G Gert Witvoet

H 2 performance analysis of reset control systems

To overcome fundamental limitations of linear controllers, reset controllers were proposed in literature. Since the closed loop system including such a reset controller is of a hybrid nature, it is difficult to determine its performance. The focus in this paper is to determine the performance of a SISO reset control system in H2 sense. The method is generally applicable in the sense that it is valid for any proper LTI plant and linear-based reset controller. We derive convex optimization problems in terms of LMIs to compute an upperbound on the H2 norm, using dissipativity theory with piecewise quadratic Lyapunov functions. Finally, by means of a simple multiobjective tracking example, we show that reset control can outperform a linear controller obtained via a standard multiobjective control design method.

Demonstration of sawtooth period locking with power modulation in TCV plasmas

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.

Systematic design of a sawtooth period feedback controller using a Kadomtsev-Porcelli sawtooth model

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.

Numerical demonstration of injection locking of the sawtooth period by means of modulated EC current drive

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.

An LMI-based L 2 gain performance analysis for reset control systems

In this paper we present a general LMI-based analysis method to determine an upperbound on the L2 gain performance of a reset control system. These computable sufficient conditions for L2 stability, based on piecewise quadratic Lyapunov functions, are suitable for all LTI plants and linear-based reset controllers, thereby generalizing the results available in literature. Our results furthermore extend the existing literature by including tracking and measurement noise problems by using strictly proper input filters. We illustrate the approach by a numerical example.

Real-time optical plasma boundary reconstruction for plasma position control at the TCV Tokamak

A dual, high speed, real-time visible light camera setup was installed on the TCV tokamak to reconstruct optically and in real-time the plasma boundary shape. Localized light emission from the plasma boundary in tangential view, broadband visible images results in clearly resolved boundary edge-features. These projected features are detected in real-time and transformed to the poloidal plane to obtain a measurement of the plasma boundary. Plasma boundary reconstructions of diverted plasma discharges are presented, showing agreement of within 1 cm compared with magnetic equilibrium reconstruction. The resulting real-time plasma shape measurement is applied in a feedback control loop for the plasma position, demonstrating effective stabilization and tracking of the plasma vertical position.

Sawtooth period control strategies and designs for improved performance

The sawtooth instability is associated with the triggering of neo-classical tearing modes, core fuelling, α-confinement and the exhaust of thermal helium. Sawtooth control is therefore important for optimal reactor performance in ELMy H-modes. Control schemes for the sawtooth period have been published in the literature, but the systematic design of high-performance controllers (yielding accurate and fast convergent responses) has not been addressed. In this work, three control strategies for high-performance sawtooth control are presented using electron cyclotron current drive (ECCD). Both degrees of freedom of the ECCD actuator will be explored and combined with advanced controller designs. First, the ECCD deposition location is used as a control variable, for which a gain-scheduled feedback controller and static feedforward control is derived. Second, the use of the driven current as a control variable is explored, and a simple controller is designed based on the identified dynamics. In the third approach both control variables are joined in an overall controller design, which enables the combination of high-performance control of the sawtooth period and control of the gyrotron power. Time-domain simulations with a combined Kadomtsev-Porcelli sawtooth model show that each strategy obtains a better closed-loop performance than standard linear feedback techniques on merely the deposition location.

Dynamic analysis and control of mirror segment actuators for the European Extremely Large Telescope

Abstract. Segmented primary mirror telescopes require dedicated piston-tip-tilt actuators for optimal optical performance. Netherlands Organisation for Applied Scientific Research (TNO) has developed various prototypes of such actuators, in particular for the E-ELT. This paper presents the dynamics analysis and feedback control results for a specific two-stage prototype. First, the dynamics of the actuator in interconnection with the to-be-positioned mass has been analyzed, both using frequency response measurements and first principles modeling, resulting in a detailed understanding of the dynamic behavior of the system. Next, feedback controllers for both the fine and the coarse stage have been designed and implemented. Finally, the feedback-controlled actuator has been subjected to a realistic tracking experiment; the achieved results have demonstrated that the TNO actuator is able to suppress wind force disturbances and ground vibrations with more than a factor 103, down to 1.4 nm root mean square, which is compliant with the requirements.

Robust sawtooth period control based on adaptive online optimization

The systematic design of a robust adaptive control strategy for the sawtooth period using electron cyclotron current drive (ECCD) is presented. Recent developments in extremum seeking control (ESC) are employed to derive an optimized controller structure and offer practical tuning guidelines for its parameters. In this technique a cost function in terms of the desired sawtooth period is optimized online by changing the ECCD deposition location based on online estimations of the gradient of the cost function. The controller design does not require a detailed model of the sawtooth instability. Therefore, the proposed ESC is widely applicable to any sawtoothing plasma or plasma simulation and is inherently robust against uncertainties or plasma variations. Moreover, it can handle a broad class of disturbances. This is demonstrated by time-domain simulations, which show successful tracking of time-varying sawtooth period references throughout the whole operating space, even in the presence of variations in plasma parameters, disturbances and slow launcher mirror dynamics. Due to its simplicity and robustness the proposed ESC is a valuable sawtooth control candidate for any experimental tokamak plasma, and may even be applicable to other fusion-related control problems.

Control oriented modeling and simulation of the sawtooth instability in nuclear fusion tokamak plasmas

Tokamak plasmas in nuclear fusion are subject to various instabilities. A clear example is the sawtooth instability, which has both positive and negative effects on the plasma. To optimize between these effects control of the sawtooth period is necessary. This paper presents a simple control oriented model, from current drive (input) to sawtooth period (output), which can mimic the most relevant aspects of the sawtooth instability. It also shows some simulation results, including a static input-output map and a comparison with experimental data.