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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 51 Bodes 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 ) is uniquely related to  G ( j )  . The relationship is that, on a loglog plot, if the slope of the magnitude plot is constant over a decade in frequency, with slope n , then G ( j ) 90 n So in the crossover region, where L ( j ) 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 52 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 53 Performance Issues II Classic question: how much phase margin do we need? Time response of a second order system gives: 1. Closedloop pole damping ratio PM/ 100 , PM < 70 2. Closedloop resonant peak M r = 1 2 1 2 1 2 sin( PM/ 2) , near r = n 1 2 2 3. Closedloop bandwidth BW = n 1 2 2 + 2 4 2 + 4 4 and c = n 1 + 4 4 2 2 Figure 29: Frequency domain performance specifications....
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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