NO, for stable system both PM & GM must be positive

Gain margins are defined at 1/|G(jw)| with w=/180, with G beeing your frequency response. So now if |G(w=-180)| is bigger than 1, you will have a value in dB which is smaller 0dB. This does not tell you about the stability of the system. For stable systems this value is called the gain margin.

To validate the stability you have to validate stability criteria like the Nyquist criterion, where you count the encirclement of of the critical point at -1 and compare if to the necessary ones to be stable. (in your case Matlab does it for you :) )

That said, you can have negative margins and a stable system, like in your case. Do not judge system stability by margins, as they are not defined for unstable systems, but of course, can be calculated if you just take the equation from the first line.

You can also simply check the closed loop poles and with real parts smaller than 0 you know that you are stable.

Hope that helps.

This is what happens when we get used to work with some systems and suddenly have to deal with a different one.

Although we got used to plot different curves, the explanation sits with the Nyquist Theorem and it is always helpful to go back and read it once more.

We must recall that, although we got used with plotting half of the curve (corresponding to 0

Thanks a lot Daniel Ossmann and Itzhak Barkana for your insightful answers.

I infer that GM and PM are not defined for open loop unstable systems!!

They may be calculated. But their values being negative or positive isn't a big deal. Stability of the open loop unstable systems should then be assessed by finding the roots and sketching the nyquist plot.

Am I right? Please suggest any modifications, if needed, in my inference.

Let's take it easy.

Your initial question is going back and rediscovering the initial problem that led Nyquist to his theorem. This is related to the fact that usually people were (and still are) dealing with the basic and common idea that, if you increase the gain, the system might get unstable and therefore, we must guarantee gain margin. Nyquist, while working with amplifiers in closed-loop, reduced the gain and was shocked to discover that the system went unstable. So, he went to study why and ended with what he ended.

Yes, you can talk about gain margin (GM) even in unstable systems, only you may have one from above and/or one from below.

Take the simple open-loop system Go=(s+1)/s-2) with one unstable pole (in RHP) and one zero in LHP and plot its Root Locus. It is clear that to move the closed-loop pole from the unstable open-loop position towards the stable zero, you must have some sufficiently large positive gain so that the closed-loop pole ends somewhere on the (stable) negative real axis. Now, if you increase the gain, the system will remain stable, so you have an infinite GM from above. Instead, if the gain decreases, the system may again become unstable, so you have some GM from below.

In more complex systems, with many poles and zeros, some closed-loop poles may move with the gain from the unstable RHP region to the stable LHP and then move again to the unstable RHP region for higher gains.

In your case, with one unstable open-loop pole, the complete Nichols plot tells you that you have one encirclement and that the closed-loop system is stable. However, if you decrease the gain you may have no encirclement, while if you increase the gain you may have two encirclements (or none again), and this again is not good.

So, instead of having no GM, now you have one from above and one from below.

The problem with Nyquist plots is that the gains may reach large values, while you must see what happens around (-1,0).

Nichols plot does same thing at logarithmic scale and allows us to see it all.

Practice taught me that nichols may give the most, yet, because these days bode, nichols, nyquist, rlocus are simple words in MATLAB, it is good to write them all and see what each one gives.

Have Fun! :-)

Thanks a lot Itzhak Barkana.

Very insightful explanation!!