In this work, model-based control techniques are used to obtain target current profiles in low-confinement-mode (L-mode) as well as high-confinement-mode (H-mode) DIII-D discharges. The control problem is formulated as a trajectory optimization problem to search for a feasible path from the expected initial condition to the desired target. The result comprises a sequence of feedforward (open-loop) control requests and a corresponding state evolution from the initial condition to the desired target. On top of this optimal feedforward control sequence a feedback (closed-loop) controller based on a linearized model and optimal control design techniques is added to track the desired state evolution. The effectiveness of the control approach is demonstrated with experiments.

We consider control of the q profile while simultaneously regulating the plasma stored energy for the DIII-D tokamak. The main objective is to improve the shot-to-shot reproducibility and facilitate the accessibility of operating conditions that have steady-state potential, i.e. plasmas with large non-inductive current drive fractions. At DIII-D, non-inductive current sources including electron cyclotron current drive (ECCD) and neutral beam injection (NBI) allow the possibility of shaping the plasma current density distribution, and therefore enabling control of the q profile. A feedback controller is designed in a model predictive control framework to regulate the q profile while simultaneously regulating the plasma stored energy. The effectiveness of the control approach is demonstrated with experiments.

The number of tokamak discharges required to carry out meaningful experiments can be significantly reduced by current profile control in the early startup phase. The tokamak is a plasma-confinement device, suitable for confining plasma at the requisite high temperature necessary for initiating fusion. It is currently the most promising device for realizing sustained fusion power generation at a commercial grade level, though it is still in the experimental stage. Presently, difficult to achieve plasma conditions, such as the shape of the plasma current profile, are achieved in a trial and error fashion, which can be a lengthy, wasteful process. In this work we make use of model-based control techniques such as optimal feedforward control via nonlinear programming and linearized feedback control to obtain a target current profile at a specified time in low-confinement-mode (L-mode) discharges. The effectiveness of the controller is demonstrated experimentally.

Control of the current profile in tokamak plasmas has been shown to play an important role in achieving advanced scenarios that could enable steady-state operation. The nonlinearity and spatially distributed nature of the current profile dynamics motivate the use of model-based control designs. In this work, we consider a control-oriented model of the current profile evolution in DIII-D high-confinement (H-mode) discharges, and the problem of regulating the current profile around a desired trajectory. The PDE model is discretized in space with a finite difference method and a backstepping design is applied to obtain a transformation from the original system into a particular target system with desirable properties. The resulting boundary condition control law is complemented with control laws for the available distributed actuators. The combined control strategy uses nonlinear combinations of the total plasma current, total power, and line averaged density as actuators. Simulation and experimental results show the ability of the controller to track desired targets and to reject input disturbances.

Active closed-loop control of the plasma safety factor profile (q-profile) and internal energy dynamics in nuclear fusion tokamak devices has the potential to significantly impact the success of the ITER project. These plasma properties are related to both the stability and performance of a given plasma operating scenario. In this work, we develop integrated feedback control algorithms to control the q-profile and internal energy dynamics in DIII-D advanced tokamak (high performance) scenarios. The feedback controllers are synthesized by embedding a nonlinear, physics-based, control-oriented partial differential equation model of the plasma dynamics into the control design and to be robust to uncertainties in the plasma electron density, electron temperature, and plasma resistivity profiles. The auxiliary heating and current-drive system and the total plasma current are the actuators utilized by the feedback controllers to control the plasma dynamics. Finally, the feedback controllers are tested both through simulations based on the physics-based model and experimentally in the DIII-D tokamak.

Abstract Extensive research has been conducted to find operating scenarios that optimize the plasma performance in nuclear fusion tokamak devices with the goal of enabling the success of the ITER project. The development, or planning, of these advanced scenarios is traditionally investigated experimentally by modifying the tokamak's actuator trajectories, such as the auxiliary heating/current-drive (H&CD) scheme, and analyzing the resulting plasma evolution. In this work, a numerical optimization algorithm is developed to complement the experimental effort of advanced scenario planning in the DIII-D tokamak. Two properties related to the plasma stability and performance are the safety factor profile ( q -profile) and the normalized plasma beta ( β N ). The optimization algorithm goal is to design actuator trajectories that steer the plasma to a target q -profile and plasma β N , such that the achieved state is stationary in time, subject to the plasma dynamics (described by a physics-based, nonlinear, control-oriented partial differential equation model) and practical plasma state and actuator constraints, such as the maximum available amount of H&CD power. This defines a nonlinear, constrained optimization problem that we solve by employing sequential quadratic programming. The optimized trajectories are then tested through simulation with the physics-based model and experimentally in DIII-D.

