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lecture14 - MAE 171A: Dynamic Systems Control Lecture 14:...

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Unformatted text preview: MAE 171A: Dynamic Systems Control Lecture 14: Loop Shaping Control Design Goele Pipeleers Loop Shaping Design: Concept feedback control configuration D ( s ) G ( s ) u y r e open-loop system L ( s ) = D ( s ) G ( s ) closed-loop system T ( s ) , Y ( s ) R ( s ) = L ( s ) 1 + L ( s ) 1 design target: T ( s ) 2 n s 2 + 2 n s + 2 n where and n are determined from specifications on t r , t s , M p this suggests that D ( s ) be chosen so that L ( s ) = T ( s ) 1- T ( s ) 2 n s ( s + 2 n ) How should the frequency response L ( j ) look like? 2 Open-loop and Closed-loop Characteristics Open-loop System L ( s ) = 2 n s ( s + 2 n ) break point b b = 2 n L ( j b ) = 1 (2 ) 2 1- 1 + j | L ( j b ) | = 1 2(2 ) 2 L ( j b ) =- 135 3 { L } { L } 1 c c b b n b 1 K v c b n b 1 PM [log] | L | [dB] [log] L [ ] PM 135 90 135 180 4 crossover frequency c | L ( j c ) | = 1 2 n j c ( j c + 2 n ) = 1 4 c + 4 2 2 n 2 c- 4 n = 0 c = n where , q p 1 + 4 4- 2 2 2 = p 1 + 4 4- 2 2 = 1 p 1 + 4 4 + 2 2 1 1 + 2 4 + 2 2 1 (1 + 2 ) 2 5 hence, c = n and 1 1 + 2 . 5 n < c < n 1 1 + 2 1 . 5 1 c / n 6 phase margin PM L ( j c ) = L ( j n ) = 1- 2 + j 2 L ( j c ) = (- 2- j 2 ) PM = 180 + L ( j c ) PM = arctan 2 100 { L } { L } 1 j 2 PM 2 L ( j c ) c 100 PM 1 100 PM 7 gain margin GM no phase crossover GM = { L } { L } 1 c 8 error constants K p = lim s L ( s ) = closed-loop perfect asymptotic tracking step K v = lim s sL ( s ) = lim s 2 n s + 2 n = n 2 low frequency asymptote of L ( j ) * L ( j ) = 2 n j ( j + 2 n ) n j 2 , L low ( j ) * hence, K v = | L low ( j 1) | 9 { L } { L } 1 c c b b n b 1 K v...
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This note was uploaded on 08/08/2011 for the course MAE 171 taught by Professor Pipeleers during the Summer '11 term at UCLA.

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lecture14 - MAE 171A: Dynamic Systems Control Lecture 14:...

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