Page 291 - Modern Control Systems

Section 4.8 Design Examples 265

We compute 5"^ as follows:
_ dG(s) p _ -2/7
G
ft =
dp G(s) s + p'
and
2
_ 1 _ S (S + Pf
, T
S'r = 4 2 2 2
1 + G c{s)G p{s)G{s) s + 2ps + (p + K D)s + K Ps + K,'
Therefore,
-r T r 2p(s + p)s 2
Si = SlS° = —. = - ^ ^ - = . (4.73)
p p 4 3 2 2 v }
° s + 2ps + (p + K D)s + K Ps + K,
7
We must evaluate the sensitivity function S p, at various values of frequency. For low
T
frequencies we can approximate the system sensitivity S p by
P
~ K,-
So at low frequencies and for a given p we can reduce the system sensitivity to varia-
tions in p by increasing the PID gain, K/. Suppose that three PID gain sets have been
proposed, as shown in Table 4.2. With p = 2 and the PID gains given as the cases 1-3 in
Table 4.2, we can plot the magnitude of the sensitivity S^ as a function of frequency for
each PID controller. The result is shown in Figure 4.26. We see that by using the PID 3
controller with the gains K P = 6, K D = 4, and Kj = 4, we have the smallest system
sensitivity (at low frequencies) to changes in the process parameter, p. PID 3 is the
controller with the largest gain K t. As the frequency increases we see in Figure 4.26
that the sensitivity increases, and that PID 3 has the highest peak sensitivity.
Now we consider the transient response. Suppose we want to reduce the MAP
by a 10% step change. The associated input is

«(,) = * = »».
s s
The step response for each PID controller is shown in Figure 4.27. PID 1 and PID 2
meet the settling time and overshoot specifications; however PID 3 has excessive
overshoot. The overshoot is the amount the system output exceeds the desired
steady-state response. In this case the desired steady-state response is a 10% decrease
in the baseline MAP. When a 15% overshoot is realized, the MAP is decreased by

Table 4.2 PID Controller Gains and System Performance Results
Input response Settling Disturbance response
PID K P K D K, overshoot (%) time (min) overshoot (%)
1 6 4 1 14.0 10.9 5.25
2 5 7 2 14.2 8.7 4.39
3 6 4 4 39.7 11.1 5.16