midterm_solution - ECE311S Dynamic Systems and Control...

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ECE311S Dynamic Systems and Control Midterm Exam Solution Winter 2004 Problem 1 (a) The poles of the system are the roots of s 4 + Ks 3 + ( Kb + K ) s 2 + K ( a + b ) s + Kab. Form the Routh array: s 4 1 Kb + K Kab s 3 K K ( a + b ) 0 s 2 Kb + K - ( a + b ) Kab 0 s [ Kb + K - ( a + b )] K ( a + b ) - K 2 ab Kb + K - ( a + b ) 0 0 s 0 Kab 0 0 Conditions for BIBO stability are obtained by imposing that all entries in the ±rst column of the Routh array be positive, i.e., (recall that we already have that K > 0 ,a > 0 ,b > 0), Kb + K - ( a + b ) > 0 [ Kb + K - ( a + b )] K ( a + b ) - K 2 ab > 0 . (b) Just apply the Final Value Theorem to obtain the following: y ss = lim s 0 sY ( s ) = 1 . 1
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Problem 2 We look at G ( s ) = ( s + 2)( s + 3) ( s + 4)( s + 5)( s + 6)( s 2 - 2 s + 2) and realize that we have 5 branches, two of which approach the zeros in - 2 and - 3, so there are n - m = 5 - 2 = 3 asymptotes with angles π/ 3, π , and 5 / 3 π . The center of the asymptotes is at α = ( - 4 - 5 - 6 + 1 + 1) - ( - 2 - 3) 3 = - 8 3 . The departure angle from the pole in 1 +
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This note was uploaded on 01/30/2012 for the course ECE ECE311 taught by Professor Francis during the Spring '11 term at University of Toronto- Toronto.

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midterm_solution - ECE311S Dynamic Systems and Control...

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