s. awad, ph.d. m. corless, m.s.e.e. e.c.e. department university of michigan-dearborn laplace...
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S. Awad, Ph.D.
M. Corless, M.S.E.E.
E.C.E. Department
University of Michigan-Dearborn
LaplaceTransform
Math Review with Matlab:
Calculating the Inverse Laplace Transform
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U of M-Dearborn ECE DepartmentMath Review with Matlab
2
Laplace Transform:X(s) Inverse Laplace Transform
Inverse Laplace Transform
Laplace Nomenclature Table Look Up Method Simple Table Look Up Example Inverse Laplace Transform General Form Distinct Pole Example Repeated Poles Example
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Laplace Transform:X(s) Inverse Laplace Transform
Laplace Nomenclature The Laplace Transform of a time domain function
x(t), will be a complex domain function X(s)
)()( sXtx LT
)()(
)()(1 sXLTtx
txLTsX
This relationship is also denoted as:
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Laplace Transform:X(s) Inverse Laplace Transform
Inverse Laplace Transform Inverse Laplace Transform is used to compute x(t) from X(s)
The Inverse Laplace Transform is strictly defined as:
Strict computation is complicated and rarely used in engineering
jc
jc
stdsesXj
sXLTtx )(2
1)()( 1
Practically, the Inverse Laplace Transform of a rational function is calculated using a method of table look-up
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Laplace Transform:X(s) Inverse Laplace Transform
Table Look Up Method If X(s) can be written as a sum of terms with known
Inverse Laplace Transforms, x(t) will be the sum of these Inverse Laplace Transforms
)()()()(
)()()()(
21
21
txtxtxtx
sXsXsXsX
n
n
Requires knowledge or reference of Laplace Transform pairs, but is much simpler than directly calculating the Inverse Laplace Transform
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Laplace Transform:X(s) Inverse Laplace Transform
Simple Table Look Up Example
Given:
3
410)(
ss
sX
Use superposition and the Table Look-Up Method determine the Inverse Laplace Transform of X(s)
Verify the result using Matlab
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Laplace Transform:X(s) Inverse Laplace Transform
Tabular Solution X(s) is written as a Linear Summation of terms with Known
Inverse Laplace Transforms
)(1
)()(
sXs
txetu t
3
410)(
ss
sX
)(4)(10)( 3 tuetutx t
LT-1 LT-1LT-1
x(t) can be directly determined using the Laplace Transform Pairs shown below to the right
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Laplace Transform:X(s) Inverse Laplace Transform
Matlab Verification
3
410)(
ss
sX
» syms X s» X=((10/s)+(4/(s+3)));» x=ilaplace(X)
x =10+4*exp(-3*t)
)(410)( 3 tuetx t
The ilaplace Matlab command can be used to quickly verify the solution
NOTE: The output of ilaplace must be interpreted
as a Causal Solution
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Laplace Transform:X(s) Inverse Laplace Transform
Inverse Laplace Transform General Form
In general X(s) can be written as a rational function
))...()((
))...()(()(
21
21
nn
mm
pspspsa
zszszsbsX
z1, z2, … , zm are the zeros of X(s)
p1, p2, … , pn are the poles of X(s)
If X(s) is written as a strictly rational function, a method using partial-fraction expansion can be used to determine the inverse Laplace transform
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Laplace Transform:X(s) Inverse Laplace Transform
Strictly Rational Function A function X(s) is strictly rational if the Degree of its
Numerator Polynomial N(s) is Less than the Degree of its Denominator Polynomial D(s)
If N(s) D(s), perform Long Division until the remainder polynomial R(s) is of lesser order than A(s)
)(
)()(
)(
)()(
sD
sRsQ
sD
sNsX
N(s) = Numerator
D(s) = Denominator
