solmidterm05_mae143b

Use solid lines to sketch the actual bode plots use

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Unformatted text preview: − arctan( = arctan( 2+ 1 1 − ω2 −ω 2jω + 1 √ √ −ω 2ω 2 lim φ(ω ) = lim arctan( ) = −90 − lim arctan( ) − arctan( ) = −270 ω →∞ ω →∞ ω →∞ 1 1 − ω2 −ω √ −ω 2ω ) = arctan(−0) − arctan(0) = 0 − 0 = 0 lim φ(ω ) = lim arctan( ) − arctan( ω →0 ω →0 1 1 − ω2 4. In the two figures below, sketch the amplitude and phase Bode plot of the transfer function model G(s) in (1). Use solid lines to sketch the actual Bode plots, use dashed lines to indicate the asymptotic behavior of the Bode plots. [10pt] 2 10 90 0 1 φ(ω ) [deg] M (ω ) 10 0 10 −90 −180 −1 10 −270 −2 10 −2 10 −1 10 0 10 1 10 −360 −2 10 2 10 ω rad/s −1 10 0 10 ω rad/s 2 1 10 2 10 Now consider the standard 2nd order system characterized by the transfer function model 2 ωn y (s) = G(s)u(s), G(s) = 2 2 s + 2βωn s + ωn (3) controlled by a PD-controller C (s) = Kp + Kd s using negative feedback: u(s) = r(s) − C (s)y (s). 5. Show that the closed-loop transfer function T (s) from r(s) to y (s) can be written as the standard 2nd order system 2 ωn with ωn = ωn ¯ ¯¯ s2 + 2β ωn s + ωn ¯2 ¯ β + ωn Kd /2 . [15pt] 1 + Kp and β = 1 + Kp 2 ωn 2 G(s) s2 + 2βωn s + ωn =2 Tyr (s) = 2 2 1 + G(s)C (s) s + 2βωn s + ωn ωn +2 2 2 Kp + Kd s s2 + 2βωn s +...
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This note was uploaded on 05/27/2010 for the course MAE MAT143B taught by Professor Linearcontrol during the Spring '10 term at UCSD.

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