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MESB 374 System Modeling and Analysis
Inverse Laplace Transform and I/O Model
Inverse Laplace Transform
• Basic steps • Partial fraction expansion (PFE)• Residue command in Matlab• Input-output model by using Laplace transform
Inverse Laplace TransformGiven an s-domain function F(s), the inverse Laplace transform is used to obtain the corresponding time domain function f (t). Procedure:
– Write F(s) as a rational function of s.
– Use long division to write F(s) as the sum of a strictly proper rational function and a quotient part.
– Use Partial-Fraction Expansion (PFE) to break up the strictly proper rational function as a series of components, whose inverse Laplace transforms are known.
– Apply inverse Laplace transform to individual components.
Partial Fraction Expansion
• Case I: Distinct Characteristic Roots
• Residual Formula
1 2
1 2
1
( )( )
( )
nn
ni
i
C s A A AY s
s p s p s ps p
• Proof
( ) ( )ii i s pA s p Y s
1 1
1 1
( ) ( ) ( )( ) ( )
i
i i i i i n i ii is p
i n
i
A p p A p p A p ps p Y s A
s p s p s p
A
1 2
1 2
1 1
1 1
( ) ( ) ( )
( ) ( ) ( ) ( )
i
ni i
n
i i i i i n i
i i n
A
A A As p Y s s p
s p s p s p
A s p A s p A s p A s p
s p s p s p s p
Partial Fraction Expansion
• Case II: Repeated Roots
• Residual Formula
• Proof
1
1
1
( ) ( ) ( )
!( ) ( )
!
!( )
!
!( )
!
1( )
( )!
nn n m
mm
n i in n m n i
mn im
im i
mm
n in
in i
s p
n in
i s pn i
s p Y s s p A
n mds p Y s s p A
ds n m n i
n ms p A
m i
n ids p Y s A
ds i i
dA s p Y s
n i ds
1 22
( )( )
( ) ( ) ( )n
n n
AC s A AY s
s p s p s p s p
1 1( ) ( ) ,
( )! !
i
i
Kn in n
i s p s p iKn ii
d dA s p Y s s p Y s K n i
n i ds K ds
Partial Fraction Expansion
• Residual Formula
1
1
1
1 1
1
1 1 211 122 2
1 1 1
( )( )
( ) ( )
( ) ( ) ( ) ( )
l
l
l
l l
nnl
lnn l lnn
l l l
n terms for p n terms for p
C sY s
s p s p
AA A AA A
s p s p s p s p s p s p
1( ) , where , 1, , 1, ,
!
jij
jji
Kn
ji j s p ji j jKji
dA s p Y s K n i i n j l
K ds
• Case III: General Case
Partial Fraction Expansion• Case IV: Order of the Numerator C(s) =
Order of the denominator D(s) : n = m
0 11
1 0
( )( )( )
( ) ( )
nn n
nn nn
c s c C sC sY s c
D s s d s d D s
1 1( )( ) ( )
( )n
n
C sy t c t
D s
£
Partial Fraction Expansion
• Case V: Complex Roots
1
2 22 2( ) i ii i
i i i i
B sBY s
s s
1( )
i i
i i
i i i i
p p
A AY s
s j s j
1( ) sin cosi it ti i i iy t B e t B e t
Residue Command in MATLAB
[A, P, K] = residue (num, den)
num: vector of coefficients of the numerator;Den: vector of coefficients of the denominator; A: vector of the coefficients in PFE, i.e.,
P: vector of the roots, i.e.,
K: constant term.1
1 1 1 2
repetitions repetitions
[ , , , , , , , , ].
l
l l
n n
P p p p p p p
11
11 12 1 n[ , , , , , , , ]l
n l lA A A A A A
Residue Command in MATLAB (Example)
Ex: Given 3 3
3 2 2
2 2( )
2 ( 1)
s sY s
s s s s s
MATLAB command: >> [A, P, K] = residue ( [1, 0, 0, 2] , [1, 2, 1, 0] )will return the following values: A = [ -4, -1, 2]T , P = [ -1, -1, 0] , K = 1
which means that
2 term
4 1 2( ) 1
1 ( 1)K
Y ss s s
( ) ( ) 4 2t ty t t e t e
Find: inverse Laplace transform
Obtaining I/O Model Using LT (Laplace Transformation Method)
– Use LT to transform all time-domain differential equations into s-domain algebraic equations assuming zero ICs
– Solve for output in terms of inputs in s-domain– Write down the I/O model based on solution in s-
domain
• Step 1: LT of differential equations assuming zero ICs
1 1 1 1 1 2 1 1 1 2
2 2 1 1 1 2 2 1 1 1 2 2 2 2
0
p p
M x B x B x K x K x
M x B x B B x K x K K x B x K x
Example – Car Suspension System
xp
1K
2K
g
2x
2B
1B
1x1M
2M
• Step 2: Solve for output using algebraic elimination method
1. # of unknown variables = # equations ?
2. Eliminate intermediate variables one by one. To eliminate one intermediate variable, solve for the variable from one of the equations and substitute it into ALL the rest of equations; make sure that the variable is completely eliminated from the remaining equations
L
21 1 1 1 1 2 1 1 1 2
22 2 1 1 1 2 2 1 1 1 2 2 2 2
0
p p
M s X s B sX s B sX s K X s K X s
M s X s B sX s B B sX s K X s K K X s B sX s K X s
21 1 1 1 1 1 2
21 1 1 2 1 2 1 2 2 2 2
0
p
M s B s K X s B s K X s
B s K X s M s B B s K K X s B s K X s
Example (Cont.)
• Step 3: write down I/O model
2
1 1 1
2 11 1
M s B s KX s X s
B s K
from first equation
Substitute it into the second equation
2
21 1 12
2 1 2 1 2 1 1 1 1 2 21 1
p
X s
M s B s KM s B B s K K X s B s K X s B s K X s
B s K
22 22 1 2 1 2 1 1 1 1 1 1 1 1 2 2 pM s B B s K K M s B s K B s K X s B s K B s K X s
4 3 2 21 2 3 4 5 1 1 2 3 pa s a s a s a s a X s b s b s b X s
1 1 2 1 3 1 4 1 5 1 1 2 3p p pa x a x a x a x a x b x b x b x
1 1 2
2 1 2 1 1 2 1 1 2
3 1 2 1 1 2 1 2 2 1 2 2 1
4 1 2 2 1 3 1 2
5 1 2
,
, ,
, ,
, .
,
a M M
a M M B M B b B B
a M M K M K B B b B K B K
a B K B K b K K
a K K
