An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators

Abstract

The aim of this paper is to provide an intrinsic Hamiltonian jormulation of the equations of motion ofnetwork models of non-resistive physical systems. A recently developed extension of the classical Hamiltonian equations of motion considers systems with state space given by Poisson manifolds endowed with degenerate Poisson structures, examples of which naturally appear in the reduction of' systems with symmetry. The link with network representations of non-resistive physical systems is established using the generalized bond graph formalism which has the essential feature of symmetrizing all the energetic network elements into a single class and introducing a coupling unit gyrator. The relation between the Hamiltonian formalism and network dynamics is then investtgated throqh the representation of the invariants of the system, either captured in the degeneracy of the Poisson structure or in the topological constraints at the ports of the gyrative type network structure. This provides a Hamiltonian formulation of dimension equal to the order of the physical system, in particular, for odd dimensional systems. A striking example is the direct Hamiltonian formulation of electrical LC networks.

4 Compositional Modelling of Distributed-Parameter Systems

“…Let us nowdefineaninterconnection structure fort his system in thes ense of network [13] [4] or port-basedm odelling [ 23] [35]. Definet he vector of flow variables to be the time variation of thes tate and denote it by:…”

Discrete port-Hamiltonian systems

“…[6,1]. Physical systems can be interconnected to each other by means of power ports, essentially meaning that the interconnection is facilitated by an exchange of power between the systems.…”

“…[6,1,14], it has been shown how port-based network modeling of complex lumped-parameter physical systems naturally leads to a generalized Hamiltonian formulation of the dynamics. In fact, the Hamiltonian is given by the total energy of the energy-storing elements in the system, while the geometric structure, defining together with the Hamiltonian the dynamics of the system, is given by the power-conserving interconnection structure of the system, and is called a Dirac structure.…”

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

“…Network modeling of lumped-parameter physical systems [7] with independent storage elements leads to the following class of dynamical systems, called port controlled Hamiltonian systems with dissipation [6], [14], [15], [ where x 2 X, an n-dimensional manifold, u; y 2 R m . The state variables x = [x1; 111; xn] > are the energy variables (i.e., the variables by which the energy of the system is defined), the smooth function H(x1; 11 1; xn): X !…”