Interconnection of port-Hamiltonian systems and composition of Dirac structures

Port-based network modeling of physical systems leads to a model class of nonlinear systems known as port-Hamiltonian systems. Port-Hamiltonian systems are defined with respect to a geometric structure on the state space, called a Dirac structure. Interconnection of port-Hamiltonian systems results in another port-Hamiltonian system with Dirac structure defined by the composition of the Dirac structures of the subsystems. In this paper the composition of Dirac structures is being studied, both in power variables and in wave variables (scattering) representation. This latter case is shown to correspond to the Redheffer star product of unitary mappings. An equational representation of the composed Dirac structure is derived. Furthermore, the regularity of the composition is being studied. Necessary and sufficient conditions are given for the achievability of a Dirac structure arising from the standard feedback interconnection of a plant port-Hamiltonian system and a controller port-Hamiltonian system, and an explicit description of the class of achievable Casimir functions is derived

On interconnections of infinite-dimensional port-Hamiltonian systems

Network modeling of complex physical systems leads to a class of nonlinear systems called port-Hamiltonian systems, which are defined with respect to a Dirac structure (a geometric structure which formalizes the power-conserving interconnection structure of the system). A power conserving interconnection of Dirac structures is again a Dirac structure. In this paper we study interconnection properties of mixed finite and infinite dimensional port-Hamiltonian systems and show that this interconnection again defines a port-Hamiltonian system. We also investigate which closed-loop port-Hamiltonian systems can be achieved by power conserving interconnections of finite and infinite dimensional port-Hamiltonian systems. Finally we study these results with particular reference to the transmission line

Discrete port-Hamiltonian systems: mixed interconnections

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. The goal of this paper is to apply a previously developed discrete modeling technique to study the interconnection of continuous systems with discrete ones in such a way that passivity is preserved. Such a theory has potential applications, in the field of haptics, telemanipulation etc. It is shown that our discrete modeling theory can be used to formalize previously developed techniques for obtaining passive interconnections of continuous and discrete systems

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

Interconnection of Dirac Structures and Lagrange-Dirac Dynamical Systems

In the paper, we develop an idea of interconnection of Dirac structures and their associated Lagrange- Dirac dynamical systems. First, we briefly review the Lagrange-Dirac dynamical systems (namely, implicit Lagrangian systems) associated to induced Dirac structures. Second, we describe an idea of interconnection of Dirac structures; namely, we show how two distinct Lagrange-Dirac systems can be interconnected through a Dirac structure on the product of configuration spaces. Third, we also show the variational structure of the interconnected Lagrange-Dirac dynamical system in the context of the Hamilton-Pontryagin-d’Alembert principle. Finally, we demonstrate our theory by an example of mass-spring mechanical systems

Interconnection structures in physical systems: a mathematical formulation

The power-conserving structure of a physical system is known as interconnection structure. This paper presents a mathematical formulation of the interconnection structure in Hilbert spaces. Some properties of interconnection structures are pointed out and their three natural representations are treated. The developed theory is illustrated on two examples: electrical circuit and one-dimensional transmission lin

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

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

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