Observer based determination of the crystal diameter in Czochralski crystal growth utilizing algorithmic differentiation

This contribution deals with the implementation of a nonlinear state observer (often referred to as smart sensor, soft sensor or virtual sensor) for the reconstruction of not directly measured system quantities in the Czochralski crystal growth process. These quantities are required in automatic control systems used to ensure proper crystal growth. The implementation of the observer on the limited computer hardware available at the growth plants is a challenging task because of the complexity of the underlying mathematical model and the necessity to deal with high order partial and total derivatives of the system equations. Using the method of algorithmic differentiation an efficient way is proposed to implement such an observer. The straightforwardness of the approach is illustrated step by step and experimental results are given.