Feedback studies on resistive wall modes in the reversed field pinch

A single fluid nonlinear magnetohydrodynamic cylindrical model is considered in this paper in order to study the problem of feedback stabilization of the nonresonant m = 1 ideal kinks in a reversed field pinch. The ideal growth rates are "passively" reduced by the presence of a resistive wall within the radius for perfectly conducting wall stabilization of these modes. In this work we consider cases with up to two resistive walls. Moreover the feedback system is assumed to be able to react to any given Fourier harmonic, with an "ideal response," in the sense that no spurious harmonics are generated. Successful feedback schemes are shown to be possible. However, a careful choice of the gains, along with the simultaneous feedback on at least 4 or 5 modes, is found to be necessary. It should be remarked that this paper is one of the first attempt to describe the active control problem in presence of a nonlinear model for the plasma. The main result is that a suitable stabilization scheme has been found also in presence of complicate nonlinear interactions between the magnetohydrodynamical modes. It is also ound that the modes are stabilized when they are dragged into slow rotation by the feedback coils by applying a phase difference between the applied perturbation and the plasma modes.