Pacing control of sawtooth and ELM oscillations in tokamaks

In tokamak plasmas, the sawtooth oscillation (ST) and the edge-localized-mode (ELM) are characterized by a phase of a slow evolution of the plasma conditions, followed by a crash-like instability that resets the plasma conditions when certain criteria of the plasma conditions are satisfied. Typically, the crashes induce losses of heat and energetic particles and may also trigger secondary instabilities. As the amplitude of the crash-like perturbation scales with the period between two crashes, period control of these oscillations is important for operations of large fusion facilities such as ITER and DEMO. In several present-day experimental facilities, a pacing control algorithm has been successfully applied for controlling the sawtooth period and the ELM period. However, a formal analysis has been lacking so far, which therefore forms the objective of the present paper. For this purpose, a reset model for the sawtooth period is introduced and, after a proper transformation a nonlinear discrete-time system is obtained, which is used for the formal analysis of pacing control. By representing the model in a Lur'e (or Lurie) form, we can derive conditions under which global asymptotic stability of the closed-loop (pacing) period control system is guaranteed. Moreover, we will show that the controller exhibits inherent robustness for model uncertainties. We envision that the analytical results in the area of pacing control of the sawtooth are also applicable to pacing period control of the ELM oscillation period. The presented reset model also explains why in recent experiments the sawtooth period locks with a periodically modulated power.