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d)
Fig. 5.36 Root-Iocii for the control system ofExample 5.10, continued.
. COnc= [r?-
1
5.8278
-. -
= Jr6~
-- 6.3587
Crdc _ -4'COnc
- -0.4 * 6.3587
.
- _2.5435
Here, crdc and tOdc are the real and imaginary parts for the dominant second-order
poles.
The transfer function of the PD compensator is given by
Gc(s) : s + zc
Now we have to determine the location of the compensating zero zc so that the
root-locus passes through the point (oZtc, COeLc). The value of zc is determined by
502 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
the angle condition of Eq. (5.124), which in this case leads to
Ozc - (OI + o2 + 03) = (2n + 1)180
Referring to Fig. 5.36b (ignoring the pole at s = pc), we find Oi = 113.5736 deg,
82 = 85.5274 deg, and 03 - 67.1488 deg.Choosing n -. -l,we obtain Ozc - 86.2498
deg and zc = 2.9261. Then, the transfer function of the PD-compensated system
is given by
2.9261) .
Gc(s) = k( ++32~+5~
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