EE 428
PROBLEM SET 7
DUE: 12 NOV 2007
Reading assignment: Reading assignment: Ch 5, sections 5.1 through 5.5
Problem 29:
(20 points)
A control engineer is designing an attitude control system for a rocket. Figure 1 shows a block diagram of the attitude
control system. The engineer finds that the inputoutput transfer function
G
p
(
s
) of the dynamics of the rocket, from
the input thrust
U
(
s
) to the output attitude
θ
(
s
), is
G
p
(
s
) =
Θ(
s
)
U
(
s
)
=
K
Js
2
where
K
[meters] is the aerodynamic gain of the rocket and
J
[kilogrammeter
2
] is the moment of inertia of the rocket
about its center of gravity. The control engineer specifies the use of a proportionalplusderivative (PD) controller
G
c
(
s
) =
U
(
s
)
E
(
s
)
=
K
o
+
K
d
s,
where the closedloop error signal
E
(
s
) = Θ
R
(
s
)

Θ(
s
)
is the difference between the desired attitude Θ
R
(
s
) and the actual attitude Θ(
s
).
Figure 1: Rocket attitude control system.
The control engineer subcontracts the following tasks to you:
1. (3 points) Assume
K
d
is fixed and sketch the root locus of the closedloop attitude control system as the
proportional
gain
K
o
is increased from zero to infinity.
2. (3 points) Assume
K
o
is fixed and sketch the root locus of the closedloop attitude control system as the
derivative
gain
K
d
is increased from zero to infinity.
Construct the PD controller gain plane of
K
o
versus
K
d
(with
K
p
on the ordinate and
K
d
on the abscissa) and show
the following regions in the plane:
3. (2 points) Stable region.
4. (2 points) Unstable region.
5. (2 points) Trajectory on which the closedloop attitude control system is critically
damped (that is, the trajec
tory on which the dimensionless damping ratio
ζ
of the closedloop attitude control system is unity).
6. (2 points) Region in which the closedloop attitude control system is overdamped
(that is, the region in which
ζ >
1).
7. (2 points) Region in which the closedloop attitude control system is underdamped
(that is, the region in which
0
< ζ <
1).
8. (2 points) Trajectory on which the underdamped natural frequency
ω
n
of the closedloop attitude control
system is 40 radians/second.
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 Fall '07
 SCHIANO
 closedloop attitude control

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