problem_11_17

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

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Unformatted text preview: Problem 11.17 of the Palm textbook Plant TF: G P s ( 29 = 6 σ 2 σ + 2 ( 29 3 σ + 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 + Κ 6 σ 2 σ + 2 ( 29 3 σ + 24 ( 29 ÷ = 1 + Κ 1 σ 3 + 9 σ 2 + 8 σ ÷ = 1 + Κ Ν σ ( 29 ∆ σ ( 29 ÷ 1. Number of branches : max N P , N Z { } = μ αξ 3,0 { } = ΤΗΡΕΕ βρανχηεσ 2. Start/end points : OL zeros : none OL poles : p 1 = 0, π 2 = -1, π 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 < σ < π 1 and s < π 3 . 5. Here we have three branches of the RL going to ∞ ; we need to determine the asymptotes for these branches: σ = p i å- z i å N P- N z = 0 - 1 - 8 3 - 0 = - 3 ¬ intercept with the real axis ( ) θ 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 TF...

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