One of the most promising devices for realizing power production through nuclear fusion is the tokamak. To maximize performance, it is preferable that tokamak reactors achieve advanced operating scenarios characterized by good plasma confinement, improved magnetohydrodynamic stability, and a largely noninductively driven plasma current. Such scenarios could enable steady-state reactor operation with high fusion gain, the ratio of produced fusion power to the external power provided through the plasma boundary. For certain advanced scenarios, control of the spatial profile of the plasma current will be essential. The complexity of the current profile dynamics, arising due to nonlinearities and couplings with many other plasma parameters, motivates the use of model-based control algorithms that can account for the system dynamics. A first-principles-driven, control-oriented model of the current profile evolution in low-confinement mode (L-mode) discharges in the DIII-D tokamak is employed to address the problem of regulating the current profile evolution around desired trajectories. In the primarily inductive L-mode discharges considered in this paper, the boundary condition, which is dependent on the total plasma current, has the largest influence on the current profile dynamics, motivating the design of a boundary feedback control law to improve the system performance. The backstepping control design technique provides a systematic method to obtain a boundary feedback law through the transformation of a spatially discretized version of the original system into an asymptotically stable target system with desirable properties. Through a nonlinear transformation of the available physical actuators, the resulting control scheme produces references for the total plasma current, total power, and line averaged density, which are tracked by existing dedicated control loops. Adaptiveness is added to the control scheme to improve upon the backstepping controller's disturbance rejection and tracking capability. Prior to experimental testing, a Simserver simulation was carried out to study the controller's performance and ensure proper implementation in the DIII-D Plasma Control System. An experimental test was performed on DIII-D to test the ability of the controller to reject input disturbances and perturbations in initial conditions and to demonstrate the feasibility of the proposed control approach.

During the tokamak discharge, especially the ramp-up phase, the plasma state equilibrium continually evolves. As a consequence, the plasma response model should evolve as well. We first identified a linear plasma response model of the rotational transform ι profile and βN around a desired equilibrium. Then, an uncertainty is introduced to the identified model to partially account for the dynamic character of the plasma state equilibrium evolution. A robust controller is designed to stabilize this family of plasma models, which are reformulated into a nominal model with uncertainty. A singular value decomposition (SVD) of the nominal identified model is carried out to decouple and identify the most relevant control channels in steady-state. The DK-iteration method, combining H∞ synthesis and μ analysis, is applied to synthesize a closed-loop controller that minimizes the tracking error and input effort. The feedback controller is then augmented with an anti-windup compensator, which keeps the given profile controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop unmodified when no saturation is present. PTRANSP simulations and experimental results in DIII-D illustrate the performance of the model-based controller.

The tokamak is a device that utilizes magnetic fields to confine a reactant gas to generate energy from nuclear fusion reactions. The next step towards the realization of a tokamak power plant is the ITER project, and extensive research has been conducted to find high performance operating scenarios characterized by a high fusion gain and plasma stability. A key property related to both the stability and performance of the plasma is the safety factor profile (q-profile). In this work, a general control-oriented physics-based modeling approach is developed, with emphasis on high performance scenarios, to convert the first-principles physics model that describes the q-profile evolution in the tokamak into a form suitable for control design, with the goal of developing closed-loop controllers to drive the q-profile to a desired target evolution. The DINA-CH& CRONOS and PTRANSP advanced tokamak simulation codes are used to tailor the first-principles-driven (FPD) model to the ITER and DIII-D tokamak geometries, respectively. The model's prediction capabilities are illustrated by comparing the prediction to simulated data from DINA-CH&CRONOS for ITER and to experimental data for DIII-D.

The paper proposes a model-based control approach for the coupled evolution of the poloidal magnetic flux profile and the normalized pressure ratio βN. The model is determined by a system identification method which is shown to sufficiently reproduce the plasma response to variations in particular actuators. Data for model identification is collected during the plasma current flattop in a large βN, high-confinement scenario (H-mode) with the actuators modulated in open loop. Using this data, a linear state-space plasma response model for the poloidal magnetic flux profile and βN dynamics around a plasma equilibrium state is identified. An optimal state feedback controller with integral action is designed for the purpose of simultaneous control of the poloidal flux profile and βN. Experimental results showing the performance of the proposed controller implemented in the DIII-D tokamak are presented.

