Delay-robust stabilization of an n + m hyperbolic PDE-ODE system

International audienceIn this paper, we study the problem of stabilizing a linear ordinary differential equation through a system of an n + m (hetero-directional) coupled hyperbolic equations in the actuating path. The method relies on the use of a backstepping transform to construct a first feedback to tackle in-domain couplings present in the PDE system and then on a predictive tracking controller used to stabilize the ODE. The proposed control law is robust with respect to small delays in the control signal

A Strict Control Lyapunov Function for a Diffusion Equation with Time-Varying Distributed Coefficients

International audienceIn this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as closely as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances

Infinite Dimensional Control and Input-to-State Stability of the Safety Factor Profile in a Tokamak Plasma

Dans cette thèse, on s'intéresse au contrôle du profil de facteur de sécurité dans un plasma tokamak. Cette variable physique est liée à plusieurs phénomènes dans le plasma, en particulier des instabilités magnétohydrodynamiques (MHD). Un profil de facteur de sécurité adéquat est particulièrement important pour avoir des modes d'opération avancés dans le tokamak, avec haut confinement et stabilité MHD. Pour cela faire, on se focalise sur la commande du gradient du profil de flux magnétique poloidal dans le tokamak. L'évolution de cette variable est donnée par une équation de diffusion avec des coefficients distribuées et temps-variants. En utilisant des techniques de type Lyapunov et les propriétés de stabilité entrée-état du système on propose une loi de commande robuste qui prend en compte des contraintes non-linéaires dans l'action imposées par la physique des actionneurs.In this thesis, we are interested in the control of the safety factor profile or q-profile in a tokamak plasma. This physical quantity has been found to be related to several phenomena in the plasma, in particular magnetohydrodynamic (MHD) instabilities. Having an adequate safety factor profile is particularly important to achieve advanced tokamak operation, providing high confinement and MHD stability. To achieve this, we focus in controlling the gradient of the poloidal magnetic flux profile. The evolution of this variable is given by a diffusion equation with distributed time-varying coefficients. Based on Lyapunov techniques and the Input-to-State stability properties of the system we propose a robust control law that takes into account nonlinear constraints on the control action imposed by the physical actuators

Delay-robust stabilization of an n + m hyperbolic PDE-ODE system

International audienceIn this paper, we study the problem of stabilizing a linear ordinary differential equation through a system of an n + m (hetero-directional) coupled hyperbolic equations in the actuating path. The method relies on the use of a backstepping transform to construct a first feedback to tackle in-domain couplings present in the PDE system and then on a predictive tracking controller used to stabilize the ODE. The proposed control law is robust with respect to small delays in the control signal

Delay-robust stabilization of an n + m hyperbolic PDE-ODE system

International audienceIn this paper, we study the problem of stabilizing a linear ordinary differential equation through a system of an n + m (hetero-directional) coupled hyperbolic equations in the actuating path. The method relies on the use of a backstepping transform to construct a first feedback to tackle in-domain couplings present in the PDE system and then on a predictive tracking controller used to stabilize the ODE. The proposed control law is robust with respect to small delays in the control signal

Backstepping-Forwarding Control and Observation for Hyperbolic PDEs With Fredholm Integrals

International audienceAn integral transform is introduced which allows the construction of boundary controllers and observers for a class of first-order hyperbolic PIDEs with Fredholm integrals. These systems do not have a strict-feedback structure and thus the standard backstepping approach cannot be applied. Sufficient conditions for the existence of the backstepping-forwarding transform are given in terms of spectral properties of some integral operators and, more conservatively but easily verifiable, in terms of the norms of the coefficients in the equations. An explicit transform is given for particular coefficient structures. In the case of strict-feedback systems, the procedure detailed in this paper reduces to the well-known backstepping design. The results are illustrated with numerical simulations

Contrôle et stabilité Entrée-Etat en dimension infinie du profil du facteur de sécurité dans un plasma Tokamak Infinite dimensional control and Input-to-State stability of the safety factor profile in a Tokamak plasma

Dans cette thèse, on s'intéresse au contrôle du profil de facteur de sécurité dans un plasma tokamak. Cette variable physique est liée à plusieurs phénomènes dans le plasma, en particulier des instabilités magnétohydrodynamiques (MHD). Un profil de facteur de sécurité adéquat est particulièrement important pour avoir des modes d'opération avancés dans le tokamak, avec haut confinement et stabilité MHD. Pour cela faire, on se focalise sur la commande du gradient du profil de flux magnétique poloidal dans le tokamak. L'évolution de cette variable est donnée par une équation de diffusion avec des coefficients distribuées et temps-variants. En utilisant des techniques de type Lyapunov et les propriétés de stabilité entrée-état du système on propose une loi de commande robuste qui prend en compte des contraintes non-linéaires dans l'action imposées par la physique des actionneurs.In this thesis, we are interested in the control of the safety factor profile or q-profile in a tokamak plasma. This physical quantity has been found to be related to several phenomena in the plasma, in particular magnetohydrodynamic (MHD) instabilities. Having an adequate safety factor profile is particularly important to achieve advanced tokamak operation, providing high confinement and MHD stability. To achieve this, we focus in controlling the gradient of the poloidal magnetic flux profile. The evolution of this variable is given by a diffusion equation with distributed time-varying coefficients. Based on Lyapunov techniques and the Input-to-State stability properties of the system we propose a robust control law that takes into account nonlinear constraints on the control action imposed by the physical actuators.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Practical output regulation and tracking for linear ODE-hyperbolic PDE-ODE systems

International audienceIn this chapter, we consider the problem of practical output regulation and output tracking for a linear 2 × 2 hyperbolic Partial Differential Equation (PDE) system with actuation and load dynamics. Indeed, it is actuated via an Ordinary Differential Equation (ODE) at one boundary and the output to be controlled is the output of an ODE at the other boundary. The main focus is on load tracking. Here, we propose to extend existing results on approximate output regulation to a class of systems similar to that considered in [8] and to extend filtering techniques to a dynamically augmented system with finite-dimensional exosystems considering possible trajectory and disturbance inputs. Issues with respect to small delays in the state reconstruction and feedback loop are considered. Due to the nature of the disturbances, the state estimation and disturbance reconstruction problems are also considered. This scenario finds applications in many systems of engineering interest, such as drilling systems [3], pneumatic systems [17], or electric transmission lines [22]

Safety Factor Profile Control in a Tokamak

2014, XI, 96 p. 29 illus., 24 illus. in color.International audienc