible matrix for a.e. ζ ∈ (0, 1). We suppose the technical assumption that S −1 , S, ∆ : [0, 1] → C n×n are continuously differentiable.Equation (1) describes a special class of port-Hamiltonian systems, which however is rich enough to cover in particular the wave equation, the transport equation and the Timoshenko beam equation, and also coupled beam and wave equations each with possibly damping on the boundary. For more information on this class of port-Hamiltonian systems we refer to the monograph [1] and the survey [2]. However, we note that here we always assume that there is no internal damping (the matrix P 0 is skew-adjoint) and that we have full boundary control ( W B is a full row rank n × 2n-matrix).Port-based network modeling of complex physical systems leads to port-Hamiltonian systems. For finite-dimensional systems there is by now a wellestablished theory [3,4,5]. The port-Hamiltonian approach has been extended to the infinite-dimensional situation by a geometric differential approach [6,7,8,9] and by a functional analytic approach [10,9,1,11,12,2]. Here we follow the functional analytic point of view. This approach has been successfully used to derive simple verifiable conditions for well-posedness [13,10,9,1,11,14], stability [1,15] and stabilization [16,17,15,18] and robust regulation [19]. For example, the port-Hamiltonian system (1) Provided the port-Hamiltonian system (1) is well-posed, we aim to characterize exact controllability. Exact controllability is a desirable property of a controlled partial differential equation and has been extensively studied, see for example [20,21,22]. We call the port-Hamiltonian system exactly controllable, if every state of the system can be reached in finite time with a suitable control input. Triggiani [23] showed that exact controllability does not hold for many hyperbolic partial differential equations. However, in this paper we prove, that the port-Hamiltonian system (1) is exactly controllable whenever it is well-posed.

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“…It has been initially introduced for finite dimensional nonlinear systems described by ODEs [13,14], and then generalized to infinite dimensional systems described by partial differential equations (PDEs) in more recent years [15][16][17]. This provides a standardized framework for control design, especially suited for energy based control strategies both for finite dimensional [18,19] and infinite dimensional systems [20,21]. The PH modelling allows to express a system as the composition of different elements that exchange energy in a power preserving way.…”

Section: Introductionmentioning

confidence: 99%

Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach

Mattioni

Gorrec

2020

Applied Mathematical Modelling

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This paper proposes a modular and control oriented model of a double flexiblelink manipulator stems from the modelling of a spatial flexible robot. The model consists of the power preserving interconnection between two infinite dimensional systems describing the beam's motion and deformation with a finite dimensional nonlinear system describing the dynamics of the actuated rotating joints. To derive the model, Timoshenko's assumptions are made for the flexible beams. Using Hamilton's principle, the dynamic equations of the system are derived and then written in the Port-Hamiltonian (PH) framework through a proper choice of the state variables. These so called energy variables allow to write the total energy as a quadratic form with respect to a state dependent energy matrix. The resulting model is shown to be a passive system, a convenient property for control design purposes.

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Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

Cited by 45 publications

References 19 publications

An operator theoretic approach to infinite‐dimensional control systems

An operator theoretic approach to infinite‐dimensional control systems

On Exact Controllability of Infinite-Dimensional Linear Port-Hamiltonian Systems

Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach

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