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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 13 (ignoring subsections marked N ). See also Belanger , Chapters 13, and Brogan , Chapters 3 & 5, or Chen Chapters 13. Problems: 4. Consider the plant described by the state space model x = [ 1 3- 3 ] x + [ 1 ] u y = [1 ] x We consider two cases: The parameter 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 , (i) Verify that d dt x 1 = x 2 , so that y = x 1 + 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 closed-loop system Y ( s ) /R ( s ), identify the closed-loop poles and zeros in a pole-zero plot, and discuss your findings. Solution : (i) From the state space model, we have x 1 = x 2 and y = x 1 + x 2 = x 1 + x 1 . (ii) First consider the open loop transfer function with = 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 left-half plane: Open loop poles:...
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This note was uploaded on 03/28/2011 for the course ECE 486 taught by Professor H during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09