HW2_W24_Q (1)
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University of California, San Diego *
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Course
674563
Subject
Aerospace Engineering
Date
Feb 20, 2024
Type
Pages
2
Uploaded by JusticeButterflyMaster3933
MAE 143A
Winter 2024
Homework 2
Due Saturday Jan 27, 2024, 11:59pm
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Problem 1 [15 points, 5 points each]
Find the inverse Laplace transform of the following
functions. Verify your results using
Matlab
’s
ilaplace
function. In all cases, assume zero initial
conditions.
a.
X
(
s
) =
s
+3
s
(
s
+1)(
s
+2)
.
b.
X
(
s
) =
10(
s
+3)
s
2
+25
.
c.
X
(
s
) =
(1
−
e
−
4
s
)(24
s
+40)
(
s
+2)(
s
+10)
Problem 2 [20 points]
Consider the LTI systems with input
x
(
t
) and output
y
(
t
), described by
the following differential equations.
(i)
d
2
y
(
t
)
d
t
2
+ 5
d
y
(
t
)
d
t
+ 4
y
(
t
) = 4
d
x
(
t
)
d
t
+ 5
x
(
t
).
(ii)
d
2
y
(
t
)
d
t
2
+ 5
y
(
t
) = 4
d
x
(
t
)
d
t
.
a.
[8 points, 4 points each]
Find the impulse response,
h
(
t
), for each system.
b.
[8 points, 4 points each]
Find the step response,
y
step
(
t
), for each system.
c.
[4 points]
Verify your answers by plotting the responses and comparing them against the
plots produced by using the
Matlab
functions
tf
,
impulse
and
step
(choose the
x
−
axis
limits of your plot appropriately).
Problem 3 [8 points]
Solve the following two simultaneous differential equations by taking
Laplace transforms and then solving a 2
×
2 linear system of equations:
dx
dt
= 2
x
(
t
)
−
3
y
(
t
)
,
with
x
(0
−
) = 8
,
dy
dt
=
−
2
x
(
t
) +
y
(
t
)
,
with
y
(0
−
) = 3
.
Problem 4 [28 points, 4 points each]
Consider the continuous-time linear time-invariant
system with transfer-function:
G
(
s
) =
1
s
+ 1
Answer the following questions:
1
a. What is the differential equation associated with the above transfer-function? Assume zero
initial condition
y
(0) = 0.
b. Calculate the poles and zeros of
G
(
s
).
c. Is
G
(
s
) BIBO stable?
d. Use the bilinear transformation (aka Tustin transformation)
s
=
2
T
s
z
−
1
z
+ 1
to calculate the corresponding
discretized
transfer-function
G
d
(
z
).
e. Calculate the poles and zeros of
G
d
(
z
).
f. Is
G
d
(
z
) asymptotically stable? (A discrete-time system is asympotically stable if the poles
satisfy
|
p
i
|
<
1.)
g. Use a computer program or calculator to sketch the magnitude and the phase of the frequency
response
G
(
jω
). Now sketch the magnitude and phase of
G
d
(
e
jωT
s
) as a function of
ω
when
T
s
=
{
0
.
01
,
0
.
1
,
1
}
s
. Compare all the obtained responses. What is the role of
T
s
?
Problem 5 [9 points]
a. Determine the stability of the causal systems with the following transfer functions.
(i)
[2 points] H
(
z
) =
3(
z
−
1
.
2)
(
z
−
1)(
z
−
0
.
9)
(ii)
[2 points] H
(
z
) =
3(
z
+0
.
9)
z
(
z
−
0
.
9)(
z
−
1
.
2)
(iii)
[2 points] H
(
z
) =
3(
z
−
0
.
9)
z
(
z
+0
.
9)(
z
+1
.
2)
Use MATLAB where required.
b.
[3 points]
For each system that is unstable, give a bounded input for which the output is
unbounded.
Problem 6 [12 points]
The response of an LTI system to input
x
(
t
) =
δ
(
t
)
−
2
e
−
t
u
(
t
) is output
y
(
t
) = 5
e
−
4
t
u
(
t
).
a.
[2 points]
Compute the transfer function.
b.
[2 points]
Is the system BIBO stable?
c.
[2 points]
Consider an input signal that containa noise
x
(
t
) =
x
s
(
t
) +
x
n
(
t
), where
x
s
(
t
) =
cos(
t
)
u
(
t
) and
x
n
(
t
) =
1
3
e
−
t
u
(
t
). Compute the signal-to-noise ratio of this input signal.
d.
[4 points]
Compute the signal-to-noise ratio of the output signal excited by the input in the
part (c).
e.
[2 points]
What can you observe? Comment
Problem 7 [8 points]
An LTI system has the LCCDE description as
d
2
y
dt
2
+ 7
dy
dt
+ 12
y
=
dx
dt
+ 2
x
Compute each of the following:
a.
[2 points]
Frequency response function
H
(
ω
).
b.
[2 points]
Poles and zeros of the system.
c.
[2 points]
Impulse response
h
(
t
).
d.
[2 points]
Response to input
x
(
t
) =
e
−
2
t
u
(
t
).
2
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