An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

Ebérard, Damien; Maschke, Bernhard; Schaft, van der Arjan

doi:10.1016/s0034-4877(07)00024-9

Cited by 150 publications

(164 citation statements)

References 24 publications

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“…Controlled contact systems are defined on the Thermodynamic Phase Space consisting of n + 1 extensive variables and n intensive variables and endowed with a contact structure associated with Gibbs' relation defining the thermodynamic properties of the system. On the Thermodynamic Phase Space, one may then define controlled contact systems which are the analogue of control Hamiltonian systems and have been introduced in [1] and further developed in [2,9,8]. After the introductory example of a 2-compartment heat exchange system we shall recall the precise definitions needed in this paper.…”

Section: On Controlled Contact Systemsmentioning

confidence: 99%

“…where I n denotes the identity matrix of order n. With this definition of contact vector fields, one may define control contact systems according to [1,2] which represent the dynamics of irreversible Thermodynamic systems [8] such as the Continuous Stirred Tank Reactor [17,18]. …”

Section: Proposition 22 [16]mentioning

confidence: 99%

“…In canonical coordinates this means that they do not depend on the x 0 coordinate associated with the Reeb vector field. For models of physical systems where the x 0 coordinate represents the generating potential of the thermodynamic system (the total energy or the total entropy), this is in general the case [2,8]. Under Assumption 2, the matching equation (12) is reduced to a relation on the feedback α and the closed-loop contact form θ d…”

Section: Feedback Equivalence With Respect To a Modified Contact Formmentioning

confidence: 99%

“…Control contact systems [1,2] have been introduced for the representation of controlled (or open) irreversible processes. They allow to represent simultaneously the energy conservation and the irreversible creation of entropy, the fundamental principles of Irreversible Thermodynamics.…”

Section: Introductionmentioning

confidence: 99%

“…Such systems are defined on the Thermodynamic Phase Space which is endowed with a contact structure (or a contact form) which is canonically associated with Gibbs' relation defining the Thermodynamic Equilibrium properties of physical systems [3,4,5,6]. Extending the work on reversible thermodynamical transformations in [7] to irreversible transformation of open thermodynamical systems leads to the definition of control contact systems [1,2] which are a strict extension of control Hamiltonian and portHamiltonian systems [8], and to the analysis of some of their dynamic properties [2,9].…”

Section: Introductionmentioning

confidence: 99%

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Feedback equivalence of input–output contact systems

Ramírez

Maschke

Sbárbaro

2013

Systems & Control Letters

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Control contact systems represent controlled (or open) irreversible processes which allow to represent simultaneously the energy conservation and the irreversible creation of entropy. Such systems systematically arise in models established in Chemical Engineering. The differential-geometric of these systems is a contact form in the same manner as the symplectic 2-form is associated to Hamiltonian models of mechanics. In this paper we study the feedback preserving the geometric structure of controlled contact systems and renders the closed-loop system again a contact system. It is shown that only a constant control preserves the canonical contact form, hence a state feedback necessarily changes the closed-loop contact form. For strict contact systems, arising from the modelling of thermodynamic systems, a class of state feedback that shapes the closed-loop contact form and contact Hamiltonian function is proposed. The state feedback is given by the composition of an arbitrary function and the control contact Hamiltonian function. The similarity with structure preserving feedback of input-output Hamiltonian systems leads to the definition of inputoutput contact systems and to the characterization of the feedback equivalence of input-output contact systems. An irreversible thermodynamic process, namely the heat exchanger, is used to illustrate the results.

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Section: On Controlled Contact Systemsmentioning

confidence: 99%

Section: Proposition 22 [16]mentioning

confidence: 99%

Section: Feedback Equivalence With Respect To a Modified Contact Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Feedback equivalence of input–output contact systems

Ramírez

Maschke

Sbárbaro

2013

Systems & Control Letters

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Integration of inventory control into the port‐Hamiltonian framework for dissipative stabilization of chemical reactors

Hoang

Ydstie

2022

Asian Journal of Control

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This work integrates passivity‐based inventory control with the port‐Hamiltonian framework to design stabilizing feedback laws for lumped parameter chemically reacting systems. A static state feedback law with admissible controls is developed using an inventory‐related storage function as a closed loop Hamiltonian function. The control gain matrix is derived from the interconnection and damping matrices of the resulting Hamiltonian representation. Consequently, the proposed controller globally and exponentially stabilizes the system at a desired set‐point (including the open loop unstable equilibrium point) without restrictions on the reaction kinetics. Numerical simulations illustrate the application of the theory to a nonisothermal, continuous stirred tank reactor with the steady‐state multiplicity. The proposed approach is compared with the thermodynamic availability‐based control approach in terms of amplitude and variation rate to show its performance and effectiveness.

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An operator theoretic approach to infinite‐dimensional control systems

Jacob

Zwart

2018

GAMM-Mitteilungen

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In this survey we use an operator theoretic approach to infinite‐dimensional systems theory. As this research field is quite rich, we restrict ourselves to the class of infinite‐dimensional linear port‐Hamiltonian systems and we will focus on topics such as well‐posedness, stability and stabilizability. We combine the abstract operator theoretic approach with the more physical approach based on Hamiltonians. This enables us to derive easy verifiable conditions for well‐posedness and stability.

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An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

Abstract: An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on mathematical physics, 60(2), 175-198.

Cited by 150 publications

References 24 publications

Feedback equivalence of input–output contact systems

Feedback equivalence of input–output contact systems

Integration of inventory control into the port‐Hamiltonian framework for dissipative stabilization of chemical reactors

An operator theoretic approach to infinite‐dimensional control systems

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