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Material Type: Notes; Professor: Zhu; Class: Control Systems; Subject: Mechanical Engineering; University: Michigan State University; Term: Spring 2009;
Typology: Study notes
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2009 Spring ME451 - GGZ
Page 1
Week 1-2: Math Review and Laplace Transformation
2 1 2 1 2 1 2 2 2 1 1 1
1 2 2 1 2 1 2 1 2 1
2
2
*^
2009 Spring ME451 - GGZ
Page 2
Week 1-2: Math Review and Laplace Transformation
e j
e
e
e
j
e
j
j
j
j
j
2
sin ,
2
cos
where ,
sin
cos
θ
θ
θ
θ
θ
θ
θ
θ
θ
−
−^
− = + = + =
θ
θ
θ^
j^
2
2
)
(
1 2
1 2
) ( 2 1 2 1 2 2 1 1 2 1
2 1
2
1
θ θ
θ θ
θ
θ
−
j j
j
j
1
−
θ
r^ θ
2009 Spring ME451 - GGZ
Page 4
Week 1-2: Math Review and Laplace Transformation
22
21
12
11
22
21
12
11
and
b
b
b
b
B
a
a
a
a
A
12
21
22
11
det
a
a
a
a
A
−
=
22
22
12
21
21
22
11
21
22
12
12
11
21
12
11
11
=
b a b a b a b a b a b a b a b a
AB
det
11
21
12
22
1
A
a
a
a
a
A
−
−
= −
2009 Spring ME451 - GGZ
Page 5
Week 1-2: Math Review and Laplace Transformation
−
=
=
0
) (
) (
)] (
[
dt
e t f s F t f L
st
The Laplace transform (LT) of f(t) is
) (
t f
) (
s
F
Re
Im
t^
s
2009 Spring ME451 - GGZ
Page 7
Week 1-2: Math Review and Laplace Transformation
0
,
) (
≥
=
t
t
t f
2
2
0
0
0
0
0
0
0
0
0
1
) 1
(^0) (
) 0 0 ( 1 1
)
(
)
(
1 , ) ( ) (
)] ( [
s s e s s e s
t
dt v t
u
vdt u t
dt
uv t
vdt u t
dt
uv t
e s
v t
u
dt
te
dt
e t f s F t f L
st
st
st
st
st
= − − + − = − ⋅ − − − =
∂^ ∂
∂ ∂
=
∂ ∂
∂ ∂
−
∂ ∂
=
− = = = = =
∞ −
∞ −
∞
∞^
∞
∞^
∞
∞
∞
−
−
−
∫
∫
∫
∫
∫
∫
∫
2 1
] [
)] (
[
=
Q
2009 Spring ME451 - GGZ
Page 8
Week 1-2: Math Review and Laplace Transformation
) (
) (
)] (
[
)] (
[
)] (
) (
[
s
bG
s
aF
t g
bL
t f
aL
t
bg
t
af L
=
=
) ( 0 ) ( 0 0
s
bG
s
aF
dt
e t g
b
dt
e t f
a
dt
e t
bg
t
af
t
bg
t
af L
s G
st
s F
st
st
∫
∫ ∫
∞
−
∞
−
∞
−
2009 Spring ME451 - GGZ
Page 10
Week 1-2: Math Review and Laplace Transformation
as −
(^
)
) (
) ( ) ( ) (
) ( ) ( ) ( ) (
)]
( )
( [
0
0
)
(
0
s F
e
a t d e f e d e u f dt
e a t u a t f
dt
e a t u a t f a t u a t f L
as
s
as
a s
a
st
st
−
∞
−
−
∞
−
∞
−
∞
−
=
−
=
=
=
− − = − − = − −
∫
∫
∫
∫
τ τ τ τ τ τ
τ
τ
2009 Spring ME451 - GGZ
Page 11
Week 1-2: Math Review and Laplace Transformation
t t f
t f
f
s
sF
t f L s F t f L
then , ) (
If
(^
