Modern Control Engineering

Example Problems and Solutions 355

Root-locus branches cross the imaginary axis at and The root-locus branch

on the real axis touches the jvaxis at v=0.A sketch of the root loci for the system is shown in

Figure 6–67(b).

Note that since this system is of third order, there are three closed-loop poles. The nature of

the system response to a given input depends on the locations of the closed-loop poles.

For 0<K<1.852,there are a set of complex-conjugate closed-loop poles and a real closed-

loop pole. For 1.852K2, there are three real closed-loop poles. For example, the closed-

loop poles are located at

For 2<K,there are a set of complex-conjugate closed-loop poles and a real closed-loop pole.

Thus, small values of K (0<K<1.852)correspond to an underdamped system. (Since the real

closed-loop pole dominates, only a small ripple may show up in the transient response.) Medium

values of K(1.852K2) correspond to an overdamped system. Large values ofK (2<K)

correspond to an underdamped system. With a large value of K, the system responds much faster

than with a smaller value of K.

A–6–6. Sketch the root loci for the system shown in Figure 6–68(a).

Solution.The open-loop poles are located at s=0, s=–1, s=–2+j3,ands=–2-j3.A root

locus exists on the real axis between points s=0ands=–1.The angles of the asymptotes are

found as follows:

Angles of asymptotes=

; 180 °(2k+1)

4

= 45 °, - 45 °, 135°, - 135 °

s =-1, s=-1, s=-2, for K= 2

s =-1.667, s=-1.667, s=-0.667, for K=1.852

v= 15 v=- 15.

(a) (b)

jv

• 6 – 53 – 4 – 3 – 2201 s

j 3

j 4

j 5

j 1

• j 1


• j 3


• j 4


• j 5


• 1


• j 2

j 2

K

s(s+ 1) (s^2 + 4 s+13)

+–

Figure 6–68
(a) Control system; (b) root-locus plot.