Systems theory of interconnected port contact systems

Port-based network modeling of a large class of complex physical systems leads to dynamical systems known as port-Hamiltonian systems. The key ingredient of any port-Hamiltonian system is a power-conserving interconnection structure (mathematically formalized by the geometric notion of a Dirac structure) linking the pairs of conjugate port variables of the various ports corresponding to energy storage (defined by a Hamiltonian function depending on energy variables), resistive effects, external interaction, etc. The interconnection of port-Hamiltonian systems defines a new port-Hamiltonian system with Dirac structure determined by the Dirac structures of the constituent parts. For thermodynamic systems this framework needs modification by extending the space of energy variables, as used for port-Hamiltonian systems, into a space of energy and co-energy variables together with an additional coordinate needed for the formulation of the energy. Geometrically this extended space is formalized as a contact manifold. The thermodynamic properties of the system are given by a Legendre submanifold of the contact manifold. Furthermore a contact Hamiltonian is defined, related to the internal power-conserving interconnection structure, whose resulting dynamics leaves invariant the Legendre submanifold. Finally, interaction contact Hamiltonians are defined together with port-conjugated pairs of input and output variables modeling the interaction of the system with its environment. Interconnection of such thermodynamic systems is shown to lead to a thermodynamic system with the same structure.