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design-lead-RL0

design-lead-RL0 - ECE 382 Lead Compensator...

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ECE 382 Lead Compensator Design (Root-Locus) G ( s ) = 4 s ( s + 2 ) Design Objective: ζ = 0 . 5 and ω n = 4 rad/sec. ( Interpret the design objective in terms of performance specification. ) Procedure : 1) General form of a lead compensator G c ( s ) = K c ( s + 1 τ ) ( s + 1 ατ ) , K c , τ > 0 , 0 < α < 1 Determine the closed-loop poles of the uncompensated system. T ( s ) = G ( s ) 1 + G ( s ) = 4 s 2 + 2 s + 4 4 = ω 2 n s 2 + 2 ζω n s + ω 2 n Closed-loop poles of the uncompensated system are at s 1 , s 2 = - ζω n ± j ω n q 1 - ζ 2 = - 1 ± j 3 = - 1 ± j 1 . 732 Also K v = lim s 0 sG ( s ) = 2. 2) Determine the desired dominant closed-loop poles location of the compensated system. T d ( s ) = G d 1 + G d ( s ) = G c ( s ) G ( s ) 1 + G c ( s ) G ( s ) = ω 2 n s 2 + 2 ζω n s + ω 2 n where, from the design objective, we have ζ = 0 . 5 and ω n = 4 rad/sec. Then we have, s ? 1 , s ? 2 = - ζω n ± j ω n q 1 - ζ 2 = - 2 ± j 2 3 = - 2 ± j 3 . 464 3) Use MATLAB’s “rlocus(num,den)” function to plot the root locus of the uncompensated system and see whether the root locus will pass through the desired dominant pole location (i.e., s ?
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