New insights in the geometry and interconnection of port-Hamiltonian systems

Abstract

We discuss a new geometric construction of port-Hamiltonian systems. Using this framework, we revisit the notion of interconnection providing it with an intrinsic description. Special emphasis on theoretical and applied examples is given throughout the paper to show the applicability and the novel contributions of the proposed framework.2 Dirac structures: forward and backward 3 Conventions and terminology. All objects in this paper are assumed to be smooth, unless otherwise stated. The maps between vector spaces and vector bundles are assumed to be linear. If π E : E → M is a vector bundle, a distribution D is an assignment of a subspace x → D x ⊂ E x for each x ∈ M . We will say that the distribution D is regular if dim(D x ) is independent of x. Regular (smooth) distributions are called subbundles. In general, our conventions agree with those on [Abraham and Marsden, 1978].Acknowledgements.

Port-Hamiltonian formulation of nonlinear electrical circuits

“…A key property of pH systems is that this class is closed under power-conserving interconnection. Different methods of how to design such interconnections are for example elucidated in [3,7,10,26]. The interconnection we will be using for the electrical circuits follows the ideas presented in [10].…”

Exergetic Port-Hamiltonian Systems: Modelling Basics

“…The essence of pH theory is to endow physical models with a geometric structure -called Dirac structure [29,30,31,32,33] -that expresses the exchange of power between the different system components and across the system boundary. Here, the Dirac structure is defined in terms of the skewsymmetric linear operator N : E → F which takes effort covectors into flow vectors.…”

Morse families and Dirac systems

“…The interconnection of simpler systems allows to describe complex systems as the theory of port-Hamiltonian systems shows[vdSJ14,vdSM95a]. Dirac structures have already been used in this context[CvdSB07,CvdSB03] (we also refer to[BLCGTAMdD18] for a recent geometric approach), but our formalism provides an intrinsic description to tackle the interconnection of Dirac systems by means of Lagrangian submanifolds and Morse families.2. To solve the generalized Dirac systems is usually challenging.…”