Page 360 - Modern Control Systems

334 Chapter 5 The Performance of Feedback Control Systems

for the disturbance is obtained by using Mason's signal-flow gain formula as
follows:

Y(s) Ptis) A!(5)
T d(s) A(s)
1
1-(1 + K&s- )
(5.43)
1 + ^1^35- 1 + K^KpS' 2
s(s + K 1K 3)
2
s + K xK 3s + K^KiK'

Typical values for the constants are K\ = 0.5 and KiK 2K p = 2.5. Then the natural
frequency of the vehicle is f n = v 2.5/(2-77-) = 0.25 cycles/s. For a unit step distur-
bance, the minimum ISE can be analytically calculated. The attitude is

/10 25K
e-° Um(^t + <A (5.44)

where /3 = V10 - K\/A. Squaring y(t) and integrating the result, we have

5K
2
/ = r^ -°- « sin (|f + A dt
e
e
5 45
2
K
= jf j2 ~^ *{\ ' \™<fr + ^)j dt ( - )
-
= ^ + 0.1tf 3 .
Differentiating I and equating the result to zero, we obtain

- ^ - = -K? + 0.1 = 0. (5.46)
OA.3
Therefore, the minimum ISE is obtained when K 3 = V10 = 3.2. This value of K 3
corresponds to a damping ratio t, of 0.50. The values of ISE and IAE for this system
are plotted in Figure 5.29.The minimum for the IAE performance index is obtained
= 4.2 and £ = 0.665. While the ISE criterion is not as selective as the IAE
when K 3
criterion, it is clear that it is possible to solve analytically for the minimum value of
ISE. The minimum of IAE is obtained by measuring the actual value of IAE for sev-
eral values of the parameter of interest. •

A control system is optimum when the selected performance index is mini-
mized. However, the optimum value of the parameters depends directly on the
definition of optimum, that is, the performance index. Therefore, in Examples 5.6