Modern Control Engineering

Example Problems and Solutions 357

A–6–7. Sketch the root loci of the control system shown in Figure 6–69(a). Determine the range of gain

Kfor stability.

Solution.Open-loop poles are located at s=1, and A root locus

exists on the real axis between points s=1ands=–q. The asymptotes of the root-locus

branches are found as follows:

The intersection of the asymptotes and the real axis is obtained as

The breakaway and break-in points can be located from Since

we have

which yields

(s+1)^2 =0

dK

ds

=-A3s^2 +6s+ 3 B= 0

K=-(s-1)As^2 +4s+ 7 B=-As^3 +3s^2 +3s- 7 B

dKds=0.

s=-

- 1 + 2 + 2

3

=- 1

Angles of asymptotes=

; 180 °(2k+1)

3

= 60 °, - 60 °, 180°

s=- 2 +j 13 , s=- 2 - j 13.

jv

s

j 3

j 2

j 1

(a) (b)

K

(s– 1) (s^2 + 4 s+ 7)

• j 3


• j 1


• 4 – 3 – 201

K= 7

K= 8

K= 16

• 1

+–

• j 2

Figure 6–69
(a) Control system; (b) root-locus plot.