Simultaneous control of effective atomic number and electron density in non-burning tokamak plasmas

The control of plasma density profiles is one of the most fundamental problems in fusion reactors. During reactor operation, the density profiles of hydrogen ions and heavier impurity ions must be precisely regulated, and while the uncontrolled open-loop behavior of these profiles is stable in non-burning plasmas, the system response time may be very slow. In this work, a controller is sought to improve the response of ion density profiles in a non-burning plasma and to track reference profiles for the electron density and effective atomic number, a set of coupled variables that are directly dependent on the hydrogen and impurity ion density profiles. A one-dimensional approximation of the transport equation for ion densities is represented in cylindrical coordinates by a partial differential equation (PDE). To control the density profiles, the PDE is discretized in space using a finite difference method and a backstepping design is applied to obtain a discrete transformation from the original system into an asymptotically stable target system. Numerical simulations of the resulting control law show that density profiles can be successfully controlled with just one step of backstepping. Tracking of electron density and effective atomic number profiles is then simulated by first transforming the profiles into corresponding ion density profiles for use by the backstepping controller. Simulations show the successful tracking of reference profiles.