Geometry of Thermodynamic Processes

Since the 1970s contact geometry has been recognized as an appropriate framework for the geometric formulation of the state properties of thermodynamic systems, without, however, addressing the formulation of non-equilibrium thermodynamic processes. In Balian & Valentin (2001) it was shown how the symplectization of contact manifolds provides a new vantage point; enabling, among others, to switch between the energy and entropy representations of a thermodynamic system. In the present paper this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, as already largely present in the literature, appears to be elegant and effective. This culminates in the definition of port-thermodynamic systems, and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.Comment: 23 page

Control of input-output contact systems.

International audienceControl input-output contact systems are the representation of open irreversible Thermodynamic systems whose geometric structure is defined by Gibbs' relation. These systems are called conservative if furthermore they leave invariant a particular Legendre submanifold defining their thermodynamic properties. In this paper we address the stabilization of controlled input-output contact systems. Firstly it is shown that it is not possible to achieve stability on the complete Thermodynamic Phase Space. As a consequence, the stabilization is addressed on some invariant Legendre submanifold of the closed-loop system. For structure preserving feedback of input-output contact systems, i.e., for the class of feedback that renders the closedloop system again a contact system, the closed-loop invariant Legendre submanifolds have been characterized. The stability of the closed-loop system has then been proved using Lyapunov's second method. The results are illustrated on the classical thermodynamic process of heat transfer between two compartments and an exterior control

Towards control by interconnection of port-thermodynamic systems

Power conserving interconnection of port-thermodynamic systems via their power ports results in another port-thermodynamic system, and the same holds for any rate of entropy increasing interconnection via the entropy flow ports. Control by interconnection seeks to control the port-thermodynamic system by the interconnection with a controller port-thermodynamic system. The stability of the interconnected port-thermodynamic system is investigated by Lyapunov functions that are based on generating functions for the submanifold characterizing the state properties, as well as additional conserved quantities. Crucial tool is the use of point transformations of the symplectized thermodynamic phase space

Irreversible port-Hamiltonian systems : a general formulation of irreversible processes with application to the CSTR.

International audienceIn this paper we suggest a class of quasi-port Hamiltonian systems called Irreversible port Hamiltonian Systems, that expresses simultaneously the first and second principle of thermodynamics as a structural property. These quasi-port Hamiltonian systems are defined with respect to a structure matrix and a modulating function which depends on the thermodynamic relation between state and co-state variables of the system. This modulating function itself is the product of some positive function and the Poisson bracket of the entropy and the energy function. This construction guarantees that the Hamiltonian function is a conserved quantity and simultaneously that the entropy function satisfies a balance equation containing an irreversible entropy creation term. In the second part of the paper, we suggest a lift of the Irreversible Port Hamiltonian Systems to control contact systems defined on the Thermodynamic Phase Space which is canonically endowed with a contact structure associated with Gibbs' relation. For this class of systems we have suggested a lift which avoids any singularity of the contact Hamiltonian function and defines a control contact system on the complete Thermodynamic Phase Space, in contrast to the previously suggested lifts of such systems. Finally we derive the formulation of the balance equations of a CSTR model as an Irreversible Port Hamiltonian System and give two alternative lifts of the CSTR model to a control contact system defined on the complete Thermodynamic Phase Space

Liouville geometry of classical thermodynamics

In the contact-geometric formulation of classical thermodynamics distinction is made between the energy and entropy representation. This distinction can be resolved by taking homogeneous coordinates for the intensive variables. It results in a geometric formulation on the cotangent bundle of the manifold of extensive variables, where all geometric objects are homogeneous in the cotangent variables. The resulting geometry based on the Liouville form is studied in depth. Additional homogeneity with respect to the extensive variables, corresponding to the classical Gibbs-Duhem relation, is treated within the same geometric framework

Feedback equivalence of input-output contact sysems.

International audienceControl 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 input-output 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

Classical Thermodynamics Revisited:A Systems and Control Perspective

Thermodynamics has been the subject of intense scientific debate throughout its long history (see "Summary"). The following famous quote from Albert Einstein's autobiographical notes expresses his admiration for the theory of classical, macroscopic thermodynamics [1

Modeling and analysis of non-isothermal chemical reaction networks:A port-Hamiltonian and contact geometry approach