System identification techniques have been successfully used to obtain linear dynamic plasma response models around a particular equilibrium in different tokamaks. This paper identifies a two-time-scale dynamic model of the rotational transform ι profile and βN in response to the electric field due to induction as well as to heating and current drive (H&CD) systems based on experimental data from DIII-D. The control goal is to regulate the plasma ι profile and βN around a particular target value. A singular value decomposition (SVD) of the plasma model at steady state is carried out to decouple the system and identify the most relevant control channels. A mixed sensitivity H∞ control design problem is solved to determine a stabilizing feedback controller that minimizes the reference tracking error and rejects external disturbances with minimal control energy. The feedback controller is augmented with an anti-windup compensator, which keeps the given controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop unmodified when no saturation is present. Experimental results illustrate the performance of the proposed controller, which is one of the first profile controllers integrating magnetic and kinetic variables ever implemented in DIII-D.

In order for ITER to be capable of operating in advanced tokamak operating regimes, characterized by a high fusion gain, good plasma confinement, magnetohydrodynamic stability and a non-inductively driven plasma current, for extended periods of time, several challenging plasma control problems still need to be solved. Setting up a suitable toroidal current density profile in the tokamak is key for one possible advanced operating scenario characterized by non-inductive sustainment of the plasma current. At the DIII-D tokamak, the goal is to create the desired current profile during the ramp-up and early flat-top phases of the plasma discharge and then actively maintain this target profile for the remainder of the discharge. The evolution in time of the toroidal current profile in tokamaks is related to the evolution of the poloidal magnetic flux profile, which is modelled in normalized cylindrical coordinates using a first-principles, nonlinear, dynamic partial differential equation (PDE) referred to as the magnetic diffusion equation. The magnetic diffusion equation is combined with empirical correlations developed from physical observations and experimental data from DIII-D for the electron temperature, the plasma resistivity and the non-inductive current drive to develop a simplified, control-oriented, nonlinear, dynamic PDE model of the poloidal flux profile evolution valid for low confinement mode discharges. In this work, we synthesize a robust feedback controller to reject disturbances and track a desired reference trajectory of the poloidal magnetic flux gradient profile by employing the control-oriented model of the system. A singular value decomposition of the static gain matrix of the plant model is utilized to identify the most relevant control channels and is combined with the dynamic response of system around a given operating trajectory to design the feedback controller. A general framework for real-time feedforward + feedback control of magnetic and kinetic plasma profiles was implemented in the DIII-D Plasma Control System and was used to demonstrate the ability of the feedback controller to control the toroidal current profile evolution in the DIII-D tokamak. These experiments constitute the first time ever a first-principles-driven, model-based, closed-loop magnetic profile controller was successfully implemented and tested in a tokamak device.

The tokamak is a high order, distributed parameter, nonlinear system with a large number of instabilities. Therefore, accurate theoretical plasma models are difficult to develop. However, linear plasma response models around a particular equilibrium can be developed by using data-driven modeling techniques. This paper introduces a linear model of the rotational transform ι profile evolution based on experimental data from the DIII-D tokamak. The model represents the response of the ι profile to the electric field due to induction as well as to heating and current drive (H&CD) systems. The control goal is to use both induction and H&CD systems to regulate the plasma ι profile around a particular target profile. A singular value decomposition (SVD) of the plasma model at steady state is carried out to decouple the system and identify the most relevant control channels. A mixed sensitivity H∞ control design problem is formulated to synthesize a stabilizing feedback controller without input constraint that minimizes the reference tracking error and rejects external disturbances with minimal control energy. The feedback controller is then augmented with an anti-windup compensator, which keeps the given profile controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop unmodified when no saturation is present. Finally, computer simulations and experimental results illustrate the performance of the model-based profile controller.

In tokamak fusion plasmas, controlling the spatial distribution profile of the toroidal current is key to achieving advanced scenarios characterized by confinement improvement and possible steady-state operation. The dynamics of the current profile are nonlinear and coupled with other plasma parameters, motivating the use of model-based control strategies. In this work, we use a control-oriented model of the current profile evolution in DIII-D to design a feedback controller for regulating the profile around a desired trajectory. Without feedback, the response of the current profile to disturbances, model uncertainty, and perturbed initial conditions can be undesirable. To improve tracking performance of the system, a nonlinear input transformation is combined with a linear-quadratic-integral (LQI) optimal controller designed to minimize a weighted combination of the tracking error and controller effort. The resulting control law utilizes the total plasma current, total external heating power, and line averaged plasma density as actuators. We present simulation and experimental results showing successful rejection of perturbed initial conditions and input disturbances.

