文档介绍：Topic #4
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 Feedback Control Systems, Fall 2007. MIT OpenCourseWare
(), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 4–2
Frequency Stability Tests
• Want tests on the loop transfer function L(s) = Gc(s)G(s) that
can be performed to establish stability of the closed-loop system
Gc(s)G(s)
Gcl(s) =
1 + Gc(s)G(s)
– Easy to determine using a root locus.
– How do this in the frequency domain? ., 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 (., imaginary axis crossing),
we know that these conditions must hold for s = jω
– So for neutral stability in the Bode plot (assume stable plant),
must have that L(jω) = ±180◦ and |L(jω)| = 1
– So for most systems we would expect to see |L(jω)| < 1 at the
◦
frequencies ωπ for which L(jωπ) = ±180
• Note that L(jω) = ±180◦ and |L(jω)| = 1 corresponds to
L(jω) = −1 + 0j
September 3, 2007
Cite as: Jonathan How, course materials for Feedback Control Systems, Fall 2007. MIT OpenCourseWare
(), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 4–3
Gain and Phase Margins
• Gain Margin: factor by which the gain is less than 1 at the
◦
frequencies ωπ for which L(jωπ) = 180
GM = −20 log |L(jωπ)|
• Phase Margin: angle by which the system phase differs from
180◦ when the loop gain is 1.
– Let ωc be the frequency at which |L(jωc)| = 1, and φ=
L(jωc) (typically less than zero), then
P M = 180◦+ φ
• Typical stable system needs both GM > 0 and P M > 0
Gain
margin
1/GM
1