Control of the Vertical Gradient Freeze crystal growth process via backstepping

TL;DR

This work develops a backstepping-based state-feedback controller for tracking in a two-phase Stefan problem arising in Vertical Gradient Freeze crystal growth, addressing the coupled solid–melt dynamics with a moving interface $\gamma(t)$. It couples a Hopf-Cole transformation to neutralize convection and a Volterra backstepping transform to map the error system to a stable target, deriving time-varying kernel equations whose existence and numerical solution are analyzed via an integral formulation and spatial discretization. A feedforward reference construction based on differential flatness supports arbitrary-tracking of the interface and temperature profiles, with a Gevrey-class smoothness requirement on the reference trajectories. Numerical simulations in a 400 mm furnace using a lumped FE model demonstrate convergence of temperature and interface errors, validating the approach and highlighting the need for further stability analysis and observer design for practical output feedback.

Abstract

This contribution presents a backstepping-based state feedback design for the tracking control of a two-phase Stefan problem which is encountered in the Vertical Gradient Freeze crystal growth process. A two-phase Stefan problem consists of two coupled free boundary problems and is a vital part of many crystal growth processes due to the time-varying extent of crystal and melt during growth. In addition, a different approach for the numerical approximation of the backstepping transformations kernel is presented.

Figures (3)

• Figure 1: Schematics of the cylindrical coordinates $(r, \varphi, z)$ and the moving coordinate frame $\tilde{z} = z - \gamma _{\mathrm{}}^{}$.
• Figure 2: Discretised kernel domain with selected kernel elements and required derivative orders for the computation of $k_{i,j}(t)$ (shaded).
• Figure 3: Trajectories of the temperature errors $L_{2}$ norm (solid) and the interface deviation $\Delta \gamma _{\mathrm{}}^{}$ (dashed).