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# topic5 - Topic#5 16.31 Feedback Control Systems Control...

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Unformatted text preview: Topic #5 16.31 Feedback Control Systems Control Design using Bode Plots • Performance Issues • Synthesis • Lead/Lag examples Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 5–1 Bode’s Gain Phase Relationship • Control synthesis by classical means would be very hard if we had to consider both the magnitude and phase plots of the loop, but that is not the case. • Theorem : For any stable, minimum phase system G ( s ) , G ( j ω ) is uniquely related to | G ( j ω ) | . • The relationship is that, on a log-log plot, if the slope of the magnitude plot is constant over a decade in frequency, with slope n , then G ( j ω ) ≈ 90 ◦ n • So in the crossover region, where L ( j ω ) ≈ 1 if the magnitude plot is (locally): s slope of 0, so no crossover possible s − 1 slope of -1, so about 90 ◦ PM s − 2 slope of -2, so PM very small • Basic rule of classical control design: Select G c ( s ) so that the LTF crosses over with a slope of -1. September 13, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 5–2 Performance Issues • Step error response e ss = 1 1 + G c (0) G p (0) and we can determine G c (0) G p (0) from the low frequency Bode plot for a type 0 system. • For a type 1 system, the DC gain is infinite, but define K v = lim s → sG c ( s ) G p ( s ) ⇒ e ss = 1 /K v – So can easily determine this from the low frequency slope of the Bode plot. September 13, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 5–3 Performance Issues II • Classic question: how much phase margin do we need? Time response of a second order system gives: 1. Closed-loop pole damping ratio ζ ≈ PM/ 100 , PM < 70 ◦ 2. Closed-loop resonant peak M r = 1 2 ζ √ 1 − ζ 2 ≈ 1 2 sin( PM/ 2) , near ω r = ω n 1 − 2 ζ 2 3. Closed-loop bandwidth ω BW = ω n 1 − 2 ζ 2 + 2 − 4 ζ 2 + 4 ζ 4 and ω c = ω n 1 + 4 ζ 4 − 2 ζ 2 Figure 29: Frequency domain performance specifications....
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topic5 - Topic#5 16.31 Feedback Control Systems Control...

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