A systematic methodology for structured design of feedback controllers for the sawtooth period is presented, based on dedicated identification of the sawtooth dynamics. Therefore, a combined Kadomtsev–Porcelli model of a sawtoothing plasma actuated by an electron cyclotron current drive system has been set-up. This is used to derive the linearized input–output relations (transfer functions) from the varying deposition location of the electron cyclotron waves (ECW) to the sawtooth period. These transfer functions are derived around a large collection of operating points. Assessment of these control-relevant transfer functions shows that a sawtooth period controller requires an integral (I) action to guarantee closed-loop stability with zero steady-state error. Additional proportional-integral (PI) action can be applied to further increase the closed-loop performance. The parameters of both the I and PII controllers have been optimized in terms of stability, performance and robustness. Moreover, the effect of the mechanical ECW launcher on the closed-loop performance is studied for realistic cases. It is shown that the launcher dynamics seriously affects the achievable closed-loop performance in present-day experiments.

In this paper we present a general linear matrix inequality-based analysis method to determine the performance of a SISO reset control system in both the L 2 gain and H 2 sense. In particular, we derive convex optimization problems in terms of LMIs to compute an upperbound on the L 2 gain performance and the H 2 norm, using dissipativity theory with piecewise quadratic Lyapunov functions. The results are applicable to for all LTI plants and linear-based reset controllers, thereby generalizing the available results in the literature. Furthermore, we provide simple though convincing examples to illustrate the accuracy of our proposed L 2 gain and H 2 norm calculations and show that, for an input constrained H 2 problem, reset control can outperform a linear controller designed by a common nonlinear optimization method. of the controller states when the switching surface is reached. The rationale behind reset control can best be understood when considering an integrating controller in particular. An integrator 'sums' the error over time in order to achieve a zero steady-state error. However, when the error becomes zero for the first time, the integrator still has the 'summed' error stored in its states, and the subsequent 'emptying' of the integrator causes the error signal to overshoot. To reduce the overshoot, it might be beneficial to reset the state of this integrator to zero as soon as its input (the error) becomes zero. In this way the integrator state, containing the 'summed' error, is emptied instantaneously, and hence the overshoot is avoided. This concept was indeed validated by simulations in recent publications [5,6], showing a significant decrease of overshoot in the step response.The concept of reset control was first introduced in 1958 by means of the resetting integrator of Clegg [7]. The describing function of the Clegg integrator has

Corroborating evidence is presented that the sawtooth period can follow the modulation frequency of an externally applied high power electron cyclotron wave source. Precise, fast and robust open loop control of the sawtooth period with a continuously changing reference period has been achieved. This period locking is not associated with the crash, but with the phase evolution of the inter-crash dynamics. This opens new possibilities of open loop control for physics studies and maybe for reactor performance control.

In this paper the sawtooth period behaviour under periodic forcing by electron cyclotron waves is investigated. The deposition location is kept constant while the gyrotron power is modulated with a certain period and duty cycle. Extensive simulations on a representative dynamic sawtooth model show that when this modulation is properly chosen, the sawtooth period quickly synchronizes to the same period and remains locked at this value. It is shown that the range of modulation periods and duty cycles over which sawtooth period locking occurs, depends on the deposition location, but is particularly large for depositions near the q = 1 surface. The simulation results reveal a novel approach to control the sawtooth period in open loop, based on injection locking, which is a well-known technique to control limit cycles of non-linear dynamic oscillators. The locking and convergence results are therefore used in a simple open-loop locking controller design, with which accurate sawtooth period tracking to any desired value is indeed demonstrated. Injection locking appears to let the sawtooth period converge to the modulation period quickly, partly because it does not suffer from slow EC mirror launcher dynamics. Moreover, simulations show that the method has a relatively large robustness against general uncertainties and disturbances. Hence, injection locking is expected to outperform conventional sawtooth control methods using a variable deposition location and constant gyrotron power. Finally, the recent result with sawtooth pacing is shown to be a special case of the general locking effect.