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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

Ch 3 Transform Methods

Laplace and Z transforms

laplace and z transform
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
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

olivier s laplace transform table
Olivier’s Laplace Transform Table
  • 1= {δ(t)}
  • n!/(s+a)n+1={ tne-atstep(t)}
  • Compare with table 3.2, pg. 61
continuous time models
Continuous time models
  • Solve CT LTI state equations
  • Free response
    • State transition matrix
    • Formal power series for (sI-A)-1
    • Laplace transform eAt
forced response
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
dt lti models
DT LTI models
  • Z-transform solution
olivier s z transform table
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
free response
Free Response
  • Compute Free response solution
  • {At}=
  • (zI-A)-1z =
  • compare
forced response1
Forced Response
  • Compute Forced Response
  • Transfer Function