Exponential stability of a class of PDE's with dynamic boundary control.

International audienceWe show that a finite dimensional strictly passive linear controller exponentially stabilizes a large class of partial differential equations which are actuated through its boundaries on a one dimensional spatial domain. This is achieved by extending existing results on exponential stability of boundary control system with static boundary control to the case with dynamic boundary control. The approach is illustrated on a physical example

Boundary port Hamiltonian control of a class of nanotweezers.

International audienceBoundary controlled-port Hamiltonian systems have proven to be of great use for the analysis and control of a large class of systems described by partial differential equations. The use of semi-group theory, combined with the underlying physics of Hamiltonian systems permits to prove existence, well-possessedness and stability of solutions using constructive techniques. On other hand, the differential geometric representation of these systems has lead to finite dimension approximation methods that conserves physical properties such as the interconnection structure and the energy. These results are applied to the modelling and control of a class of nanotweezers used for DNA-manipulation. The Nanotweezer may be modelled as a flexible beam interconnected with a finite dimensional dynamical system representing the manipulated object. A boundary controlled-port Hamiltonian model for the ensemble and an exponentially stabilizing controller are proposed. A geometric approximation scheme is used to reduce the infinite dimensional system and numerical simulations of the closed-loop system presented

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

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

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

Modelling and control of multi-energy systems : An irreversible port-Hamiltonian approach.

International audienceIn recent work a class of quasi port Hamiltonian system expressing the first and second principle of thermodynamics as a structural property has been defined : Irreversible port-Hamiltonian system. These systems are very much like port-Hamiltonian systems but differ in that their structure matrices are modulated by a non-linear function that precisely expresses the irreversibility of the system. In a first instance irreversible port-Hamiltonian systems are extended to encompass coupled mechanical and thermodynamical systems, leading to the definition of reversible-irreversible port Hamiltonian systems. In a second instance, the formalism is used to suggest a class of passivity based controllers for thermodynamic systems based on interconnection and Casimir functions. However, the extension of the Casimir method to irreversible port-Hamiltonian systems is not so straightforward due to the "interconnection obstacle". The heat exchanger, a gas-piston system and the non-isothermal CSTR are used to illustrate the formalism

Passivity based control of irreversible port Hamiltonian Systems.

International audienceThe frameworks of thermodynamic availability function and irreversible port Hamiltonian systems are used to derive passivity based control strategies for irreversible thermodynamic systems. An energy based availability function is defined using as generating function the internal energy. This is a variation with respect to previous works where the total entropy usually corresponds to the generating function. The specific structure of irreversible port-Hamiltonian systems then permits to elegantly derive stability conditions for open and closed thermodynamic systems. The results are illustrated on two classical thermodynamic examples: The heat exchanger and the continuous stirred tank reactor

Exponential stability of boundary control port Hamiltonian systems with dynamic feedback.

International audienceIn this paper it is shown that an input strictly passive linear finite dimensional port-Hamiltonian controller exponentially stabilizes a large class of boundary control systems. This follows since the finite dimensional controller dissipates the energy owing through the boundaries of the infinite dimensional system. The assumptions on the controller is that it is input strictly passive and that it is exponentially stable. The result is illustrated on the model of a DNA-manipulation process, that is used to show that the interconnection of the DNA-bundle+the controller (finite dimensional part of the system) and a micro-gripper (infinite dimensional part) is exponentially stable

A Lyapunov Approach to Robust Regulation of Distributed Port Hamiltonian Systems

International audienceThis paper studies robust output tracking and disturbance rejection for boundary controlled infinite-dimensional port-Hamiltonian systems including second order models such as the Euler-Bernoulli beam. The control design is achieved using the internal model principle and the stability analysis using a Lyapunov approach. Contrary to existing works on the same topic no assumption is made on the external well-posedness of the considered class of PDEs. The results are applied to robust tracking of a piezo actuated tube used in atomic force imaging

An irreversible port-Hamiltonian model for a class of piezoelectric actuators

International audience<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtAn irreversible port-Hamiltonian system formulation of a class of piezoelectric actua-</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gttor with non-negligible thermodynamic behavior is proposed. The proposed model encompasses</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtthe hysteresis and the irreversible thermodynamic changes due to mechanical friction, electrical</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtresistance and heat exchange between the actuator and the environment. The electromechanical</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtdynamic of the actuator is modeled using a generalized Maxwell resistive-capacitive model</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtcoupled with a mass-spring-damper system, while the non-linear hysteresis is characterized using</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gthysterons. The thermodynamic behavior of the model is constructed by making the electro-</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtmechanical coupling temperature dependent, and by characterizing the entropy produced by</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtthe irreversible phenomena. By means of numerical simulations it is shown that the proposed</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtmodel is capable of reproducing the expected behaviors and is in line with reported experimental</span&gt</font&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtresults.</span&gt</font&gt</div&g