Bode criterion for stability - applicability conditions

Question

While checking Bode criterion for stability, I found several sources disagreeing on what is needed to be able to use it.

In general, every source I checked agrees that the open-loop system must be stable and minimum-phase; however, this is were I found different additional conditions:

• the gain diagram must have one and only one crossover frequency

• the phase diagram must have one and only one crossing at -180 ° (while another source not only didn't use this condition, it calculated the margin for the first crossing without saying why)

• the gain diagram must be decreasing (i.e., diagrams like this and this are out, since the diagram has a increasing part); this one I think makes sense given what I understood about the criterion, since a phase margin less than zero means - if the diagram is decreasing - that going towards smaller frequencies there must be a frequency with φ = -180 ° and gain > 1

• the crossover slope must be decreasing (i.e., from positive to negative gain in the direction of increasing frequency: this is fine, this is not)

• the open loop function can't have imaginary poles

• the function's gain must be positive

Some are linked (like having a decreasing diagram means there can't be more than one crossover frequency and it has to be with negative slope; but a diagram can have an increasing part and a negative slope crossover, so they aren't the same condition), but in general I wasn't able to understand under which conditions the criterion can be used. I even found some books that didn't list any condition at all - they just explain briefly the criterion and went on on the next topic.

Which of those additional conditions - beside being open-loop stable and minimum-phase - are needed to be able to use Bode criterion?

A couple of examples I'm not sure can be analyzed with Bode (first, is increasing and has a positive slope crossover; second, it has two crossovers at -180 °):

I found this paper that I think helps: it redefines Bode criterion by adding conditions and some analysis of the results (in the last page there is a summary).

Answer 1

I think, there is nothing like a "Bode criterion for stability". Most probably, what you are referring to is the "Nyquist stability criterion" applied to a Bode plot.

In this context, it should be pointed out that there is no fundamental difference between the Nyquist plot for the loop gain and the corresponding Bode plot (magnitide and phase). Both plots contain exactly the same information.

More than that, it should be pointed out that there is

(a) a general Nyquist criterion which applies to all kinds of functions for the loop gain, and

(b) a simplified version of this criterion which applies to loop gain functions only which have no zeroes in the right half of the s-plane.

In both cases, we also can use the Bode diagram for stability analyses.

• However, in some cases (where the general Nyquist diagram is required), we must not use the simplified version of the Bode diagram analysis (phase margin at the 0 dB crossing).

  In such a case, the number and the direction (slope) of the 180deg - phase crossings is relevant and must be analyzed - dependent on the number "p" of poles in the right half of the s-plane (p=0, 1 or 2)

• In the Bode diagram, the simplified version of the criterion can be used only when (a) the loop gain has no pole in the right half of the s-plane, (b) there is only one 0dB crossing for the magnitude function and (c) there is only one 180deg crossing of the phase response.

Answer 2

You can gain some insights if you look at it from a different view and ask yourself: what are the conditions under which the system might be oscillating? The Barkhausen criteria for oscillation provides two simultaneous conditions for sustained oscillation: unity loop gain (0 dB gain) and positive feedback which could also mean 180° phase shift in a negative feedback system.

Using the bode plot and the stability criteria you can gage how large the margin of the system to this oscillation condition is. Phase margin meaning the margin in phase when the oscillation criterion for gain is met. Gain margin meaning how large the margin in gain is when the phase condition for oscillation is met.

To gage a systems stability you need to know the frequency range that is susceptable to oscillation. Bode plots only show a limited frequency range. So the question you have to ask is: Is the Bode plot showing the relevant frequency range to gage stability? Or in other words: Are there frequencies beyond the displayed frequency range that are susceptable to oscillation?

The conditions that you mention aim towards a reasoning why there cannot be any frequency susceptable to oscillation beyond the displayed frequency range. E.g. decreasing gain makes sure that there is no "hidden" frequency point of unity gain.

So in my book, these conditions can be considered soft conditions that actually mean: you might have to increase the frequency range you are looking at to say anything about stability.

This is one of your examples which has 20dB gain at the upper frequency limit of the Bode diagram. But any amplifier has some bandwidth limit where gain will start to decrease and will eventually drop to unity gain and below. This frequency is not shown so you cannot tell if the system might be oscillating or not.