In dit proefschrift worden verschillende benaderingen gebruikt voor de meetkundige modellering en analyse van chemische reactienetwerken met varierende temperatuur. deze benaderingen kunnen in twee klassen worden verdeeld: de ene gebaseerd op poort-Hamiltonse systeemtheorie, en de ander gebaseerd op de theorie van contactsystemen. De eerste aanpak is de irreversibele poort-Hamiltonse formulering op basis van de interne energie. Beginnend met een overzicht van de wiskundige structuur van chemische reactienetwerken in het niet-isothermische geval wordt een irreversibele poort-Hamiltonse formulering van niet-isothermische reactienetwerken gegeven. Daarna volgt een thermodynamische analyse, inclusief de voorwaarden voor het bestaan van een thermodynamisch evenwicht en de asymptotische stabiliteit van de verzameling van thermodynamische evenwichtspunten. De tweede benadering betreft de quasi poort-Hamiltonse modellering met behulp van de totale entropie. In dit poort-Hamiltonse systeem wordt niet alleen de energiebalans maar ook de entropiebal- ansvergelijking gebruikt. Ook de thermodynamische analyse wordt in dit kader uitgevoerd, in het bijzonder de karakterisatie van evenwichtspunten en hun asymptotische stabiliteit.Gebaseerd op deze nieuwe quasi poort-Hamiltonse formulering wordt verder de interconnectie van chemische reactienetwerken bestudeerd. Tenslotte wordt de regeling van contactsystemen door middel van structuurbe-houdende terugkoppeling bestudeerd. Een aantal regelontwerpen die hierop gebaseerd zijn worden bestudeerd. Een lokale stabiliteitsanalyse wordt uitgevoerd om de structuurbehoudende terugkoppeling te bepalen, op basis van evenwichtsvoorwaarden en de Jacobimatrix van het teruggekoppelde systeem. Verder worden voorwaarden voor lokale en gedeeltelijke stabiliteit ten opzichte van de gesloten-lus invariant Legendre deelvarieteit gegeven, alsmede de gesloten-lus contact Hamiltonfunctie

Commande des Systèmes Hyperboliques décrits par des Equations aux Dérivées Partielles

This work is part, from a theoretical point of view, of the control of systems describedby partial differential equations (PDE). The other aspect is the application of those resultsto real process’s applications.Great developments have been done on modelization technics, identification and thecontrol for systems in finite dimension since a long time. Those technics have reached amaturity level, and are applied to numerous applications. Nevertheless, the developmentof advanced technologies have for consequence to increase the size of the controlledmodels, rising which is the symbol of the passage from finite dimension system to infinitedimension system.In recent decades, a real work on the development of infinite dimension tools hasemerged. This work originally dedicated to rather academic cases are being extendedtoday to practical cases.My work has found its place at this level : since 10 years I am interested in stabilityproblems and in the development of controls for systems described by hyperbolic PDE.For this I use mathematical structures such as semigroups, "natural" invariants like theRiemann invariants, energy structures like the Hamiltonian functional, or by the extensionof existing results in finite dimension to the infinite dimension such for the LMI (LinearMatrices Inequalities) to LOI (Linear Operator Inequalities).All these theoretical results have no interest if they are not applied, at least that’sthe goal I would like to maintain. To this end, all results have been developed on realprocesses : irrigation channels, navigable waterways, extrusion process, and more tocome. The issue of water described by the shallow water equations is a central examplein my work, but this is simply because I have access to benchmarks allowing me tovalidate the developed approaches.All of my works has been published internationally, but also broadcasted on lessonsfrom doctoral schools, in training of masters students and PhD students.Ce travail s’inscrit, d’un point de vue théorique, dans le domaine du contrôle dessystèmes décrits par des équations aux dérivées partielles (EDP). L’autre versant de cetravail est l’application concrète à des procédés.Un grand effort de développement des techniques de modélisation, d’identification etde commande a été réalisé pour les systèmes de dimension finie depuis des années.Ces techniques ont atteint un certain degré de maturité et sont utilisées dans de nombreusesapplications. Néanmoins, les développements des technologies de pointes ontentrainé une hausse considérable de la taille des modèles de commande, hausse quiest le reflet dans beaucoup de cas, du passage de la commande d’un vrai système dedimension finie vers un système de dimension infinie.Depuis quelques décennies, un réel travail de développement des outils en dimensioninfinie a donc vu le jour. Ces travaux initialement dédiés à des cas plutôt académiquesse voient aujourd’hui étendus à des cas pratiques.Mes travaux se posent à ce niveau : depuis 10 ans je m’intéresse aux problèmes destabilité et au développement de commandes de systèmes décrits par des EDP hyperboliques.Pour cela, j’utilise des structures mathématiques telles que les semigroupes,des invariants "naturels" comme ceux de Riemann, des structures énergétiques commeles Hamiltoniens, ou par l’extension de résultats existants en dimension finie à la dimensioninfinie comme pour les LMI (linear matrices inequalities) en LOI (linear operatorinequalities).Tous ces résultats théoriques n’ont d’interêt que s’ils sont appliqués, du moins c’estl’objectif que je souhaite maintenir. A cette fin, tous les résultats ont été développés surde réels process : les canaux d’irrigation, les voies naviguables, l’extrusion, et d’autres àvenir. La problématique de l’eau décrite par les équations de Saint-Venant est certe unexemple central dans mon travail, mais cela est dû simplement au fait que j’ai accès àdes bancs d’essais me permettant de valider les approches développées.L’ensemble de mes travaux a été publié au niveau international, mais aussi diffusé enlocal lors d’enseignements auprès d’écoles doctorales, lors d’encadrement de mastersrecherche et de thésards