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.