Controlling Czochralski Crystal Growth

Institut für Regelungs- und Steuerungstheorie

1. Project Overview	2. Process Overview	3. Control Tasks	4. Observers
5. Pulling Velocity Control	6. Temperature Control	7. References	Jobs, Student's Jobs etc.

Nonlinear Observer for Estimation of and $\alpha _i$

This page is just intended to explain the real basics of the designed observer. It will be discussed in detail in a article to be published in a few months when experimental results are available.

Aim: Conclusion from measurements (e.g., force

) to unmeasurable variables (e.g. crystal radius

, growth angle $\alpha _i$ ).

Introduction of auxiliary variables , :

$\displaystyle z = \begin{pmatrix}z_1 \\ z_2 \end{pmatrix}:= \begin{pmatrix}r_i \\ r_i' \end{pmatrix} + \begin{pmatrix}\Psi_1(F) \\ \Psi_2(F) \end{pmatrix}$

where $\Psi_1(F)$ and $\Psi_2(F)$ are nonlinear functions of the force .
Observer state:

$\displaystyle \hat z = \begin{pmatrix}\hat z_1 \\ \hat z_2 \end{pmatrix}:= \beg... ...\ \hat r_i' \end{pmatrix} + \begin{pmatrix}\Psi_1(F) \\ \Psi_2(F) \end{pmatrix}$
Observer equations can be derived from a process model: $\hat z' = g(\hat z, v_z, F)$
Observation error: $\tilde z_1 := \hat z_1 - z_1 = \hat r_i - r_i, \qquad \tilde z_2 := \hat z_2 - z_2 = \hat r_i' - r_i'$
Choose $\Psi_1(F)$ and $\Psi_2(F)$ in such a way that the observation error converges towards zero fast enough.
A first method consists in parameterizing $\Psi_1$ and $\Psi_2$ as functions of the desired crystal radius and its derivatives and of the measurements.
Conclusion:

$\displaystyle \begin{pmatrix}\tilde z_1 \\ \tilde z_2 \end{pmatrix} \to \begin{... ...hat r_i \\ \hat r_i' \end{pmatrix} \to \begin{pmatrix}r_i \\ r_i' \end{pmatrix}$
Problems with singularities (related to $\Psi_1(F),\Psi_2(F)$ ) can be overcome by a proper choice of the desired crystal shape $\tilde r(\lambda)$ .

Last modified: Mon Jul 1 20:04:10 2002