Rec 7 | MIT 6.01SC Introduction to Electrical Engineering and Computer Science I, Spring 2011

Rec 7 | MIT 6.01SC Introduction to Electrical Engineering and Computer Science I, Spring 2011


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.

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