An Overview on Irreversible Port-Hamiltonian Systems

Abstract

A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible thermodynamic systems for finite and infinite dimensional systems. This is achieved by including, in an explicit manner, the coupling between irreversible mechanical and thermal phenomena with the thermal domain as an energy-preserving and entropy-increasing operator. Similarly to Hamiltonian systems, this operator is skew-symmetric, guaranteeing energy conservation. To distinguish from Hamiltonian systems, the operator depends on co-state variables and is, hence, a nonlinear-function in the gradient of the total energy. This is what allows encoding the second law as a structural property of irreversible port-Hamiltonian systems. The formalism encompasses coupled thermo-mechanical systems and purely reversible or conservative systems as a particular case. This appears clearly when splitting the state space such that the entropy coordinate is separated from other state variables. Several examples have been used to illustrate the formalism, both for finite and infinite dimensional systems, and a discussion on ongoing and future studies is provided.

Geometric Modeling for Control of Thermodynamic Systems

“…The Hamiltonian formulation of the dynamics of thermodynamic systems is, however, much more elusive. This has already studied and elaborated upon in, e.g., [ 16 , 24 , 41 , 46 ]. The present formulation emphasizes the factorization ( 15 ) of the irreversible entropy production.…”

“…(Chemical reaction network) . A third, more involved, example of irreversible thermodynamics are the dynamics of chemical reaction networks [ 34 , 41 ]. Consider an isolated (no incoming or outgoing chemical species, and no external heat flow) reaction network, with m chemical species and r reactions.…”

“…(Gas-piston-damper system continued) . The dynamics of the gas-piston-damper system ( 22 ) can be written into the quasi-Hamiltonian form as (see also [ 41 ]) with as the velocity of the piston and the temperature. The thermodynamic flow and force are and , respectively.…”

“…This caused by the fact that, in this example, we have two entropies, and , corresponding to the two compartments (and not a total entropy as in the previous two examples). In fact, the quasi-Hamiltonian formulation of the heat exchanger is given as (see [ 41 ]) since , and the heat flow from compartment 1 to 2 is given by . Here, we recognize as the thermodynamic force .…”

“…A further structured form of quasi-Hamiltonian modeling of irreversible thermodynamic systems, called irreversible port-Hamiltonian systems , was introduced in [ 24 ]; see [ 41 , 46 , 47 ] for more developments and references.…”