Ch 3 Transform Methods

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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

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
• Notation
• Definition
• Linearity
• Derivative/left shift
• Integral/delay

Notation DefinitionTheorem

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
• Solve CT LTI state equations
• Free response
• State transition matrix
• Formal power series for (sI-A)-1
• Laplace transform eAt
Forced response
• Transfer Function
• Characteristic polynomial
• Dimensions of H(s)
• Properness
• Proper/improper
• Strictly proper
• Co-proper
• Long division and Markov parameters
DT LTI models
• Z-transform solution
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
• Compute Free response solution
• {At}=
• (zI-A)-1z =
• compare
Forced Response
• Compute Forced Response
• Transfer Function