On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

Abstract

In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

Structure-preserving model reduction of complex physical systems

“…Remark 2.2: It can be shown [? ], [6], [5] that in the case of a finite-dimensional linear space F a Dirac structure D is equivalently characterized as a subspace such that e T f = < e | f >= 0 for all (f, e) ∈ D, together with dim D = dim F. The property < e | f >= 0 for all (f, e) ∈ D corresponds to power conservation. A port-Hamiltonian system is defined as follows.…”

“…We start with a Dirac structure D on the space of all flow and effort 1 For the definition of Dirac structures on manifolds we refer to e.g. [5]. …”

“…For example, linear resistive elements are given as f R = −Re R , R = R T > 0. Definition 2.3: Consider a Dirac structure (1), a Hamiltonian H : X → R with constitutive relations (2), and a resistive relation f R = −F (e R ) as in (5). Then the dynamics of the resulting port-Hamiltonian system is given as (−ẋ(t), ∂H ∂x (x(t)), −F (e R (t)), e R (t), f P (t), e P (t)) ∈ D (6) It follows [15], [6] from the power-conservation property of Dirac structures and (3) and (5) that…”

“…Indeed, any Dirac structure D ⊂ F x × E x × F R × E R × F P × E P can be represented by a linear set of equations involving all the effort and flow variables [13], [5], [15], [6] …”

“…For other useful representations of port-Hamiltonian systems, as well as for the way to transform one representation into another we refer to [5], [15], [6].…”