Monte-Carlo analysis vs phase-margin or gain-margin

Question

It's my understanding that phase and gain margins are used to see how much margin there is for the stability of a circuit. In other words, how much gain or phase would need to change in order to make the circuit unstable.

However, I don't think I've ever heard of using Monte-Carlo analysis to determine a stability metric. This approach would seem to me to have merit in that it would examine the actual variation of the components to look at the effects, rather than just some generic notion of margin.

On the other hand, I suspect that a generic margin has merit in the sense that it could capture parameters that are difficult to model (i.e., opamp open-loop gain).

So my question is why Monte-Carlo analysis isn't used for stability analysis?

Answer 1

Ideally you need both analyses. They do different things and both are used together.

Gain and phase margin not only show that a system is stable (e.g. a yes/no answer), but also how stable it is -- e.g. is there ringing or overshoot; how long does it take to settle.

Having some margin does also allow for (unanalyzed) variability in the system without it becoming actually unstable, but without MC analysis, this is often a judgement call, and may not have clear limits.

So in the end, for full confidence, you need MC analysis applied to gain/phase margin analysis.

Answer 2

You cannot guarantee stability just by doing a Monte-Carlo analysis, but that analysis can find parameters for which the control loop will be unstable or give you some confidence that the current parameters are less susceptible to become unstable due to some parameters being slightly off.

Another thing to point out is that the phase and gain margin on their own might be deceiving regarding how stable a system is. In Feedback systems, by Astrom and Murray Example 10.8 shows a system with good margins (phase and gain) that is actually close to being unstable, and that can be detected by looking at the Nyquist plot. Overall the book goes over a bunch of interesting concepts, theory- and practice-wise.

Another aspect that they cover is the part on Robust performance (which will give the guarantees to small changes in the parameters of linear systems) and the chapter on Fundamental limits. Having an actual procedure to determine how robust a system is to changes in its parameters might be why you do not see MC being done in the linear case.

Overall, doing MC might be a good starting option to look for parameters that can cause the system to become unstable but will not guarantee, if you don't find any such parameters within the range you looked for, that the system will be stable. While there are mathematical analysis that will give you such guarantees, and even procedures to determine parameters that lead to a robust controller. But the math used for that can be a bit daunting so it is not as popular as some heuristic techniques.