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### Abstract

The control of the current, position and shape of an elongated cross-section tokamak plasma is complicated by the instability of the plasma vertical position. In this case the control becomes a significant problem when saturation of the power supplies is considered. Current saturation is relatively benign due to the integrating nature of the tokamak, resulting in a reasonable time horizon for strategically handling this problem. On the other hand, voltage saturation is produced by the feedback controller itself, with no intrinsic delay. In practice, during large plasma disturbances, such as sawteeth, ELMs and minor disruptions, voltage saturation of the power supply can occur and as a consequence the vertical position control can be lost. If such a loss of control happens the plasma displaces vertically and hits the wall of the vessel, which can cause damage to the tokamak. The consideration and study of voltage saturation is especially important for ITER. Due to the size and therefore the cost of ITER, there will naturally be smaller margins in the Poloidal Field coil power supplies implying that the feedback will experience actuator saturation during large transients due to a variety of plasma disturbances. The next generation of tokamaks under construction will require vertical position and active shape control and will be fully superconducting. When the magnetic transverse field in superconducting magnets changes, the magnet generates two types of heat loss, the so-called coupling loss and the so-called hysteresis loss, grouped together as AC losses. Superconducting coils possess superconducting properties only below a critical temperature around a few K. AC losses are detrimental since they heat up the superconducting material. Thus, if AC losses are too large, the cryogenic plant can no longer hold the required temperature to maintain the superconductivity properties. Once the superconductivity is lost, the electric currents in the coils produce an enormous heat loss due to the ohmic resistivity, which can lead to a possible damage to the coils. In general, the coils are designed with enough margin to absorb all likely losses. A possible loss reduction could allow us to downsize the superconducting cross section in the cables, reducing the overall cost, or simply increase the operational cooling margin for given coils. In this thesis we have tried to take into consideration these two major problems. The thesis is therefore focused on the following main objectives: i) the stability analysis of the tokamak considering voltage saturation of the power supplies and ii) the proposition of a new controller which enhances the stability properties of the tokamak under voltage saturation and iii) the proposition of a controller which takes into consideration the problem of reducing the AC losses. The subject of the thesis is therefore situated in an interdisciplinary framework and as a result the thesis is subdivided into two principal parts. The first part is devoted to tokamak physics and engineering, while the second part focuses on control theory. In the tokamak physics and engineering part we present the linear tokamak models and the nonlinear tokamak code used for the controller design and the validation of the new proposed controller. The discussion is especially focused on the presence of a single unstable pole when the vertical plasma position is unstable since this characteristic is essential for the work presented in the control theory part. In order to determine the enhancement of the stability properties we have to bring the new proposed controller to its stability limits by means of large disturbances. Validation by means of simulations with either linear or nonlinear tokamak models are imperatively required before considering the implementation of the new controller on a tokamak in operation. A linear tokamak model will probably be inadequate since large disturbances can move its state outside its validity regions. A full nonlinear tokamak evolution code like DINA is indispensable for this purpose. We give a detailed description of the principal plasma physics implemented in the DINA code. Additionally, validation of DINA is provided by comparing TCV experimental VDE responses with DINA code simulations. To allow a study of the AC losses reduction, the nature of the AC losses has to be reduced to a simplified form. We analyse to what extent the accumulated AC losses in ITER could be reduced by taking into account the losses themselves when designing the feedback control loops. In order to be able to carry out this investigation a simple and fast AC loss model, referred to as "AC-CRPP" model, is proposed. In the control theory part we study the stability region in state space, referred to as the region of attraction, for linear tokamak-like systems with input saturation (voltage saturation) and a linear state feedback. Only linear systems with a single unstable pole (mode) and a single saturated input are considered. We demonstrate that the characterisation of the region of attraction is possible for a second order linear system with one unstable and one stable pole. For such systems the region of attraction possesses a topological bifurcation and we provide an analytical condition under which this bifurcation occurs. Since the analysis relies on methodologies like Poincaré and Bendixson's theorems which are unfortunately only valid for second order systems it is evident that there is no way to apply the results for second order systems to higher order systems. It turned out that the search for characterising the region of attraction for higher order systems was illusory and thus this research direction had to be abandoned. We therefore focused on controllers for which the region of attraction is the maximal region of attraction that can be achieved under input saturation. This region is referred to as the null controllable region and its characterisation is simple for any arbitrary high order system possessing a single unstable pole. We present a new globally stabilising controller for which its region of attraction is equal to the null controllable region. This result is obtained by incorporating a simple continuous nonlinear function into a linear state feedback controller. There are several advantages linked to this new controller: i) the stability properties are enhanced, ii) the performance, AC loss reduction and fast disturbance rejection, can be taken into account, iii) the controller can be applied to any arbitrary high order system and iv) the controller possesses a simple structure which simplifies the design procedure. We close the control theory part by focusing on the application of the proposed new controller to tokamaks. Since this controller is a state feedback controller one of the major problems is linked to the state reconstruction. Other pertinent topics are: i) the study of the effect of the disturbances on the closed-loop system stability, ii) the problem inherent to the nature of a state feedback controller when we want an output of the system to track a reference signal and iii) the discussion of the detrimental effects on stability if a pure time delay or a limited bandwidth are added to the closed-loop system, as is the case in reality. The validation of the proposed controller is carried out by means of simulations. We present results for ITER-FEAT and JET using the linear tokamak model CREATE-L. Finally, we present a validation for the case of TCV using the nonlinear DINA-CH code.