I got very confused with the nyquist plot.
I have a basic question that would suffice:
say I got a general nyquist plot of some transfer function $ G(s) $:
Is $ G(s) $ stable? what information apart from the evolution of Phase and magnitude I can get from the plot?
I Know that the Nyquist plot gives information about $\frac{1}{1+G(s)} $ stability. But what else?
Best Answer
Nyquist's Theorem is about closed loop stability using open loop frequency response information. If you want to know if the TF $G(s)$ is open loop stable, you do need to work backwards from the phase/mag information to a (crude) bode plot, to a TF in the s-domain. In your Nyquist plot of $G(s),$ we can see the magnitude is approximately constant with an increasing phase of $180^\circ$ for low frequencies. This suggests an unstable pole and stable zero at low frequencies.
As we pass the phase of $180^\circ,$ the magnitude drops rapidly suggesting another pole activates. The phase continues to increase for a brief period. This suggests another unstable pole. The magnitude continues to decrease, but the phase begins to decrease back down to a phase of $90^\circ.$ This suggests an additional two stable poles at high frequencies.
Roughly,
$$G(s) = 0.9\,\frac{\left(\frac{s}{z_1} + 1\right)}{\left(\frac{s}{p_1} + 1\right)\left(\frac{s}{p_2} + 1\right)\left(\frac{s}{p_3} + 1\right)^2},$$
where $z_1 > 0,$ $p_1 \approx -z_1 < 0,$ $p_2 \approx 10\,p_1,$ $p_3 \approx -10\,p_2.$
Example, consider the TF,
$$G(s) = 0.9\,\frac{\left(\frac{s}{1} + 1\right)}{\left(\frac{s}{-1} + 1\right)\left(\frac{s}{-10} + 1\right)\left(\frac{s}{100} + 1\right)^2}.$$
The Nyquist plot of this TF is,
Pretty close. It isn't perfect, but you get the idea. You can use the real axis intersections to get more precise information. I didn't do that. However, we can see that $G(s)$ is not BIBO stable.