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

Jochen Schmid*, Hans Zwart

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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Abstract

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order 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 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.

Original languageEnglish
Article number53
Number of pages37
JournalESAIM - Control, Optimisation and Calculus of Variations
Volume27
DOIs
Publication statusPublished - 4 Jun 2021

Keywords

  • Actuator disturbances
  • Infinite-dimensional systems
  • Input-to-state stability
  • Nonlinear boundary control
  • Port-Hamiltonian systems

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