1
Final Exam
AerE355
Fall 2011 (Take

Home)
Name _____________________________________
***DUE by 2pm Monday @ the front desk of the AerE main office ***
PROBLEM 1 (15
pts)
The lateral mode dynamics are described by:
r
a
r
p
v
r
p
v
r
p
v
r
r
r
a
a
N
L
u
Y
N
L
r
p
N
N
N
L
L
L
u
g
u
Y
u
Y
u
Y
r
p
0
/
0
0
0
0
1
0
0
0
/
cos
1
/
/
/
0
0
0
0
0
0
(1.1)
Equation (1.1) can be expressed in compact notation as:
u
B
x
A
x
.
(a) (8pts)
Give the explicit definition of
the response variable
x
, and define in aerodynamics term
s
what each of its
elements is.
Solution
:
tr
x
(b) (4pts)
Give the explicit definition of the input
u
, and define in aerodynamics terms what each of its elements is.
Solution
:
tr
u
(c
) (3
pt
s)
The complex

valued scalar
i
s
is said to be an
eigenvalue
of
A
if for some
v
, we have the equality
v
s
v
A
.
The homogeneous equation associated with (1.1) is
)
(
)
(
t
x
A
t
x
. Assume a
comp
limentary
solution of the
form
o
st
c
x
e
t
x
)
(
, Insert this solution into the homogeneous equation. Then use the result to explain why the scalar,
s,
must be an eigenvalue of
A,
and
o
x
must b
e its corresponding eigenvector,
v
.
Solution
:
2
PROBLEM 2 (15
pts)
Consider a general aviation plane whose lateral dynamics described by (1.1) include:
0
0
1
0
0
76
.
35
.
49
.
4
0
19
.
2
40
.
8
02
.
16
18
.
1
0
25
.
A
.
(a)
(3
pts)
Use
Matlab
to f
ind the eigenvalu
e
s of the 4

D system (1.1).
Solution
:
(b)(3
pts)
You should have obtained two real eigenvalues and a pair of complex eigenvalues. Hence, there are three
modes. Use the eigenvalues to argue whether or not the system is stable.
Argument
:
(c)(6
pts)
For eac
h mode, obtain numerical values for the following dynamic parameters, as appropriate:
(i) time constant
, (ii) damped natural frequency, (iii) damping ratio.
Solution
:
mode #1:
mode #2:
mode #3:
(d)(3pts)
Each of the three lateral modes has a name. Give the name for each mode.
Answer
:
mode #1
.
mode #2
mode #3
3
PROBLEM 3 (3
5
pts)
The longitudina
l dynamics of a plane are described by:
e
q
w
u
w
u
w
u
e
e
e
M
Z
X
q
w
u
M
M
M
u
Z
Z
g
X
X
q
w
u
0
0
1
0
0
0
0
0
0
.
(3.1)
The phugoid mode has the approximate 2

D dynamics described by:
e
u
u
u
Z
X
u
u
Z
g
X
u
e
e
0
0
/
0
/
.
(3.2)
Suppo
se that for a given plane, we have:
0
1
0
0
0
3
04
.
0
0
180
2
4
.
2
.
32
0
04
.
04
.
0
1
0
0
0
0
0
0
q
w
u
w
u
w
u
M
M
M
u
Z
Z
g
X
X
(3.3)
(a)(10
pts)
Compute the eigenvalues of the systems (3.1) and (3.2). Then compute the % error of the approximation (3.2)
to in relation to (i) the
4
settling time, and (ii) the damped natural period
d
d
T
/
2
. [Note: define a % error to be
%
100
]
/*
)
*
[(*
4
4
2
D
D
D
.]
Answer
:
(b)
(5pts)
Consider the initial condition
0
3
/
0
10
0
tr
x
. use
the
Matlab
command sequence below
to obtain
plots of
the initial condition responses
,
)
(
t
u
and
)
(
t
.
x0= [10; 0;pi/3
;0];
dt = 5*tau/1000;
tvec = 0:dt:5*tau;
C=eye(4);
sys4 = ss(A,[],C,[]);
G
= initial(sys4,x0,tvec);
figure(1)
subplot(2,1,1), plot(tvec
,G
(:,1),
'k'
,
'LineWidth'
,2)
title(
'u(t) phugoid response'
)
grid
subplot(2,1,2), plot(tvec,G
(:,4),
'k'
,
'LineWidth'
,2)
title(
'theta(t) phugoid response'
)
xlabel(
'Time (sec)'
)
grid
Plots
:
4
(c)(5pts)
Modify the above code to obtain plots of
)
(
t
u
and
)
(
t
associated with the 2

