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Finding gain and phase margins from Nyquist plots is basically finding the margin of change in gain and phase beyond which the plot encircles -1, respectively, with the assumption that the Nyquist plot should have 0 encirclements of -1 for stability.(Reference link)

But what happens if the encirclement does not have to be zero for Closed-Loop Stability (because of Nyquist Criterion)? How can one find Phase and Gain Margin if the encirclements around -1 are not zero for a Nyquist plot?

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It is the same margin of change. If the Nyquist plot for open-loop gain just touches -1, then there is one or more poles exactly on the stability boundary (i.e., the real part of the pole in the s-domain is zero, or the z-domain pole's absolute value is 1). At that point the system is metastable, and nudging the pole ever so slightly beyond that point will render it unstable.

If you're starting with a stable system, and you don't let your poles cross the stability boundary, you'll end with a stable system -- that's the idea behind phase and gain margins.

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  • \$\begingroup\$ If I have an open loop pole on the RHP, it would mean that there should be a counterclockwise encirclement around -1 for the Closed Loop system to be stable (Nyquist Criterion). In this case, at 180 degrees, the radius of plot will be greater than 1, so this would mean Gain Margin is negative and system is unstable. But Nyquist Criterion says it is stable. In this sense, for every encircled Nyquist plot with RHP open loop pole, gain margin goes negative. Am I right? \$\endgroup\$ Commented Oct 16, 2019 at 21:47
  • \$\begingroup\$ Um, yes, no, and keep thinking... It may help to look at this on a Bode plot, but if there's an unstable open-loop pole, it means that there is a gain that you cannot go below, as well as one that you cannot go above. So there is a "negative" gain margin in the sense that you could measure such a margin in negative dB -- but it's not negative in the signed-number sense. (Pardon me for what is undoubtedly a confused explanation -- just make a Bode plot and notice that there will be phase crossings at gains both higher and lower than unity!). \$\endgroup\$
    – TimWescott
    Commented Oct 16, 2019 at 22:56
  • \$\begingroup\$ Aniket Sharma...if there is an open-loop pole in the RHP you must apply the GENERAL Nyquist criterion - not the simplified one (which is valid for stable open loops only). \$\endgroup\$
    – LvW
    Commented Oct 17, 2019 at 13:07

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