problem_11_20 - Problem 11.20 of the Palm textbook Plant...

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Problem 11.20 of the Palm textbook Plant TF: G P s ( ) = 1 s 2 + 3 s + 2 Sketch the root locus. Design a PI controller that minimizes the dominant time constant with ! =1/ 2 . SOLUTION PI controller TF : G C s ( ) = K P + K I s = K P s + K I s Closed-loop characteristic equation : 0 = 1 + G C s ( ) G P s ( ) = 1 + K P s + K I s ! " # $ % & 1 s 2 + 3 s + 2 ! " # $ % & In order to produce a root locus plot, we need to have our CLCE in terms of a single parameter. For this problem, we have two parameters: K P and K I . Here we will write the CLCE in terms of K = K P and a = K I / K P (see footnote below) 1 : 0 = 1 + K s + a s s 2 + 3 s + 2 ( ) = 1 + K N s ( ) D s ( ) ! s s 2 + 3 s + 2 ( ) + K s + a ( ) = s 3 + 3 s 2 + 2 + K ( ) s + Ka = 0 1. Number of branches : max N P , N Z { } = max 3,1 { } = THREE branches 2. Start/end points : OL zeros : z 1 = ! a OL poles : p 1 = 0, p 2 = ! 1, p 3 = ! 2 Therefore, the RL START at poles p 1 , p 2 , p 3 . One RL branch ENDS at z 1 , and two RL branches end at ! . 1 If this problem were given to you on an exam as a root locus construction problem, we would likely give you a numerical value of K P , K I , K P / K I OR K I / K P , and ask you to construct the RL based on the remaining free parameter.
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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_20 - Problem 11.20 of the Palm textbook Plant...

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