When designing feedback systems, I often evaluate stability by thinking about phase margin: the closed loop system $$T(s) = \frac{L(s)}{1+L(s)}$$ is stable if $L(s)$ has positive phase margin, i.e., $$\angle L(j\omega) > -\pi/2$$ at all frequencies for which $$|L(j\omega)|>1$$. This is easy to visualize graphically using a Bode plot.
But for some systems -- typically involving right-half-plane poles and/or zeros in $L(s)$ -- this criterion doesn't work and we need to use the Nyquist criterion to check stability. That is, the number of unstable closed-loop poles is equal to the number of encirclements of $-1+j0$ by the polar plot of $L(s)$ when $s$ is evaluated along an appropriate contour. My understanding is that this Nyquist criterion is more general, and the "positive phase margin" rule is a simplification that only applies in certain cases.
I'm looking for a criterion that will let me decide when these two criteria (positive phase margin and the Nyquist test) will give me the same results. I suspect that it involves right-half-plane poles and/or zeros in $L(s)$, but I'm not sure.