First-principles predictive models based on flux-averaged transport equations often yield complex expressions not suitable for real-time control implementations. It is however always possible to reduce these models to forms suitable for control design while preserving the dominant physics of the system. If further model simplification is desired at the expense of less model accuracy and controller capability, data-driven modeling emerges as an alternative to first-principles modeling. System identification techniques have the potential of producing low-complexity, linear models that can capture the system dynamics around an equilibrium point. This paper focuses on the control of the poloidal magnetic flux profile evolution in response to the heating and current drive (H&CD) systems and the total plasma current. Open-loop data for model identification is collected during the plasma current flattop in a high-confinement scenario (H-mode). Using this data a linear state-space plasma response model for the poloidal magnetic flux profile dynamics around a reference profile is identified. The control goal is to use the H&CD systems and the plasma current to regulate the magnetic profile around a desired target profile in the presence of disturbances. The target profile is defined close enough to the reference profile used for system identification in order to stay within the range of validity of the identified model. An optimal state feedback controller with integral action is designed for this purpose. Experimental results showing the performance of the proposed controller implemented in the DIII-D tokamak are presented.

Efficient, high-gain operation of a tokamak device requires the achievement of certain radial shapes for the toroidal current profile. The evolution in time of the toroidal current profile in tokamaks is related to the evolution of the poloidal magnetic flux profile. A model-based control approach for the regulation of the poloidal magnetic flux profile at the DIII-D tokamak is proposed in this work. The model describing the poloidal flux evolution is based on a controloriented formulation of the magnetic diffusion equation. Auxiliary heating and current drive (H&CD) systems including electron cyclotron (EC) and neutral beam injection (NBI) along with the total plasma current are used as actuators to manipulate the profile shape. Optimal state feedback control with integral action is used to design a controller to regulate the profile around a target while rejecting disturbances. Combining the profile controller with control of the plasma stored energy is found to improve tracking performance. Simulations and experimental results are presented to demonstrate the controller’s effectiveness.

Control of the spatial profile of plasma current in tokamak plasmas has been demonstrated to be a key condition for achieving advanced scenarios with improved confinement and possible steady-state operation. The dynamics of the current profile are nonlinear and coupled with several other plasma parameters, motivating the design of model-based controllers that can account for these complexities. In this work, we consider a control-oriented model of the current profile evolution in DIII-D and the problem of regulating the current profile around a desired feed-forward trajectory. In open-loop, the response of the system to disturbances and perturbed initial conditions may be undesirable. To improve the performance of the system, the PDE model is discretized in space using a finite difference method and a backstepping design is applied to obtain a discrete transformation from the original system into an asymptotically stable target system with desirable properties. Through a nonlinear transformation, the resulting boundary control law utilizes the total plasma current, total power, and line averaged density as actuators. A Simserver simulation study is done to test the controller's performance and its implementation in the DIII-D plasma control system. Finally, experimental results showing the ability of the controller to reject input disturbances and perturbations in initial conditions are presented.

Setting up a suitable toroidal current profile in a fusion tokamak reactor is vital to the eventual realization of a commercial nuclear fusion power plant. Creating the desired current profile during the ramp-up and early flat-top phases of the plasma discharge and then actively maintaining this target profile for the remainder of the discharge is the goal at the DIII-D tokamak. The evolution of the toroidal current profile in toka-maks is related to the evolution of the poloidal magnetic flux profile, which is modeled by the magnetic diffusion equation. A simplified first-principles-driven, nonlinear, dynamic, control-oriented, partial differential equation model of the poloidal flux profile evolution is obtained by combining the magnetic diffusion equation with empirical correlations obtained from experimental data at DIII-D and is used to synthesize a robust H∞ feedback controller to track a desired reference trajectory of the poloidal magnetic flux gradient profile. We employ a singular value decomposition of the static gain matrix of the plant model to identify the most relevant channels which we control with the feedback controller. A framework for real-time feedforward + feedback control was implemented in the DIII-D Plasma Control System and experimental results in the DIII-D tokamak are presented to illustrate the capabilities of the feedback controller. These experiments mark the first time ever a first-principles-driven model-based magnetic profile controller was successfully implemented and tested in a tokamak device.