# Optimization Methods for Robust Control with Application to Resistive Wall Modes in Tokamaks Doctoral thesis, 2003

In many research areas physical systems are described by mathematical models. We focus here on systems that are regulated by a controller, typically realized by software, whose design involves certain parameters. Inevitably, there will be differences between the behavior of the physical system and the model (uncertainties), and it is desirable that the control system meet certain demands in the face of all likely uncertainties. Achieving such reliability is the purpose of robust control. This thesis aims at solving the robust control problem by combining techniques for numerical optimization with theory for robust control. For the latter, w e consider PID and H∞ controllers, and the design problem is formulated as a nonlinear optimization problem with controller parameters as unknowns, and closed loop robustness and performance measures entering as constraints (criteria) and as objective function. Robustness to model uncertainties is guaranteed by use of quantitative feedback theory and structured singular values. In particular, a conceptually simple, and sometimes efficient, methodology is presented for computing an upper bound of the supremum of the structured singular value with respect to frequency. In solving the robust control problem, we make use of local optimization algorithms (using sequential quadratic programming) and semi-definite programming (using interior point methods to satisfy Linear Matrix Inequalities). Since all problems do not have unique solutions, global optimization also plays an important role. The design method is applied to two different systems. The first is the addition of external carbon to the Rya wastewater treatment plant in Göteborg. Uncertainty models are derived and it is shown in a feedback/feedforward design that both parts of the controller have to be designed jointly for the performance to be optimal, and that the feedforward controller is particularly sensitive to the uncertainties. The second application is one of the bottleneck problems in the advanced tokamak (the leading candidate for a fusion reactor); for general scenarios and for the DIII-D tokamak at General Atomics. Plant models from perturbed magnetic fields at various sensor locations to voltage and current in feedback coils are developed to stabilize non-axisymmetric, pressure driven, resistive wall modes (RWMs). The models for DIII-D are validated with recent experimental data and show good agreement with reality. A Kalman filter is applied to discriminate unwanted contributions in the sensor signals, improving reliability of the control system. Control analyses show that simple controllers in combination with a surrounding wall can achieve stabilization of the RWMs, and although the resulting closed loop performance is not always ideal, it is shown that closed loop behavior can be significantly improved in DIII-D, and that pressure limits in advanced scenarios can be increased.

H?

fusion

QFT

performance

tokamak

optimization

robustness

structured singular value

control

plasma

resistive wall modes

LMI

PID control

## Author

### Carl-Magnus Fransson

Chalmers, Signals and Systems

### Subject Categories

Electrical Engineering, Electronic Engineering, Information Engineering

### ISBN

91-7291-331-2

Technical report - School of Electrical Engineering, Chalmers University of Technology, Göteborg, Sweden: 454

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 213