One more on Nyquist plots this time for a non-minimum-phase transfer-function with a pole at the origin.
In a previous post we sketched the magnitude and phase of the frequency response of the third-order non-minimum phase transfer-function with a pole at the origin:
$$G(s) = \frac{s+2}{s(s^2-9s -10)}= \frac{s+2}{s(s+1)(s-10)} .$$
The diagrams obtained are in the following figures:
We start the Nyquist plot at $90^\circ$ and infinite radius with a decreasing phase:
From there the phase crosses the imaginary axis ($90^\circ$) at about $0.05$ ($\approx -25$dB) with a growing phase:
to finally reach $180^\circ$ with a zero radius:
We connect all these points:
and draw the mirror image
complete with markings denoting the limits at $0^+$ and $0^-$. As discussed in detail in Section 7.6, because there is only one pole at the origin these two large magnitudes should be connected by a clockwise $180^\circ$ arc starting at $0^-$ and ending at $0^+$ as in:
to complete the Nyquist plot. Compare this hand sketch with the one produced by the Matlab command nyquist:
It amazes me that a hand sketch can be much more informative than a plot produced by a professional software! But this seems to be the case in almost all interesting Nyquist plots, which are hard to scale well.
For closed-loop analysis with $L(s)=G(s)$, note that $P_\Gamma = 1$ and that for any $\alpha> 0$ there will always be one clockwise encirclement, therefore
$$Z_\Gamma=P_\Gamma\,-\, \frac{1}{2 \pi} \Delta_\Gamma^{-1/\alpha} L(s) = 1 – (-1) = 2$$
poles on the right-hand side of the complex plane. As before, the 2 poles on the right-hand side can be visualized in the root-locus:
If one wants to stabilize the above system in closed-loop then one needs more than a gain.