Control of a non-linear PDE system arising from non-burning tokamak plasma transport dynamics

Abstract

The control of kinetic profiles is among the most important problems in fusion reactor research. It is strongly related to a great number of other problems in fusion energy generation such as burn control, transport reduction, confinement time improvement, MHD instability avoidance and high-β or high-confinement operating modes access. We seek a controller which is able to make the kinetic profiles converge to their desired equilibrium profiles. We are interested in constructing a stabilizing controller that achieves stability for unstable equilibrium profiles and increases performance for stable equilibrium profiles. As a first approach, we consider in this work a set of non-linear partial differential equations (PDEs) describing approximately the dynamics of the density and energy profiles in a non-burning plasma. This nonlinear PDE model represents the one-dimensional transport equations for the kinetic variables, density and energy, in cylindrical geometry. The transport coefficients in this model are in turn non-linear functions of the kinetic variables. The original set of PDEs is discretized in space using a finite difference method which gives a high order set of coupled nonlinear ordinary differential equations (ODEs). Applying a backstepping design we obtain a discretized coordinate transformation that transforms the original system into a properly chosen target system that is asymptotically stable in l2-norm. To achieve such stability for the target system, convenient boundary conditions are chosen. Then, using the property that the discretized coordinate transformation is invertible for an arbitrary (finite) grid choice, we conclude that the discretized version of the original system is asymptotically stable and obtain a non-linear feedback boundary control law for the energy and density in the original set of coordinates. Numerical simulations show that the feedback control law designed using only one step of backstepping can successfully control the kinetic profiles.