Port Hamiltonian formulation of infinite dimensional systems I. Modeling

Abstract

Abstract-In this paper, some new results concerning the modeling 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 multivariable 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.

Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection

“…Consider the following multi-variable distributed port Hamiltonian system with spatial domain Z ⊂ R d (closed and compact), [3]:…”

“…Both X either W are spaces of vector value smooth functions of proper dimension. It is possible to prove that the following energy balance relation holds, [3]:…”

“…where the integral over ∂Z represent the power exchanged with the environment through the boundary and the B {J, R} is a constant operator depending on the differential operators J and R. See [3] for more details. Suppose that (13) has to be stabilized in the configuration x * ∈ X by means of the finite dimensional controller (2) that has to be interconnected to the system (13) in power conserving way.…”

“…The port Hamiltonian representation of a finite dimensional system [1], [2] has been recently generalized to the infinite dimensional case, [3], [4] by extending the notion of Dirac structure in order to cope with an infinite dimensional space of power variables. From the modeling point of view, the port Hamiltonian formulation of an infinite dimensional system with spatial domain Z provides a deep insight of the structure of the system, whose dynamics can be interpreted as the result of the interaction among (at least) two energy domains within Z and/or between the system and its environment through ∂Z.…”

Port-based modelling of manipulators with flexible links

“…The approach has been generalized to the distributed parameter case by extending the notion of interconnection structure to infinite dimensions, [3], [4]. The system is characterized by a spatial domain Z and the dynamics is the result of the exchange of power between different energy (sub-)domains in Z.…”