System Control
2. The Laplace Transform p
P f K Yi
Professor Kyongsu Yi
©2009 VDCL
Vehicle Dynamics and Control Laboratory Vehicle Dynamics and Control Laboratory
Seoul National University
Laplace Transformation
Definition: £ [f(t)] =
0
( )
st( )
f t e dt F s
£: f(t) F(s) , s= +j (complex variable)
f(t) : a time function such that f(t)=0 for t<0
Inverse Laplace Transformation
1
[ ( )] F f t ( )
L
1[ ( )] F s
f t ( )
Existence of Laplace Transformation
f(t) Laplace transformable
f(t) Laplace – transformable
if i) f(t) piecewise-continuous
ii) f(t) of exponential order as t approaches infinity ii) f(t) of exponential order as t approaches infinity
- bounded, exist.
- or et f t( ) approaches zero as t approaches infinity )
(t et f
or approaches zero as t approaches infinity.
If
0
lim
t( )
ct
e f t for
for
( ) e f tthe : the abscissa of convergence
t
for
c
cExistence of Laplace Transformation
, sin , sin ...
t t t t
example : 1)
0 0
lim sin
0
t t
e t t if
if
the abscissa of convergence
c 0
0 if c
, , sin , .
ct ct ct
e
te
e
t c const
2)
lim
t ct0
t
if c
e te
if c
the abscissa of convergence
c c
2 2
t t
e te
does not possess L T2
0
( )
0 0, T< t
e
tfor t T
f t
for t
, e te
does not possess L. T.L[f(t)] exists.
,
f
- The signals that can be physically generated always have corresponding Laplace transforms
Laplace Transformation of simple function
0 0
( ) for t
f
Exponential function
( ) t 0
f t f
Ae for t
A,
a
: constants j
0
[ Ae
t] Ae
te dt
st
L
( )
0
( ) 0
1 1
[ ]
s t
s
Ae dt
A e e
( )
[ ]
1 1
[
j]
A e e
s s
A e
s s
(s j)[0 1 ] A
A s s
Laplace Transformation of simple function
step function
step function
( ) 0
0 0
f t A t
t
0 0
1 1
[ ( )] [ ]
1
st s s
f t Ae dt A e e
s s
L
unit step input function
[0 1 ] Re[ ] 0
A if s
s
1
01( ) t t
t t
p p 0
0
1( )
0
1 0
1( ) 0 0
t t
t t t t
t
0 0 [1( )] 1
t
t s
L
Laplace Transformation of simple function
Sinusoidal Functions
[ sin ] A (
j t j t)
stA t
e
e
e dt
L
0
[ sin ] ( )
2
1 1 1
( )
2
A t e e e dt
j
j s j s j
sin 0 L
( ) 0 0
A t t
f t t
cos 1 ( )
2
sin 1 ( )
j t j t
j t j t
t e e
t e e
2 2
j j j
A s
As
sin ( )
t 2 e e
j
2 2
[ cos ] As
A t
s
L
Ramp function
0[ ]
stst st
At A te dt
e e
A t dt
0
L
( ) 0 0
At t
f t t
0 0
0 2
1
st1
s s
A e dt A
s s
0s s
Laplace Transformation of simple function
Pulse function
0 0
0
( ) 0
0 0, < t
A for t t
f t t
for t t
0
0 0
[ ( )] 1( ) 1( )
1
A A
f t t t t
t t
A
L L L
0
0 0
( ) A 1( ) A 1( )
f t t t t
t t
0
0
1 (1
t s)
A e
t s
0
0 0
[ ( )] lim 1(1 t s)
t
f t A e
t s
Impulse function L
0
0
0 0
(1 )
t
t s
t s
d A e
dt A
0
0 0
lim 0
( )
tA for t t
f t
t
0
0 0
0 0
lim
( )
t
dt A s
d s A
dt t s
0 for t 0, t
0t
Laplace Transformation of simple function
Unit impulse function ; impulse function of magnitude 1
0
0 0
0
lim 1 0
( )
0 0,
t for t t
f t t
for t t t
0
0
0 0
( ) 0 ( ) ( ) ( ) 1
0 0
t st
t t t e dt t dt
t
L
0 0
The unit-impulse function occuring at
t t
0
0 0 00
0
0 0 0
0 0
( ) ( ) ( ) ( ) 1
0
( ) 1( )
t
st t s
st t
t t
t t t t t e dt t t e dt e
t t d
L
0 0
(t t ) 1(t t )
dt
Useful Theorems
Theorem 1. Linearity
[ af t ( )]
aF s ( ) L
Theorem 2. Superposition
[ ( )f t f t( )] F s( ) F s( )
L
Theorem 3. Translation in time.
