In this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as close as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances.

https://hal.archives-ouvertes.fr/hal-00719502/document

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

Real-time simultaneous control of several radially distributed magnetic and kinetic plasma parameters is being investigated on JET, in view of developing integrated control of advanced tokamak scenarios. This paper describes the new model-based profile controller which has been implemented during the 2006–2007 experimental campaigns. The controller aims to use the combination of heating and current drive (H&CD) systems—and optionally the poloidal field (PF) system—in an optimal way to regulate the evolution of plasma parameter profiles such as the safety factor, q(x), and gyro-normalized temperature gradient, ρ ∗ (x). In the first part of the paper, a technique for the experimental identification of a minimal dynamic plasma model is described, taking into account the physical structure and couplings of the transport equations, but making no quantitative assumptions on the transport coefficients or on their dependences. To cope with the high dimensionality of the state space and the large ratio between the time scales involved, the model identification procedure and the controller design both make use of the theory of singularly perturbed systems by means of a two-time-scale approximation. The second part of the paper provides the theoretical basis for the controller design. The profile controller is articulated around two composite feedback loops operating on the magnetic and kinetic time scales, respectively, and supplemented by a feedforward compensation of density variations. For any chosen set of target profiles, the closest self-consistent state achievable with the available actuators is uniquely defined. It is reached, with no steady state offset, through a near-optimal

http://iopscience.iop.org/article/10.1088/0029-5515/48/10/106001/pdf

A two-time-scale dynamic-model approach for magnetic and kinetic profile control in advanced tokamak scenarios on JET

A real-time model-based controller is developed for the tracking of the distributed safety-factor profile in a tokamak plasma. Using relevant physical models and simplifying assumptions, theoretical stability and robustness guarantees were obtained using a Lyapunov function. This approach considers the couplings between the poloidal flux diffusion equation, the time-varying temperature profiles and an independent total plasma current control. The actuator chosen for the safety-factor profile tracking is the lower hybrid current drive, although the results presented can be easily extended to any non-inductive current source. The performance and robustness of the proposed control law is evaluated with a physics-oriented simulation code on Tore Supra experimental test cases.

/pdf/lyapunov-based-distributed-control-of-the-safety-factor-5926oes3ap.pdf

Lyapunov-based distributed control of the safety-factor profile in a tokamak plasma

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

Backstepping Control of the Toroidal Plasma Current Profile in the DIII-D Tokamak

Receding-horizon optimal control of the current profile evolution during the ramp-up phase of a tokamak discharge

This paper describes several different tokamak plasma instabilities and the control approaches for their stabilization. It also describes the progress achieved at tokamak devices around the world

T.H. Osborne

Papers

Emerging applications in tokamak plasma control