MN 507 Process Modelling and Control Homework‘
0 In Golten & Verwer Chapter 4
1. In working out a control system for a crystal growth puller, we came across
some strange results. Using a simple proportional controller, we examined that
the input was to follow a step command signal and found a strange result. This
problem examines why that happens and what we should do about it.
The transfer function is presented in dimensionless units, where the input and
output variables are normalized by their nominal operating values. The output
of the system is the crystal radius and the input is pulling rate. Thus, the
“scaled” or normalized output is represented as g = ”i? and the scaled input
is 21 = 9-3-9, where ya = .02 meters, and ac = 5 x 10“5 meters/sec (ie this
is the values of crystal radius and pull rate used to run the process). These
normalized units are used since they provide a basis for determining which of
several actuators will achieve good peformance (stay tuned for details).
(a) The scaled (or dimensionless) transfer function is given by:
s = – 1
g( ) s + 2
Determine the transfer function with units.
(b) Using a proportional controller (ie k(s) = kp), find the control gain that
will make the system 10 times faster.
(c) Sketch the response to a unit step change in diameter for the closed loop
system. Also sketch the response of the input (here it is pull rate). Note,
you should first derive the appropriate closed loop transfer functions. Try
to reason what the response will look like (ie consider the zero location)
and do a simulation to confirm your intuition.
(d) Can you explain why the input has its particular shape. Use both transfer
function reasoning as well as physical ideas of what the closed loop sys-
tem is trying to do (think about what the error signal is and the type of
controller you are using).
(e) Would you recommend this control be implemented as is? If not, what
suggestions do you have for improving the situation.
Modelling / Design
2. Thermal design for CD case. In class I described the new CD n‘iarmfacturing
process that was deveIOped. The question was why does the temperature rise so