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:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/bdm4b.jpg?resize=512%2C304&ssl=1)
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/bdsp9.jpg?resize=512%2C284&ssl=1)
We start the Nyquist plot at $90^\circ$ and infinite radius with a decreasing phase:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq10.png?resize=512%2C345&ssl=1)
From there the phase crosses the imaginary axis ($90^\circ$) at about $0.05$ ($\approx -25$dB) with a growing phase:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq11.png?resize=512%2C345&ssl=1)
to finally reach $180^\circ$ with a zero radius:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq12.png?resize=512%2C347&ssl=1)
We connect all these points:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq13.png?resize=512%2C343&ssl=1)
and draw the mirror image
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq14.png?resize=512%2C346&ssl=1)
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:
![](https://i0.wp.com/linearcontrol.info/fundamentals/wp-content/uploads/2019/06/nyq15.png?resize=512%2C344&ssl=1)
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.