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Automatic Control EEM 3117 Laplace Transformation Dr. Sezai Taskin Department of Electrical&Electronics Engineering Faculty of Engineering, Manisa Celal Bayar University
Automatic ControlEEM 3117
Laplace Transformation
Dr. Sezai TaskinDepartment of Electrical&Electronics Engineering
Faculty of Engineering, Manisa Celal Bayar University
Laplace Transform
Introduction
The study of signals and systems can be carried out in terms of either a time-domain or a
transform-domain formulation.
Both approaches are often used together in order to maximize our ability to deal with a
particular problem arising in applications.
This is very much the case in controls engineering where both time-domain and transform-
domain techniques are extensively used in analysis and design.
The transform-domain approach to signals and systems is based on the transformation of
functions using the Fourier, Laplace, and z-transforms.
10/13/2020
10/13/2020
Laplace Transform
10/13/2020
Stability is assessed by determining the location of
the poles in the Laplace domain (s-plane). Values in
the s-plane are made up of real (σ) and imaginary
(jω) components. If the real portions of system poles
are negative, the system is considered to be stable.
Laplace Transformation
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by: L
F(s)
0
[ ( )] ( ) ( ) stf t F s f t e dt
L
Why s-domain?
• We can transform an ordinary differential equation into an algebraic equation which is easy to solve.
• It is also easy to analyze and design interconnected (series, feedback etc.) systems.
6
Unit step function
Slides 2 7
1 0( ) ( )
0 0
tf t u t
t
0
0
[1]
0 1
1
st
st
e dt
e
s
s s
s
L
•
1[1]
sL
Exponential function
Slides 2 8
( )
0 0
( )
0
( )
[ ]
( )
0 1
( ) ( )
1
at
at at st s a t
s a t
f t e
e e e dt e dt
e
s a
s a s a
s a
L
1[ ]ate
s a
L
Frequency shift
Slides 2 9
1[1]
sL 1
[ 1]ates a
L
[ ( )] ( )ate f t F s a L
Sine and cosine functions
Slides 2 10
2 2[cos ]
st
s
L 2 2
[sin ]ts
L
Impulse function
Slides 2 11
( )f t
0 0
[ ( )] ( ) 1st st
t
t t e dt e
L
Unit ramp
Slides 2 12
0( ) ( )
0 0
t tf t u t
t
udv uv vdu (integration by parts)
0
2
0 00
[ ]
1
st
st st st
t e t dt
e t e et d dt
s s s s
L
1
![ ]n
n
nt
s L
similarly
Differentiated function
Slides 2 13
[ ( )] ( )f t F sL
( )( ) (0)
df tsF s f
dt
L
1
2 ( 1)
( )( ) (0)
(0)..... (0)
nn n
n
n n
d f ts F s s f
dt
s f f
L
Integrated function
Slides 2 14
0
( )( )
t F sf t dt
s
L
2nd shifting theorem
( ) ( )asf t a e F s L
Solution of differential equations using Laplace Transformation
Slides 2 15
DifferentialEquation
TransformedEquation
Transformed Solution
Solution
Laplace Transformation
Algebraic Manipulation
Inverse Laplace
ParticularIntegral
Complementary function
Example
Slides 2 16
2 2
00
2
2 5
5( ) (0) 2 ( )
5( )
( 2)
5( ) 5
2
2.5(1 )
t tt t
t
dxx
dt
sx s x x ss
x ss s
x t e dt e
e
Transformed Solution
Transformed Equation
Convolution integral
Slides 2 17
1 2 1 2( ) ( ). ( ) ( ) ( ) ( )F s F s F s f t f t f t
Ex:
2
2
1 20
2( ) 2
0 0
2 2
1 1 1( )
3 2 ( 1) ( 2)
( )
( ) ( )
( 1)
t t
t
t tt t
t t t t
F ss s s s
f t e e
f f t d
e e d e e d
e e e e
