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.


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.

        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.

Now, examine a blow-up of this plot.

       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 -180o crossing.
Conversely,
A closed loop system is unstable if the unit circle crossing is at a higher
 frequency than the -180o 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 -180o 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 -180o crossing of the open loop frequency response.
        What does that mean on a Bode' plot?  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 -180o crossing.
        Here's a Nyquist plot (positive frequency only) for an unstable system.

        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.

        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.

        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:

        We're going to change how we plot our Bode' plots.         Here's the same plot of  KG(jw)H(jw) as above.         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:

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