)
) 0 ( ) ( ) ( ) (
)
(
)
(
) (
,
) (
)] ( [
) (
0
0
0
0
0
0
0
0
0
f
s
sF
dt
e t f s e t f
dt v t
u
vdt u t
dt
uv t
vdt u t
dt
uv t
t f
v
e
u
dt v t
u
dt
e t f t f L
s F
st
st
st
st
−
=
−
−
=
∂^ ∂
∂ ∂
=
∂ ∂
∂ ∂
−
∂ ∂
=
=
=
∂ ∂
=
=
∫
∫
∫
∫
∫
∫
∫
∫
∞
−
∞ −
∞
∞
∞
∞
∞
−
∞
∞
−
4
3
4
2
1
&
&
2
f
sf
s F s t f L
2009 Spring ME451 - GGZ
Page 13
Week 1-2: Math Review and Laplace Transformation
t
t f^
G(s)
find ,
If
2
t g L e t g
t^
2
2
2
− →
− →
s
s
e L
s
G
s s
s s
t
Therefore
2t
s
e L
2
2
s s s s t L L t L t f L
2009 Spring ME451 - GGZ
Page 14
Week 1-2: Math Review and Laplace Transformation
Then, . 3
and ,
Let
t a t f t t f a
where
Find
t u
t
t g
t g L
s
G
2
3
s
e
t u t L t u t L t g L s G
s −
3
3s-
G(s)
Therefore,
e
6
9
3 −
6 −
6 3
) ( t g
t
2009 Spring ME451 - GGZ
Page 16
Week 1-2: Math Review and Laplace Transformation
Cosine
Sine
Exponential n
th
Unit stepUnit ramp order ramp
t f
t
1^ s
t u
t δ
t f
L
s
F
(^112) s
n t^
!^1 + n n s
at e
a s^
(^1) −
wt
sin
2
2
w
s
w +
wt
cos
2
2
w
s
s +
2009 Spring ME451 - GGZ
Page 17
Week 1-2: Math Review and Laplace Transformation
Convolution
Transforms of integrals
Transforms of
derivatives
“s” shifting properties
Linearity
t g
bL
t f
aL
t
bg
t
af L
f
s
sF
t f
L
)] ( [
) (
)]
( )
( [^
t f L e s F e a t u a t f L
as
as
−
−^
=
=
−
−
a s s
at
t f L a s F t f e L
2
f
sf
s F s t f L
∫
=
=
t
t f L s s F s d f L
0
)] ( [ 1 ) ( 1 ] ) ( [
τ
τ
∫
∫
−
=
−
=
−
t
t
d t f g d t g f s G s F L
0
0
1
) ( ) ( ) ( ) (
)] ( ) ( [
τ τ τ τ τ τ
“t” shifting properties
2009 Spring ME451 - GGZ
Page 19
Week 1-2: Math Review and Laplace Transformation
)] (
[
) (
Find ,
6 1
) (
Given
1
2
3
s Y L t y s s s
s
s
Y
−
=
−
=
3 15 2
2 10 3 6 1 - 6 1 ) (
next it
discuss
will
We
2
3
4
4
4
3
4 4
4
2
1
− + − + = − +
=
s s s s s s
s
s
Y
t
t^
e
e
s L s L s L t y
3
2
1
1
1
15 2
10
3
6
1
] 3 1
[
15 2
] 2 1
[
10 3
] 1 [
6
1
) (
−
−
−
−
−
−
=
2009 Spring ME451 - GGZ
Page 20
Week 1-2: Math Review and Laplace Transformation
C
A, B
s
C
s
B
A s
s
s s
s
s
Y
and ,
Find , 3
2
) 3
)( 2
(
1
) (
=
) 2 ( ) 3 ( ) 3
)( 2
(
1
− + + + + − = + ⇒ s
Cs
s
Bs
s
s
A
s
A way to expand general rational functions into forms thatappears in the LT table
15
2
) 5
)( 3 (
2
; 3
10 3
) 5 )( 2 (
3
; 2
6
1
) 3 )( 2 (
1
; 0
⇒
=
=
− = ⇒ − = =
C
C
s
B
B
s
A
A
s