Q(s) = Quotient
R(s) = Remainder
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Laplace Transform:X(s) Inverse Laplace Transform
Partial-Fraction Expansion based on Poles
Once X(s) is written in a strictly rational form, Partial Fraction Expansion can be performed
Partial-Fraction Expansion of of X(s) can be classified into two categories based on the Poles of X(s)
CASE I: All Poles are Distinct CASE II: All or Some Poles are Repeated
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Laplace Transform:X(s) Inverse Laplace Transform
CASE1: Distinct Poles Assume that all Poles are Different ( pipj if i j )
Assume numerator order is less than denominator order, then the Partial Fraction Expansion of X(s) is given by:
n
n
ps
c
ps
c
ps
csX
...)(
2
2
1
1
where: nisXpscipsii ,...,2,1,)(
))...()((
))...()(()(
21
21
nn
mm
pspspsa
zszszsbsX
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Laplace Transform:X(s) Inverse Laplace Transform
Building from previous work, x(t) is the summation of unit steps multiplied by exponentials
LT-1 of Distinct Terms X(s) can be written as the sum of terms Due to the Linearity Property, x(t) will be the sum of the
Inverse Laplace Transform of the terms
n
i
tpi tuectx i
1
)()(
n
i i
iLTn
i i
i txps
cLT
ps
csX
1
1
1
)()(
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Laplace Transform:X(s) Inverse Laplace Transform
CASE2: Repeated Pole
))...(()(
)(
)(
)()(
11 nrr pspsps
sN
sD
sNsX
n
n
r
rr
r
ps
c
ps
c
ps
c
ps
c
ps
csX
...)()(
...)(
)(1
1
12
1
2
1
1
Assume that Some or All Poles (Roots) are Repeated For the case below, Pole p1 is Repeated r Times
Partial Fraction Expansion shows Repeated and Distinct poles
Repeated Pole Distinct Poles
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Laplace Transform:X(s) Inverse Laplace Transform
Determine Coefficients Find Distinct Poles Coefficients, i = r+1, r+2, ...,n
nrrisXpscipsii ,...,2,1,)()(
ipsr
rr sXpsc )()(
Find General Repeated Poles, i = 1, 2, ... , r-1
1
)( )!(
11)(
)(
ps
r
ir
ir
i sXpsds
d
irc
Find First Repeated Pole Coefficient, i = r
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Laplace Transform:X(s) Inverse Laplace Transform
Partial-Fraction Expansion Method for Determining Inverse Laplace Transform
Put Rational Function into Strictly Rational Form where the degree of the numerator polynomial less than that of the denominator polynomial
Factor the Denominator Polynomial
Perform Partial-Fraction Expansion
Use Laplace Transform Pair Table to obtain the inverse Laplace transform
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Laplace Transform:X(s) Inverse Laplace Transform
Distinct Pole Example Given:
Use the Partial-Fraction Expansion and the Table Look-Up Method determine the Inverse Laplace Transform of X(s)
Verify the result using Matlab
)3)(2)(1(
52)(
2
sss
sssX
Notice all Poles are Distinct
Order of Numerator (2) < Order of Denominator (3)
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Laplace Transform:X(s) Inverse Laplace Transform
Partial Fraction Expansion
)3)(2)(1(
52)(
2
sss
sssX
Start by finding Partial Fraction Expansion of X(s)
321)( 321
s
c
s
c
s
csX
Poles of
p1=-1, p2=-2, p3=-3
Partial Fraction Expansion
Find ci coefficients
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Laplace Transform:X(s) Inverse Laplace Transform
c1 is explicitly evaluated as:
Find Coefficient C1
C1 = -2
2)31)(21(
5)1()1(2
)3)(2(
52
)1()3)(2)(1(
52)1)((
2
1
2
1
2
11
S
Ss
ss
ss
ssss
ssssXc
Each coefficient is determined by evaluating:
nisXpscipsii ,...,2,1,)(
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Laplace Transform:X(s) Inverse Laplace Transform
Find Coefficient C2 and C3
3)32)(12(
5)2()2(2
)2()3)(2)(1(
52
)2)((
2
2
2
2
2
22
c
ssss
ssc
ssXc
s
s
C2 = 3
1)23)(13(
5)3()3(2
)3()3)(2)(1(
