homework4_solution.pdf - Solutions to Homework 4 ECE 460...

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Solutions to Homework 4 ECE 460: Automatic Control Due: Monday, February 8, at 2:10pm Note 1. Your solutions should be neat and legible; always show your work. Problem 1. (20 points) Reduce the block diagram given in Figure 1 to find the transfer function G ( s ) = Y ( s ) /R ( s ). Figure 1 Solution: Suppose G ( s ) = 1 s + 1 · 1 s and H ( s ) = 1 s + 2 . E ( s ) = R ( s ) - H ( s ) Y ( s ) Y ( s ) = G ( s ) E ( s ) = G ( s )( R ( s ) - H ( s ) Y ( s )) Y ( s ) + G ( s ) H ( s ) Y ( s ) = G ( s ) R ( s ) Y ( s )(1 + G ( s ) H ( s )) = G ( s ) R ( s ) T ( s ) = Y ( s ) R ( s ) = G ( s ) 1 + G ( s ) H ( s ) = 1 s ( s +1) 1 + 1 s ( s +1) 1 s +2 = s + 2 s ( s + 1)( s + 2) + 1 = s + 2 s 3 + 3 s 2 + 2 s + 1 Problem 2. (20 points) Use the Routh’s criterion to determine how many roots with positive real parts the following equations have. 1
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(a) s 3 + 4 s 2 + 8 s + 12 = 0 Solution: s 3 1 8 s 2 4 12 s 1 - 1 8 4 12 4 = 5 s 0 - 4 12 5 0 5 = 12 There is no root in the RHP. (b) s 4 + s 3 - s - 1 = 0 Solution: s 4 1 0 -1 s 3 1 -1 s 2 - 1 0 1 - 1 1 = 1 - 1 - 1 1 0 1 = - 1 s 1 - 1 - 1 1 - 1 1 = 0 We have a zero on the first column and there is nothing on the next column, which is basically the same as having zeros on the entire row. So, we need to take the row of s 2 to find its derivative, i.e.,
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