# Control Systems/Nyquist Stability Criteria

## Nyquist Stability Criteria

The **Nyquist Stability Criteria** is a test for system stability, just
like the
Routh-Hurwitz
test, or the Root-Locus
Methodology. However, the Nyquist Criteria can also give us additional
information about a system. Routh-Hurwitz and Root-Locus can tell us
where the poles of the system are for particular values of gain. By
altering the gain of the system, we can determine if any of the poles
move into the RHP, and therefore become unstable. The Nyquist Criteria,
however, can tell us things about the *frequency characteristics* of the
system. For instance, some systems with constant gain might be stable
for low-frequency inputs, but become unstable for high-frequency inputs.

Also, the Nyquist Criteria can tell us things about the phase of the
input signals, the time-shift of the system, and other important
information.

## Contours

A **contour** is a complicated mathematical construct, but luckily we
only need to worry ourselves with a few points about them. We will
denote contours with the Greek letter Γ (gamma). Contours are lines,
drawn on a graph, that follow certain rules:

1.  The contour must close (it must form a complete loop)
2.  The contour may not cross directly through a pole of the system.
3.  Contours must have a direction (clockwise or counterclockwise,
    generally).
4.  A contour is called \"simple\" if it has no self-intersections. We
    only consider simple contours here.

Once we have such a contour, we can develop some important theorems
about them, and finally use these theorems to derive the **Nyquist
stability criterion**.

## Argument Principle

Here is the argument principle, which we will use to derive the
stability criterion. Do not worry if you do not understand all the
terminology, we will walk through it:

When we have our contour, Γ, we transform it into $\Gamma_{F(s)}$ by
plugging every point of the contour into the function F(s), and taking
the resultant value to be a point on the transformed contour.

### Example: First Order System

### Example: Second-Order System

## The Nyquist Contour

The Nyquist contour, the contour that makes the entire nyquist criterion
work, must encircle the entire unstable region of the complex plane. For
analog systems, this is the right half of the complex s plane. For
digital systems, this is the entire plane outside the unit circle.
Remember that if a pole to the closed-loop transfer function (or
equivalently a zero of the characteristic equation) lies in the unstable
region of the complex plane, the system is an unstable system.

Analog Systems:The Nyquist contour for analog systems is an infinite semi-circle that encircles the entire right-half of the s plane. The semicircle travels up the imaginary axis from negative infinity to positive infinity. From positive infinity, the contour breaks away from the imaginary axis, in the clock-wise direction, and forms a giant semicircle.\
Digital Systems:The Nyquist contour in digital systems is a counter-clockwise encirclement of the unit circle.

## Nyquist Criteria

Let us first introduce the most important equation when dealing with the
Nyquist criterion:

$$N = Z - P$$

Where:

-   N is the number of encirclements of the (-1, 0) point.
-   Z is the number of zeros of the characteristic equation.
-   P is the number of poles in the of the open-loop characteristic
    equation.

With this equation stated, we can now state the **Nyquist Stability
Criterion**: `{{TextBox|1=
;Nyquist Stability Criterion:A feedback control system is stable, if and only if the contour <math>\Gamma_{F(s)}</math> in the F(s) plane does not encircle the (-1, 0) point when P is 0.

:A feedback control system is stable, if and only if the contour <math>\Gamma_{F(s)}</math> in the F(s) plane encircles the (-1, 0) point a number of times equal to the number of poles of F(s) enclosed by &Gamma;.}}`{=mediawiki}

In other words, if P is zero then N must equal zero. Otherwise, N must
equal P. Essentially, we are saying that Z must always equal zero,
because Z is the number of zeros of the characteristic equation (and
therefore the number of poles of the closed-loop transfer function) that
are in the right-half of the s plane.

Keep in mind that we don\'t necessarily know the locations of all the
zeros of the characteristic equation. So if we find, using the nyquist
criterion, that the number of poles is not equal to N, then we know that
there must be a zero in the right-half plane, and that therefore the
system is unstable.

## Nyquist ↔ Bode

A careful inspection of the Nyquist plot will reveal a surprising
relationship to the Bode plots of the system. If we use the Bode phase
plot as the angle θ, and the Bode magnitude plot as the distance r, then
it becomes apparent that the Nyquist plot of a system is simply the
polar representation of the Bode plots.

To obtain the Nyquist plot from the Bode plots, we take the phase angle
and the magnitude value at each frequency ω. We convert the magnitude
value from decibels back into gain ratios. Then, we plot the ordered
pairs (r, θ) on a polar graph.

## Nyquist in the Z Domain

The Nyquist Criteria can be utilized in the digital domain in a similar
manner as it is used with analog systems. The primary difference in
using the criteria is that the shape of the Nyquist contour must change
to encompass the unstable region of the Z plane. Therefore, instead of
an infinitesimal semi-circle, the Nyquist contour for digital systems is
a counter-clockwise unit circle. By changing the shape of the contour,
the same N = Z - P equation holds true, and the resulting Nyquist graph
will typically look identical to one from an analog system, and can be
interpreted in the same way.

