lecture5

lecture5 - MAE 171A Dynamic Systems Control Lecture 5...

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Unformatted text preview: MAE 171A: Dynamic Systems Control Lecture 5: Dynamical System Properties effect of zeros Goele Pipeleers Recall from Lecture 4 . . . System Poles and Step Response • first-order system: G ( s ) = σ s + σ = 1 τs + 1 – step response: y ( t ) = 1- e- t/τ t τ 1 y 1 − e − 1 ℑ{ p } ℜ{ p } − σ 1 • second-order system: G ( s ) = ω 2 n s 2 + 2 ξω n s + ω 2 n = σ 2 + ω 2 ( s + σ ) 2 + ω 2 – step response: y ( t ) = 1- e- σt p 1- ξ 2 cos( ωt- β ) t 1 y 1 − e − σt / ò 1 − ξ 2 T ℑ{ p } ℜ{ p } ω − σ ω n β =arcsin ξ 2 Performance Measures • performance measures on step response t 1 y . 9 . 1 M p t s t r ± 1% t p t r : rise time t s : settling time t p : peak time M p : overshoot • heuristic formulas for second-order system 3 In This Lecture • effect of zeros on system response 4 Real Zero ! Real Pole Partial Fraction Expansion • consider G ( s ) = 1- s/z ( s- p 1 )( s- p 2 ) = 1- p 1 /z p 1- p 2 1 s- p 1 + 1- p 2 /z p 2- p 1 1 s- p 2 • if p i /z ≈ 1 : the contribution of p i in the sum vanishes • hence: a zero near a pole reduces the contribution of that pole in the system response • example: G ( s ) = 1 + s/ 1 . 01 ( s + 1)( s + 2) = . 01 1 . 01 1 s + 1 + . 99 1 . 01 1 s + 2 5 Effect on Impulse Response • alternative analysis: set...
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lecture5 - MAE 171A Dynamic Systems Control Lecture 5...

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