ME375 Dynamic System Modeling and Control

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MESB 374 System Modeling and AnalysisInverse
Laplace Transform and I/O Model
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Inverse Laplace Transform

• Basic steps
• Partial fraction expansion (PFE)
• Residue command in Matlab
• Input-output model by using Laplace transform

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Inverse Laplace Transform

• Given 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.

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Partial Fraction Expansion

• Case I Distinct Characteristic Roots

• Residual Formula

• Proof

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Partial Fraction Expansion

• Case II Repeated Roots

• Residual Formula

• Proof

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Partial Fraction Expansion

• Case III General Case

• Residual Formula

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Partial Fraction Expansion

• Case IV Order of the Numerator C(s)
• Order of the denominator D(s) n m

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Partial Fraction Expansion

• Case V Complex Roots

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Residue Command in MATLAB

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Residue Command in MATLAB (Example)
Ex Given
Find inverse Laplace transform
MATLAB command gtgt A, P, K residue ( 1,
0, 0, 2 , 1, 2, 1, 0 ) will return the
following values A -4, -1, 2T , P -1,
-1, 0 , K 1 which means that

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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

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Example Car Suspension System

• Step 1 LT of differential equations assuming
  zero ICs

• Step 2 Solve for output using algebraic
  elimination method

• 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
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Example (Cont.)
from first equation
Substitute it into the second equation

• Step 3 write down I/O model