problem_11_18 - Problem 11.18 of the Palm textbook Plant...

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Problem 11.18 of the Palm textbook Plant TF: G P s ( ) = s + 10 s 2 + 5 s + 6 Sketch the root locus. Determine the smallest damping ratio possible for positive values of proportional gain K. Determine the value of proportional gain K that is required to minimize the dominant time constant with ζ = 0.707, and determine this time constant. SOLUTION Closed-loop characteristic equation: 0 = 1 + K s + 10 s 2 + 5 s + 6 ! " # $ % & = 1 + K N s ( ) D s ( ) ! " # $ % & OR D CL s ( ) = D s ( ) + KN s ( ) = s 2 + 5 s + 6 + K s + 10 ( ) = s 2 + 5 + K ( ) s + 6 + 10 K 1. Number of branches : max N P , N Z { } = max 2,1 { } = TWO branches 2. Start/end points : OL zeros : z 1 = ! 10 OL poles : p 1 = ! 2, p 2 = ! 3, Therefore, the RL START at poles p 1 , p 2 . One RL branch ENDS at p 1 , and one RL 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 following regions on the real axis can contain the RL: p 2 < s < p 1 and s < z 1 . 5. Here we have one branch of the RL going to ! ; we need to determine the asymptotes for these branches. We could use Rule 5 to determine this asymptote. Alternately, we can
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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_18 - Problem 11.18 of the Palm textbook Plant...

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