gain margin and phase margin - significance

Question

While gain margin(s) and phase margin(s) in Bode Plots of open loop transfer functions are defined at particular frequencies (when they exist) --- such as phase margin is defined at frequencies where the dB gain cuts through 0 dB ------ may I ask if these particular frequencies (ie. gain margin frequencies and phase margin frequencies) are points of interest only because we can conveniently define them at these frequencies?

I was thinking that, if we choose a particular frequency in a Bode Plot where the dB gain curve does not cut through 0 dB, and where the phase curve does not cut through -180 degrees, then there would simply be TWO degrees of freedom, such that a particular combination of additional gain and additional phase would push the system to an unstable condition (or onset of instability).

Do gain margins and phase margins also (in some way) cover other regions of the Bode Plot - aside from the very specific frequencies where the gain margin(s) and phase margin(s) are defined?

Thanks all!

Answer 1

First of all I find it more intuitive to find the gain and phase margins using the Nyquist plot instead of the Bode plot. However, you are correct that the gain and phase margins do give a bit of a narrow minded view on stability margins. Namely, one could construct a transfer function that is arbitrarily close to crossing the minus one point while still having great gain and phase margins, so a small change to both the gain and phase could make the system unstable. One such example would be the openloop transfer function

$$ L(s) = \frac{0.21\,(2\,s + 1)}{s\,(s^2 + 0.8\,s + 1)} \cdot \frac{s^2 + 0.273\,s + 1.43}{s^2 + 0.135\,s + 1.43} \cdot \frac{s^2 + 0.04\,s + 1.56}{s^2 + 0.029\,s + 1.56} \cdot \frac{s^2 + 0.01155\,s + 1.592}{s^2 + 0.0091\,s + 1.592}, $$

where one could get closer to the minus one point by adding more and more inverse notch filters. From the Nyquist plot below it can be seen that $L(s)$ has an infinite gain margin and 106 degrees of phase margin:

However, when applying both a gain and phase shift of only 1.05 and 8.11 degrees respectively the closed loop system (using unitary negative feedback) does already become unstable.

An additional robustness measure one could use is the modulus margin, which is defined as the shortest distance from the minus one point to the open loop transfer function in the Nyquist plot. A more in-depth explanation can be found in this paper in section 4. It can also be shown that the inverse of the modulus margin is also equal to the peak of the magnitude of the sensitivity transfer function and that a given non-zero modulus margin also gives a lower bounds for the gain and phase margin.

At my university they did teach the modulus margin, however I believe many other universities/control theory books only cover the gain and phase margin. Maybe this is they think it is easier for people to imagine how just the gain or phase of a system is changed for example due some amplification or time delay respectively. Or maybe it is because only these two terms became industry standards and the modulus margin did not. It can also be noted that in practice one probably does not often encounter systems like the above $L(s)$, so for most systems it does hold that a good gain and phase margin also implies a good modulus margin. However, your reasoning is correct that the gain and phase margin can sometimes fail and I would encourage you to also use the modulus margin.