K. Erik J. Olofsson

Vector dither experiment design and direct parametric identification of reversed-field pinch normal modes

Magnetic confinement fusion (MCF) research ambitiously endeavours to develop a major future energy source. MCF power plant designs, typically some variation on the tokamak, unfortunately suffer from magnetohydrodynamic (MHD) instabilities. One unstable mode is known as the resistive-wall mode (RWM) which is a macroscopically global type of perturbation that can degrade or even terminate the plasma in the reactor if not stabilized. In this work the topic of RWMs is studied for the reversed-field pinch (RFP), another toroidal MCF concept, similar to the tokamak. The problem of identifying RWM dynamics during closed-loop operation is tackled by letting physics-based parametric modeling join forces with convex programming experiment design. An established MHD normal modes description is assessed for the RFP by synthesizing a multivariable dither signal where spatial Fourier modes are spectrally shaped, with regard to real experiment constraints, to yield minimum variance parameter estimates in the prediction-error framework. The dithering is applied to the real RFP plant EXTRAP-T2R, and experimental MHD spectra are obtained by an automated procedure.

Stability analysis and model-based control in EXTRAP-T2R with time-delay compensation

In this paper, we investigate the stability problems and control issues that occur in a reversed-field pinch (RFP) device, EXTRAP-T2R (T2R), used for research in fusion plasma physics and general plasma (ionized gas) dynamics. The plant exhibits, among other things, magnetohydrodynamic instabilities known as resistive-wall modes (RWMs), growing on a time-scale set by a surrounding non-perfectly conducting shell.We propose a novel model that takes into account experimental constraints, such as the actuators dynamics and control latencies, which lead to a multivariable time-delay model of the system. The open-loop field-error characteristics are estimated and a stability analysis of the resulting closed-loop delay differential equation (DDE) emphasizes the importance of the delay effects. We then design a structurally constrained optimal PID controller by direct eigenvalue optimization (DEO) of this DDE. The presented results are substantially based on and compared with experimental data.

Synthesis and operation of an FFT-decoupled fixed-order reversed-field pinch plasma control system based on identification data

Recent developments and applications of system identification methods for the reversed-field pinch (RFP) machine EXTRAP T2R have yielded plasma response parameters for decoupled dynamics. These data sets are fundamental for a real-time implementable fast Fourier transform (FFT) decoupled discrete-time fixed-order strongly stabilizing synthesis as described in this work. Robustness is assessed over the data set by bootstrap calculation of the sensitivity transfer function worst-case -gain distribution. Output tracking and magnetohydrodynamic mode m = 1 tracking are considered in the same framework simply as two distinct weighted traces of a performance channel output-covariance matrix as derived from the closed-loop discrete-time Lyapunov equation. The behaviour of the resulting multivariable controller is investigated with dedicated T2R experiments.

Cascade and multibatch subspace system identification for multivariate vacuum-plasma response characterisation

A particular cascade structure system identification problem is formulated for the purpose of characterising the vacuum-plasma response for a magnetic confinement fusion experiment. A predictor-form closed-loop subspace system identification approach is advocated due to (i) plant instability (ii) sizes of input-output vectors and (iii) inherent multivariate eigenmodes of the physical system. Since experiment data come in relatively short batches, specialised means for data merging for subspace identification are developed. A batchwise delete-group jackknife procedure is utilised to estimate the standard error of the estimate of the dominant unstable empirical plasma response eigenvalue.

Experimental modal analysis of resistive wall toroidal pinch plasma dynamics

The linearized magnetofluid dynamics of a magnetically confined reversed-field pinch plasma is extracted from a set of perturbative randomized nondestructive experiments. The spectrum of the generically estimated linear time-invariant system is compared to solutions of the corresponding eigenvalue problems formed by the ideal magnetohydrodynamic model. Stable and unstable nonresonant resistive wall modes are accurately identified.

A first attempt at few coils and low-coverage resistive wall mode stabilization of EXTRAP T2R

The reversed-field pinch features resistive-shell-type instabilities at any (vanishing and finite) plasma pressure. An attempt to stabilize the full spectrum of these modes using both (i) incomplet ...

Closed-loop MIMO ARX estimation of concurrent external plasma response eigenmodes in magnetic confinement fusion

Detailed experimental MIMO models of plasma stability behaviour are becoming increasingly important in magnetic confinement fusion (MCF) energy research as an assortment of magnetohydrodynamic (MHD) instabilities develop when fusion performance is pushed. Some of these problems could perhaps be handled by magnetic feedback. We here show a practical method for experimental closed-loop multi-input multi-output (MIMO) characterisation of the macroscopic stability of toroidal MCF devices. It is demonstrated, by application to the MCF experiment EXTRAP T2R, that MHD eigenmodes can be detected using the workhorse MIMO autoregressive exogeneous (ARX) model structure. Plausibly, the presented methodology could significantly improve highly-desired magnetic feedback accuracy in MCF.

Extended benchmark results for impulse response estimation

Recent attention has been given to the use of regularised least-squares methods for improving the quality of classical impulse response estimates from short/noisy datasets. This paper significantly augments and contextualises previously reported benchmark results by inclusion of a larger set of standard and non-standard methods in the comparison. The numerical experiments are summarised using several metrics. There is no obvious winner method.

A Practical Approach to Input Design for Modal Analysis Using Subspace Methods

A basic class of multivariate system identification input design methods is proposed. Only reliable numerical linear algebra is used. The underlying idea is to inject energy into the invariant eigenspace for a subset of preestimated plant eigenvalues. Standard Schur pseudotriangular factorisation is used to pretarget a subsequent singular value decomposition. Explicit state-space formulas are given. Examples indicate that the approach may be useful in some practical applications. The approach may be considered user-friendly.