CONTROL SYSTEM ENGINEERING (6ET2) 4/6/2021
Inverse Laplace Transform
Inverse Laplace Transform: Definition
• Inverse Laplace simple transforms a function in s-domain back to
the time domain: →
• If then the Inverse Laplace Transform is
•
• (.) is Inverse Laplace transform operator
• Note: Inverse Laplace Transform need not exists for all
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Properties Inverse Laplace Transform
• Finding the Laplace transform of x(t) following
the basic definition is cumbersome in many
cases
• To simplify the process, certain properties of
Laplace transform have been derived from
definition
• In practice, Laplace transforms are obtained by
the application of one or more of these derived
properties
Properties Inverse Laplace Transform
1. Linearity Property
2. Time Shifting Property
! " # "
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Properties Inverse Laplace Transform
3. Time Scaling Property
4. Time Reversal Property
! !
Properties Inverse Laplace Transform
5. Multiplication by S Property
$
…… " "
$
6. Division by S Property
& $
"
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Properties Inverse Laplace Transform
6. Frequency Differentiation Property
$
!
$
$' ' '
!
$ '
7. Frequency Integration Property
)
& ( $(
Properties Inverse Laplace Transform
8. Convolution Property
*+',+-( .+' . /# 0 /#0 .1 - 2 3 +4 1+0 .'.'5 2+ .5' - +
5# /.6$. 8 . ' .' #56 - / 0# (6# /# 6# +,#6- 9 +4 +'#
.5' - . . .0# /.4 #$ +,#6 /# 3
) )
∗3 & ; 3 ; $; & ; 3 ; $; 3∗
) )
8 . ' .09+6 ' ++- .' ($3.'5 /# 6# 9+' # +4 /# 3 #0 4+6
5.,#' .'9( .5' -
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Example: Convolution Property
*+',+-( .+' +4 6#1 '5(- 6 4('1 .+' 2. / . #-4
.5' - 4 6#1 ,5 6#1
4∗5 6.
Convolution in S-domain
.' .0# $+0 .', $# #60.'.'5 /# 1+',+-( .+' +4 2+ .5' - #1+0#
,#63 1+09-.1 #$ $#9#'$.'5 +' /# .5' -.
2/#' /# .5' - 6# 6 ' 4+60 + ! $+0 .', 1+',+-( .+' #1+0#
,#63 # 3
3 1+',+-( .+' /#+6#0
∗3 =
= ∗3
.. #.
/# 1+',+-( .+' .' .0# $+0 .' #1+0# /# >6+$(1 .' ! $+0 .'
?/. 6(-# /#-9 .' +-,.'5 0 '3 96+ -#0 6#- #$ + 1+',+-( .+'
'$ 8',#6 # 9- 1# ?6 ' 4+60.
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Example: Convolution in S-domain
@.'$ /# 1+',+-( .+' # 2##' ( '$ ABC (
∴ ABC ( &3 (
) )
∴ ∗3 & ; 3 ; $; & ABC ; ( ; !; ( ! ; $;
) )
… … 1+09-# 1 -1(- .+'
3 ! $+0 .'
= !
= ! ABC ∗3
… … .09-# 1 -1(- .+'
Advantages of Laplace Transform
• Solving ODEs in s-domain is much simpler
compare to solving ODEs in time domain, because
ODEs become algebraic equations in s-domain
FGH FH J
• e.g. F G 2F ! ↔ 2 !1
• Mℎ O 0, Q
0
• Laplace transform easily helps us in Convolution
where Convolution is just simple multiplication
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Advantages of Laplace Transform
• Laplace transform is applicable to Continuous,
Piecewise Continuous, Periodic, Step and
Impulse functions
• Properties of Laplace transform enable its
calculations to be very easy
• Properties of Inverse Laplace transform make it
convenience to transform back to time domain
after necessary analysis in s-domain
Ex.1 Solving ODEs in S-domain
@.'$ 8 4+6 R "
5.,#'
8 " "
S* " "
S (
T Ω, V, * @
W99-3 XS + 5# $3' 0.1 +4 /# 3 #0
$8
S T8 &8 $
$ * "
8
? Y.'5 9- 1# ?6 ' 4+60S T8 8 -8 "Z
*
S
8
T [ *
'+2, S ( ,∴ S T Ω, V, * @
[
8
T [ *
8 8 # #
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Ex.2 Solving ODEs in S-domain
\+-,# ( .'5 9- 1# 6 ' 4+60 3] ! 3^ ! _ " ……3 " `, 3^ " a
? Y.'5 9- 1# ?6 ' 4+60
= ! 3^ " ! 3 " ! = !3 " ! _= "
` `
=
! !_ !b
99-3.'5 9 6 . - 46 1 .+'
=
!b
99-3.'5 8',#6 # 9- 1# ?6 ' 4+60
3 =
!b
3 #b #
Transfer Function: Motivation
V+2 + 4.'$ /# 6# 9+' # +4 /# 3 #0 4+6 5.,#' .'9( .5' -.
