On the port-Hamiltonian representation of systems described by partial differential equations

We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissipation mapping and the in/output mapping. In contrast to the well-known representation on the basis of the underlying Stokes-Dirac structure our approach is not necessarily based on using energy-variables which leads to a different port-Hamiltonian representation of the analyzed partial differential equations.Comment: A definitive version has been published in ifac-papersonline.ne

Port Hamiltonian formulation of infinite dimensional systems I. Modeling

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

Multi-variable port Hamiltonian model of piezoelectric material

In this paper, the dynamics of a piezoelectric material is presented within the new framework of multi-variable distributed port Hamiltonian systems. This class of infinite dimensional system is quite general, thus allowing the description of several physical phenomena, such as heat conduction, elasticity, electromagnetism and, of course, piezoelectricity. 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, the dynamics of the system results from the interconnection of a proper set of elements, each of them characterized by a particular energetic behavior, while the interaction with the environment is described in terms of mechanical and electrical boundary ports

Trajectory Tracking for Discrete-Time Port-Hamiltonian Systems

This letter presents a regulator for nonlinear, discrete-time port-Hamiltonian systems that lets the state track a reference signal. Similarly to continuous-time approaches, the synthesis is based on the mapping via state-feedback of the open-loop error system to a target one in port-Hamiltonian form, and with an asymptotically stable origin that corresponds to the perfect tracking condition. The procedure is formally described by a matching equation that, in continuous-time, turns out to be a nonlinear partial differential equation (PDE). This is not the case for sampled-data systems, so an algebraic approach is proposed. The solution is employed to construct a dynamical regulator that performs an “approximated” mapping. The stability analysis relies on Lyapunov arguments

Bridging the gap between passivity and transparency

In this paper a structure will be given which in a remarkably simple way offers a solution to the implementation of different telemanipulation schemes for discrete time varying delays by preserving passivity and allowing the highest trans- parency possible. This is achieved by splitting the communication channel in two separate ones, one for the energy balance which will ensure passivity and one for the haptic information between master and slave and which will address transparency. The authors believe that this structure is the most general up to date which preserves passivity under discrete time varying delays allowing different control schemes to address transparency

Stabilization of Unstable Distributed Port-Hamiltonian Systems in Scattering Form

In this letter, we consider the exponential stabilization of a distributed parameter port-Hamiltonian system interconnected with an unstable finite-dimensional linear system at its free end and control input at the opposite one. The infinite-dimensional system can also have in-domain anti-damping. The control design passes through the definition of a finite-dimensional linear system that “embeds” the response of the distributed parameter model, and that can be stabilized by acting on the available control input. The conditions that link the exponential stability of the latter system with the exponential stability of the original one are obtained thanks to a Lyapunov analysis. Simulations are presented to show the pros and cons of the proposed synthesis methodology