University of Maryland at College Park
Dept. of Aerospace Engineering
ENAE 432: Aerospace Control Systems
Problem Set #9
Issued:
23 Apr 2011
Due By:
29 Apr 2011
Question 1:
Prove that:

S
(
jω
γ
)

=
1
2 sin(
γ/
2)
Question 2:
Prove that if the peak of the sensitivity diagram satisfies
max
ω
≥
0

S
(
jω
 ≤
6
dB
then the feedback system is guaranteed to have gain margin in the range 0
.
5
≤
a
≤
1
.
5 and
the phase margin is guaranteed to satisfy

γ

>
29
◦
.
Hint:
this is not so easy as applying
the result in Question #1, since

S
(
jω
γ
)

is not generally the peak of the sensitivity diagram.
We will show in class that

1 +
L
(
jω
)

is the distance from the 1 point to the polar plot of
L
at each frequency
ω
. The statement above is equivalent to a minimum guaranteed value
of this distance. Use this fact, and the definition of phase and gain margin, to establish the
claimed properties.
Question 3:
A tracking antenna has azimuthal pointing dynamics given by
¨
θ
(
t
) + 2
˙
θ
(
t
) = 5
u
(
t
)
where
θ
(
t
) is the azimuth angle, and
u
(
t
) is the current commanded to the driving servo
motor by the tracking computer. The quantity 5
u
(
t
) is the actual torque developed by the
servomotor in response to these commands.
The system is required to perfectly track targets moving so that
θ
d
(
t
) =
A
0
+
A
1
t
where
A
0
and
A
1
can be of arbitrary sign and magnitude. The tracking bandwidth of the
system should also be at least 2 rad/sec. The goal of this problem is to design a feedback
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 Spring '09
 DR.SANNER
 ωf, Aerospace Control Systems

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