problem_1120

problem_1120 - Problem 11.20 of the Palm textbook Plant TF:...

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Unformatted text preview: Problem 11.20 of the Palm textbook Plant TF: G P s ( 29 = 1 2 + 3 + 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 ( 29 = + = + Closed-loop characteristic equation : = 1 + ( 29 ( 29 = 1 + + 1 2 + 3 + 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 = and a = / (see footnote below) 1 : = 1 + + 2 + 3 + 2 ( 29 = 1 + ( 29 ( 29 2 + 3 + 2 ( 29 + + ( 29 = 3 + 3 2 + 2 + ( 29 + = 0 1. Number of branches : max N P , N Z { } = 3,1 { } = 2. Start/end points : OL zeros : z 1 = - 1 = 0, 2 = -1, 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 . 4. Since the RL can exist on the real axis only to the left of an ODD number of real OL zeros/poles, the following regions on the real axis can contain the RL based on the value 1 If this problem were given to you on an exam as a root locus construction problem, we...
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problem_1120 - Problem 11.20 of the Palm textbook Plant TF:...

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