Root Locus
Q
UESTION
11
For the system shown in Figure 11 with
(
29
3
1
C
s
G
s
s
+
=
+
and
(
29
2
3
2
s
G s
s

=
+
, sketch the Root
Locus Diagram by hand and using Matlab. How many zeros does the system contain and what
are their values? How many poles does the system contain and what are their values? What are
the total number of branches and how many branches will go to infinity? Where is the Root
Locus on the real axis? If necessary, calculate the asymptote angles and real axis intercept,
imaginary axis crossing, and the value of
K
at the imaginary axis crossing.
G(s)
KG
C
(s)
R(s)
+

Y(s)
Figure 11
There are
m
= 2 zeros located at –3 and 3.
There are
n
= 3 poles located at –1,
2
j

, and
2
j
.
There are
n
= 3 branches and
n
–
m
= 1 branch goes to infinity. Since one branch goes to infinity,
the asymptotic angle is 180
0
and the real axis intercept is not required.
The Root Locus is on the real axis between 3 and –1, and from –3 to infinity.
The closed–loop transfer function is
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
2
3
2
9
1
1
2
2
9
C
C
K s
KG
s G s
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 Spring '11
 LANDERS
 Root Locus, Complex number, imaginary axis

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