52
)3)((
2
3
3
2
3
33
c
ssss
ssc
ssXc
s
s
C3 = 1
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Laplace Transform:X(s) Inverse Laplace Transform
)3)(2)(1(
52)(
2
sss
sssX
Partial Fraction Expanded X(s)
Original Expression
3
1
2
3
1
2)(
sss
sX
321)( 321
s
c
s
c
s
csX
Expansion
Replace with Coefficients
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Laplace Transform:X(s) Inverse Laplace Transform
)()(3)(2)( 32 tuetuetuetx ttt
3
1
2
3
1
2)(
sss
sX
Inverse Laplace Transform x(t) = Sum of the Inverse Laplace Transform of
the individual terms of X(s)
)(32)( 32 tueeetx ttt
LT-1LT-1 LT-1LT-1
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Laplace Transform:X(s) Inverse Laplace Transform
» syms Xnum Xden s» Xnum = 2*s^2 -s +5; % Numerator X(s)» Xden = (s+1)*(s+2)*(s-3); % Denominator» x = ilaplace( Xnum/Xden ) x =-2*exp(-t)+3*exp(-2*t)+exp(3*t)
Distinct Pole Matlab Verification This can easily be verified in
Matlab using ilaplace
)(32)( 32 tueeetx ttt
)3)(2)(1(
52)(
2
sss
sssX
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Laplace Transform:X(s) Inverse Laplace Transform
Repeated Pole Example Given:
Use the Partial-Fraction Expansion and the Table Look-Up Method determine the Inverse Laplace Transform of X(s)
Verify each step using Matlab
)1()2(
22)(
3
34
ss
ssssX
Order of Numerator (4) = Order of Denominator (4) therefore X(s) is NOT Strictly Rational
Notice Pole s = -2 is Repeated 3 times
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Laplace Transform:X(s) Inverse Laplace Transform
Solution Steps Convert X(s) to a strictly rational function
Perform long division by hand Verify long division using Matlab deconv
Perform partial fraction expansion on rational part of X(s) Calculate coefficients by hand Calculate coefficients in Matlab using diff and subs
Verify conversion to strictly rational function using combination of int and diff commands
Use table method to determine Inverse Laplace Transform
Verify entire process using ilaplace
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Laplace Transform:X(s) Inverse Laplace Transform
Repeated Pole Example X(s) must be decomposed into a constant plus a Strictly Rational
Function
)1()2(
)()(
)1()2(
22)(
33
34
ss
sRsQ
ss
ssssX
)1()2(
)()(
3
ss
sNcsX o )(lim sXc
so
Q(s) will just be a constant since the order of the numerator and the denominator are the same
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Laplace Transform:X(s) Inverse Laplace Transform
Convert to Rational Functions
Use Long Division to Normalize the Transfer Function such that the Highest Order of the Numerator is Less Than the Highest Order of the Denominator
ssssssssX
sNumsDensX
22820187)(
)()()(
34234
820187
22
)1()2(
22)(
234
34
3
34
ssss
sss
ss
ssssX
Must Decompose X(s) into Strictly Rational Functions of s
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Laplace Transform:X(s) Inverse Laplace Transform
Long Division Perform long division to decompose X(s) into a
constant plus a purely rational function
Thus:
)1()2(
164236132)(
3
23
ss
ssssX
2
16423613
164036142
0 2s 0s s2s820187
22
234
23 4234
sss
ssss
ssss
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Laplace Transform:X(s) Inverse Laplace Transform
Matlab Polynomial Division: deconv
The Matlab command deconv can be used to perform polynomial division
deconv operates on arrays of polynomial coefficients, NOT symbolic variables
A
RQ
A
B
[Q, R] = deconv( B, A)B = Numerator polynomial coefficients
A = Denominator polynomial coefficients
Q = Quotient of B/A
R = Remainder of B/A
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Laplace Transform:X(s) Inverse Laplace Transform
Long Division Verification Use Matlab to verify the polynomial long division:
820187
820187
164236132
820187
22
234
234
