Y. Le Gorrec

Exponential Stability of a Class of Boundary Control Systems

We study a class of partial differential equations (with variable coefficients) on a one dimensional spatial domain with control and observation at the boundary. For this class of systems we provide simple tools to check exponential stability. This class is general enough to include models of flexible structures, traveling waves, heat exchangers, and bioreactors among others. The result is based on the use of a generating function (the energy for physical systems) and an inequality condition at the boundary. Furthermore, based on the port Hamiltonian approach, we give a constructive method to reduce this inequality to a simple matrix inequality.

Stability and Stabilization of a Class of Boundary Control Systems

We study a class of partial differential equations on a one dimensional spatial domain with control and observation at the boundary. For this class of systems we describe how to obtain an impedance energy-preserving system, as well as scattering energy-preserving system. For the first type of systems we consider (static and dynamic) feedback stabilization by means of boundary control. For the scattering energy-preserving systems we give conditions for which the system is either asymptotically or exponentially stable.

A robust self-scheduled missile autopilot: design by multi-model eigenstructure assignment

Abstract This paper describes the synthesis of a self-scheduled controller robust with respect to parametric uncertainties. Traditionally, scheduling is done a posteriori by interpolation of gains computed for several points within the parameter space. The method proposed here, named (Mu-μ)-iteration, is based on worst-case analysis and multi-model eigenstructure assignment. It permits the designer to synthesize a robust low order controller by considering a priori its dependence of some scheduling parameters. The method is then applied to derive a robust missile autopilot.

Lyapunov based control for non isothermal continuous stirred tank reactor

Abstract In this contribution we apply the approach of passivity proposed by Ydstie [M. Ruszkowski, V. Garcia-Osorio, and B.E. Ydstie. AIChE Journal, 2005] for physico-chemical processes. The originality of this work lies in the fact we consider a thermodynamically nonlinear consistent model for a continuous stirred tank reactor to built the appropriate Lyapunov function for stabilization purpose. Indeed the kinetics of reaction modelled by Arhenius law leads to non linear model with multiple steady state. We propose to stabilize the reactor around the unstable point. In order to apply the Ydstie approach, we assume that the fluid remains homogeneous. This assumption permits to use the concavity property of the entropy function to build the Lyapunov function. We propose feedback laws in order to ensure the closed loop properties of the Lyapunov function. Finally we propose some simulation results.

Thermodynamics based stability analysis and its use for nonlinear stabilization of the CSTR

Abstract We show how the availability function as defined from the entropy function concavity can be used for the stability analysis and derivation of control strategies for non-isothermal Continuous Stirred Tank Reactors (CSTRs). We first propose an overview of the required thermodynamic concepts. Then, we show how the availability function restricted to the thermal domain can be used as a Lyapunov function. The derivation of the control law and the way the strict entropy concavity is insured are discussed. Numerical simulations illustrate the application of the theory to the open loop stability analysis and the closed loop control of liquid-phase non-isothermal CSTRs. The proposed approach is compared with the classical proportional control strategy. Two chemical reactions are studied: the acid-catalyzed hydration of 2-3-epoxy-1-propanol to glycerol subject to steady state multiplicity and the production of cyclopentenol from cyclopentadiene by acid-catalyzed electrophilic addition of water in dilute solution exhibiting a non-minimum phase behavior.

Thermodynamics based stabilitization of CSTR networks

In this work, we show that any potential function fulfilling certain thermodynamic stability criteria can be used as a storage function for the Port based modeling of the non isothermal Continuous Stirred Tank Reactor (CSTR) model using the Brayton-Moser formulation without any restriction on the chemical reaction kinetics. By means of an extended IDA-PBC (Interconnection and Damping Assignment-Passivity Based Control) approach, the closed loop energy is then shaped to be equal to the thermal part of the availability with the heat flowrate as the only control input. Some numerical simulations illustrate the theoretical developments.

Thermodynamic approach for Lyapunov based control

Abstract Abstract This paper focuses on non linear control of non isothermal Continuous Stirred Tank Reactors (CSTRs). The model of the CSTR is thermodynamically consistent in order to apply the control strategy based on the concavity of the entropy function and the use of thermodynamic availability as Lyapunov function. More precisely the stabilization problem of continuous chemical reactors is addressed operated at an unstable open loop equilibrium point. The chosen control variable is the jacket temperature. In this paper we propose a state feedback strategy to insure asymptotic stability with physically admissible control variable solicitations. Theoretical developments are illustrated on a first order chemical reaction.

Passivity-based nonlinear control of CSTR via asymptotic observers☆

This work makes use of a passivity-based approach (PBA) and tools from Lyapunov theory to design a nonlinear controller for the asymptotic stabilization of a class of nonisothermal Continuous Stirred Tank Reactors (CSTR) around any desired stationary point. The convergence and stability proofs are derived in the port Hamiltonian framework. Asymptotic observers that do not require knowledge of reaction kinetics are also proposed for a system with incomplete state measurement. Numerical simulations are given to illustrate the application of the theoretical results to a CSTR with multiple steady states.

Port Hamiltonian based modeling and control of exothermic Continuous Stirred Tank Reactors

Abstract This paper proposes a thermodynamically consistent pseudo Hamiltonian formulation of Continuous Stirred Tank Reactor model in which takes place one general reaction scheme. This is done both in the isothermal and non isothermal cases. It is shown that Gibbs free energy and opposite of entropy can be chosen as Hamiltonian function respectively. For the non isothermal case, the so called Interconnection and Damping Assignment Passivity Based Control method is applied. Even for a general reaction the control problem is shown to be easy to solve as soon as closed loop hamiltonian function is chosen to be proportional to the so called thermodynamic Availability function.

Port based modelling of a multiscale adsorption column

A port-based distributed parameter model of an adsorption column under isothermal conditions is presented. This model is made using bond graph terminology and basic axioms of irreversible thermodynamics. The particularity of this model is to split each phenomenon in basic elements with particular energetic behaviour, such as accumulation or dissipation. These basic elements are interconnected using the port variables and some interconnection power conserving structures, named Dirac structures. These energetic elements are presented in terms of the geometry of the different scales of the adsorption column. Moreover it is shown that each scale of the model has the same bond graph structure. The different scales of the model are also connected by a power conserving structure.