Gray-box model-based predictive control of Czochralski process

The present study proposes a gray-box (GB) model-based predictive control method to produce high-quality 300 mm silicon ingots in the commercial Czochralski (CZ) process. The GB model consists of an energy transfer, hydrodynamic, and geometrical model and a statistical model, predicts three controlled variables, i.e., crystal radius, growth rate, and melt position, and represents the time-varying and nonlinear characteristics of the CZ process. Solving an optimization problem with the GB model requires heavy computational load; therefore, the proposed method derives the prediction model by successive linearization of the GB model to compute optimal manipulated variables in several seconds. The proposed method was compared with the conventional method using PID controllers in disturbance rejection performance through control simulations. The results have demonstrated that the integral absolute error (IAE) of the proposed method was reduced by 60% on average and 89% at maximum even when a plant-model mismatch exists.