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Unformatted text preview: ECE 486 Assignment # 2 Issued: January 27 Due: February 3, 2011 Reading Assignment: Continue reading ... FPE , 5th or 6th ed., Chapters 1–3 (ignoring subsections marked “ N ”). See also Belanger , Chapters 1–3, and Brogan , Chapters 3 & 5, or Chen Chapters 1–3. Problems: 4. Consider the plant described by the state space model ˙ x = [ 1 3 3 ] x + [ 1 ] u y = [1 α ] x We consider two cases: The parameter α takes the value 1 or 1. Consider a “state feedback control” of the form u = Kx + v = K 1 x 1 K 2 x 2 + K 3 r , where r is a reference input. For each value of α , (i) Verify that d dt x 1 = x 2 , so that y = x 1 + α d dt x 1 . (ii) Find K such that the resulting closed loop system with input r and output y is BIBO stable, and the DC gain of the transfer function Y ( s ) /R ( s ) is unity. Include step response plots in your solution — This can only be done using Matlab! (iii) Compute the transfer function of your closedloop system Y ( s ) /R ( s ), identify the closedloop poles and zeros in a polezero plot, and discuss your findings. Solution : (i) From the state space model, we have ˙ x 1 = x 2 and y = x 1 + αx 2 = x 1 + α ˙ x 1 . (ii) First consider the open loop transfer function with α = 0. Since ˙ x 1 = x 2 we have, ¨ x 1 = 3 x 1 3 ˙ x 1 + u = ⇒ X 1 U = 1 s 2 + 3 s 3 . The open loop poles are located at 3 ± √ 21 / 6 ≈ (4 . 1 , 5 . 1). The pole at 4.1 will cause instability. We wish to move the poles into the lefthalf plane: Open loop poles:...
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