lecture14

# 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 ) ≈ ω 2 n s 2 + 2 ξω n s + ω 2 n – where ξ and ω 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 ) ≈ ω 2 n s ( s + 2 ξω n ) • How should the frequency response L ( jω ) look like? 2 Open-loop and Closed-loop Characteristics Open-loop System • L ( s ) = ω 2 n s ( s + 2 ξω n ) • break point ω b – ω b = 2 ξω n – L ( jω b ) = 1 (2 ξ ) 2 · 1- 1 + j ⇒ | L ( jω b ) | = 1 √ 2(2 ξ ) 2 ∠ L ( jω b ) =- 135 ◦ 3 ℑ{ L } ℜ{ L } – 1 ω c ω c ω b ω b ω n ω b 1 K v ω c ω b ω n ω b 1 PM ω [log] | L | [dB] ω [log] Ó L [ ◦ ] PM – 135 ◦ – 90 – 135 – 180 4 • crossover frequency ω c – | L ( jω c ) | = 1 ⇒ ω 2 n jω c ( jω c + 2 ξω n ) = 1 ⇒ ω 4 c + 4 ξ 2 ω 2 n ω 2 c- ω 4 n = 0 ⇒ ω c = κω n where κ , q p 1 + 4 ξ 4- 2 ξ 2 – κ 2 = p 1 + 4 ξ 4- 2 ξ 2 = 1 p 1 + 4 ξ 4 + 2 ξ 2 ≈ 1 1 + 2 ξ 4 + 2 ξ 2 ≈ 1 (1 + ξ 2 ) 2 5 – hence, ω c = κω n and κ ≈ 1 1 + ξ 2 → . 5 ω n < ω c < ω n κ 1 1 + ξ 2 1 . 5 1 ω c /ω n ξ 6 • phase margin PM – L ( jω c ) = L ( jκω n ) = 1- κ 2 + j 2 ξκ → ∠ L ( jω c ) = ∠ (- κ 2- j 2 ξκ ) – PM = 180 ◦ + L ( jω c ) → PM = arctan 2 ξ κ ≈ 100 ξ ℑ{ L } ℜ{ L } – 1 – j 2 κξ PM – κ 2 Ó L ( jω c ) ω c 100 ξ PM 1 100 ξ PM 7 • gain margin GM – no phase crossover → GM = ∞ ℑ{ L } ℜ{ L } – 1 ω c 8 • error constants – K p = lim s → L ( s ) = ∞ → closed-loop perfect asymptotic tracking step – K v = lim s → sL ( s ) = lim s → ω 2 n s + 2 ξω n = ω n 2 ξ – low frequency asymptote of L ( jω ) * L ( jω ) = ω 2 n jω ( jω + 2 ξω n ) ≈ ω n jω 2 ξ , L low ( jω ) * hence, K v = | L low ( j 1) | 9 ℑ{ L } ℜ{ L } – 1 ω c ω c ω b ω b ω n ω b 1 K v ω...
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lecture14 - MAE 171A Dynamic Systems Control Lecture 14...

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