On the use of Dirac structures on Hilbert spaces in the synthesis of boundary control laws for port-Hamiltonian systems

Aim of this paper is to show how the Dirac structure properties can be exploited in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilisation of non-zero equilibria has been achieved by looking at, or generating, a set of structural invariants, namely Casimir functions, in closed-loop. Since this approach fails when an infinite amount of energy is required at the equilibrium (dissipation obstacle), this paper illustrates that the class of stabilising controllers is enlarged if the synthesis relies on the parametrisation of the dynamics provided by the image representation of the Dirac structure, able to show the effects of the boundary inputs on state evolution. The theoretical results are discussed with the help of a simple but illustrative example, i.e. a transmission line with RLC load in both serial and parallel configurations.