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Stability of the closed loop system (with -1 feedback) can be assessed via Bode plot gain/phase margin analysis of its open loop system.
How can one deduce stability of the closed loop system directly its Bode plot?
One approach would be to fit a transfer function to the Bode (Frequency Response) and examine the poles' location of the fitted transfer function. But I'm looking for a rather intuitive approach using directly the Bode (frequency Response) plot of the closed loop system.
You can easily find the phase margin by simply solving for the open loop gain from the closed loop gain. If \$F(s)\$ is the closed loop transfer function, and \$G(s)\$ is the open loop, then:
$$ F(s)=\frac{G(s)}{1+G(s)}$$ so $$ G(s)=\frac{F(s)}{1-F(s)}$$
You can work out the closed loop gain that corresponds to the open loop unity gain with a particular phase margin.
and just look at the closed loop gain bode plot to see where your loop is. The one you posted seems to have a phase margin of about 70 degrees.
The bode plot is not the easiest way to do this, if you plotted the closed loop gain on a polar plot, then the table above would be a curve and you could just find the point of intersection.
I am not sure, if this answer is what you expect. Nevertheless:
Whether a system with feedback is stable (or not) can best be checked using the loop gain plot (BODE diagram). Now the well-known stability criterion can be applied.
Simulation programs are able to calculate magnitude and phase functions also for closed-loop systems which are unstable. Now - the question arises if we can derive from these BODE plots (for an unstable system) some information about stability properties. And the answer is Yes.
For stable closed-loop systems the phase function has - in general - a falling characteristic (in particular, in the frequency region where the pole frequency of the dominant pole pair is located.) In contrast, for unstable sysytems this phase function will exhibit a rising characteristic (posive slope). In case of an oscillator (pole on the imag. axis) this function will have a vertical form (phase jump).
More than that, the finite (positive) slope of this rising phase function is a kind of "measure" for the degree of instability - that means: This slope is also an indication how far the system is away from the stability limit - hence an indication for the amount of the negative phase margin. Of course, for stable systems, the negative slope of the phase function is a measure for the positive phase margin.
It should be noted that the max. phase slope can conveniently be expressed with the peak of the corresponding group delay figure. Therefore, we have quasi-linear relationship between the phase margin and the inverse group delay peak.