23
234
34
ssss
RQ
ssss
sss
ssss
sss
» polynum = [2 1 0 -2 0]; » polyden = [1 7 18 20 8];» [Q, R]=deconv(polynum, polyden)Q = 2R = 0 -13 -36 -42 -16
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Laplace Transform:X(s) Inverse Laplace Transform
Expand Rational Part of X(s) The rational part of X(s) will be referred to as Xo(s)
)1()2(
16423613)(
3
23
ss
ssssX o
1)2()2(2)( 4
33
221
s
c
s
c
s
c
s
csX o
Xo(s) has 3 repeated roots and one distinct root
Partial Fraction Expansion
)(2)1()2(
164236132)(
3
23
sXss
ssssX o
Partial Fraction Expansion must be performed on Xo(s)
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Laplace Transform:X(s) Inverse Laplace Transform
Evaluating Coefficient C4 C4 is evaluated using the distinct pole expression as
shown in the previous example
14 )()1( so sXsc
3
23
1
3
23
1
3
23
4
)21(
16142136113
)2(
16423613
)2)(1(
16423613)1(
s
s
s
sss
ss
ssssc
34 c
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Laplace Transform:X(s) Inverse Laplace Transform
Matlab C4 Verification c4 can verified in Matlab by creating the symbolic
expression Xo and evaluating for c4
» Xo =( -13* s^3 -36*s^2 -42*s -16)/((s+1)*(s+2)^3);
» c4 = (Xo)*(s+1)c4 =(-13*s^3-36*s^2-42*s-16)/(s+2)^3
» c4=subs(c4,s,-1)c4 = 3
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U of M-Dearborn ECE DepartmentMath Review with Matlab
34
Laplace Transform:X(s) Inverse Laplace Transform
Evaluating Coefficient C3 C3 is evaluated similar to C4
28
)21(
16242236213
)2)(1(
16423613)2(
)()2(
3
23
2
3
233
23
3
s
so
ss
ssss
sXsc
283 c
» c3 = (Xo)*((s+2)^3);» c3 = subs(c3,s,-2)c3 = -28
C3 is easily verified in Matlab
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U of M-Dearborn ECE DepartmentMath Review with Matlab
35
Laplace Transform:X(s) Inverse Laplace Transform
Repeated Pole Coefficients To find C2 and C1, the following expression must be evaluated
for each case ( r = 3, p1 = -2, i = 2, 1 )
2
23
)3(
)3(
23
233
)3(
)3(
1)(
)(
)1(
16423613
)!3(
1
)2)(1(
164236132
)!3(
1
)( )!(
1
1
si
i
si
i
ps
r
ir
ir
i
s
sss
ds
d
i
ss
ssss
ds
d
i
sXpsds
d
irc
)1(
16423613)(
23
s
ssssY
For simplicity, let Y(s) be the expression to be differentiated
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36
Laplace Transform:X(s) Inverse Laplace Transform
Repeated Pole Coefficients
Before differentiating, Y(s) can be rewritten as:
222)23(
)23(
2 )( )( !1
1)(
)!23(
1
sss
sYsYds
dsY
ds
dc
12323
)1)(16423613()1(
16423613)(
sssss
ssssY
21
222
2
2)13(
)13(
1
2
)(
)(2
1 )(
!2
1)(
)!13(
1
s
sss
sYc
sYds
dsY
ds
dsY
ds
dc
Similarly, C1 equals the second derivative of Y(s) / 2
C2 equals the first derivative of Y(s)
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37
Laplace Transform:X(s) Inverse Laplace Transform
Calculating First Derivative Calculating First Derivative by Hand
123 )1)(16423613()( sssssY
2
232
21223
)1(
16423613
)1(
427239)(
)427239()1()1)(1)(16423613()(
s
sss
s
sssY
ssssssssY
» Y =(Xo)*((s+2)^3);» dY = diff(Y); pretty(dY) 2 3 2 -39 s - 72 s - 42 -13 s - 36 s - 42 s - 16 ------------------ - -------------------------- s + 1 2 (s + 1)
Matlab Verification of First Derivative
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38
Laplace Transform:X(s) Inverse Laplace Transform
Simplify Y’(s) Since Y’(s) will eventually have to be calculated, it will be helpful to simplify its terms
» dY=simplify(dY); pretty(dY) 3 2 26 s + 75 s + 72 s + 26 - ------------------------- 2 (s + 1)
2
23
2
232
)1(
26727526
)1(
16423613
)1)(1(
)1)(427239(
s
sss
s
sss
ss
sssY
Matlab can also be used to perform this simplification
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39
Laplace Transform:X(s) Inverse Laplace Transform
Evaluate C2 C2 can now be directly evaluated from Y’(s)
» c2 = subs(dY,s,-2)c2 = 26
Matlab Verification of C2
