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

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Presentation on theme: "S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Laplace Transform Math Review with Matlab: Calculating the Inverse."— Presentation transcript:

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

3 Laplace Transform: X(s) Inverse Laplace Transform 2 Laplace Nomenclature Table Look Up Method Simple Table Look Up Example Inverse Laplace Transform General Form Distinct Pole Example Repeated Poles Example

4 Laplace Transform: X(s) Inverse Laplace Transform 3 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:

5 Laplace Transform: X(s) Inverse Laplace Transform 4 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

6 Laplace Transform: X(s) Inverse Laplace Transform 5 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

7 Laplace Transform: X(s) Inverse Laplace Transform 6 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

8 Laplace Transform: X(s) Inverse Laplace Transform 7 Tabular Solution X(s) is written as a Linear Summation of terms with Known Inverse Laplace Transforms LT -1 x(t) can be directly determined using the Laplace Transform Pairs shown below to the right

9 Laplace Transform: X(s) Inverse Laplace Transform 8 Matlab Verification » syms X s » X=((10/s)+(4/(s+3))); » x=ilaplace(X) x = 10+4*exp(-3*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

10 Laplace Transform: X(s) Inverse Laplace Transform 9 Inverse Laplace Transform General Form In general X(s) can be written as a rational function z 1, z 2, …, z m are the zeros of X(s) p 1, p 2, …, p n 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

11 Laplace Transform: X(s) Inverse Laplace Transform 10 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) N(s) = Numerator D(s) = Denominator Q(s) = Quotient R(s) = Remainder

12 Laplace Transform: X(s) Inverse Laplace Transform 11 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

13 Laplace Transform: X(s) Inverse Laplace Transform 12 CASE1: Distinct Poles Assume that all Poles are Different ( p i p j if i j ) Assume numerator order is less than denominator order, then the Partial Fraction Expansion of X(s) is given by: where:

14 Laplace Transform: X(s) Inverse Laplace Transform 13 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

15 Laplace Transform: X(s) Inverse Laplace Transform 14 CASE2: Repeated Pole Assume that Some or All Poles (Roots) are Repeated For the case below, Pole p 1 is Repeated r Times Partial Fraction Expansion shows Repeated and Distinct poles Repeated PoleDistinct Poles

16 Laplace Transform: X(s) Inverse Laplace Transform 15 Determine Coefficients Find Distinct Poles Coefficients, i = r+1, r+2,...,n Find General Repeated Poles, i = 1, 2,..., r-1 Find First Repeated Pole Coefficient, i = r

17 Laplace Transform: X(s) Inverse Laplace Transform 16 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

18 Laplace Transform: X(s) Inverse Laplace Transform 17 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 Notice all Poles are Distinct Order of Numerator (2) < Order of Denominator (3)

19 Laplace Transform: X(s) Inverse Laplace Transform 18 Partial Fraction Expansion Start by finding Partial Fraction Expansion of X(s) Poles of p 1 =-1, p 2 =-2, p 3 =-3 Partial Fraction Expansion Find c i coefficients

20 Laplace Transform: X(s) Inverse Laplace Transform 19 c 1 is explicitly evaluated as: Find Coefficient C 1 C 1 = -2 Each coefficient is determined by evaluating:

21 Laplace Transform: X(s) Inverse Laplace Transform 20 Find Coefficient C 2 and C 3 C 2 = 3 C 3 = 1

22 Laplace Transform: X(s) Inverse Laplace Transform 21 Partial Fraction Expanded X(s) Original Expression Expansion Replace with Coefficients

23 Laplace Transform: X(s) Inverse Laplace Transform 22 Inverse Laplace Transform x(t) = Sum of the Inverse Laplace Transform of the individual terms of X(s) LT -1

24 Laplace Transform: X(s) Inverse Laplace Transform 23 » 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

25 Laplace Transform: X(s) Inverse Laplace Transform 24 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 Order of Numerator (4) = Order of Denominator (4) therefore X(s) is NOT Strictly Rational Notice Pole s = -2 is Repeated 3 times

26 Laplace Transform: X(s) Inverse Laplace Transform 25 Solution Steps Convert X(s) to a strictly rational function u Perform long division by hand Verify long division using Matlab deconv Perform partial fraction expansion on rational part of X(s) u 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

27 Laplace Transform: X(s) Inverse Laplace Transform 26 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

28 Laplace Transform: X(s) Inverse Laplace Transform 27 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 Must Decompose X(s) into Strictly Rational Functions of s

