hw3soln_F2009 - MAE170 Homework 3 Solution Fall 2009 AP4.5...

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Unformatted text preview: MAE170 Homework 3 Solution Fall 2009 AP4.5 (a) The transfer function from the disturbance D ( s ) to the output Y ( s ) is Y ( s ) D ( s ) =- s s 3 + 4 s 2 + 4 s + K The steady-state error (when D ( s ) = 1 /s ) is e ss = lim s s [- Y ( s )] = lim s s s s 3 + 4 s 2 + 4 s + K 1 s = lim s s s 3 + 4 s 2 + 4 s + K = 0 5 10 15 20 25-0.15-0.1-0.05 0.05 0.1 Response due to a step disturbance Time (sec) Amplitude Figure 1: Error plot with a step disturbance input and K = 8 for problem AP4.5 (b) the closed-loop transfer function is Y ( s ) R ( s ) = K s 3 + 4 s 2 + 4 s + K The steady-state error (when R ( s ) = 1 /s 2 ) is 1 e ss = lim s s [ R ( s )- Y ( s )] = lim s s 1- K s 3 + 4 s 2 + 4 s + K 1 s 2 = lim s s 3 + 4 s 2 + 4 s s ( s 3 + 4 s 2 + 4 s + K ) Apply LHopitals Rule to compute the limit e ss = lim s 3 s 2 + 8 s + 4 4 s 3 + 12 s 2 + 8 s + K = 4 K (c) Let K = 8. Then Y ( s ) D ( s ) =- s s 3 + 4 s 2 + 4 s + 8 The response due to the step disturbance for...
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hw3soln_F2009 - MAE170 Homework 3 Solution Fall 2009 AP4.5...

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