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For a minimum phase system where there are multiple 180deg phase crossings, is there a detailed, unambiguous, and rigorous rule that can be applied in Bode Plot analysis, for using Gain Margin as a test for stability?

Illustrative example: standard negative-feedback scenario, Loop TF having low frequency triple pole, plus double-lead compensator, such as:

\$\frac{100000}{(1+s)(1+s)(1+s)}\frac{(1+s/30)(1+s/30)}{(1+s/10000)(1+s/10000)}\$

with OL bode plot showing multiple 180deg-phase-crossover (vertical white lines): enter image description here

image generated using Wolfram Alpha

(This came up when trying to answer this question: Having trouble understanding the stability of following system

I know this can by done with Nyquist, RL, or algebraic methods, specifically asking about Bode Plot analysis method and application of Phase Margin / Gain Margin test)

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2 Answers 2

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  1. You can always make a Nyquist plot from a Bode plot (if it extends far enough). A Bode plot contains all the information in a Nyquist plot, and in addition contains frequency information. To construct a Nyquist plot from a Bode plot do as follows. For each frequency, find the magnitude and phase from the Bode plot. Place a point on the Nyquist plot using the magnitude and phase angle as polar coordinates.

  2. To properly interpret a Nyquist plot, one must know the number of RHP zeros.

  3. The assumption that the transfer function is minimum phase implies that the number of RHP zeros is 0. With that assumption, it is possible to rigorously determine the number of RHP poles from the Bode plot, and thus stability. However, it may be simplest to convert the Bode plot to a Nyquist plot.

  4. Simple phase margin and gain margin tests are inconclusive for demonstrating instability. One must know the number of encirclements of the -1 +0j (or 1 +0j) point and their directions in the Nyquist plot. However, once again, one can construct a Nyquist plot from a sufficiently extended Bode plot.

  5. The reason why phase and gain margin tests are inconclusive in determining instability is that knowing that for some \$\omega\$, \$|F(j\omega)|\ge 1\$ and \$\angle F(j\omega) = 180^{\circ}\$ (or \$0^{\circ}\$), says nothing conclusive about whether or not there is a right half plane pole in the closed loop transfer function. Neither the phase angle nor the magnitude on a Bode plot needs to be monotone. The magnitude may fall below 0 dB only to rise again at a higher frequency. The phase may fall to 180 degrees only to rise again and then fall again. Thus, the indication of poles not in the left half plane is a global property of a Bode plot, not something rigorously discernible from local phase and magnitude information. Put another way, knowing that there is a particular phase/magnitude point in the Nyquist plot does not tell us whether the plot encircles another point, i.e. -1 +0j (or 1 +0j).

  6. In the special case where both magnitude and phase decline monotonically, gain margin is a reliable indicator of stability/instability. Positive gain margin indicates stability. Negative gain margin indicates instability.

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  • \$\begingroup\$ I like this answer. I'm particularly interested in point (3): "With that assumption, it is possible to rigorously determine the number of RHP poles from the Bode plot, and thus stability" ... is there a procedure for doing this? The multiple phase-crossover would imply that the Nyquist shape has additional loops in it, and ofc we are asking whether those loops encircle (-1, 0j) or create a cut-out avoiding the encirclement within a larger loop. Does this translate into some kind of odd/even pattern in the bode plot GM? \$\endgroup\$ Commented yesterday
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    \$\begingroup\$ @PeteW unfortunately, I don't know of any odd/even pattern in the Bode plot that is going to give you encirclements in the Nyquist plot. My approach, although there may be better, would be to replot the phase/magnitude information as a Nyquist plot. Often, there is obvious global information that one can just see, for example, one might notice that the phase crosses 180 3 times before the magnitude falls permanently below 0 dB. That kind of global information that we can just "see", might make it easier to reason about encirclements without actually plotting the Nyquist (cont.) \$\endgroup\$ Commented yesterday
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    \$\begingroup\$ @PeteW (cont.) And, in many cases, the Bode plot is simple enough that such reasoning will work. But if you want something rigorous, that works in every case, no matter how complex, I don't know that you can do any better than replotting the phase/magnitude info as a Nyquist. Maybe someone else can give a "direct from Bode" algorithm that is both rigorous and doesn't involve replotting. \$\endgroup\$ Commented yesterday
  • \$\begingroup\$ @MathKeepsMeBusy - please see my EDIT in my detailed answer (Supplement) \$\endgroup\$ Commented 9 hours ago
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Question: I know this can by done with Nyquist, RL, or algebraic methods, specifically asking about Bode Plot analysis method and application of Phase Margin / Gain Margin test)

Short answer: Yes - all the information contained in the Nyqist plot are also available in Bodes magnitude and phase diagram.

If we want to transfer the stability requirements from the Nyquist to the Bode plot for non-minimum systems and/or for loop gain poles in the RHP we have to analyze the number as well the directions of 180deg-crossings. Then, a special form of the criterion must be applied.

But (as mentioned already in another answer) it makes no sense to define a phase margin or a gain margin.

However, it seems that evaluation of the Nyquist plot is somewhat simpler (if compared with the Bode plot).

Supplement:

When the encirclements and axis crossings in the Nyquist plot are transferred to the Bode plots we get the following stability criterion (for more than a single phase crossing of the 180deg-line):

1.) Definitions:

pr=number of poles of L(s) with a positive real part;

po=number of poles of L(s) on the imag. axis and/or the origin.

Cp=Number of positive 180deg crossings (from the upper to the lower half plane);

Cn=Number of negative 180deg crossings (from the lower to the upper HP).

  • Note that L(s) is the loop gain without consideration of the sign inversion due to negative feedback.

  • Note that only phase crossings C are to be evaluated in a frequency region for which the loop gain magnitude is greater than 0 dB.

2.) Criterion:

The closed-loop system will be stable when the following condition is fulfilled:

Cp-Cn=pr/2 (if po=0 or 1),

Cp-Cn=(pr+1)/2 (if po=2),

Cp-Cn=0 (if pr=0 and po=0)

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  • \$\begingroup\$ @MathKeepsMeBusy - thank you very much for pointing my attention to a typing error. I have corrected the symbols. Thanks, \$\endgroup\$ Commented 7 hours ago
  • \$\begingroup\$ "The closed-loop system will be stable when the following condition is fulfilled:" Will it be unstable otherwise? Or is the situation ambiguous if the condition is not fulfilled? \$\endgroup\$ Commented 7 hours ago
  • \$\begingroup\$ thanks so far, though I'm struggling to apply the algorithm to the example I gave. Pr=0, Po=0, Cp=2?? Cn=1?? ; Intuitively, I would think crossings must be paired such that one happens with |gain|>1 and another with |gain|<1 to be relevant, and the acceptable total indeed depending on where the Nyquist phase "starts" at |gain|=maximum , I.e. like PO is doing? \$\endgroup\$ Commented 6 hours ago
  • \$\begingroup\$ @MathKeepsMeBusy - I think that the sentence you have quoted is identical to the classical formulation which always is used in connection with Nyquists stability criterion, is it not? \$\endgroup\$ Commented 5 hours ago
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    \$\begingroup\$ @ Pete W - Your comment reminded me that I forgot to mention that only the phase crossings C in the frequency region with “magnitude>0dB” may be evaluated. In your case: Cp-Cn=1-1=0. I will correct my answer correspondingly. \$\endgroup\$ Commented 3 hours ago

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