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 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.

Partial Fraction Expansion Case I: Distinct Characteristic Roots Residual Formula Proof

Partial Fraction Expansion Case II: Repeated Roots Residual Formula Proof

Partial Fraction Expansion Residual Formula Case III: General Case

Partial Fraction Expansion Case IV: Order of the Numerator C(s) = Order of the denominator D(s) : n = m

Partial Fraction Expansion Case V: Complex Roots

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.

Residue Command in MATLAB (Example) Ex: Given 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 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 Example – Car Suspension System x p 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

Example (Cont.) Step 3: write down I/O model from first equation Substitute it into the second equation