Hi,
The poles and zeros of a function lie in the complex plane. It's called the complex
plane because it maps a complex number to an x and y axis. The x axis is the 'real'
part and the y axis is the 'complex' part. The y axis component of the number is
denoted by placing a "j" in front of the number like: j5. The complex number is
then represented like for example a complex number (1,2) would be 1+2j. That maps
to the complex plane 1 unit to the right of the y axis and 2 units up from the x
axis. It's just like any other plane where we plot things like a point (x,y).
The y axis is sometimes called the "jw" axis or the "imaginary" axis.
With a system there is a lot of significance where the point maps to. For example,
the point 1+2j maps to the right half plane (the right half plane is the first and
fourth quadrants) and is said to be 'unstable' because the real part (1) is positive.
The point -1+2j maps to the left half plance (the left half plane is the second and
third quadrants) and is said to be stable because the real part (-1) is negative.
So you see just by knowing which side the pole is on we know something about the
system. Sometimes there is pole zero cancelation, where the pole gets canceled by
the zero, so that's something to consider as well.
As you are now reading this page, you see before you a two dimensional field of graphics.
Imagine if you only had a one dimensional field...you wouldnt be able to make out any
characters. Add that second dimension however, and things change radically. Suddenly you
can read entire paragraphs and that in turn leads to immense communications which allows
you to convey information about almost anything you than think of, as well as receive
information about almost anything we can think of. Well, the complex plane, because it has
two dimensions, allows us to 'see' a system in much more detail than if we looked at a
single number like the amplitude. That's how the complex plane helps, by allowing us to
visualize more than what we could ever see with just numbers alone.
The reason we consider up or down is because that shows us the frequency, because the
y axis is jw which is j*w. Thus the point (0,1) oscillates more slowly than the point
(0,2), because the '1' is less than the '2'. Since that '2' is really the angular
frequency 'w', the frequency in Hertz is 2/2pi=1/pi in this case.
So now you know the basic reason we consider left or right, because in most cases we know
something about the stability of the system. You also now know that the up and down
tells us something about the frequency. And there's more to it than that as indicated above.
If the point is in the left half plane, that means that the response is decreasing
exponentially. If the point is in the right half plane that means the response is
increasing exponentially, and so is considered to be unstable. If the point is right on
the jw (imaginary) axis then the system is an oscillator unless it is right at (0,0).
If the point is not on the real axis then the frequency is non zero, and closer to the
real axis the frequency is less than the frequency when it is farther away from the
real axis. The frequency in the left half plane is often the 'ringing' we see on a
waveform, where it jumps up and then oscillates for a while, then dies down to
some steady DC level.
So what does all this do for us?
We can look at a system to see if it will break into oscillation when certain parameters
change. For example, if we have a feedback system that has a forward gain K, we might
want to know if increasing that gain will ever cause oscillation or if it will be stable
for any gain K. What we do in this case is look at the roots of the characteristic
equation and see if any roots cross the jw axis while the K increases. I've included a
small diagram of a typical system where the roots, for some gain K, cross the jw axis
(the vertical axis in the center) and since they do that would mean that we would have
to find a way to prevent the gain K from increasing beyond that point in the real life
system or else it would break into oscillation or simply max out and saturate. That would
make the system unusable at that point.
In the diagram, the gain K increases as shown by the arrows. There are three roots to
this system so we track the path of all three roots. Two of the roots cross the jw axis, and
only one of them doesnt. The gain K starts out at zero where the very small white dots
appear, and end at the red dots at some large value of gain.
The plot in the diagram is called a "Root Locus" diagram, and you can look that up on the
web and find out how to do the "Root Locus Procedure" and that will help you analyze systems
that have feedback.
Other methods for investigating stability are Phase Margin, and Gain Margin. There are numerous
other methods too though.