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

Authors

Jochen Schmid, Hans Zwart

Abstract

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary orderN∈ ℕ on a bounded 1-dimensional spatial domain (a,b). In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary pointsa,bof the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed back into the system. What we show here is that the resulting nonlinear closed-loop system is input-to-state stable w.r.t. square-integrable disturbance inputs. In particular, we obtain uniform input-to-state stability for systems of orderN= 1 and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary orderN∈ ℕ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to 0 of all solutions ast→∞. Applications are given to vibrating strings and beams.

Citation

• Journal: ESAIM: Control, Optimisation and Calculus of Variations
• Year: 2021
• Volume: 27
• Issue:
• Pages: 53
• Publisher: EDP Sciences
• DOI: 10.1051/cocv/2021051

BibTeX

@article{Schmid_2021,
  title={{Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances}},
  volume={27},
  ISSN={1262-3377},
  DOI={10.1051/cocv/2021051},
  journal={ESAIM: Control, Optimisation and Calculus of Variations},
  publisher={EDP Sciences},
  author={Schmid, Jochen and Zwart, Hans},
  year={2021},
  pages={53}
}

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