Stability
Q
UESTION
11
For the system shown in Figure 11 with
(
29
5
4
3
2
18
7
7
18
G s
s
s
s
s
s
=
+



, create the Routh table
and determine how many closed–loop poles are in the left s–plane, in the right s–plane, and on
the imaginary axis. Without the use of the Routh table, what can be said about the stability of this
closed–loop system? Explain. Numerically determine the location of the closed–loop poles.
+

R(s)
G(s)
Y(s)
E(s)
Figure 11
The closed–loop transfer function is
(
29
(
29
(
29
5
4
3
2
18
1
7
7
18
18
G s
T s
G s
s
s
s
s
s
=
=
+
+



+
. The
closed–loop characteristic polynomial is
5
4
3
2
7
7
18
18
s
s
s
s
s
+



+
. Since the signs of the
coefficients are not the same, there is at least one unstable pole. Therefore, the closed–loop
system is unstable.
The Routh table is
5
4
3
2
2
1
0
1
7
18
1
7
18
0
36
0
36
7
18
0
18
36
0
0
36
7
18
0
0
s
s
s
s
s
s
ε
+



+
+

+
+

+


+






+
+
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Since there are two sign changes in the first column, there are two poles in the right s–plane.
Since there are five total poles, the other three poles are in the left s–plane.
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 Spring '11
 LANDERS
 French Revolution, Complex number

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