problem_11_15 - Problem 11.15 of the Palm textbook...

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Plant TF: G P s ( ) = 2 s 2 ! s ! 26 s s + 2 ( ) s + 3 ( ) Sketch the root locus. Determine the range of proportional control gains K for which the system response is stable. Determine the value of K required for a time constant of ! = 2 / 3 . SOLUTION Closed-loop characteristic equation: 0 = 1 + K 2 s 2 ! s ! 26 s s + 2 ( ) s + 3 ( ) " # $ % ' = 1 + K 2 s 2 ! s ! 26 s 3 + 5 s 2 + 6 s " # $ % ' = 1 + K N s ( ) D s ( ) " # $ % ' 1. Number of branches : max N P , N Z { } = max 3,2 { } = THREE branches 2. Start/end points : OL zeros : z 1 = ! 3.36, z 2 = 3.86 OL poles : p 1 = 0, p 2 = ! 2, p 3 = ! 3 Therefore, the RL START at poles p 1 , p 2 , p 3 . Two RL branches ENDS at z 1 and z 2 while one branch ends at ! . 4. Since the RL can exist on the real axis only to the left of an ODD number of real OL zeros/poles, the regions on the real axis containing the RL are: p 1 < s < z 2 , p 3 < s < p 2 and s < z 2 . 5.
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This note was uploaded on 12/23/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue University-West Lafayette.

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problem_11_15 - Problem 11.15 of the Palm textbook...

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