Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

Schmid, Jochen; Zwart, Hans

doi:10.1051/cocv/2021051

Cited by 9 publications

(9 citation statements)

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“…Also, weak input-to-state stability can be established for a rather large class of semilinear systems (both in the case of inputs entering in the domain and in the case of inputs entering at the boundary of the domain on which the partial differential equation describing the system lives). See [16], [17], [19]. It is therefore natural to study the property of weak input-to-state stabilityand especially its relation to other common stability properties -from a general point of view.…”

Section: Introductionmentioning

confidence: 99%

Weak input-to-state stability: characterizations and counterexamples

Schmid

2019

Math. Control Signals Syst.

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We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also answer, by means of suitable counterexamples, two open questions concerning weak input-to-state stability (and its relation to other common stability concepts) raised in the aforementioned paper.

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Section: Introductionmentioning

confidence: 99%

Weak input-to-state stability: characterizations and counterexamples

Schmid

2019

Math. Control Signals Syst.

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“…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)…”

Section: Introductionmentioning

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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“…to attractors of semilinear systems [26], but that result is a theoretical characterization of input-to-state practical stability in terms of rather abstract conditions which are probably not easily verifiable in practice. All input-to-state stability results for concrete pde systems -like those from [12], [16], [17], [18], [22], [23], [25], [27], [28], [35], [37], [40], [38], [44], [45], [46] -however, establish input-to-state stability only w.r.t. an equilibrium point θ of the respective undisturbed system and, without loss of generality, this equilibrium point is then assumed to be θ = 0.…”

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confidence: 99%