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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 The Nyquist Stability Criterion (NSC)  Using Bode' Plots to Evaluate
Stability with the NSC
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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
loglog 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 blowup 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