Optimal control of a class of nonlinear parabolic PDE systems arising in fusion plasma current profile dynamics

The need for new sources of energy is expected to become a critical problem within the next few decades. Nuclear fusion arises as a potential source of energy with sufficient energy density to supply the world population with its steadily increasing energy demands.;The need to optimize the tokamak concept for the design of an economical, possibly steady state, fusion power plant have motivated extensive international research aimed at finding the so-called "advanced tokamak (AT) operation scenarios." It has been demonstrated that simultaneous real-time control of the current and pressure profiles could lead to the steady state sustainment of an internal transport barrier (ITB), and so to a stationary optimized plasma regime. It has also been suggested that global current profile control, eventually combined with pressure profile control, can be an effective mechanism for neoclassical tearing mode (NTM) control and avoidance.;The control of linear or quasi-linear parabolic diffusion-reaction partial differential equations (PDE) has been extensively studied using interior control (see [1] and references therein) or boundary control (see [2] and references therein). Recently, the control of bilinear parabolic partial differential equations via actuation of the diffusive coefficient term, named diffusivity control here, has caught increasing interest. The diffusive coefficient term in a parabolic PDE is not necessary fixed or uncontrollable. For example, the diffusivity control problem arises in the control of the current density profile in magnetically confined fusion plasmas [3], where physical actuators such as plasma total current, line-averaged density and non-inductive total power are used to steer the plasma current density to a desired profile in a designated time period. By modulating these physical actuators it is possible not only to vary the amount of non-inductive current driven into the system (interior control) and the total plasma current (boundary control) but also to modify the resistivity of the plasma (diffusivity control).;Motivated by the current profile control problem in nuclear fusion reactors, we study in this thesis a particular class of nonlinear parabolic PDEs that admit interior, boundary and diffusivity actuation. We make in this way theoretical and practical contributions to control systems and nuclear fusion respectively. First, a simplified dynamic PDE model describing the evolution of the poloidal flux, and therefore the q profile, during the inductive phase of the discharge is introduced. Simulation results show qualitative agreement with experiments. Then, a multi-parameter, extremum-seeking, non-model-based, open-loop, optimal controller is designed, successfully tested in simulations, and implemented experimentally in the DIII-D tokamak, to match a desired q profile within a predefined time window during the flattop phase of the tokamak discharge. The controller is shown to be effective to deal with an optimal control problem defined for a nonlinear PDE system subject to many constraints in its actuators. Next, using the Proper Orthogonal Decomposition (POD) and Galerkin Projection techniques, we derive a finite dimensional ODE (Ordinary Differential Equation) dynamical system that preserves the dominant dynamics of the original infinite dimensional PDE system. This low dimensional model is used to design several closed-loop controllers, which have been tested successfully in simulations and are being implemented in the DIII-D tokamak: (i) we propose a convergent successive scheme based on the quasi-linear approximation to compute an optimal tracking control for the reduced order system; (ii) we formulate the problem as an abstract bilinear-quadratic regulator (BQR) problem. A receding horizon control (RHC) algorithm to solve the problem based on the infinite-dimensional system is proposed and stability of the algorithm for the solution of the BQR problem is studied; (iii) we present a robust control scheme to regulate the poloidal magnetic flux profile in tokamaks in the presence of model uncertainties. These uncertainties come mainly from the resistivity term of the magnetic diffusion equation. Finally, we prove that the system is not completely controllable and we provide an estimate for the unreachable region. (Abstract shortened by UMI.)...