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.
Highlights
We consider linear port-Hamiltonian systems of arbitrary order N ∈ N on a bounded 1dimensional spatial domain (a, b)
Such systems are described by a linear partial differential equation of the form
With the above lemmas at hand, we can conclude the proof of our input-to-state stability result in two steps
Summary
We consider linear port-Hamiltonian systems of arbitrary order N ∈ N on a bounded 1dimensional spatial domain (a, b). We establish the input-to-state stability for the closed-loop system Sw.r.t. square-integrable disturbance inputs d. In [51], sensor – instead of actuator – disturbances are considered, that is, disturbances do occur in [51] but they corrupt the input of the controller instead of its output It is shown in [51] that for a port-Hamiltonian system of the special form (1.3) with negative definite P and a linear dynamic boundary controller, the resulting closed-loop system is uniformly input-to-state stable w.r.t. essentially bounded disturbance inputs d (meaning that the perturbed stability and attractivity estimates (1.5) and (1.6) are satisfied with the 2-norm d 2 replaced by the ∞-norm d ∞ of the disturbance).
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