L
l
T
f
Laplace Transforms
1.
Standard notation
in dynamics and control
(shorthand notation) = how you
communicate
with other engineers
r 3
with other engineers.
2. Converts
differential calculus
mathematics to
hapte
algebraic
operations, very useful for:
Initial Value Problems (IVPs).
Ch
3. Advantageous for
block diagram analysis
= a
“systems” way to analyze more complex systems.
(study subsystems then combine to large system)
(study subsystems, then combine to large system)
r 3
hapte
For “simple”
Ch
For simple
Y(s)
-- maybe
a short cut !
Laplace Transform Definition
-
0
[
]=
( )
( )
( )
st
e
F
t
dt
f
s
f t
∞
=
∫
L
Two derivations
(“a” is a constant, f ’ is derivative
(“a” is a constant, f ’ is derivative wrt
wrt time):
time):
-st
st
a
a
a
[ ]=
e
dt
e
0
a
a
∞
∞
−
⎡
⎤
⎡
⎤
= −
=
− −
=
⎣
⎦
⎢
⎥
∫
L
3
0
0
-st
-(b+s)t
(b
s)t
-bt
-bt
[ ]
s
s
s
1
1
[
]=
e
dt
e
dt
-e
b
b
e
e
∞
∞
∞
−
+
⎣
⎦
⎡
⎤
=
=
=
⎣
⎦
∫
∫
L
hapte
0
0
0
-st
b+s
s+b
df
df
and a theorem ...
[f ]
e
dt
( )
(
0)
dt
dt
sF s
f t
∞
⎡
⎤
′
=
=
=
−
=
⎢
⎥
⎣
⎦
∫
L
L
Ch
0
Very convenient for zero initial condition
f(0) = 0
An example is when
f
= the
“error”
in a process control IVP
An example is when
= the
error
in a process control IVP,
i.e at
t
= 0 the process is at a “nominal design” steady-state.
Other Transforms
2
where we assign
( )
d
f
dg
df
g
f
t
⎡
⎤
⎡
⎤
′
⎢
⎥
⎢
⎥
L
L
[
]
2
where we assign
( ) -
(0)
dt
dt
dt
sG s
g
=
=
=
⎣
⎦
⎣
⎦
=
2
( ) -
(0)
(0)
( )
(0)
(0)
s sF s
f
f
s F s
sf
f
′
=
−
′
=
−
−
n
r 3
etc. for higher orders
-
[
(
)]
sin
2
j
t
j
t
e
e
t
j
ω
ω
ω
+
−
⎡
⎤
=
⎢
⎥
⎣
⎦
L
L
n
d
f
dt
-
j
t
j
t
e
e
ω
ω
+
⎡
⎤
+
hapte
1
1
1
⎛
⎞
+
⎜
⎟
2
2
s
ω
ω
=
+
cos
[
(
)]
2
t
ω
=
⎢
⎥
⎣
⎦
L
L
Ch
Note:
2
2
2
2
2
1
2
s
j
s
j
s
j
s
j
ω
ω
ω
ω
=
+
−
⎝
⎠
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- Winter '08
- Staff
- dt, Complex number, Impulse response
-
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