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
