Quasi-Online Disturbance Rejection for Nonlinear Parabolic PDE using a Receding Time Horizon Control

Abstract

Null controllability of nonlinear partial differential equation is a very complex challenge. The context underlying this study is to improve the behavior of the plasma in a tokamak reactor in order to lengthen the duration of the nuclear fusion process. Considering the class of a specific parabolic PDE, the well known heat equation is nonlinear if thermal properties are temperature dependent. In such a context a numerical method based on the resolution of inverse heat conduction problem is proposed. It aims to provide identified control laws quasi-online in order to guarantee that the thermal state is kept close to its equilibrium state at zero. The iterative conjugate gradient method is implemented in order to control the temperature in the one-dimensional spatial domain despite several disturbances (time-dependent or thermo-dependent). The proposed strategy is based on successive numerical resolutions of ill-posed inverse problem on receding time horizons which are adapted considering the previous evolution of the system. Numerical results in the investigated configuration highlight that identified control laws are able to reject disturbances and to ensure null controllability.