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Figure 6.
PI control, 1 sample per day, 1 day delay
Case no 5: Proportional control, 1 sample per 2 days, 2 days delay
As we can see in Figure 7, the pure proportional feedback control
loop is not too much affected by the decrease of the sampling
frequency and the corresponding increase of the time lag. On the
long term, it is stabilizing to the same permanent error after a more
significant overshoot. Of course, as it is leaving a permanent offset,
the proportional only feedback control is not the best option.
Figure 7. Proportional control, 1 sample per 2 days, 2 days delay
Case no 6: PI control, 1 sample per 2 days, 2 days delay
By comparing Figure 8 to Figure 6, we can see that the doubling of
the time lag had some destabilizing effect on the feedback
response. But considering that the laboratory bath sample analysis
expenses have been cut by half, that little negative impact on the
process may well be small enough to lead to a positive economical
impact for the smelter.
Case no 7: Proportional control, 1 sample per 3 days, 3 days delay
For some reason, this is no longer the case when the sampling
frequency is further decreased to 1 sample per 3 days with a
corresponding 3 days time lag in reporting the results to the cell
controller.
Figure 8. PI control, 1 sample per 2 days, 2 days delay
By keeping the same 1 kg/hr% proportional adjustment constant, as
we can see in Figure 9, even the pure proportional feedback control
mode is destabilized enough to exhibit permanent oscillations in
addition to a permanent error. That proportional adjustment
constant is now too big considering the 3 days time lag in the
feedback control loop.
Figure 9. Proportional control, 1 sample per 3 days, 3 days delay
Case no 8: PI control, 1 sample per 3 days, 3 days delay
Of course, adding integral feedback to a constantly overshooting
proportional controller only contributes to destabilize more the
system as we can see in Figure 10. This time, we have a truly
unstable feedback control that will need to be reinitialized
frequently before it diverges completely!
Case no 9: Proportional control, 1 sample per 3 days, 3 days delay
Fortunately, when a proportional controller overshoots, there is a
simple remedy to the problem, the proportional feedback constant
simply needs to be reduced. While keeping the 3 days sampling
frequency, simply reducing the proportional feedback constant to
0.5 kg/hr% completely re-stabilizes the system, obviously the price
to pay is the doubling of the permanent error as we can see in
Figure 11.