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Integrated control of the toroidal current density profile, or alternatively the q-profile, and plasma stored energy is essential to achieve advanced plasma scenarios characterized by high plasma confinement, magnetohydrodynamics stability, and noninductively driven plasma current. The q-profile evolution is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The MDE prediction depends heavily on the chosen models for the electron temperature, plasma resistivity, and non-inductive current drives. To aid control synthesis, control-oriented models for these plasma quantities are necessary to make the problem tractable. However, a relatively large deviation between the predictions by these control-oriented models and experimental data is not uncommon. For this reason, the electron temperature, plasma resistivity, and non-inductive current drives are modeled for control synthesis in this work as the product of an “uncertain” reference profile and a nonlinear function of the different auxiliary heating and current-drive (H&CD) source powers and the total plasma current. The uncertainties are quantified in such a way that the family of models arising from the modeling process is able to capture the q-profile and plasma stored energy dynamics from a typical EAST shot. A control-oriented nonlinear PDE model is developed by combining the MDE with the “uncertain” models for the electron temperature, plasma resistivity, and non-inductive current drives. This model is then rewritten into a control framework to design a controller that is robust against the modeled uncertainties. The resulting controller utilizes EAST's H&CD powers and total plasma current to regulate the q profile and plasma stored energy even when mismatches between modeled and actual dynamics are present. The effectiveness of the controller is demonstrated through nonlinear simulations.

Active control of the toroidal current density profile is critical for the upgraded National Spherical Torus eXperiment device (NSTX-U) to maintain operation at the desired high-performance, MHD-stable, plasma regime. Initial efforts towards current density profile control have led to the development of a control-oriented, physics-based, plasma-response model, which combines the magnetic diffusion equation with empirical correlations for the kinetic profiles and the non-inductive current sources. The developed control-oriented model has been successfully tailored to the NSTX-U geometry and actuators. Moreover, a series of efforts have been made towards the design of model-based controllers, including a linear-quadratic-integral optimal control strategy that can regulate the current density profile around a prescribed target profile while rejecting disturbances. In this work, the tracking performance of the proposed current-profile optimal controller is tested in numerical simulations based on the physics-oriented code TRANSP. These high-fidelity closed-loop simulations, which are a critical step before experimental implementation and testing, are enabled by a flexible framework recently developed to perform feedback control design and simulation in TRANSP.

There is an increasing need for integrating individual plasma-control algorithms with the ultimate goal of simultaneously regulating more than one plasma property. Some of these integrated-control solutions should have the capability of arbitrating the authority of the individual plasma-control algorithms over the available actuators within the tokamak. Such decision-making process must run in real time since its outcome depends on the plasma state. Therefore, control architectures including supervisory and/or exception-handling algorithms will play an essential role in future fusion reactors like ITER. However, most plasma-control experiments in present devices have focused so far on demonstrating control solutions for isolated objectives. In this work, initial experimental results are reported for simultaneous current-profile control, normalized-beta control, and Neoclassical Tearing Mode (NTM) suppression in DIII-D. Neutral beam injection (NBI), electron-cyclotron (EC) heating & current drive (H&CD), and plasma current modulation are the actuation methods. The NBI power and plasma current are always modulated by the Profile Control category within the DIII-D Plasma Control System (PCS) in order to control both the current profile and the normalized beta. EC H&CD is utilized by either the Profile Control or the Gyrotron categories within the DIII-D PCS as dictated by the Off-Normal and Fault Response (ONFR) system, which monitors the occurrence of an NTM and regulates the authority over the gyrotrons. The total EC power and poloidal mirror angles are the gyrotron-related actuation variables. When no NTM suppression is required, the gyrotrons are used by the Profile Control category, but when NTM suppression is required, the ONFR transfers the authority over the gyrotrons to the NTM stabilization algorithm located in the Gyrotron category. Initial experimental results show that simultaneous control of different aspects of the plasma dynamics may improve the overall control and plasma performances. Also, the potential of the ONFR system to successfully integrate competing control algorithms is demonstrated.

