Controllability and Reduced State Space Models for Feedback Control of the Resistive Wall Kink Mode

Controllability conditions are derived for the resistive wall kink mode using analytic single circuit theory. Using feedback coils that couple more strongly to the plasma than the wall allows controllability up to the ideal wall limit of performance. Conversely, it is found that the controllability of the unstable resistive wall mode is lost at some value of instability growth rate between the no-wall and ideal wall limits when feedback coils are placed outside the passive stabilizing wall. The controllability criterion of the mode can be written in terms of the sign of a dimensionless coupling number that characterizes the magnitude of system inductances. Similar coupling numbers can be defined and used to characterize mode observability conditions as well as the interaction of sensor and feedback coils with the passive wall and plasma. A method is developed based upon contraction of a finite element inductance and resistance matrix model of the system conducting structures that allows quantitative calculation of these coupling numbers for a given system eigenmode. The construction of simple reduced order ordinary differential equation models based upon these coupling numbers accurately calculates mode eigenvalues and serves as a useful starting point for developing few mode approximations of the full eigenmode spectrum for use in advanced controller algorithm development