Midterm  MAE143B, Fall 2005
Name:
Student #:
Solutions
November 15, 2005 11:00am11:50pm, HSS1330
openbook and opennotes midterm exam
use the available space to derive your results, attach extra paper if necessary
use of any electronic equipment (calculator, phone, PDA) not allowed during exam
Consider a linear dynamic system
G
characterized by the transfer function model
y
(
s
) =
G
(
s
)
u
(
s
)
, G
(
s
) =
1
−
s
s
2
+
√
2
s
+ 1
(1)
1. Show that the system is stable and has a right half plane zero.
Derive the value of the
undamped natural frequency and the damping ratio of this system. [15pt]
The poles are found by solving
s
2
+
√
2
s
+ 1 = 0
,
and by using the quadratic equation the roots can be shown to be located at
s
=
−
1
√
2
±
1
√
2
j
.
Therefore, the system is stable since the real part of the poles is negative. The dampening
ratio
β
and the undamped natural frequency
ω
n
can be found via the following comparison:
ω
2
n
=
1
2
βω
n
=
√
2
.
This gives
β
=
1
√
2
. The zero is found by solving
1
−
s
= 0
,
and therefore a zeros is located are the point
s
= 1, which is in the RHP.
2. Compute the impulse response
y
(
t
) of this dynamcal system. [15pt]
y
(
s
)
=
1
−
s
s
2
+
√
2
s
+ 1
=
√
2
1
√
2
(
s
+
1
√
2
)
2
+
1
2
−
√
2
s
1
√
2
(
s
+
1
√
2
)
2
+
1
2
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 Spring '10
 LinearControl
 undamped natural frequency

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