In future burning tokamaks, active control of the plasma temperature and density will be necessary to produce a determined amount of fusion power. Conventional actuation methods, such as auxiliary power modulation and fueling rate modulation, are usually employed to tackle this so-called burn control problem. Fueling rate modulation can be used not only to directly control the plasma density but also to indirectly control the plasma energy by means of isotopic fuel tailoring. Moreover, based on recent experiments, the in-vessel coil system arises as a possible actuation method to decrease the plasma energy by generating non-axisymmetric fields in the plasma that may reduce the energy confinement time. In this work, a nonlinear burn controller that integrates the three mentioned actuators is proposed. The controller performance is tested via simulations for an ITER-like scenario.

Ongoing work in the fusion community focuses on developing advanced plasma scenarios characterized by high plasma confinement, magnetohydrodynamic (MHD) stability, and noninductively driven plasma current. The toroidal current density profile, or alternatively the q profile, together with the normalized beta, are often used to characterize these advanced scenarios. The development of these advanced scenarios is experimentally carried out by specifying the devices’ actuator trajectory waveforms, such as the total plasma current, the plasma density, and the auxiliary heating and current-drive (H&CD) sources based on trial-and-error basis. In this work, a model-based numerical optimization approach is followed to complement the experimental effort on actuator trajectory planning in the EAST tokamak. The evolution of the q profile is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). In this work, the MDE is combined with physics-based correlations obtained from EAST experimental data for the plasma density, temperature, resistivity and non-inductive current drives to develop a control-oriented nonlinear PDE model. The optimization objective is to design feedforward trajectories for the plasma current, plasma density, electron cyclotron heating power, neutral beam injection power and lower hybrid current drive power that steer the plasma to desired q profile and βN such that the achieved state is stationary in time. The optimization is subject to the plasma dynamics (described by the physics-based PDE model) and plasma state and actuator constraints, such as the maximum available amount of H&CD power and MHD stability limits. This defines a nonlinear, constrained optimization problem that is solved by employing sequential quadratic programming. The optimized actuator trajectories are assessed in nonlinear transport simulations in preparation for experimental tests in EAST.

Active control of the toroidal current density profile is among those plasma control milestones that the National Spherical Torus eXperiment-Upgrade (NSTX-U) program must achieve to realize its next-step operational goals. Motivated by the coupled, nonlinear, multivariable, distributed-parameter plasma dynamics, the first step towards control design is the development of a physics-based, control-oriented model for the current profile evolution in response to non-inductive current drives and heating systems. The evolution of the toroidal current density profile is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The proposed control-oriented model predicts the spatial-temporal evolution of the current density profile by combining the nonlinear MDE with physics-based correlations obtained at NSTX-U for the electron density, electron temperature, and non-inductive current drives (neutral beams). The resulting first-principles-driven, control-oriented model is tailored for NSTX-U based on predictions by the time-dependent transport code TRANSP. Main objectives and possible challenges associated with the use of the developed model for the design of both feedforward and feedback controllers are also discussed.

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.

In future tokamaks, the control of burning plasmas will require careful regulation of the plasma density and temperature. Along with the design of effective burn-control systems, understanding how the fusion power varies in the density-temperature space is vital for the operation of fusion power plants. In this work, the steady-state operational space of ITER is studied using a control-oriented core-plasma model coupled to a two-point model of the scrape-off-layer (SOL) and divertor regions. The two models are coupled through the exchange of input-output parameters. The deuterium and tritium recycling from the wall are output parameters of the SOL-divertor model that are used as input parameters in the core-plasma density balance. Furthermore, the separatrix temperature, which is an output parameter of the SOL-divertor model, is incorporated into the radial core-plasma temperature profiles. Therefore, the temperature-dependent power balance of the plasma core is intimately linked to the SOL-divertor model. Both the power entering the SOL from the core, as determined by the core-plasma power balance, and the separatrix density, as dictated by the core-plasma density balance, are input parameters to the SOL-divertor model. They are control knobs in the SOL-divertor model that can be regulated using the core-plasma actuators: auxiliary power and pellet injection. There are various operational limitations, such as the saturation of the aforementioned actuators, that will prevent ITER from accessing certain high-fusion plasma regimes. The achievable tritium concentration in the fueling lines and the maximum sustainable heat load on the divertor will impose further restrictions. By accounting for these limitations, the ITER operational space is computed based on the coupled core-SOL-divertor model and visualized using Plasma Operation Contour (POPCON) plots that map performance metrics, such as the fusion to auxiliary power ratio, over the density-temperature space. Comparisons are drawn between plasmas with different recycling, confinement, and SOL-divertor conditions.

