KENDRA PUGH: Today I'd like to talk to about poles.
Last time I ended up talking to you about LTI
representations and manipulations.
And, in particular, I want to emphasize the relationship
between feedforward and feedback systems, in order to
segway us into the poles..
Using what we know about that relationship, we can find the
base of a geometric sequence.
And that geometric sequence actually represents the
long-term response of our system to a unit sample
response or a delta.
That value is also what we refer to the poles.
At this point, I'll go over how to solve for them and very
basic properties of them.
So that we can use the information that we have about
the poles to try to actually predict the future or look at
the long-term behavior of our system.
First a quick review.
Last time we talked about feedforward systems.
And, in particular, I want to emphasize the fact that if you
have a transient input to your feedforward system, you're
going to end up with a transient response.
There's no method by which a feedforward system can retain
information over more than the amount of time steps that you
feed information into it.
Feedback systems, on the other hand, represent a persistent
response to a transient input.
Because you're working with a feedback system information
that you put in can be reflected in more than one
time step and possibly multiple time steps, depending
upon how many delays you have working in
your feedback system.
Last time I also drew out the relationship between
feedforward and feedback systems.
You can actually turn a feedback system or talk about
a feedback system in terms of a feedforward system that
takes infinite samples of the input and feeds them through a
summation that takes infinite delays through your system.
You can represent that translation
using a geometric sequence.
The basis of that geometric sequence is the object that
we're going to use in order to predict the future.
And that's what we're talking about when
we talk about poles.
You can have multiple geometric sequences involved
in actually determining the long-term
behaviors of your system.
If you only have one, then things are pretty simple.
You find your system function.
You find the value associated with p0 in this expression.
It's OK if there's some sort of scalar on the outside of
this expression.
We're working with linear time and variance systems, so that
scalar is going to affect the initial
response to your system.
But in terms of a long term behavior, it
doesn't matter as much.
So don't worry about it right now.
Relatedly, if you're solving for these expressions in
second or higher order systems, you're going to end
up having to solve partial fractions.
You can do this.
And in part, one of the reasons that you would want to
do this is so that you can get out those scalars, if you're
going to be talking about the very short-term response to
something, like a transient input.
We're not going to be too interested in
those in this course.
We're mostly going to be talking
about long-term response.
So we can get around the fact the we're dealing with a
higher order of systems and not solving partial fractions
by substituting in for an expression called z, which
actually represents the inverse power of R and then
solving for the roots of that equation.
If you substitute z in for 1 over R in this denominator and
then solve for the root associated with that
expression, you'll get the same result.
You'll actually end up out with p0.
All right.
So now we know how to find the pole or multiple poles, if
we're interested in multiple poles.
What do we do now?
I still haven't gone over how to figure out the long-term
behavior of your system.
The first thing you do is look at the magnitude of all the
poles that you've solved for and select the poles with the
largest magnitude.
If there are multiple poles with the same magnitude, then
you'll end up looking at all of them.
If you have different properties than the ones here,
you can end up with some, you know, complex behavior.
I would not worry about that too much.
Or I would ask a professor or TA when that happens.
But in the general sense, if your dominant pole has a
magnitude greater than 1, then you're going to see long-term
divergence in your system.
This make sense if you think about it.
If at every time step your unit sample response is
multiplied by a value that is greater than 1, then it's
going to increase.
And, in fact, the extent to which the magnitude of your
dominant pole is greater than 1 is going to determine your
rate of increase and also determine how fast your
envelope explodes.
Similarly, if the magnitude of your dominant pole is less
than 1, then in response to a unit sample input or a delta,
your system's going to converge.
This also makes sense intuitively.
If you are progressively multiplying the values in your
system by a scalar that is less than 1, then eventually
you're going to end up converging to 0.
To cover the only category we haven't talked about, if your
dominant pole is actually equal in magnitude to 1, then
you're not going to see convergence or divergence.
And this is one of the places in which the magnitude of the
scalar that you end up multiplying your system by can
become relevant.
We're not going to focus on this situation too much.
But it's good to know what actually happens when the
magnitude of your dominant pole is equal to 1.
The other feature that we're interested in when we're
looking at the dominant pole of a system is if we were to
represent the dominant pole in this form, what the angle
associated with that pole is, if you were to graph that pole
on the complex plane using polar coordinates.
If your pole stays on the real axis, or if your pole does not
have a complex component, then you'll see one of two things.
The first thing that it's possible for you to see is
that you'll get absolutely non-alternating behavior.
Your system response stays on one side of the x-axis and
either converges, diverges, or remains constant as a
consequence of input of the unit sample.
And you won't see any sort of alternating
or oscillating behavior.
This only happens when your dominant
pole is real and positive.
If you're dominant pole is real and negative, this also
means that it's still on the real axis, but
its value is negative.
So if you're looking at polar coordinates, it's going to
have an angle pi associated with it.
This means you get alternating behavior.
And what I mean when I say alternating behavior is that
your unit sample response is going to jump across the
x-axis at every time step.
This is also equivalent to having a period of 2.
The other situation you can run into is that this angle is
neither 0 nor pi.
And at that point you're going to be talking about
oscillatory behavior or a sinusoidal response that
retains its edges at the envelope of your function.
In order to find the period, or in order to find the amount
of time it takes for your unit sample response to complete
one period, you're going to take the angle associated with
your dominant pole and divide 2pi by it.
This is the general equation for a period.
This covers the basics of what you want to do once you
already have your poles.
Next time I'm actually going to solve a pole problem and
show you what the long-term response looks like and also
talk about some things about poles that I've pretty much
skimmed over.
And at that point you should be able to solve and look at
poles for yourself.