On structure-preserving discretization of distributed parameter port-Hamiltonian systems
The underlying structure of port-Hamiltonian systems considered in this paper is a Stokes-Dirac structure [1] and as such is defined on a certain space of differential forms on a smooth finite-dimensional orientable manifold with a boundary. The Stokes-Dirac structure generalizes the framework of the Poisson and symplectic structures by providing a theoretical account that permits the inclusion of varying boundary variables in the boundary problem for partial differential equations. From an interconnection and control viewpoint, such a treatment of boundary conditions is essential for the incorporation of energy exchange through the boundary, since in many applications the interconnection with the environment takes place precisely through the boundary. For numerical integration, simulation and control synthesis, it is of paramount interest to have finite approximations that can be interconnected to one another or via the boundary coupled to the other systems, be they finite- or infinite-dimensional.