Page 531 - Mechanical Engineers' Handbook (Volume 2)

522 Control System Performance Modification

K 0 0
and
K 20 4.9

Example 4 Consider the unity negative-feedback system shown in Fig. 19a. Obtain the
loci of the closed-loop poles as is varied from 0 to , that is, obtain the root loci for 0
.

Solution: The root loci are the points s satisfying the CLCE:
750
1 0
(s 5)(s 10)(s )
To utilize the rules for constructing the root loci, the CLCE is rearranged in the form
of Eq. (8).
By expanding and rearranging, we get
CLCE s(s 5)(s 10) (s 5)(s 10) 750 0
or

(s 5)(s 10)
1 0 (13)
(s 15)(s 50)
2
Equation (13) has three poles at 15, j 50, j 50 and two finite zeros at 5, 10.

Figure 19 (a) Closed-loop system of Example 4; (b) root loci of system of Example 4 where G(s)
(s 5)(s 10)/(s 15)(s 50).