Nonlinear model-based control of the Czochralski process III: Proper choice of manipulated variables and controller parameter scheduling

This contribution continues an article series [1] and [2] about the nonlinear model-based control of the Czochralski crystal growth process. The key idea of the presented approach is to use a sophisticated combination of nonlinear model-based and conventional (linear) PI controllers for tracking of both, crystal radius and growth rate. Using heater power and pulling speed as manipulated variables several controller structures are possible. The present part tries to systematize the properties of the materials to be grown in order to get unambiguous decision criteria for a most profitable choice of the controller structure. For this purpose a material specific constant MM called interface mobility   and a more process specific constant SS called system response number are introduced. While the first one summarizes important material properties like thermal conductivity and latent heat the latter one characterizes the process by evaluating the average axial thermal gradients at the phase boundary and the actual growth rate at which the crystal is grown. Furthermore these characteristic numbers are useful for establishing a scheduling strategy for the PI controller parameters in order to improve the controller performance. Finally, both numbers give a better understanding of the general thermal system dynamics of the Czochralski technique.

► Interface mobility “M” represents material specific thermal system dynamics.
► System Response Number “S” represents general thermal system dynamics of Czochralski assemblies.
► “S” can be applied for PI feedback controller parameter scheduling.
► “S” can be applied for appropriate choice of manipulated variables.