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Unformatted text preview: Topic #4 16.31 Feedback Control Stability in the Frequency Domain Nyquist Stability Theorem Examples Appendix (details) This is the basis of future robustness tests. 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 42 Frequency Stability Tests Want tests on the loop transfer function L ( s ) = G c ( s ) G ( s ) that can be performed to establish stability of the closedloop system G cl ( s ) = G c ( s ) G ( s ) 1 + G c ( s ) G ( s ) Easy to determine using a root locus. How do this in the frequency domain? i.e., what is the simple equivalent of the statement does root locus go into RHP? Intuition : All points on the root locus have the properties that L ( s ) = 180 and  L ( s )  = 1 So at the point of neutral stability (i.e., imaginary axis crossing), we know that these conditions must hold for s = j So for neutral stability in the Bode plot (assume stable plant), must have that L ( j ) = 180 and  L ( j )  = 1 So for most systems we would expect to see  L ( j )  < 1 at the frequencies for which L ( j ) = 180 Note that L ( j ) = 180 and  L ( j )  = 1 corresponds to L ( j ) = 1 + 0 j September 3, 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 43 Gain and Phase Margins Gain Margin: factor by which the gain is less than 1 at the frequencies for which L ( j ) = 180 GM = 20 log  L ( j )  Phase Margin: angle by which the system phase differs from 180 when the loop gain is 1. Let c be the frequency at which  L ( j c )  = 1 , and = L ( j c ) (typically less than zero), then PM = 180 + Typical stable system needs both GM > and PM > Figure 1: Gain and Phase Margin for stable system in a polar plot September 3, 2007 Gain margin Positive phase margin 1/GM 11 L(j ) Figure by MIT OpenCourseWare. 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 44 Figure 2: Gain and Phase Margin in Polar plots Figure 3: Gain and Phase Margin in Bode plots Can often predict closedloop stability looking at the GM and PM September 3, 2007 Positive gain margin Negative gain margin Positive phase margin Negative phase margin 1/GM 1/GM 1 11 L(j ) L(j ) Re Re Im Im Stable system Unstable system Positive gain margin Negative gain margin Positive phase margin Negative phase margin Stable system Unstable system L db 090 o180 o270...
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 Fall '07
 jonathanhow

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