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ece580 lecture 13 - Lecture 13 ECE 580 Feedback Control...

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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 1 Lecture 13 ECE 580 Feedback Control Systems (I) MIE 444 Automatic Controls Doug Looze
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 2 Announce Ø PS 5 available Due Tuesday, Oct. 25 Ø Midterm exam Wednesday, Oct. 26 7-9 PM Hasbrouck Lab Add 124 Open book, notes No electronic devices Material through lecture 12 Practice exam available
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 3 Nyquist Plot Analysis Ø Nyquist criterion Then the closed loop system is stable if and only if The locus does not pass through - 1 ( 29 ( 29 # open loop ORHP poles of # CW encirclements of 1 by locus P L s N L s - D ( 29 L s D N P = - - ( 29 L s
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 4 Nyquist plot Re s Im s - 1 x x x - 2 0 r R Re L Im L 0 ϖ + = 0 ϖ - = 0 ϖ 0 ϖ < ϖ = ±∞ 1 ~ r
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 5 Last Time Ø Gain and phase margins m M γ γ γ < < Gain decrease margin Gain increase margin 0 M φ φ < Phase margin Margins from Nyquist diagram ( 29 G s ( 29 D s - j e φ - γ
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 6 - 1 Ø Information from Nyquist locus Re L ( s ) Im L (s) gL gR Assumed stable M φ 1 m L g γ = - 1 M R g γ = - m M γ γ γ < < 0 M φ φ < Unit circle
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 7 Today Ø Example Ø Transient response objectives Step response of unity feedback system Measures of response Two-dominant pole model
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 8 Ø Example: ( 29 ( 29 ( 29 ( 29 2 25 1 2 2 16 s L s s s s s + = + + + Construct Nyquist: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 16 2 25 1 2 2 16 2 16 2 j j j j L j j j j j j j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - - + - - + = + - + + - - + - - g ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 25 1 2 16 2 4 16 4 j j j j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - + - + - - = + - + ÷ ( 29 ( 29 ( 29 ( 29 2 2 2 4 2 25 2 16 2 4 28 256 j j j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - + + - - = + - + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 16 2 25 1 2 2 16 2 16 2 j j j j L j j j j j j j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - - + - - + = + - + + - - + - - g - ( 29 L s
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 9 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 4 2 4 2 2 4 2 25 12 3 Re 4 28 256 25 400 800 Im 4 28 256 L j L j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - = + - + - - = + - + ( 29 ( 29 ( 29 ( 29 ( 29 2 2 4 2 4 2 25 12 3 800 400 25 4 28 256 j L j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - - + - = + - + ( 29 ( 29 ( 29 ( 29 4 2 2 4 2 25 400 800 Im 4 28 256 L j ϖ ϖ ϖ ϖ ϖ ϖ ϖ - - = + - +
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Oct. 20, 2011 Feedback Control Systems (I) © Douglas Looze 10 Quantitative analysis Limits ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 300 lim Re 0.293 4 256 800 lim Im 4 256 800 lim Im 4 256 L j L j L j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ = = - = = +∞ - = =-∞ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 4 2 4 2 2 4 2 25 12 3 Re 4 28 256 25 400 800 Im 4 28 256 L j L j ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ - = + - + - - = + - + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 300
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