Towards model based control of the Vertical Gradient Freeze crystal growth process

TL;DR

This work addresses tracking control for a one-dimensional two-phase Stefan problem modeling Vertical Gradient Freeze crystal growth, a regime where a moving solid–liquid interface governs process quality. It develops both collocated and distributed feedback strategies, augmented by an observer, and leverages differential flatness to construct feedforward references and finite-dimensional state representations. The main contributions are a Lyapunov-based stability analysis for collocated control, a flatness-based distributed control using a truncated state, and a practical observer design with Riccati-based gains, all validated by simulations showing effective reference tracking and robustness to disturbances. The results advance model-based control for industrial crystal growth, offering methodologies to regulate interface position and growth-rate while mitigating remelting risks.

Abstract

In this contribution tracking control designs using output feedback are presented for a two-phase Stefan problem arising in the modeling of the Vertical Gradient Freeze process. The two-phase Stefan problem, consisting of two coupled free boundary problems, is a vital part of many crystal growth processes due to the temporally varying extent of the solid and liquid domains during growth. After discussing the special needs of the process, collocated as well as flatness-based state feedback designs are carried out. To render the setup complete, an observer design is performed, using a flatness-based approximation of the original distributed parameter system. The quality of the provided approximations as well as the performance of the open and closed loop control setups is analysed in several simulations.

Figures (8)

• Figure 1: Sketch of a crystal growth furnace.
• Figure 2: Schematics of the cylindrical coordinate system $(r, \varphi, z, t)$, a meridional plane (blue) and the shifted coordinate $\tilde{z} = z - \gamma _{\mathrm{}}^{}$.
• Figure 3: Reference trajectories for gradients and phase boundary (top) as well as the generated heater trajectories for the system inputs (bottom) of the feedforward control.
• Figure 4: Calculated reference temperature profile $T_{\mathrm{r}}(z, t)$ with the reference phase boundary trajectory $\gamma _{\mathrm{r}}^{}$ (blue).
• Figure 5: Comparison of the tracking behaviour of the reference (blue) between collocated (orange) and distributed feedback (green) using the original (solid) or modified error (dashed), respectively. Note that the orange and green dashed lines are nearly equal.