Dirac Structures on Hilbert Spaces and Boundary Control of Distributed Port-Hamiltonian Systems

The scope of this paper is to show how to exploit the properties of the Stokes–Dirac structure in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilisation of non-zero equilibria has been achieved by shaping the open-loop Hamiltonian function by interconnecting a finite-dimensional port-Hamiltonian controller to the boundary of the distributed parameter system. The procedure is based on the presence of structural invariants, namely Casimir functions, in closed-loop, but this approach fails when a non-zero power flow from the controller is required at the equilibrium (dissipation obstacle). This paper illustrates that the class of stabilising controllers can be enlarged by relying on the parametrisation of the system dynamics provided by the image representation of the Stokes–Dirac structure, that is able to show the effects of the boundary inputs on the state evolution in a simple and effective way.