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Nyquist Plots / Nyquist Stability Criterion

Given

Nyquist plot is a polar plot for vs using the Nyquist contour (K=1 is assumed)

Applying the Nyquist criterion to the Nyquist plot we can determine the stability of the closed-loop system.

Nyquist Criterion

Let be the number of poles of inside s-plane contour and denote the net number of clockwise (CW) encirclements of the point

If a contour that encircles the entire right half plane (say in CW direction) is mapped through the open loop transfer function, then the number of poles of the closed loop transfer function, , in the right half plane equals to the number of open loop poles plus number of CW encirclements of the that is

Skecthing Nyquist Plot

Example :

Draw the Nyquist plot for for the following s-plane contour

Re{s}

Im{s}

RR→∞

R

Solution :

First rewrite the open loop transfer function in the form

Then plot a pole/zero plot for on the s-plane contour and include measurement scheme to the axis along the necessary critical points

Re{s}

Im{s}

A

B

C-1

Then form a Table for the magnitude and angle entries of the open loop transfer function for each critical point

Use these critical points to skecth the Nyquist plot. Note that Nyquist plot is symetrical with respect to the real axis

Re{H(j)}

Im{H(j)}

B,C A

r

More Examples

Example :

Draw the Nyquist plot for using the given s-plane contour

The pole/zero plot along the necessary critical points

When we form a table of the magnitude and phase values at critical points;

Stability with Nyquist Plot

Given

use Nyquist plot, contour and criterian to discuss quantitatively how the control gain K effects closed loop stability

with

Select We can now select critical points on the plot and use zero/pole grapgh to identify the mag/phase table

Re{s}

Im{s}

-1-10A

B

C

D

Using the table we can form the magnitude vs phase plot as

For K=1 As K increases the actual shape of the plot does not change

But the encirclement of the -1 point can be achieved using K

Back to Nyquist Criterion

For our special case we have :

Apply Nyquist Criterion :

1- Find (number of poles of inside the contour)

2- Determine the sign notations for the encirclements

Contour is CW so, CW encirclements are positive and CCW encirclements are negative

3- Find N (the number of encirclements of (-1,0) for different values of K) and

For small K no closed loop are inside the s-plane contuour, hence system is stable.

For big values of K, two closed-loop poles are inside the s-plane contour; hence system is unstable

Check the result with root locus

Nyquist Performance Specification Parameters

phase margin

gain crossover frequency

phase crossover frequency

Gain Margin :

d

unit circle

pm Re{H(j)}

Im{H(j)}

-1

-1

H(j)

Example

For the system given below

Use Nyquist plot, contour and criterian to discuss quantitively how the control gain K effects the closed loop stability

Note that

with

and root locus is

Plot the zero/pole plot with CW Nyquest contour to form the table

Now using the table let plot the Nyquist plot (note that it is symettrical with respect to real axis)

Re{H(j)}

Im{H(j)}

B,C-1

A

r

As K decreases below 1 the Nyquist plot intersects the real axis before -1

As K increases beyod 1, the real axis intersection points goes beyond -1

using

Another Example

Same system as before but this time

- Start with the Nyquist contour which does not include the origin. And from the zero/pole plot form a magnitude/pahse table

Re{s}

Im{s}

A

B

C

D-1-2

Using the table form the Nyquist plot and comment on the stability using

Re{H(j)}

Im{H(j)}

B

C,D A

r

r=∞

-1

Using Nyquist plot to draw Root Locus

This time consider the same block diagram with

looking at the pole/zero locations we may draw the root locus (by making a big mistake !!!)

Re{s}

Im{s}

-3-5 -2 -1-4

However when we use the Nyquist plot

Re{s}

Im{s}

AB

C

D-3-5 -2 -1-4

-1

Re{H(j)}

Im{H(j)}

r

r

Stable region

-b -1/a

What does the above plot tells us ?

Forming a table using the plot we obtain

The initial root locus we have plotted must be wrong !!!

There is a region with “medium“ values of K that the closed loop system is stable !!!

The actual root locus plot for the given system should be as follows :

From the “correct“ version of the root locus we can conclude that for values of K

system is stable. And from Nyquist criteria