26)12(
26)2(72)2(75)2(26
)1(
26727526)(
2
23
2
2
23
22
s
s s
ssssYc
262 c
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40
Laplace Transform:X(s) Inverse Laplace Transform
Calculate Second Derivative The second derivative of Y(s) must be calculated to find C1
3
23
2
2
22323
2232
23
)1(
267275262
)1(
7215078
)7215078()1()1)(2)(26727526()(
)1)(26727526()1(
26727526)(
s
sss
s
ss
ssssssssY
sssss
ssssY
» ddY = diff(dY); pretty(ddY) 2 3 2 78 s + 150 s + 72 26 s + 75 s + 72 s + 26 - ------------------ -+ 2 ------------------------- 2 3 (s + 1) (s + 1)
Matlab Verification of Y’’(s)
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U of M-Dearborn ECE DepartmentMath Review with Matlab
41
Laplace Transform:X(s) Inverse Laplace Transform
Calculate C1
The following expression can be calculated by hand
16
)1(
267275262
)1(
7215078
2
1
2
)(
2
3
23
2
2
21
s
s
s
sss
s
ss
sYc
Matlab Verification
» c1 = subs( ddY/2 ,s ,-2)c1 = -16
161 c
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42
Laplace Transform:X(s) Inverse Laplace Transform
Result of Expansion The Rational Part of X(s) is expanded to:
1
3
)2(
28
)2(
26
2
16
1)2()2(2)1()2(
16423613)(
32
43
32
213
23
ssss
s
c
s
c
s
c
s
c
ss
ssssX o
2
16
)2(
26
)2(
28
1
32)(2)(
230
sssssXsX
Thus X(s) can be rewritten as:
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43
Laplace Transform:X(s) Inverse Laplace Transform
Matlab Partial Fraction Expansion
As of Matlab 5.3.x, there is currently no function to directly convert a symbolic expression to a strictly rational function, the following “trick” can be performed
Integrate the function to be expanded Differentiate the result of the integration The result of the symbolic differentiation will be expressed in
strictly rational form Thus the result of the entire process is the original function
expanded into strictly rational form
Remember that the integration process MUST occur first so that no constant data is lost
Vector expressions can be evaluated using residues
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44
Laplace Transform:X(s) Inverse Laplace Transform
Expansion Verification By hand, X(s) was converted to a strictly rational function
» num = 2*s^4 + s^3 -2*s;» den = (s+1)*( (s+2)^3 );
2
16
)2(
26
)2(
28
1
32
)2)(1(
22)(
233
34
ssssss
ssssX
This can be verified in Matlab
» X = diff(int( num/den ));» pretty(X) 3 28 26 16 2 + ----- - -------- + -------- - ----- s + 1 3 2 s + 2 (s + 2) (s + 2)
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45
Laplace Transform:X(s) Inverse Laplace Transform
Table Method Solution
)(1626143)(2)( 2222 tueteetettx tttt
2
16
)2(
26
)2(
28
1
32)(
23
sssssX
The following transform pairs can be used to evaluate x(t)
)(1
)()(
sX
txet t
LT-1
The Inverse Laplace Transform can be calculated directly
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46
Laplace Transform:X(s) Inverse Laplace Transform
Verification Using Matlab
»x = ilaplace( X )
x =
2*Dirac(t)+3*exp(-t)-14*t^2*exp(-2*t)
+26*t*exp(-2*t)-16*exp(-2*t)
»pretty(x)
2
2 Dirac(t) + 3 exp(-t) - 14 t exp(-2 t) +
26 t exp(-2 t) - 16 exp(-2 t)
)(1626143)(2)( 2222 tueteetettx tttt
Of course all of our hard work can be easily done in one step by using the ilaplace command in Matlab
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47
Laplace Transform:X(s) Inverse Laplace Transform
Summary Direct calculation of Inverse Laplace Transform is
difficult
Practically, the Inverse Laplace Transform of a rational function is calculated using a table look-up method
Use long division and partial fraction expansion to put X(s) in strictly rational form
Two general types of poles: distinct and repeated
Matlab can be used to verify each step by hand or quickly perform the entire Inverse Laplace Transformation using ilaplace