In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order N ∈ 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 points a, b of 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 order N = 1 and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary order N ∈ ℕ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to 0 of all solutions as t →∞. Applications are given to vibrating strings and beams.
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Keywords: Input-to-state stability, infinite-dimensional systems, port-Hamiltonian systems, nonlinear boundary control, actuator disturbances
@article{COCV_2021__27_1_A55_0,
author = {Schmid, Jochen and Zwart, Hans},
title = {Stabilization of {port-Hamiltonian} systems by nonlinear boundary control in the presence of disturbances},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021051},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021051/}
}
TY - JOUR AU - Schmid, Jochen AU - Zwart, Hans TI - Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021051/ DO - 10.1051/cocv/2021051 LA - en ID - COCV_2021__27_1_A55_0 ER -
%0 Journal Article %A Schmid, Jochen %A Zwart, Hans %T Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021051/ %R 10.1051/cocv/2021051 %G en %F COCV_2021__27_1_A55_0
Schmid, Jochen; Zwart, Hans. Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 53. doi: 10.1051/cocv/2021051
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