HW11_solutions

# HW11_solutions - EECS 460 Winter 2008 Homework#11 Solutions...

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9-30 (a) Bode Plot: For stability: 166 (44.4 dB) < K < 7079 (77 dB) Phase crossover frequencies: 7 rad/sec and 85 rad/sec Nyquist Plot:
9-30 (b) Root Loci. Note that at -180 o , ω = 7.16 rad/s and ω = 85.3 rad/s which match the results calculated in part (a). 9-39 (a) Forward-path Transfer Function: Gs Ys Es e ss s () () ( ) ( ) == ++ 2 11 0 12 5 From the Bode diagram, phase crossover frequency = 0.21 rad/sec GM = 21.55 dB gain crossover frequency = 0 rad/sec PM = infinite (b) Approximate the time delay using a second order approximation to obtain: 2 2 2 1 1 2 PM = GM=25dB 1 1 2 s s e s ⎛⎞ ⎜⎟ ⎝⎠ ≈⇒ +

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(c) Approximate the time delay using a first order approximation to obtain the following results: 2 1 PM = GM=25dB (around 0.3 rad/s) 1 s s e s ≈⇒ + -150 -100 -50 0 Magnitude (dB) 10 -3 10 -2 10 -1 10 0 10 1 10 2 -180 0 180 360 Phase (deg) Bode Diagram Frequency (rad/sec) Plant 1st order Delay 2nd order Delay

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Problem 10-20 () 2 100 10 100 Gs ss
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HW11_solutions - EECS 460 Winter 2008 Homework#11 Solutions...

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