29 Laplace Transform: X(s) Inverse Laplace Transform 28 Long Division Perform long division to decompose X(s) into a constant plus a purely rational function Thus:

30 Laplace Transform: X(s) Inverse Laplace Transform 29 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 [Q, R] = deconv( B, A) B = Numerator polynomial coefficients A = Denominator polynomial coefficients Q = Quotient of B/A R = Remainder of B/A

31 Laplace Transform: X(s) Inverse Laplace Transform 30 Long Division Verification Use Matlab to verify the polynomial long division: » polynum = [ ]; » polyden = [ ]; » [Q, R]=deconv(polynum, polyden) Q = 2 R =

32 Laplace Transform: X(s) Inverse Laplace Transform 31 Expand Rational Part of X(s) The rational part of X(s) will be referred to as X o (s) X o (s) has 3 repeated roots and one distinct root Partial Fraction Expansion Partial Fraction Expansion must be performed on X o (s)

33 Laplace Transform: X(s) Inverse Laplace Transform 32 Evaluating Coefficient C 4 C 4 is evaluated using the distinct pole expression as shown in the previous example

34 Laplace Transform: X(s) Inverse Laplace Transform 33 Matlab C 4 Verification c 4 can verified in Matlab by creating the symbolic expression Xo and evaluating for c 4 » 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

35 Laplace Transform: X(s) Inverse Laplace Transform 34 Evaluating Coefficient C 3 C 3 is evaluated similar to C 4 » c3 = (Xo)*((s+2)^3); » c3 = subs(c3,s,-2) c3 = -28 C 3 is easily verified in Matlab

36 Laplace Transform: X(s) Inverse Laplace Transform 35 Repeated Pole Coefficients To find C 2 and C 1, the following expression must be evaluated for each case ( r = 3, p 1 = -2, i = 2, 1 ) For simplicity, let Y(s) be the expression to be differentiated

37 Laplace Transform: X(s) Inverse Laplace Transform 36 Repeated Pole Coefficients Before differentiating, Y(s) can be rewritten as: Similarly, C 1 equals the second derivative of Y(s) / 2 C 2 equals the first derivative of Y(s)

38 Laplace Transform: X(s) Inverse Laplace Transform 37 Calculating First Derivative Calculating First Derivative by Hand » Y =(Xo)*((s+2)^3); » dY = diff(Y); pretty(dY) s - 72 s s - 36 s - 42 s s (s + 1) Matlab Verification of First Derivative

39 Laplace Transform: X(s) Inverse Laplace Transform 38 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) s + 75 s + 72 s (s + 1) Matlab can also be used to perform this simplification

40 Laplace Transform: X(s) Inverse Laplace Transform 39 Evaluate C2 C 2 can now be directly evaluated from Y(s) » c2 = subs(dY,s,-2) c2 = 26 Matlab Verification of C 2

41 Laplace Transform: X(s) Inverse Laplace Transform 40 Calculate Second Derivative The second derivative of Y(s) must be calculated to find C 1 » ddY = diff(dY); pretty(ddY) s s s + 75 s + 72 s (s + 1) (s + 1) Matlab Verification of Y(s)

42 Laplace Transform: X(s) Inverse Laplace Transform 41 Calculate C 1 The following expression can be calculated by hand Matlab Verification » c1 = subs( ddY/2,s,-2) c1 = -16

43 Laplace Transform: X(s) Inverse Laplace Transform 42 Result of Expansion The Rational Part of X(s) is expanded to: Thus X(s) can be rewritten as:

44 Laplace Transform: X(s) Inverse Laplace Transform 43 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 u Integrate the function to be expanded u Differentiate the result of the integration u The result of the symbolic differentiation will be expressed in strictly rational form u 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

45 Laplace Transform: X(s) Inverse Laplace Transform 44 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 ); This can be verified in Matlab » X = diff(int( num/den )); » pretty(X) s s + 2 (s + 2) (s + 2)

46 Laplace Transform: X(s) Inverse Laplace Transform 45 Table Method Solution The following transform pairs can be used to evaluate x(t) LT -1 The Inverse Laplace Transform can be calculated directly

47 Laplace Transform: X(s) Inverse Laplace Transform 46 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) Of course all of our hard work can be easily done in one step by using the ilaplace command in Matlab

48 Laplace Transform: X(s) Inverse Laplace Transform 47 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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