Nyquist Stability

Using Bode' Plots To Evaluate Stability With The Nyquist Stability Criterion.

Why Not Just Use Nyquist Plots?
Interpreting The Nyquist Stability Criterion On A Bode' Plot
Using The Nyquist Criterion On Bode' Plots
Going Beyond Stability

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Why Not Just Use Nyquist Plots?

Bode plots are the most widely used means of displaying and communicating frequency response information. As we discussed in the lesson on Bode' plots, there are many reasons for that.

Bode' plots are really log-log plots, so they collapse a wide range of frequencies (on the horizontal axis) and a wide range of gains (on the vertical axis) into a viewable whole.

Bode' plots are widely used, designers are used to them, and measures of stability are more easily computed on Bode' plots than they are on Nyquist plots. We haven't yet tried to answer the question of "How stable is a system?" but sooner or later we will need to, and Bode' plots are the instrument of choice when we do.

Goals For This Lesson

The focus of this lessons is to learn how to interpret stability of a system - using the Nyquist Stability Criterion - using a Bode' plot instead of a Nyquist plot. That means the goal looks like this.

Given a linear feedback system,

Be able to apply the Nyquist Stability Criterion to the system using Bode' plot information.

You will need to have a good working knowledge of the Nyquist stability criterion and how to apply it, particularly how to apply it using Bode' plot data.

Interpreting Tne Nyquist Stability Criterion On A Bode' Plot

In this section we will revisit the Nyquist stability criterion and examine how to interpret it on a Bode' plot. Here's a sample Nyquist plot. We've taken the liberty of drawing a unit circle on this Nyquist plot - in blue.

Remember that we are concerned with encirclements of the -1 point in the Nyquist plane. We'll try to rephrase what that means.

The point at -1 is on the unit circle. It has a magnitude of 1, and an angle of -180 degrees.
In other words, it's one unit away from the origin on the negative real axis.

Now, examine a blow-up of this plot.

On this plot you can see how -1 gets encircled.
Notice two more interesting points.

As frequency increases the plot crosses - 180^o to the left of the point at -1 on the unit circle.
At a higher frequency, the plot crosses the unit circle.

The amazing thing about these two point is that we can draw a fairly general conclusion that applies to a lot of systems. If you did the problems in the Nyquist stability lesson, you should have concluded that many Nyquist plots have many general properties in common, even for a large variety of systems.

The general conclusion that we can make is this:

A closed loop system is stable if the unit circle crossing is at a lower frequency than the -180^o crossing.

Conversely,

A closed loop system is unstable if the unit circle crossing is at a higher
frequency than the -180^o crossing. - as in the example.

You can find counterexamples to the general conclusions we just drew, but they tend to be more complex systems - often special cases. For many - in fact most - systems you will encounter, the Nyquist stability criterion can be interpreted as the following.

A closed loop system is stable if the unit circle crossing of the open loop frequency response is at a lower frequency than the -180^o crossing of the open loop frequency response.

Now, that's getting pretty simple to apply. We can even apply this criterion easily on a Bode' plot.

A closed loop system is stable if the unit circle crossing of the open loop frequency response is at a lower frequency than the -180^o crossing of the open loop frequency response.

What does that mean on a Bode' plot?

The unit circle crossing occurs when the magnitude of G(jw) is one. At that point, the gain - expressed in db - is zero db.
The -180^o crossing is just that. The -180^o crossing is the frequency at which the phase becomes -180^o.

A closed loop system is stable if the unit circle crossing (zero db crossing on a Bode' plot!) of the open loop frequency response is at a lower frequency than the -180^o crossing of the open loop frequency response.

What does that mean on a Bode' plot?

A closed loop system is stable if the zero db crossing of the magnitude plot occurs at a lower frequency - to the left on the Bode' plot - than the -180^o crossing.

Here's a Nyquist plot (positive frequency only) for an unstable system.

The -180^o crossing occurs at -2.4 or thereabouts.
The zero db crossing is not far to the left of the -1 point - shown with a blue dot.

Notice also the scale problems we have with these plots. Clearly, the plot at the right doesn't show the entire plot, but the detail near the unit circle is still not good. (The entire plot has a DC gain of 20!)

Here's a Bode' plot for the same system.

The -180^o crossing occurs at a frequency of 0.028 Hz. That's shown with a blue dot.
The zero db crossing is shown with a red dot at a frequency of .04 Hz.

Notice that the scale problem we had with the Nyquist plots is better here. You can see all the plot, and with a reasonable size plot, you can see all the detail you need.

We can summarize how to evaluate the Nyquist stability criterion when the frequency response data is plotted on a Bode' plot.

Plot the amplitude and phase plots, - in Bode' plot form - for KG(jw)H(jw).
The closed loop system is stable if the zero db crossing occurs at a lower frequency than the -180o crossing.

And that just about sums it all up.

There are some caveats. If the Bode' plot looks really complex, then consider sketching the Nyquist plot from the Bode' plot data. That should shed some light on the situation. Now, let's take a look at how to use this criterion.

Using The Nyquist Criterion On Bode' Plots.

In this section, we'll learn to apply the criterion we developed in the last section. Recall that criterion was:

Plot the amplitude and phase plots, - in Bode' plot form - for KG(jw)H(jw).
The closed loop system is stable if the zero db crossing occurs at a lower frequency than the -180^o crossing.

We're going to change how we plot our Bode' plots.

We will plot the amplitude and phase plots for KG(jw)H(jw) on the same plot.
For this plot, the red is the magnitude plot, and the blue is the phase plot.

Here's the same plot of KG(jw)H(jw) as above.

We want to determine the values of gain for which the system is stable.

The phase crosses -180^o around a frequency of f = 0.3
At f = 0.3, the magnitude is about -18 db, so we can increase the gain by 18 db and the system will just be on the edge of instability.
A gain of 18 db corresponds to some K value, K_max. To compute K_max note:

20 log10(Kmax) = 18
K_max = 10.9 = 7.94

At this point you should get the point. It's relatively easy to determine the gain for instability. If you need to, you can compare this to the viewpoint using Nyquist plots. Click here.

How stable is this system?

It's not enough to know when a system is stable. In many cases you have to answer a difficult question:

How stable is this system?

Even more difficult, you may have to design a system to achieve a particular level of stability.

Clearly, we need to develop some way of measuring how stable a system is. The lesson on relative stability (Click here to open that lesson.) will help you with that. Once we have an idea of what we mean by relative stability, and we have precise definitions that will give numerical measures of relative stabiltiy, we are going to find that relative stability can also be best interpreted using Bode' plots.

Problems