Contributions on the control of the VGF crystal growth process

The VGF method is a modern process for the growth of single crystals, the basis for almost all electronic applications. The gradual solidification of the melt into a single crystal is realized by suitably controlling the heaters that serve as actuators in the plant. The resulting control problem is the subject of this article that aims at an illustrative introductory presentation of the topic, with mathematical details taking a background role. Based on a model of the crystal growth process that is referred to as a Stefan problem and involves partial differential equations for the heat diffusion in the crystal and melt as well as a condition at the phase boundary, two designs of tracking controllers are presented. A flatness-based approach uses a finite-dimensional approximation of the model, while the state feedback resulting from a backstepping design is infinite-dimensional. Both controllers are validated and compared to one another in simulation studies.