@.'$ /# Sc 4+6 /# 5.,#' .'9( .5' - S.
W99-3 XS + 5# $3' 0.1 +4 /# 3 #0
$8
S. T8 &8 $
$ * "
S+ &8 $
* "
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Transfer Function: Motivation
• To find the time response, we need to solve Ordinary Differential
Equations (ODE) (Integration, Differentiation)
• When the model equations are transferred to the s-domain, they
turn out to be Algebraic equations, which are easy to solve
• The transform model in s-domain is called Transfer Function model
• It is a model which is applicable for all kinds of input signals
Transfer Function
• For LTI (SISO) system, Transfer Function is the ratio of Laplace
transform of the output (Response) to the Laplace transform of
the input (Excitation), with the initial conditions being zero
• Mathematically, if the U(s) is the Laplace transform of the input
function and Y(s) is the Laplace transform of the output function,
=
the Transfer Function G(s) is given by d
e
Input Output
System
u(t) Y(t)
3 = ? +4 c( 9(
?. @. d , 8 "
( e ? +4 8'9(
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Transfer Function: as an Impulse Response
809(- # .5' - f . .'4.'. # .0 --3 ' 66+2
'$ .'4.'. #-3 -- 3# .' #56 .'5 + +'#
8 Y# g#6+ , -(# #,#63 2/#6# # 1#9 "
)
& f
)
84 /# .'9( + /# 3 #0 . /# ('. .09(- #,
/#' /# +( 9( . 1 --#$ 809( # T# 9+' # .. #.
( f →e →d =
?/ 0# ' /# ?6 ' 4#6 @('1 .+' . /# 9- 1# 6 ' 4+60 +4 .09(- #
6# 9+' # +4 ' ?8 3 #0, 2/#' /# .'. . - 1+'$. .+' 6# # + g#6+
809(- # 6# 9+'1# . - + 1 --#$ 3 #0 6# 9+' #, ' (6 - 6# 9+'1# '$
46## 4+61# 6# 9+' #
Steps to find Transfer Function
• Find the model equation of the given system
(by KVL,KCL, Newton's Theorem etc.)
• Identify the system input and output
variables
• Take Laplace transform of model equations,
assuming initial conditions are zero
• Find the ratio of Laplace transform of output
to the Laplace transform of input
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Example 1: Transfer Function
V+2 + 4.'$ /# 6# 9+' # +4 /# 3 #0 4+6 5.,#' .'9( .5' -.
@.'$ /# Sc 4+6 /# 5.,#' .'9( .5' - S.
\ #9 : W99-3 XS + 5# $3' 0.1 +4 /# 3 #0
$8
S. T8 &8 $
$ * "
S+ &8 $
* "
\ #9 : 8$#' .43 .'9( '$ +( 9( , 6. -#
.'9( S.
+( 9( Sc
Example 1: Transfer Function
\ #9 `: 9- 1# 6 ' 4+0 +4 0+$#- #i( .+'
(0.'5 .'. . - 1+'$. .+' 6# g#6+
S. T8 8 8
*
S+ 8
*
\ #9 b: ?6 ' 4#6 @('1 .+'
Sc 8
d *
S. T 8
*
Sc *
d
S. T * T*
*
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Example 1: Transfer Function, if the current I(t) is output
\ #9 : W99-3 XS + 5# $3' 0.1 +4 /# 3 #0
$8
S. T8 &8 $
$ * "
\ #9 : 8$#' .43 .'9( '$ +( 9( , 6. -#
.'9( S.
+( 9( 8
\ #9 `: 9- 1# 6 ' 4+0 +4 0+$#- #i( .+'
(0.'5 .'. . - 1+'$. .+' 6# g#6+
S. T 8
*
\ #9 b: ?6 ' 4#6 @('1 .+'
8 *
d
S. T * T*
*
Example 2: Transfer Function
@.'$ /# 6 ' 4#6 4('1 .+' +4 /# 3 #0 $# 16. # 3 4+--+2.'5 #i( .+'
2. / g#6+ .'. . - 1+'$. .+'
$` 3 $ 3 $3 $(
" !j 3 " (
$ ` $ $ $
\ #9 : k+ '##$#$ /#6#
\ #9 : 8$#' .43 .'9( '$ +( 9( , 6. -#
.'9( ( +( 9( 3
\ #9 `: 9- 1# 6 ' 4+0 +4 0+$#- #i( .+'
(0.'5 .'. . - 1+'$. .+' 6# g#6+
= ` " !j e "
\ #9 b: ?6 ' 4#6 @('1 .+'
= "
d
e ` " !j
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Properties Transfer Function
• Transfer function of any system is independent of the magnitude
and nature of an input signal
• Using the Transfer function, the response can be studied for various
inputs to understand the nature of the system
• Transfer Function does not provide any information concerning the
physical structure of the system. i.e. Two different physical system
can have the same transfer functions (e.g.)
• MSD System d … n o p 1
@ l Zm ZX Z Z
Sc
• RLC Circuit d … q r 1
S. *Z T*Z Z Z
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