Root Locus
Q
UESTION
4
For the system shown in Figure 4 with
(
29
1
C
G
s
=
and
(
29
(
29
(
29
(
29
2
3
2
2
5
s
G s
s
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 4
There is
m
= 1 zero located at –3.
There are
n
= 4 poles located at
2
j

,
2
j
, 2, and –5.
There are
n
= 4 branches and
n
–
m
= 3 branches go to infinity.
The Root Locus is on the real axis between 2 and –3, and from –5 to infinity.
The asymptote real axis intercept is
[
]
2
2
2
5
3
0
4
1
a
j
j
σ

+
+

 
=
=

.
The asymptote angles are
[
]
0
2
1 180
0,
1,
2,
a
i
i
n
m
θ
+
=
=
±
±

K
. There are
n
–
m
= 3 distinct
asymptotes with angles
0
0
0
60 ,
60 ,180
a
θ
=

.
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Root Locus
The closed–loop transfer function is
(
29
(
29
(
29
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 Spring '11
 LANDERS
 Complex number, 3k, 2 j, 3 j, 4 poles, 27K

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