D system (3.2). Overlay these
pl
ots on those in (b). In the event that a response obtained using the 2

D approximation is poor, offer an explanation.
Answer
:
(d)(5pts)
Recall that for matrices
A
and
B
,
if both of the following two conditions hold, then
B
is the inverse of
A
:
(C1)
I
AB
, and (C2)
I
BA
. Consider the matrices
d
b
c
a
A
and
)
/(
bc
ad
a
b
c
d
B
.
Prove that
B
is the inverse of
A
(i.
e.
1
A
B
).
Solution
: (C1):
(C2):
(e)(5
pts)
Use the formula in (d
)
in relation
to (3.2) to obtain the Laplace domain expression for
tr
s
s
U
)
(
)
(
for
the initial condition given (b) [Assume no forcing functio
n.]
Solution
:
(f)(5pts)
For
your expression for
tr
s
s
U
)
(
)
(
use a table of Laplace transforms to obtain the expressions for
tr
t
t
u
)
(
)
(
.
Identify the transform used in your table and reference the table.
Solution
:
5
PROBLEM 4 (20pts)
The approximate 2

D phugoid and short period models are:
e
u
u
u
Z
X
u
u
Z
g
X
u
e
e
0
0
/
0
/
&
e
q
e
e
M
u
Z
q
M
M
u
Z
q
0
0
/
1
/
Each of these systems is a 1

input/2

output system. Hence each has two transfer functions.
(a)(5pts)
Use the result
in (d
) of PROBLEM 3 to obtain the transfer functions for the phugoid model.
Solution
:
(b
)(5pts)
Use the resul
t in (d
) of PROBLEM 3 to obtain the transfer functions for the short period model.
Solution
(c)(5
pts)
Supp
ose that the transfer function
0483
.
04
.
0
002
.
01
.
)
(
)
(
)
(
2
s
s
s
s
s
U
s
H
e
u
.
Use the ‘bode’
command to obtain a
plot
of the fr
equency response function (FRF)
)
(
i
H
u
.
Solution
:
(d)(5pts)
Suppose that the FRF in (c) is:
80
60
40
20
0
20
Magnitude (dB)
10
2
10
1
10
0
10
1
10
2
90
180
270
360
Phase (deg)
Bode Diagram
Frequency (rad/sec)
Use the magnitude plot to obtain estimates of (i) the system
static ga
in
(in dB) and (ii) the damped natural frequency.
Solution
:
dB
g
s
_____
and
s
rad
d
/
_____
6
PROBLEM 5 (20pts)
DC motors have traditionally been used to activate various control surfaces of an aircraft. Suppose
that a given moto
r has the transfer function
volt
s
s
V
s
s
G
sec
deg/
1
5
)
(
)
(
)
(
, where
)
(
&
)
(
s
s
V
are the Laplace
transforms of
)
(
&
)
(
t
t
v
, respectively.
Since
)
(
)
(
t
t
, it follows
that (ignoring i.c.
s)
)
(
)
(
s
s
s
. Hence,
volt
s
s
s
V
s
s
G
deg
)
1
(
5
)
(
)
(
)
(
.
(a)(5pts)
Use the
Matlab
command sequence ‘tf’ and ‘step’ to obtain the response of the angular position system,
)
(
s
G
to a unit step voltage.
Solution
:
(b)(5pts)
In (a), you should have found that a 1

volt step inp
ut to the motor resulted in
)
(
t
. This makes sense,
because this input will result in constant
speed.
Hence, as a positioning system, a DC motor is
unstable.
Consider the
following block diagram of a feedback control system that is desig
ned to make the motor a
stable
positioning system:
Show that the closed loop system transfer function,
)
(
)
(
)
(
s
s
s
W
r
is equal to
)
(
)
(
1
)
(
)
(
s
G
s
G
s
G
s
G
c
c
.
Solution
:
(c)(5pts)
Consider a
proportional controller
,
p
c
K
s
G
)
(
. For th
is controller, obtain the expression for
)
(
s
W
. Your
expression should be simplified to be a ratio of only two polynomials.
Solution
:.
(d)(5pts)
You should have found that
)
(
s
W
is a 2
nd
order system. Find the value of
p
K
so that the damping ratio
2
/
1
(i.e. the system is ‘optimally’ damped).
Solution
:
)
(
s
G
c
)
(
s
G
)
(
t
r
)
(
t
e
)
(
t
)
(
t
v
7
APPENDIX
Matlab
code
for PROBLEM 3
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