| Nuclear fusion is the energy-producing process that takes place continuously in the sun and stars. It presents an economically feasible, environmentally sustainable, and politically acceptable source of energy. The most promising magnetic-confinement-based fusion reactor is the tokamak. Tokamaks are high order, nonlinear, distributed parameter systems (modeled by partial differential equations (PDEs)) with a large number of instabilities. There are many extremely challenging mathematical modeling and control problems that must be solved before fusion power systems become viable entities. This dissertation focuses on several PDE control problems arising in magnetic confinement fusion (MCF) reactors, including the regulation of plasma spatial profiles, the boundary control of magnetohydrodynamic (MHD) flows and other theoretical PDE control/estimation problems motivated by nuclear fusion research.;The regulation of plasma spatial profiles of quantities such as toroidal current density, pressure and temperature may be the enabler of a steady-state, instability-free, high-confinement tokamak fusion reactor. By integrating model order reduction (MOR) techniques, numerical nonlinear optimization, the minimal surface theory, and sequential linear quadratic (SLQ) feedback synthesis methods, optimized control of the plasma profile dynamics during the ramp-up phase of the discharge is obtained in both open loop and closed-loop, respectively.;Due to the lack of globally accurate mathematical models to describe complicated plasma profile dynamics in different plasma discharge experiments, a data-driven modeling approach associated with the model structures provided by the first principle, is studied by using experimental data to reconstruct or identify transport coefficients based on the powerful Kalman filtering theory.;Model reduction techniques are critical in distributed signal processing and control synthesis of distributed parameter systems. The proper orthogonal decomposition method, as a widely used model reduction technique, has some shortcomings. The most notable one is the lack of approximation accuracies when systems behave beyond the pre-collected historic data ensembles. Recursive proper orthogonal decomposition is incorporates newly collected data snapshots in a recursive way to modify dominant subspace bases to adaptively represent the system dynamics. This method has the potential to be used in the processing of high resolution diagnostic data in fusion plasmas, such as soft X-ray tomography and reconstruction of MHD equilibrium.;Different distributed parameter systems in fusion plasmas exhibit unstable dynamics. Designing feedback controllers to stabilize unstable linearized distributed parameter systems maybe sometimes necessary in plasma control. Much existing work in this field is based on the discretize-then-design approach to synthesize stabilizing controllers. One frequently used discretization technique in plasma control synthesis is the Fourier decomposition. A design-then-discretize method is developed in this dissertation by using the Sturm-Liouville theory to provide an alternative approach for the controller synthesis.;Boundary control of MHD flows arising in nuclear reactor cooling blankets is a very challenging topic in both fluid mechanics and control engineering. By using a PDE backstepping synthesis approach, a strict feedback controller is obtained such that the MHD equilibrium is exponentially stabilized in the square integrable function space. |