An integrated-control architecture for simultaneous regulation of the plasma internal inductance and normalized beta has been designed and tested in simulations using COTSIM (Control-Oriented Transport SIMulator). As present-day tokamaks evolve into nuclear-fusion reactors capable of producing net energy, a significant control-engineering challenge must be solved: regulating a wide variety of plasma variables, often simultaneously, by employing only a reduced number of actuators. As a contribution towards this objective, the present work tackles the problem of controlling the plasma internal inductance, which is a proxy for the broadness of the current-density profile, simultaneously with the plasma normalized beta. Based on zero-dimensional, control-oriented models of the plasma dynamics, individual Lyapunov-theory-based controllers for the internal inductance and normalized beta have been developed. These controllers are integrated by means of an actuator manager that decides, in real time, how the available actuators are utilized in order to fulfill as many control objectives as possible. In addition, the actuator manager is designed to achieve a particular performance metric defined by the control engineer. This metric could be, for example, prioritizing a particular control task over the others and/or minimizing the use of a particular actuator during certain phases of the plasma discharge. Using COTSIM, which includes one-dimensional models of the plasma current-density and electron-temperature dynamics, the performance of the integrated-control framework has been tested in a steady-state scenario for the DIII-D tokamak. These simulation results yield illustrative insights into the plasma current-density and electron-temperature controllability with the current actuation capabilities in DIII-D. Moreover, these simulations show that the way in which the different actuators are employed during the discharge (based on the choice of the aforementioned actuator-manager performance metric) highly determines the value of internal inductance and normalized beta achieved in steady-state conditions, and therefore, the final current-profile shape.

A control-oriented, two-timescale, linear, dynamic, response model of the rotational transform ι profile and the normalized beta βN is proposed based on experimental data from the DIII-D tokamak. Dedicated system-identification experiments without feedback control have been carried out to generate data for the development of this model. The data-driven dynamic model, which is both device-specific and scenario-specific, represents the response of the ι profile and βN to the electric field due to induction as well as to the heating and current drive (H&CD) systems during the flat-top phase of an H-mode discharge in DIII-D. The control goal is to use both induction and the H&CD systems to locally regulate the plasma ι profile and βN around particular target values close to the reference state used for system identification. 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 robust control design problem is formulated based on the dynamic model to synthesize a stabilizing feedback controller without input constraints 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 controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop system unmodified when no saturation is present. The proposed controller represents one of the first feedback profile controllers integrating magnetic and kinetic variables ever implemented and experimentally tested in DIII-D. The preliminary experimental results presented in this work, although limited in number and constrained by actuator problems and design limitations, as it will be reported, show good progress towards routine current profile control in DIII-D and leave valuable lessons for further advancements in the field.

A neural network model of the effects of neutral beam injection on DIII-D has been developed. The training and testing data used by the model have been generated by the NUBEAM module of TRANSP for experimental discharges from the 2018 DIII-D campaign. Using a principle component analysis to reduce the dimensionality of profile data, the model has been shown to reproduce the results of the Monte Carlo code NUBEAM with a high level of accuracy and an execution time orders of magnitude faster than the execution time of NUBEAM. This makes the neural network model uniquely suited to applications in model-based scenario planning (off-line) and active control (on-line), where a large number of simulation runs are required by the associated optimization tasks that need to be performed before and during the discharge.

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.

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.

The next experimental step in the development of nuclear fusion reactors is the ITER tokamak. It is designed to explore the burning plasma regime in which the plasma temperature is sustained mostly by self-heating from fusion reactions. Burn control, the control of fusion power and other reactor parameters through modulation of fueling and heating, will be essential for achieving and maintaining desired operating points and ensuring stability. Design of burn control strategies is made challenging by the multi-variable, highly nonlinear, uncertain nature of the system. Furthermore, due to the extreme conditions in fusion reactors, diagnostic systems may be limited. To deal with these challenges, we propose the use of a nonlinear, multi-variable output feedback control strategy with a proportional-integral observer. A simulation study is carried out to illustrate the performance of the scheme using a set of diagnostics likely to be available in ITER.

