Root Locus
Q
UESTION
14
For the system shown in Figure 14 with
(
29
1
2
C
G
s
s
=
+
,
(
29
(
29
(
29
2
2
2
4
5
s
s
G s
s
s

+
=
+
+
, and
(
29
1
6
H
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)
H(s)
Figure 14
There are
m
= 2 zeros located at
1
j
±
.
There are
n
= 4 poles located at –2, –4, –5, and –6.
There are
n
= 4 branches and
n
–
m
= 2 branches go to infinity.
The Root Locus is on the real axis between –2 and –4, and between –5 and –6.
The asymptote real axis intercept is
[
]
[
]
2
4
5
6
1
1
9.5
4
2
a
j
j
σ





+
+

=
= 

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

K
. There are
n
–
m
= 2 distinct
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 Spring '11
 LANDERS
 Root Locus, Complex number

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