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

TL;DR

This paper develops a nonlinear boundary control method to stabilize port-Hamiltonian systems of any order on a bounded domain, ensuring stability and convergence despite disturbances, with applications to vibrating structures.

Contribution

It introduces a nonlinear boundary control approach for port-Hamiltonian systems that guarantees input-to-state stability under disturbances, extending to systems of arbitrary order.

Findings

Input-to-state stability achieved for first-order systems with specific controllers.

Weak input-to-state stability established for higher-order systems.

Solutions converge to zero as time approaches infinity.

Abstract

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ 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 $N \in \mathbb{N}$ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to $0$ of all solutions as $t \to \infty$. Applications are given to vibrating strings and beams.