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 control-oriented 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.

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.

Setting up a suitable current profile, characterized by a weakly reversed magnetic shear, has been demonstrated to be a key condition for one possible advanced tokamak operating scenario with improved confinement and possible steady-state operation. Experiments at DIII-D focus on creating the desired current profile during the plasma current ramp-up and early flat-top phases with the aim of maintaining this target profile throughout the subsequent phases of the discharge. The evolution in time of the current profile, or alternatively the safety factor q, is related to the evolution of the poloidal flux, which is modeled in normalized cylindrical coordinates using a partial differential equation referred to as the magnetic flux diffusion equation. A control-oriented model of the current profile evolution in DIII-D was recently developed for the plasma current ramp-up and early flat-top phases and used to synthesize both open-loop and closed-loop control schemes. In this work, we report on the implementation of an advanced model-based current profile controller in the DIII-D Plasma Control System (PCS) and on the assessment of this controller implementation in closed-loop Simserver (simulation server) simulations.

It is sometimes incorrectly assumed that, because superconducting tokamaks already have significant intrinsic or imposed sources of control delay, introducing extra delays/lags into the axisymmetric control loops will have negligible detrimental impact on the plasma control. This study exposes and quantifies the detrimental effects imposed by time delays/lags in the control loop in superconducting tokamaks, using as an example the plasma current control and radial position control in a vertically stable circular plasma in the KSTAR tokamak. Delays and lags in the power supplies, data acquisition, and vessel structure are taken into account. Optimal tuning of PID controllers in combination with an ohmic-flux control strategy is proposed as a possible method for remediating the negative effects of time delays/lags. In addition, an augmentation of the control loop by the introduction of a robust predictor has been proposed to improve the performance of the time-delayed closed-loop system when the amount of delay/lag in the loop is unknown. The Nyquist dual locus technique based on the Argument Principle in complex theory is employed to assess stability of the optimally tuned closed-loop system in the presence of time delays.

Plasma current, position and shape control is a challenging problem due to the strong coupling between the different parameters describing the shape of the plasma. By leveraging the availability of rtEFIT, this paper proposes a robust model-based multi-input–multi-output (MIMO) controller to provide real-time shaping, position stabilization and current regulation in NSTX. The proposed controller is composed of three loops: the first loop is devoted to plasma current regulation, the second loop is dedicated to plasma radial and vertical position stabilization, and the third loop is used to control the plasma shape. This control approach transforms the shape control problem into an output tracking problem. The goal is the minimization of a quadratic cost function that describes the tracking error in steady state. A singular value decomposition (SVD) of the nominal plasma model is carried out to decouple and identify the most relevant control channels. The H ∞ technique is used to minimize the tracking errors and optimize input efforts. Computer simulation results illustrate the performance of the robust model-based shape controller, showing potential for improving the performance of present non-model-based controllers.

We present a framework to solve a finite-time optimal control problem for parabolic partial differential equations (PDEs) with diffusivity-interior actuators, which is motivated by the control of the current density profile in tokamak plasmas. The proposed approach is based on reduced order modeling (ROM) and successive optimal control computation. First we either simulate the parabolic PDE system or carry out experiments to generate data ensembles, from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained reduced order model corresponds to a bilinear control system. Based on quasi-linearization of the optimality conditions derived from Pontryagin's maximum principle, and stated as a two boundary value problem, we propose an iterative scheme for suboptimal closed-loop control. We take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers. The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative schemes for optimal control of bilinear systems, the proposed scheme avoids repeated numerical computation of the Riccati equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical simulation study shows the effectiveness of this approach.

