Nonlinear state estimation for the Czochralski process based on the weighing signal using an extended Kalman filter

Abstract

The paper presents real-time estimation of not directly measurable quantities in the Czochralski process using an extended Kalman filter. The Kalman filter is proposed as an alternative to the nonlinear tracking observer used in the control concept of Winkler et al. (2010) [1,2]. Within the scope of this paper an approach of optimal adjustment of the filter parameters is worked out and the results of its successful application are presented. The hydrodynamical-geometrical model based on the weighing signal out of Winkler et al. (2010) [1] is adopted for the Kalman filter. Avoiding complex thermodynamic modeling, different methods of model free estimation of the growth rate are proposed and evaluated. The designed Kalman filter is compared to the nonlinear observer. It can be shown that the observer׳s divergence in consequence of an unobservable point at one particular crystal shape trajectory can be avoided using a Kalman filter with model-free estimation of the growth rate. Furthermore there is a material-specific problem of the nonlinear observer in case of silicon only – a systematic phase-shift between the crystal radius and the slope angle. This problem could likewise be removed by the designed Kalman filter.