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**Math Review with Matlab: Calculating the Inverse Laplace Transform**

4/1/2017 Math Review with Matlab: Laplace Transform Calculating the Inverse Laplace Transform S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn

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**Inverse Laplace Transform**

4/1/2017 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 Nomenclature The Laplace Transform of a time domain function x(t), will be a complex domain function X(s) This relationship is also denoted as:

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**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 Practically, the Inverse Laplace Transform of a rational function is calculated using a method of table look-up

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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 Requires knowledge or reference of Laplace Transform pairs, but is much simpler than directly calculating the Inverse Laplace Transform

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**Simple Table Look Up Example**

Given: 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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Tabular Solution X(s) is written as a Linear Summation of terms with Known Inverse Laplace Transforms x(t) can be directly determined using the Laplace Transform Pairs shown below to the right LT-1

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Matlab Verification The ilaplace Matlab command can be used to quickly verify the solution » syms X s » X=((10/s)+(4/(s+3))); » x=ilaplace(X) x = 10+4*exp(-3*t) NOTE: The output of ilaplace must be interpreted as a Causal Solution

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**Inverse Laplace Transform General Form**

In general X(s) can be written as a rational function 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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**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) ³ D(s), perform Long Division until the remainder polynomial R(s) is of lesser order than A(s) N(s) = Numerator D(s) = Denominator Q(s) = Quotient R(s) = Remainder

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**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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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: where:

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**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 Building from previous work, x(t) is the summation of unit steps multiplied by exponentials

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**CASE2: Repeated Pole Repeated Pole Distinct Poles**

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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**Determine Coefficients**

Find Distinct Poles Coefficients, i = r+1, r+2, ...,n Find First Repeated Pole Coefficient, i = r Find General Repeated Poles, i = 1, 2, ... , r-1

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**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 1 2 Factor the Denominator Polynomial 3 Perform Partial-Fraction Expansion Use Laplace Transform Pair Table to obtain the inverse Laplace transform 4

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**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 Order of Numerator (2) < Order of Denominator (3) Notice all Poles are Distinct

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**Partial Fraction Expansion**

Start by finding Partial Fraction Expansion of X(s) Poles of p1=-1, p2=-2, p3=-3 Partial Fraction Expansion Find ci coefficients

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Find Coefficient C1 Each coefficient is determined by evaluating: c1 is explicitly evaluated as: C1 = -2

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**Find Coefficient C2 and C3**

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**Partial Fraction Expanded X(s)**

Original Expression Expansion Replace with Coefficients

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**Inverse Laplace Transform**

x(t) = Sum of the Inverse Laplace Transform of the individual terms of X(s) LT-1

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**Distinct Pole Matlab Verification**

This can easily be verified in Matlab using ilaplace » 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)

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**Repeated Pole Example Given:**

Order of Numerator (4) = Order of Denominator (4) therefore X(s) is NOT Strictly Rational Notice Pole s = -2 is Repeated 3 times Use the Partial-Fraction Expansion and the Table Look-Up Method determine the Inverse Laplace Transform of X(s) Verify each step using Matlab

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**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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Repeated Pole Example X(s) must be decomposed into a constant plus a Strictly Rational Function Q(s) will just be a constant since the order of the numerator and the denominator are the same

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**Convert to Rational Functions**

Must Decompose X(s) into Strictly Rational Functions of s 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

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Long Division Perform long division to decompose X(s) into a constant plus a purely rational function Thus:

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**Matlab Polynomial Division: deconv**

The Matlab command deconv can be used to perform polynomial division [Q, R] = deconv( B, A) B = Numerator polynomial coefficients A = Denominator polynomial coefficients Q = Quotient of B/A R = Remainder of B/A deconv operates on arrays of polynomial coefficients, NOT symbolic variables

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**Long Division Verification**

Use Matlab to verify the polynomial long division: » polynum = [ ]; » polyden = [ ]; » [Q, R]=deconv(polynum, polyden) Q = 2 R =

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**Expand Rational Part of X(s)**

The rational part of X(s) will be referred to as Xo(s) Partial Fraction Expansion must be performed on Xo(s) Partial Fraction Expansion Xo(s) has 3 repeated roots and one distinct root

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**Evaluating Coefficient C4**

C4 is evaluated using the distinct pole expression as shown in the previous example

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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) 3

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**Evaluating Coefficient C3**

C3 is evaluated similar to C4 C3 is easily verified in Matlab » c3 = (Xo)*((s+2)^3); » c3 = subs(c3,s,-2) c3 = -28

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**Repeated Pole Coefficients**

To find C2 and C1, the following expression must be evaluated for each case ( r = 3, p1 = -2, i = 2, 1 ) For simplicity, let Y(s) be the expression to be differentiated

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**Repeated Pole Coefficients**

C2 equals the first derivative of Y(s) Similarly, C1 equals the second derivative of Y(s) / 2 Before differentiating, Y(s) can be rewritten as:

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**Calculating First Derivative**

Calculating First Derivative by Hand Matlab Verification of First Derivative » Y =(Xo)*((s+2)^3); » dY = diff(Y); pretty(dY) -39 s s s s s - 16 s (s + 1)

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Simplify Y’(s) Since Y’(s) will eventually have to be calculated, it will be helpful to simplify its terms Matlab can also be used to perform this simplification » dY=simplify(dY); pretty(dY) 26 s s s + 26 2 (s + 1)

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**Evaluate C2 C2 can now be directly evaluated from Y’(s)**

Matlab Verification of C2 » c2 = subs(dY,s,-2) c2 = 26

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**Calculate Second Derivative**

The second derivative of Y(s) must be calculated to find C1 Matlab Verification of Y’’(s) » ddY = diff(dY); pretty(ddY) 78 s s s s s + 26 (s + 1) (s + 1)

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**Calculate C1 The following expression can be calculated by hand**

Matlab Verification » c1 = subs( ddY/2 ,s ,-2) c1 = -16

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**Result of Expansion The Rational Part of X(s) is expanded to:**

Thus X(s) can be rewritten as:

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**Matlab Partial Fraction Expansion**

Vector expressions can be evaluated using residues 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

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**Expansion Verification**

By hand, X(s) was converted to a strictly rational function This can be verified in Matlab » num = 2*s^4 + s^3 -2*s; » den = (s+1)*( (s+2)^3 ); » X = diff(int( num/den )); » pretty(X) s s + 2 (s + 2) (s + 2)

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Table Method Solution The following transform pairs can be used to evaluate x(t) The Inverse Laplace Transform can be calculated directly LT-1

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**Verification Using Matlab**

Of course all of our hard work can be easily done in one step by using the ilaplace command in 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)

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**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

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