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.

Flatness Based Tracking Controller

Aim: Reducing the influence of fast actuating disturbances on the crystal radius trajectory, which is designed in order to obtain crystals with excellent quality.

Concept

desired trajectories for the crystal radius can be freely chosen
stabilizing feedback: $r_i'' = r_{i0}'' - \epsilon_1(r_i' - r_{i0}') - \epsilon_0(r_i - r_{i0})$
$\Rightarrow$ determines the transition between two stationary points by choosing $\epsilon_1$ and $\epsilon_2$ adequately.
inverse process: $v_{p} = \tilde v_p(r_{i}, r_{i}', r_{i}'')$ includes feedback of and
and estimated by observer

A general introduction into the theory of flatness based controllers can be found in [Fliess et.al.].

Desired behaviour of the control error $r_i - r_{i0}$ (linear oscillator):

$\displaystyle (\ast) \quad (r_i'' - r_{i0}'') + \epsilon_1(r_i' - r_{i0}') + \epsilon_0(r_i - r_{i0}) = 0, \qquad \epsilon_0, \epsilon_1 \,$ free

$\displaystyle r_{i0}, r_{i0}', r_{i0}'':$	given
$\displaystyle r_{i}, r_i':$	observed
$\displaystyle r_i'':$	to be calculated

Equation ( $\ast$ ) leads to:

$\displaystyle r_i'' = r_{i0}'' - \epsilon_1(r_i' - r_{i0}') - \epsilon_0(r_i - r_{i0})$

$\Rightarrow$ control input can be calculated: $\quad v_{p} = \tilde v_p(r_{i}, r_{i}', r_{i}'')$

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