Distributed port-Hamiltonian formulation of infinite dimensional systems

In this paper, some new results concerning the modeling and control of distributed parameter systems in port Hamiltonian form are presented. The classical finite dimensional port Hamiltonian formulation of a dynamical system is generalized in order to cope with the distributed parameter and multi-variable case. The resulting class of infinite dimensional systems is quite general, thus allowing the description of several physical phenomena, such as heat conduction, piezoelectricity and elasticity. Furthermore, classical PDEs can be rewritten within this framework. The key point is the generalization of the notion of finite dimensional Dirac structure in order to deal with an infinite dimensional space of power variables. In this way, also in the distributed parameter case, the variation of total energy within the spatial domain of the system can be related to the power ow through the boundary. Since this relation deeply relies on the Stokes theorem, these structures are called Stokes-Dirac structures. As far as concerns the control problem, it seems natural that also finite dimensional control methodologies developed for finite dimensional port Hamiltonian systems can be extended in order to cope with infinite dimensional systems. In this paper, the control by interconnection and energy shaping methodology is applied to the stabilization problem of a distributed parameter system by means of a finite dimensional controller interconnected to its boundary. The key point is the generalization of the definition of Casimir function to the hybrid case, that is the dynamical system to be considered results from the power conserving interconnection of an infinite and a finite dimensional part. A simple application concerning the stabilization of the one-dimensional heat equation is presented