The control of the toroidal current density spatial profile in tokamak plasmas will be absolutely critical in future commercial-grade reactors to enable high fusion gain, non-inductive sustainment of the plasma current for steady-state operation, and magnetohydrodynamic (MHD) instability-free performance. The evolution in time of the current profile is related to the evolution of the poloidal magnetic flux, which is modeled in normalized cylindrical coordinates using a partial differential equation (PDE) usually referred to as the magnetic flux diffusion equation. The control objective during the ramp-up phase is to drive an arbitrary initial profile to approximately match, in a short time windows during the early flattop phase, a predefined target profile that will be maintained during the subsequent phases of the discharge. Thus, such a matching problem can be treated as an optimal control problem for a PDE system. A distinctive characteristic of the current profile control problem in tokamaks is that it admits interior, boundary and diffusivity actuation. A receding-horizon control scheme is proposed in this work to exploit this unique characteristic and to solve the associated open-loop finite-time optimal control problem using different optimization techniques. The efficiency of the proposed scheme is shown in simulations.

One of the major non-axisymmetric instabilities under study in the DIII-D tokamak is the resistive wall mode (RWM), a form of plasma kink instability whose growth rate is moderated by the influence of a resistive wall. One of the approaches for RWM stabilization, referred to as magnetic control, uses feedback control to produce magnetic fields opposing the moving field that accompanies the growth of the mode. These fields are generated by coils arranged around the tokamak. One problem with RWM control methods used in present experiments is that they predominantly use simple non-model-based proportional-derivative (PD) controllers requiring substantial derivative gain for stabilization, which implies a large response to noise and perturbations, leading to a requirement for high peak voltages and coil currents, usually leading to actuation saturation and instability. Motivated by this limitation, current efforts in DIII-D include the development of model-based RWM controllers. The General Atomics (GA)/Far-Tech DIII-D RWM model represents the plasma surface as a toroidal current sheet and characterizes the wall using an eigenmode approach. Optimal and robust controllers have been designed exploiting the availability of the RWM dynamic model. The controllers are tested through simulations, and results are compared to present non-model-based PD controllers. This comparison also makes use of the μ structured singular value as a measure of robust stability and performance of the closed-loop system.

An integrated combustion optimization approach is presented for the combined considering the trade offs in optimization of coal-fired boiler and selective catalyst reaction (SCR) system, to balance the unit thermal efficiency, SCR reagent consumption and NOx emissions. Field tests were performed at a 160 MW coal-fired unit to investigate the relationships between process controllable variables, and optimization targets and constraints. Based on the test data, a modified on-line support vector regression model was proposed for characteristic function approximation, in which the model parameters can be continuously adapted for changes in coal quality and other conditions of plant equipment. The optimization scheme was implemented by a genetic algorithm in two stages. Firstly, the multi-objective combustion optimization problem was solved to achieve an optimal Pareto front, which contains optimal solutions for lowest unit heat rate and lowest NOx emissions. Secondly, best operating settings for the boiler, and SCR system and air preheater were obtained for lowest operating cost under the constraints of NOx emissions limit and air preheater ammonium bisulfate deposition depth.

A nonlinear Lyapunov-based boundary feedback control law is proposed for mixing enhancement in a 2D magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting, incompressible, and subject to an external transverse magnetic field. The MHD model is a combination of the Navier-Stokes PDE and the Magnetic Induction PDE, which is derived from the Maxwell equations. Pressure sensors, magnetic field sensors, and micro-jets embedded into the walls of the flow domain are employed for mixing enhancement feedback. The proposed control law, designed using passivity ideas, is optimal in the sense that it maximizes a measure related to mixing (which incorporates stretching and folding of material elements), while at the same time minimizing the control and sensing efforts. A DNS code is developed, based on a hybrid Fourier pseudospectral-finite difference discretization and the fractional step technique, to numerically assess the controller.

a b s t r a c t We present a boundary control law that stabilizes the Hartman profile for low magnetic Reynolds numbers in an infinite magnetohydrodynamic (MHD) channel flow. The proposed control law achieves stability in the L 2 norm of the linearized MHD equations, guaranteeing local stability for the fully

We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier-Stokes equations and a Poisson equation for the electric potential under the so-called inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure, and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and current measurements from one of the walls are used for output injection. For zero magnetic field or nonconducting fluid, the design reduces to an observer for the Navier-Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation. We show that for the linearized MHD model the estimation error converges to zero in the L^2 norm. Despite being a subject of practical interest, the problem of observer design for nondiscretized 3-D MHD or Navier-Stokes channel flow has so far been an open problem.

