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sample_RL_solns

# sample_RL_solns - Sample Root Locus Plots U(s R(s K plant...

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Sample Root Locus Plots Poles and zeros of the plant TF G P s ( ) shown below as “X”s and “O”s, respectively. Assume that all plant TFs G P s ( ) have a static gain of “1”. Sketch the loci of the closed-loop poles from K = 0 to K → ∞ for each plant. From plot and given static gain of G P 0 ( ) = 1 : G P s ( ) = 12 s + 2 ( ) s + 6 ( ) = 12 s 2 + 8 s + 12 = N D From Rule #4: The RL contains the real axis between the two OL poles. From Rule #6: d ds N D = D dN ds N dD ds D 2 0 = 0 12 2 s + 8 ( ) D 2 2 s + 8 = 0 s = 4 From Rule #5: σ 0 = 2 6 0 2 0 = 4 θ k = 2 k 1 ( ) 180 2 = 2 k 1 ( ) 90 = 90 ° ,270 ° From Rule #2: CL root loci start at OL poles ( s = 2, 6 ) and end at . Y s ( ) R s ( ) K G P s ( ) U s ( ) plant

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G P s ( ) = 4 / 3 ( ) s + 6 ( ) s + 2 ( ) s + 4 ( ) = 4 / 3 ( ) s + 6 ( ) s 2 + 6 s + 8 = N D From Rule #4: The RL contains the real axis between the two OL poles and to the left of the OL zero. From Rule #6: d ds N D = D dN ds N dD ds D 2 0 = 4 3 s 2 + 6 s + 8 ( ) s + 6 ( ) 2 s + 6 ( ) D 2 = 4 3 s 2 12 s 28 D 2 s 2 + 12 s + 32 = 0 s 1,2 = 12 ± 12 2
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sample_RL_solns - Sample Root Locus Plots U(s R(s K plant...

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