Root Locus
Q
UESTION
8
For the system shown in Figure 8 with
(
29
2
C
G
s
s
=
+
,
(
29
2
6
25
s
G s
s
s
β
+
=
+
+
, and
(
29
10
10
H
s
s
=
+
sketch the Root Locus Diagram by hand and using Matlab.
Note:
the free parameter is
β
. 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)
G
C
(s)
R(s)
+

Y(s)
H(s)
Figure 8
The closed–loop transfer function is
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
3
3
2
3
2
2
3
2
10
2
6
10
2
6
20
105
370
10
1
20
105
370
10
1
20
105
370
C
C
s
s
G
s G s H
s
s
s
s
s
s
T
s
s
s
G
s G s H
s
s
s
s
s
s
s
s
s
β
β
+
+
+
+
+
+
+
=
=
=
+
+
+
+
+
+
+
+
+
+
+
.
The effective open–loop transfer function with
β
as the free variable is
(
29
2
3
2
10
20
105
370
eff
s
s
G
s
s
s
s
+
=
+
+
+
There are
m
= 2 zeros located at 0 and –10.
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Root Locus
There are
n
= 3 poles located at –14.52 and –2.738 ± 4.240
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 Spring '11
 LANDERS
 Root Locus, Complex number

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