Conditions for stabilization of the tokamak plasma vertical instability using only a massless plasma analysis

This paper describes the problem of feedback control for stabilization of the plasma vertical instability in a tokamak. Such controllers are typically designed based on a model that assumes the plasma mass m is identically zero while in reality the mass is small but positive. The assumption that m is zero can lead to a controller that appears to be stabilizing according to the massless analysis but in fact can increase the instability of the physical system. In this work, we consider a general class of controllers, which contains as a special case the type of controller most commonly used in operating tokamaks to stabilize the vertical instability, a proportional-derivative controller. Suppose C is a controller in this class which stabilizes the vertical instability with plasma mass assumed to be zero. We give easy-to-check necessary and sufficient conditions for C to also stabilize the physical system, in which the plasma actually has a small mass. We allow for the possibility that the tokamak could have both superconducting and resistive conductors. The practical implications of the results presented provide substantial insight into some long-standing issues regarding feedback stabilization of the vertical instability with PD controllers and also provide a rigorous foundation for the common practice of designing controllers assuming m = 0. For controllers that operate only on the plasma vertical position, we settle the question: when are m = 0 models predictive of actual plasma behavior?