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

A Finite Dimensional Approximation of the shallow water Equations: The port-Hamiltonian Approach

We look into the problem of approximating a distributed parameter port-Hamiltonian system which is represented by a non-constant Stokes-Dirac structure. We here employ the idea where we use different finite elements for the approximation of geometric variables (forms) describing a infinite-dimensional system, to spatially discretize the system and obtain a finite-dimensional port-Hamiltonian system. In particular we take the example of a special case of the shallow water equations.

Port-Hamiltonian formulation of shallow water equations with Coriolis force and topography

We look into the problem of approximating the shallow water equations with Coriolis forces and topography. We model the system as an infinite-dimensional port-Hamiltonian system which is represented by a non-constant Stokes-Dirac structure. We here employ the idea of using different finite elements for the approximation of geometric variables (forms) describing a distributed parameter system, to spatially discretize the system and obtain a lumped parameter port-Hamiltonian system. The discretized model then captures the physical laws of its infinite-dimensional couterpart such as conservation of energy. We present some preliminary numerical results to justify our claims

Discrete-time port-Hamiltonian systems: A definition based on symplectic integration

We introduce a new definition of discrete-time port-Hamiltonian systems (PHS), which results from structure-preserving discretization of explicit PHS in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a discrete-time energy balance - expressed in terms of the discrete-time Dirac structure - extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order is consistent with the order of the numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.Comment: 12 pages. Preprint submitted to Systems & Control Letter

Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds

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

Tools for analysis of Dirac structures on Hilbert spaces

In this paper tools for the analysis of Dirac structures on Hilbert spaces are developed. Some properties are pointed out and two natural representations of Dirac structures on Hilbert spaces are presented. The theory is illustrated on the example of the ideal transmission line. \u

Structure Preserving Spatial Discretization of a 1-D Piezoelectric Timoshenko Beam

In this paper we show how to spatially discretize a distributed model of a piezoelectric beam representing the dynamics of an inflatable space reflector in port-Hamiltonian (pH) form. This model can then be used to design a controller for the shape of the inflatable structure. Inflatable structures have very nice properties, suitable for aerospace applications, e.g., inflatable space reflectors. With this technology we can build inflatable reflectors which are about 100 times bigger than solid ones. But to be useful for telescopes we have to achieve the desired surface accuracy by actively controlling the surface of the inflatable. The starting point of the control design is modeling for control. In this paper we choose lumped pH modeling since these models offer a clear structure for control design. To be able to design a finite dimensional controller for the infinite dimensional system we need a finite dimensional approximation of the infinite dimensional system which inherits all the structural properties of the infinite dimensional system, e.g., passivity. To achieve this goal first divide the one-dimensional (1-D) Timoshenko beam with piezoelectric actuation into several finite elements. Next we discretize the dynamics of the beam on the finite element in a structure preserving way. These finite elements are then interconnected in a physical motivated way. The interconnected system is then a finite dimensional approximation of the beam dynamics in the pH framework. Hence, it has inherited all the physical properties of the infinite dimensional system. To show the validity of the finite dimensional system we will present simulation results. In future work we will also focus on two-dimensional (2-D) models.

Port-Hamiltonian discretization for open channel flows

A finite-dimensional Port-Hamiltonian formulation for the dynamics of smooth open channel flows is presented. A numerical scheme based on this formulation is developed for both the linear and nonlinear shallow water equations. The scheme is verified against exact solutions and has the advantage of conservation of mass and energy to the discrete level