LMI-Based $\mathcal{H}_{\infty }$ Synthesis of the Current Profile in Tokamak Plasmas

Abstract

The $\mathcal{H}_{\infty }$ stabilization of the plasma current profile in a tokamak is invoked in this chapter to illustrate the linear matrix inequality (LMI) extension to the partial differential equation (PDE) setting. The subsequent control design is based on a one-dimensional resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. While being of a parabolic type, the underlying PDE contains a non-self-adjoint infinitesimal operator in the state equation, a feature that is not typical in the existing literature on distributed parameter system (DPS) control. The proposed distributed control is a proportional-integral state feedback that takes into account both interior and boundary engineering actuators. A target profile, which should constitute the steady state of the closed-loop system, is designed a priori, using manipulatable system inputs such as the loop voltage, the lower hybrid power, and the wave refractive index. A model-based optimization procedure is then applied at the simulation stage to derive the engineering plant inputs related to both inductive and noninductive current drive means. Numerical simulations are performed using typical Tore Supra values and yield quite positive results with promising robustness properties.