Composition of Dirac structures and control of Port-Hamiltonian systems

Key feature of Dirac structures (as opposed to Poisson or symplectic structures) is the fact that the standard composition of two Dirac structures is again a Dirac structure. In particular this implies that any power-conserving interconnection of port-Hamiltonian systems is a port-Hamiltonian system itself. This constitutes a fundamental property in the port-Hamiltonian approach to modeling, simulation and control of complex physical systems. Furthermore, the composed Dirac structure directly determines the algebraic constraints of the interconnected system, as well as its Casimir functions. Especially the Casimirs are of prime importance in the set-point regulation of port-Hamiltonian systems. It is therefore of importance to characterize the set of achievable Dirac structures when a given plant port-Hamiltonian system is intercon-nected with an arbitrary controller port-Hamiltonian system. The set of achievable Dirac structures in a restricted sense has been recently characterized in [1, 2]. Here we extend this theorem to the present situation occurring in the interconnection of a plant and controller Hamiltonian system. Furthermore, wegive an insightful procedure for the construction of the controller Dirac structure. This procedure works for the general case of (non-closed) Dirac structures on manifolds. In this way we also fully characterize the set of achievable Casimir functions of the interconnected ("closed-loop") system. This yields a fundamental limitation to the design of stabilizing controllers for underactuated mechanical systems by interconnection with a port-Hamiltonian controller