Step-by-step Nyquist plot example. Part II

Second part of our posts on the Nyquist plot. This time with a non-minimum-phase transfer-function.

In this post we have sketched the Bode plot magnitude and phase for the non-minimum-phase second-order transfer-function

$$G(s) = \frac{s+2}{s^2-9s -10}=\frac{s+2}{(s+1)(s-10)}$$

and obtained the following diagrams:

In order to sketch the Nyquist plot we start at $-180^\circ$ with an arc of radius of $0.2$ ($-13$dB) and a decreasing phase:

Next the phase increases and the plot crosses $180^\circ$ near $0.11$ ($\approx -19$dB) and an increasing phase:

Finally, the phase approaches $-90^\circ$ with a $0$ radius ($-\infty$dB):

Connecting all these parts

and drawing the mirror image

completes the sketch.

Now for closed-stability analysis with $L(s)=G(s)$, note that $P_\Gamma = 1$ because of the pole at $s=10$. Because of that, for any $1/\alpha > 0.2$ there will be no encirclements and

$$Z_\Gamma=P_\Gamma\,-\, \frac{1}{2 \pi} \Delta_\Gamma^{-1/\alpha} L(s) = 1 – 0 = 1.$$

The closed-loop system will remain unstable with $Z_\Gamma=1$ poles on the right-hand side of the complex plane. For $0.11 < \frac{1}{\alpha} < 0.2$ ($0.11$ is an approximation) there will be one clockwise encirclement, that is

$$Z_\Gamma=P_\Gamma\,-\, \frac{1}{2 \pi} \Delta_\Gamma^{-1/\alpha} L(s) = 1 – (-1) = 2$$

so that not only the system remains unstable but it now has $Z_\Gamma=2$ poles on the right-hand side. Finally, if $1/\alpha< 0.11$ there will be one counterclockwise encirclement so that

$$Z_\Gamma=P_\Gamma\,-\, \frac{1}{2 \pi} \Delta_\Gamma^{-1/\alpha} L(s) = 1 – (+1) = 0$$

and the closed-loop system is now asymptotically stable.

This behavior can also be seen on the corresponding root-locus

in which we see a stable root first going unstable before both roots becoming stable for large values of gain.

Note how the Bode + Nyquist plot approach provides also information on the values of the gain at which roots cross from one side of the complex plane to the other, something you could not directly obtain from the root-locus. On the other hand, the root-locus gives you the location of the closed-loop poles, which translates into very useful natural frequency and damping ratio information.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.