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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 4–2 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 ω – So for neutral stability in the Bode plot (assume stable plant), must have that L ( j ω ) = ± 180 ◦ and  L ( j ω )  = 1 – So for most systems we would expect to see  L ( j ω )  < 1 at the frequencies ω π for which L ( j ω π ) = ± 180 ◦ • Note that L ( j ω ) = ± 180 ◦ and  L ( j ω )  = 1 corresponds to L ( j ω ) = − 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 4–3 Gain and Phase Margins • Gain Margin: factor by which the gain is less than 1 at the frequencies ω π for which L ( j ω π ) = 180 ◦ GM = − 20 log  L ( j ω π )  • Phase Margin: angle by which the system phase differs from 180 ◦ when the loop gain is 1. – Let ω c be the frequency at which  L ( j ω c )  = 1 , and φ = L ( j ω c ) (typically less than zero), then PM = 180 ◦ + φ • 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 4–4 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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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
 jonathanhow

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