Distributed Regulation of the Safety Factor Profile in Tokamaks Using Nonlinear Infinite-dimensional Control

Tokamaks are toroidal devices that use helical magnetic fields to confine a plasma (hot ionized gas). Such confinement increases the probability of ionic collisions. When the colliding ions have high enough kinetic energy, they can overcome the Coulombic forces of repulsion and fuse to form a heavier ion. The difference in mass between reactant and product ions is turned into energy, which can potentially be harvested to meet the growing world's energy demands. In tokamaks, the safety factor profile is a plasma property that characterizes the pitch of the helical magnetic field. Experiments have shown that the safety factor is related to the magnetohydrodynamic (MHD) stability of the confined plasma as well as to the capability of achieving highly-confined steady-state operation. Thus, active control of the safety factor profile or related plasma properties is critical for achieving MHD-stable, high-performance plasma operation. The evolution of the safety factor profile is governed by a nonlinear nonautonomous partial differential equation (PDE). The most commonly used control design approach reduces the governing PDE to a set of ordinary differential equations (ODEs) before synthesizing a control law. In this work, a distributed nonlinear safety-factor control law that does not require a finite-dimensional approximation of the governing PDE is proposed. Rigorous analysis shows that the proposed control law can drive the error between the safety factor profile and target profile to zero. The effectiveness of the proposed control law is demonstrated using nonlinear simulations for the DIII-D tokamak.