A key goal in the control of a magnetic fusion reactor is to maintain current profiles that are compatible with a high fraction of self-generated non-inductive current as well as with magnetohydrodynamic (MHD) stability at high plasma pressure. This enables high fusion gain and noninductive sustainment of plasma current for steady-state operation. The approach taken toward establishing such plasma current profiles at the DIII-D tokamak is to create the desired profile during the plasma current ramp-up and early flattop phases. The evolution in time of the current profile is related to the evolution of the poloidal flux, which is modeled in normalized cylindrical coordinates using a nonlinear partial differential equation (PDE) usually referred to as the magnetic diffusion equation. We propose, and test in simulations, an extremum-seeking-based, receding-horizon, diffusivity-interior-boundary control scheme designed to match as close as possible a desired current profile within a prespecified time interval.

In this work, µ-synthesis is employed to stabilize a model of the resistive wall mode (RWM) instability in the DIII-D tokamak. The GA/Far-Tech DIII-D RWM model is used to derive a linear state space representation of the mode dynamics. The key term in the model characterizing the magnitude of the instability is the time-varying uncertain parameter cpp, which is related to the RWM growth rate γ. Taking advantage of the structure of the state matrices, the model is reformulated into a robust control framework, with the growth rate of the RWM modeled as an uncertain parameter. A robust controller that stabilizes the system for a range of practical growth rates is proposed and tested through simulations.

The matching problem for a low energy transport system in a charged particle accelerator is approached using the extremum seeking method for non-model based optimization and adaptive control. The beam dynamics used for numerical simulations are given by the KV (Kapchinsky–Vladimirsky) envelope equations. Extremum seeking is employed for the lens tuning in both a matching and a periodic channel. Numerical simulations illustrate the effectiveness of this approach. r 2007 Elsevier B.V. All rights reserved. PACS: 29.27.� a; 29.27.Eg; 02.30.Yy; 41.85.Lc

A key goal in control of an advanced tokamak (AT) discharge is to maintain safety factor (q) and pressure profiles that are compatible with both MHD stability at high toroidal beta and a high fraction of the self-generated bootstrap current. This will enable high fusion gain and non-inductive sustainment of the plasma current for steady-state operation. In this work we report progress towards enabling model-based active control of the current profile at DIII-D. Initial results on modeling-for-control and simulation of the dynamic evolution of the poloidal flux profile during and just following the ramp-up of the plasma current are presented. The magnetic diffusion equation is combined with empirical correlations obtained at DIII-D for the density, temperature and non-inductive current to introduce a simplified dynamic model describing the evolution of the poloidal flux, and therefore the q profile, during the inductive phase of the discharge. The physical model is rewritten in a control-oriented formulation and the control challenges asocciated with the problem are discussed.

Equilibrium reconstruction codes calculate the distributions of flux and toroidal current density over the plasma and surrounding vacuum region that best fit the external magnetic measurements in a least square sense, and that simultaneously satisfy the MHD equilibrium equation (Grad-Shafranov equation). Although these codes often use direct measurements of the currents in the plasma and poloidal coils, they sometimes neglect the current induced in the tokamak vessel due to the fact that they cannot be directly measured. Kalman filtering theory is employed in this work to optimally estimate the current in the tokamak vessel. The real-time version of the EFIT code is modified to accept the estimated vessel currents with the goal of improving the equilibrium reconstruction for the DIII-D tokamak.

In the advanced tokamak (AT) operating mode of the DIII-D tokamak, an integrated multivariable controller takes into account highly coupled influences of plasma equilibrium shape, profile, and stability control. Time-scale separation in the system allows a multi-loop design: the inner loop closed by the nominal vertical controller designed to control a linear exponentially unstable plant and the outer loop closed by the nominal shape controller designed to control a linear stabilized plant. Due to actuator constraints, the nominal vertical controller fails to stabilize the vertical position of the plasma inside the tokamak when large or fast disturbances are present or when the references coming from the shape controller change suddenly. Anti-windup synthesis is proposed in this paper to find a nonlinear modification of the nominal vertical controller that prevents vertical instability and undesirable oscillations but leaves the inner loop unmodified when there is no input saturation.