Unified Engineering II
Spring
2004
Problem S11 (Signals
and Systems)
Consider
an aircraft
ﬂying in cruise at 250
knots, so
that
v
0
=
129 m/s
Assume
that
the
aircraft
has
lifttodrag
ratio
L
0
=
15
D
0
Then
the
transfer
function from
changes
in
thrust
to
changes
in
altitude is
2
g
1
G
(
s
) =
(1)
mv
0
s
(
s
2
+
2
ζω
n
s
+
ω
2
n
)
where
the
natural frequency
of
the phugoid
mode is
g
ω
n
=
√
2
(2)
v
0
the
damping ratio
is
1
ζ
=
(3)
√
2(
L
0
/D
0
)
and
g
= 9
.
82 m/s
is
the
acceleration
due to
gravity.
The transfer
function
can
be
2
g
normalized by the
constant
factor
mv
0
, so
that
1
¯
G
(
s
) =
(4)
s
(
s
2
+
2
ζω
n
s
+
ω
2
n
)
is the
normalized transfer
function, corresponding
to
normalized
input
2
g
u
(
t
) =
δT
mv
0
¯
1. Find
and plot
the
impulse
response corresponding
to
the transfer
function
G
(
s
),
using partial
fraction expansion
and
inverse Laplace techniques. Hint: The poles
of the
system
are
complex,
so
you
will have to
do
complex
arithmetic.
2. Suppose
we
try to control
the
altitude through
a
feedback
loop, as
shown
below
G(s)
+
u(t)
e(t)
r(t)
h(t)
k
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That
is,
the
control
input
u
(
t
)
(normalized
throttle) is
a
gain
k
times
the error,
e
(
t
),
which is
the
difference
between
the altitude
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 Fall '05
 MarkDrela
 Ωn, normalized transfer function

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