[ (f )1( )] f( )1( ) std
L
1 2 1 2
[ ( )f t f t( )] F s( ) F s( )
L
(a0)
0
( )
[ ( )1( )] ( )1( )
( )1( ) ( . )
st
s a
f t a t a f t a t a e dt
f e d let t a
L
0
( )1( ) ( ) ( ( )1( ) 0 0)
a
s sa sa
f e e d e F s f for
Useful Theorems
Theorem 4. Complex differentiations [ ( )] d ( )
tf t F s
ds L
[1] 1
s
L 12
[ 1] d ( )
t F s
ds s
L
2 3
2 1 2
[ ] [ ] [ ( )] ( )
) ( ) ( ) st
d d
t t t tf t F s
ds s s ds
proof let F s f t e dt
L L L
0
0 0
) . ( ) ( )
( ) ( ) st ( ) st [ ( )]
proof let F s f t e dt
d F s f t e dt tf t e dt tf t
ds s
LTh 5 T l ti i th d i
0 0
2
2 2
, [ ( )] ( 1) 2 ( ) [ ( )] ( 1) ( )
n
n n
n
d d
similarly t f t F s t f t F s
ds ds
L L
Theorem 5. Translation in the s-domain
2 2
( )
[ cos ]
( )
at
s a
e t
s a
[ e f t
at( )] F s a ( ) L
L ( )
Useful Theorems
( ) d ( ) Df t f t Theorem 6. Real Differentiation
( ) ( )
) [ ( )] ( ) st
Df t f t
dt
d d
proof let f t f t e dt
L
0
0 0
) . [ ( )] ( )
( ) st ( ) st ( )
proof let f t f t e dt
dt dt
f t e f t e dt s
L
(0) ( )
( ) (0)
f s F s
s F s f
2
2
, [ ( )] [ ( )] [ ( )]
( ) (0) ( ) (0) '(0)
[ ( )] ( )
n
n
similarly D f t D Df t Df t
s s F s f s F s s f f
d f t s F s s
L L L
L[ n f t( )] s F s( ) sn1 f (0) sn2 f '(0)
dt
L
( 1)
(0) '(0)
n (0)
f s f
f
Useful Theorems
Theorem 7. Real Integration 0t f (t)dt D1f (t) D1f (0)
0 0 0
( ) ( )
1 1
t t
st
t
f t dt f d e dt
L
0 0 0
1
1 1
( ) ( )
1 1 ( ) (0)
( ) ( )
st st
t
st
e f d e f t dt
s s
F s f
f d e f t dt
s s s s
Theorem 8. Final value Theorem
0 t 0 0
s
s
s s0
( ) ( )
lim lim
t
f t sF s
Theorem 9. Initial value Theorem
0
t s
0
( ) ( )
lim lim
t s
f t sF s
Theorem 10. Complex Integration f t( ) t
L ( )
sF s ds
Inverse Laplace Transformation
1
: ( ) ( ) ( ) ( ) f t F s
F f
L L
-1
-1
: ( ) ( )
( ) ( ) 1 ( )
c j
st
F s f t
f t F s F s e ds
L
L
( c : real constant )( ) ( ) ( )
2 c j
f t F s F s e ds
j
L ( c : real constant )
* Inverse Laplace Transformation by Partial Fraction Method
1
1 0
1
1 1 0
( ) ( )
( )
m m
m m
n n
n
a s a s a
F s P s
Q s s b s b s b
(nm)
1
1 1 0
2
1 2 1 2
real complex conjugate, a , b : real num.
( )( ) ( )
n n
s b s
nb s b
s c s c s d s d
1 2 1 2
2
1 2 1 2
( ) s
F s s c s c s d s d
Examples of Inverse Laplace Transformation
2 2
) ( ) 1
( 2) ( 3) ( 2) ( 2) ( 3)
a b c
ex F s
s s s s s
2
d ds
( 2)( 3) ( 3) ( 2) 1
2, 1, 3, 1
2 1 ( 3) ( 2) 1
( 2)
a s s b s c s
let s than b let s than c
s s s
a s b c a c
2
( 2)
3 3 3 ( 3)
1, 1
a s b c a c
s s s s
a b
1 1 2 1
, 1, ( )
( 2) ( 2) ( 3)
c F s
s s s
2 2 3
( ) t t t
f t t
=> Inverse Laplace Transformation
( ti l f ti th d)
2 2 3
( )f t e t te t e t (partial fraction method)
2 2 2 2 2
10 1 4
10 4 10 4
) ( )
ex F s
2 2 2 2 2
3
) ( )
6 25 ( 3) 4 4 ( 3) 4
( ) 10sin 4 4
t
s s s s
f t te
4
Solution of Differential Equation by Laplace Transformation
2 4 1 (0) 0 , (0) 2 y y y y y
2 1
L.T: s Y s( ) sy(0) y (0) 2 sY s( ) y(0) 4 ( )Y s
s
2 1 2 1
( 2 4) ( ) 2 s
s s Y s
s s
22 2
2 1 1 1 1 1
( )
( 2 4) 4 4 ( 1) 3
s s
Y s s s s s s
1 1 1
( ) cos 3 sin 3
4 4 4 3
t t
y t te te
Dynamic System Mathematical Model (Differential eq.) S-domainL T
Solution in Time Domain S-domain
L. T.
Inverse L.T.
Solution in Time Domain S domain
