problem_11_17

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

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Problem 11.17 of the Palm textbook Plant TF: G P s ( ) = 6 s 2 s + 2 ( ) 3 s + 24 ( ) 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: 0 = 1 + K 6 s 2 s + 2 ( ) 3 s + 24 ( ) ! " # \$ % & = 1 + K 1 s 3 + 9 s 2 + 8 s ! " # \$ % & = 1 + K N s ( ) D s ( ) ! " # \$ % & 1. Number of branches : max N P , N Z { } = max 3,0 { } = THREE branches 2. Start/end points : OL zeros : none OL poles : p 1 = 0, p 2 = ! 1, p 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 < s < p 1 and s < p 3 . 5. Here we have three branches of the RL going to ! ; we need to determine the asymptotes for these branches:

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! 0 = 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 ° 6. BA/RE points d ds N s ( )
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problem_11_17 - Problem 11.17 of the Palm textbook Plant TF...

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