3,891 research outputs found
Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
We present the mixed Galerkin discretization of distributed parameter
port-Hamiltonian systems. On the prototypical example of hyperbolic systems of
two conservation laws in arbitrary spatial dimension, we derive the main
contributions: (i) A weak formulation of the underlying geometric
(Stokes-Dirac) structure with a segmented boundary according to the causality
of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac
structure by a finite-dimensional Dirac structure is realized using a mixed
Galerkin approach and power-preserving linear maps, which define minimal
discrete power variables. (iii) With a consistent approximation of the
Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models.
By the degrees of freedom in the power-preserving maps, the resulting family of
structure-preserving schemes allows for trade-offs between centered
approximations and upwinding. We illustrate the method on the example of
Whitney finite elements on a 2D simplicial triangulation and compare the
eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the
CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0
Port-Hamiltonian systems: an introductory survey
The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models
The wave equation as a port-Hamiltonian system and a finite-dimensional approximation
The problem of approximating a distributed parameter system with free boundary conditions is solved for the 2-dimensional wave equation. To this end we first model the wave equation as a distributed-parameter port-Hamiltonian system. Then we employ the idea that it is natural to use different finite elements for the approximation of di?erent geometric variables (forms) describing a distributed-parameter system, to spatially discretize the system and we show that we obtain a ?nite-dimensional port-Hamiltonian system, which also preserves the conservation laws
Geometric Scattering in Robotic Telemanipulation
In this paper, we study the interconnection of two robots, which are modeled as port-controlled Hamiltonian systems through a transmission line with time delay. There will be no analysis of the time delay, but its presence justifies the use of scattering variables to preserve passivity. The contributions of the paper are twofold: first, a geometrical, multidimensional, power-consistent exposition of telemanipulation of intrinsically passive controlled physical systems, with a clarification on impedance matching, and second, a system theoretic condition for the adaptation of a general port-controlled Hamiltonian system with dissipation (port-Hamiltonian system) to a transmission line
Geometry of Thermodynamic Processes
Since the 1970s contact geometry has been recognized as an appropriate
framework for the geometric formulation of the state properties of
thermodynamic systems, without, however, addressing the formulation of
non-equilibrium thermodynamic processes. In Balian & Valentin (2001) it was
shown how the symplectization of contact manifolds provides a new vantage
point; enabling, among others, to switch between the energy and entropy
representations of a thermodynamic system. In the present paper this is
continued towards the global geometric definition of a degenerate Riemannian
metric on the homogeneous Lagrangian submanifold describing the state
properties, which is overarching the locally defined metrics of Weinhold and
Ruppeiner. Next, a geometric formulation is given of non-equilibrium
thermodynamic processes, in terms of Hamiltonian dynamics defined by
Hamiltonian functions that are homogeneous of degree one in the co-extensive
variables and zero on the homogeneous Lagrangian submanifold. The
correspondence between objects in contact geometry and their homogeneous
counterparts in symplectic geometry, as already largely present in the
literature, appears to be elegant and effective. This culminates in the
definition of port-thermodynamic systems, and the formulation of
interconnection ports. The resulting geometric framework is illustrated on a
number of simple examples, already indicating its potential for analysis and
control.Comment: 23 page
SystÚmes hamiltoniens à ports de dimension infinie : réduction et propriétés spectrales.
National audienceDans ce papier on s'intĂ©resse aux propriĂ©tĂ©s spectrales des systĂšmes hamiltoniens Ă ports de dimension infinie. On montre que le spectre de tels systĂšmes peut ĂȘtre dĂ©duit du spectre associĂ© Ă une structure canonique, la structure de Stokes-Dirac, Ă l'aide de transformations gĂ©omĂ©triques simples telles que des homothĂ©ties, translations ou dilatations. On montrera en particulier que les spectres des systĂšmes d'Ă©quations d'onde et de diffusion peuvent ĂȘtre dĂ©duits du mĂȘme spectre canonique. Enfin on Ă©tudiera les propriĂ©tĂ©s de convergence d'une mĂ©thode de rĂ©duction structurĂ©e de type Ă©lĂ©ments finis mixtes sur une problĂšme de diffusion. Pour cela on montrera la convergence du spectre de la structure rĂ©duite vers le spectre de la structure canonique de Stokes-Dirac, puis on en dĂ©duira les propriĂ©tĂ©s de convergence finale par transformation gĂ©omĂ©trique. ABSTRACT. This paper deals with spectral properties of infinite dimensional Port Hamiltonian systems. It is shown that the spectra of these systems can be deduced from the spectrum associated to a canonical structure, the Stokes Dirac structure, thanks to geometric transformations such as homothety, translations or dilations. In particular, it is shown that the spectrum of the wave equation system and of the diffusion system can be deduced from the same canonical spectrum. Finally the spectral convergence properties of a mixed finite element based spatial reduction methods is studied of a diffusion system. To this purpose, the convergence of the spectrum of the reduced structure to the spectrum of the canonical Stoke Dirac structure is proved. It is obtained from the convergence properties of the canonical structure through geometric transformations
Port contact systems for irreversible thermodynamical systems
In this paper we propose a definition of control contact systems, generalizing input-output Hainiltonian systems, to cope with models arising from irreversible Thermodynamics. We exhibit a particular subclass of these systems, called conservative, that leaves invariant some Legendre submanifold (the geometric structures associated with thermodynamic properties). These systems, both energy-preserving and irreversible, are then used to analyze the loss-lessness of these systems with respect to different generating functions
Transparency in Port-Hamiltonian-Based Telemanipulation
After stability, transparency is the major issue in the design of a telemanipulation system. In this paper, we exploit the behavioral approach in order to provide an index for the evaluation of transparency in port-Hamiltonian-based teleoperators. Furthermore, we provide a transparency analysis of packet switching scattering-based communication channels
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