problem_11_17

problem_11_17 - Problem 11.17 of the Palm textbook Plant...

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Unformatted text preview: Problem 11.17 of the Palm textbook Plant TF: G P s ( 29 = 6 2 + 2 ( 29 3 + 24 ( 29 Sketch the root locus. Show that it is not possible to achieve a dominant time constant of less than 2.07 seconds for the plant with proportional control. Use PD control to produce a dominant time constant of 0.5 seconds and a damping ratio of 0.707. SOLUTION Closed-loop characteristic equation: = 1 + 6 2 + 2 ( 29 3 + 24 ( 29 = 1 + 1 3 + 9 2 + 8 = 1 + ( 29 ( 29 1. Number of branches : max N P , N Z { } = 3,0 { } = 2. Start/end points : OL zeros : none OL poles : p 1 = 0, 2 = -1, 3 = -8 Therefore, the RL START at poles p 1 , p 2 , p 3 . All 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: p 2 < < 1 and s < 3 . 5. Here we have three branches of the RL going to ; we need to determine the asymptotes for these branches: = p i - z i N P- N z = 0 - 1 - 8 3 - 0 = - 3 intercept with the real axis ( ) k = 2 k- 1 ( ) 180 N P- N z = 120 k- 60 = 60,180, 300...
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problem_11_17 - Problem 11.17 of the Palm textbook Plant...

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