Page 186 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

An Underdamped Process 161

6-4-1 Complete Cancellation
Perhaps the reader is wondering: what would happen if the zeros of
the PID controller were chosen to exactly match those of the process?
That is, what if:
-1±~ =-{±~{2-1

1 1
D=- I=-
2~ 2~

This would cause the open-loop transfer function to become

2
G G = 1 K s +I+ Ds Kc
2
P c s + 2{ s + 1 c s s
and the closed-loop transfer function would be

Kc
s - 1
--K-- s +
1+-c 1
S Kc

which means that the response to a step in the set point would look
like a unity-gain first-order process with a time constant of 1 I Kc.
In general, using controller zeros to cancel process poles can be
dangerous. If a zero in the controller is used to cancel an unstable
process pole, problems could occur if the cancellation is not exact. For
this case, the perfect cancellation values for D and I are much larger
than those used in the simulation. As an exercise you might want to
use the Matlab script and simulink model that I used to generate
Fig. 6-14 to see what happens when these "perfect cancellation" val-
ues are applied.

6-4-2 Adding Sensor Noise
At this point, as a manager, you might be impressed to the point
where you would conclude that the addition of derivative was the
best thing since sliced bread (aside from the preceding comments
about the extreme response to set-point steps). However, when the
process output is noisy, troubles arise. For the purposes of this simu-
lation exercise, we will add just a little white sensor noise (to be
defined later) to the PI and the PID simulations. Figures 6-18 and 6-19
show the impact of adding a small amount of sensor noise on the
process output signal for PI and PID. The added noise is barely
discernible when PI control is used but when the same amount of