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Ch 3 Transform Methods

Ch 3 Transform Methods. Laplace and Z transforms. LaPlace and Z transform. Laplace Transform definition Transform of a vector Z transform definition t is a non negative integer. Table 3.1, pg 60. * time * exponential/power Time shift Convolution Initial value Final value. Function

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Ch 3 Transform Methods

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  1. Ch 3 Transform Methods Laplace and Z transforms

  2. LaPlace and Z transform • Laplace Transform definition • Transform of a vector • Z transform definition • t is a non negative integer

  3. Table 3.1, pg 60 • * time • * exponential/power • Time shift • Convolution • Initial value • Final value • Function • Notation • Definition • Linearity • Derivative/left shift • Integral/delay Notation DefinitionTheorem

  4. Olivier’s Laplace Transform Table • 1= {δ(t)} • n!/(s+a)n+1={ tne-atstep(t)} • Compare with table 3.2, pg. 61

  5. Continuous time models • Solve CT LTI state equations • Free response • State transition matrix • Formal power series for (sI-A)-1 • Laplace transform eAt

  6. Forced response • Transfer Function • (sI-A)-1=adj(sI-A)/det(sI-A) • Characteristic polynomial • Dimensions of H(s) • Properness • Proper/improper • Strictly proper • Co-proper • Long division and Markov parameters

  7. DT LTI models • Z-transform solution

  8. Olivier’s Z-transform Table • 1 = {δk}; δk={1, 0, 0, …} • z/(z-a)q = {0 for k < q-1, comb(k,q-1)ak-q+1} • Compare with Table 3.3 pg. 68

  9. Free Response • Compute Free response solution • {At}= • (zI-A)-1z = • compare

  10. Forced Response • Compute Forced Response • Transfer Function

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