Page 123 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

98 Chapter Four

disturbances of different frequencies are controlled. From Chap. 3,
the "comer" frequency of

m cor = 1 I r = 1 I 10 = 0.1 rad I sec
or

2
fcor = l.Ocor I (2n) = 0.0159 = 1.5910- Hz
denoted a point on the Bode plot where the process magnitude and
phase plot showed a change. It is a key variable in the error transmis-
sion plot. Disturbances having frequencies below the comer frequency
in Fig. 4-20 are attenuated. Figure 4-21 shows that zero frequency dis-
turbances, that is, constant offsets, are completely removed. At the cor-
ner frequency things start to change. Disturbances having frequencies
above the comer frequency are passed with little effect. Physically, this
is to be expected because disturbances with low frequencies would be
relatively easy to control whereas high-frequency disturbances would
be beyond the capabilities of the controller. Disturbances having fre-
quencies around 0.08 Hz are actually amplified slightly. Sometimes it is
difficult for one to grasp the reality that only a small part of the distur-
bance spectrum is actually controlled when feedback control is applied.
This suggests that process improvement and process problem solving
is the best way to improve performance.

4-3-3 Partial Summary and a Rule of Thumb Using Phase
Margin and Gain Margin
Based on the example in this section it looks like we can gain some
insight into the controllability of a process by looking at the Bode
plot for the open-loop transfer function G,GP. We want to avoid
design situations where the phase lag of G,GP is near 180° when the
amplitude ratio is unity. Experience suggests that a phase margin of
at least 45° is required for good control performance. That is, when
the amplitude ratio is unity we would like the phase lag to be no
more than 135°.
Conversely, we want to avoid situations where the amplitude
ratio is near unity when the phase lag is 180°. Again, experience has
shown that a gain margin of at least 6 dB is desirable. That is, when the
phase lag is 180° we would like the gain to be less than 0.5 where
20 log (.5) = -6.02 dB.
10
We can't seem to find a way to make our first-order process go
unstable when we put it under PI control. This situation will be changed
when we study some new processes in the upcoming sections.

Question 4-7 Why are we having trouble making first-order processes go
unstable by adding controUers?