

Type 
Article 
Author(s) 
A. Macchelli, A. van der Schaft, C. Melchiorri 
Title 
Port Hamiltonian Formulation of Infinite Dimensional Systems.
Modeling and Boundary Control by Interconnection 
Editor 
submitted to Automatica 
Keywords 
PortHamiltonian systems, Infinitedimensional systems, Modeling,
Energyshaping 
Abstract 
In this paper, some new results on 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 multivariable case. The
resulting class of infinite dimensional systems is quite general,
thus allowing the description of several physical phenomena, such
as diffusion, 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
flowing through the boundary. Since this relation deeply relies on
the Stokes theorem, these structures are called
StokesDirac 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. Central
issue is the generalization of the definition of Casimir
function to the hybrid case, i.e. when the dynamical
system to be considered results from the power conserving
interconnection of an infinite and a finite dimensional part. The
stabilization of the onedimensional diffusion equation is
presented as simple example. 

Year 
2004 
