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PROFESSOR: Welcome.

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In this session, we're going
to examine pole diagrams.

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So let's consider the pole
diagrams for the linear

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time-independent systems
of the form p(D)y equals f.

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So here, we have a
few pole diagrams

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to examine with the poles
marked with the red crosses.

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The axes are imaginary
axis and the real axis.

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And the questions are list
all the stable systems.

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Choose the systems
with the fastest decay.

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And last, choose fastest
decay without oscillation.

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So you have to think of how
to interpret these diagrams

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and what are the meanings of the
position of the different poles

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to address these questions.

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So pause the video.

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Take a few minutes.

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And I'll be right back.

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Welcome back.

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So the first question asks us
to list all the stable systems.

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So if you recall, all
the stable systems

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would be the ones for which
the real part of the pole

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are all negative.

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So if we have all the poles
with real part negative,

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we would have a stable
system for which

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the solution after a long
time would decay, basically.

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It would be transient.

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So here, if we look
at our pole diagrams,

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we have diagram B for which
the real part of the two poles

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are minus 3 and minus 2.

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So this would
definitely generate

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a decaying exponential,
exponential minus 2t,

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exponential minus 3t type
function at large time

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for this system.

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In A, we have a minus 1 and a 1.

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So the real part
here would give us

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a decay, an exponential
decay, exponential minus t.

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But we would have
here a term that will

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diverge with exponential t.

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So this would have basically
behavior at long time

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that diverges and
doesn't go to 0.

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So this would not, basically,
be a stable system.

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For C, we have all the
real parts negative.

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So we're in the stable case.

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For D, we also have the same.

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All the real parts are negative.

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So these are imaginary,
3i-- minus 2 plus 3i.

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So both of them have
also negative real parts.

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Everything is on the
left side of the plane.

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Same thing for E.
And for F, we're

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right on the
imaginary axis, which

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means that we can't
tell if it's going

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to be an exponential--
it's not going

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to be an exponential decay
or growth at this point.

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So to answer the first
question, the stable systems

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would be B, C, D,
E, and that's it.

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So now, for the second
part, choose the system

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with the fastest decay.

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So here, again, as we just
saw, the real part of the poles

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gives us the speed at which
the system is going to decay.

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And so we have to
look for a system that

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has its pole on the
rightmost part of the plane.

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So for example, if we
restrict ourselves here,

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the fastest decay here
is going to be governed

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by the rightmost pole,
so exponential minus 2t,

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that we need to examine.

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Here, we have an exponential
minus 3t decay time.

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Here, we would have an
exponential minus t decay.

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Here, we would have something
a little bit-- minus 0.5t.

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And here, again, this
is not stable system,

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with a decay at large time.

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So which of the system
has the fastest decay?

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We would need to select the one
with the rightmost pole that

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is the most on the left.

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So we would need to compare
this minus 2, with this minus 3,

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with this minus 1,
with this minus 0.5.

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And from all these poles, the
one that has the minus 3 here

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would be the one that would
be decaying the fastest.

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So we would have system C
decaying as exponential minus

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3t.

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And I'm going to just
write down the others.

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And these would be
the dominant terms

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that would basically subscribe
the behavior at large t.

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So this is the fastest.

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For our system B, which
is not the fastest,

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I'm just going to
write it, it's going

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to be dominated by a minus 2t.

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For the system D, it would
be dominated by a minus 1t.

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For the system E, it would be
dominated by the minus 0.5t.

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So the fastest decay would
definitely be system C.

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So the last question,
choose the fastest decay

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without oscillations.

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So now, we need to pay attention
to whether the poles are

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on the imaginary axis,
on the real axis,

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or somewhere in the middle.

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So let's again, examine
only the stable cases.

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So here, the two poles
are in the real axis.

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So it's going to give
us a behavior that

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is only pure decay, exponential
decay, exponential minus

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2t, exponential minus
3t, with no oscillation.

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Because basically,
these two people

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don't have any imaginary parts.

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Here, we have a term in
exponential minus 3t.

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But we would have a term
that would show oscillations

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with the circular
frequencies of 3.

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So you would have an
exponential minus 3t cosine 3t

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plus exponential minus
3t sine 3t, for example.

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So here, we would have a
system that shows oscillation.

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So we don't want that.

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For this case, that
would be similar.

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So you see the pattern.

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Here, we have an
exponential decay.

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Here, we would have
decaying oscillations.

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So there is oscillations.

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Same thing here.

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And so we're left with only
system B, which basically

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does not show any oscillation
but only an exponential decay.

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Pure decay.

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And it's dominated by
the exponential minus 2t

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without oscillations.

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So that ends this first
part of the problem.

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I'm just going to take
a minute and come back

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to show you the second
part of the problem.

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Welcome back for the
second part of the problem.

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So here, another pole
diagram that we're going

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to label J for this system.

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And the poles are
basically here,

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0.5 plus 4i, 0.5 minus 4i.

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And then here, we
have almost a minus 3.

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And here, it would be a minus 4.

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So the question is to consider
now the previous system we had,

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p of D y equals f of t.

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But now, with f of t equals
F_0 cos omega*t, and omega,

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the circular frequency
going between 1 and 5.

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And the question is for
which values of omega

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do we have the largest
response from the system G?

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So basically, we're looking
at phenomenon of resonance

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where we want to
find the frequency

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omega for which the
response of the system

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is going to be the biggest.

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So I give you just
a few minutes,

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and we'll be right back.

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Welcome back for
this second part.

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So here, if you recall,
the amplitude response from

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the previous recitations, the
amplitude is governed by 1 over

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the characteristic polynomial
evaluated at i*omega.

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Where does i omega come from?

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If you recall, it comes from the
exponential response formula.

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Because on the right-hand side
here, we have a cosine omega*t,

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then we complexify, exponential
i*omega*t, et cetera,

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to get the amplitude response.

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So the amplitude is going to
be the largest for basically

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values of i*omega that are
the closest to the poles

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of the polynomial-- to the
poles of this function.

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Basically, the poles of the 1
over characteristic polynomial

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is going to give
us the 1 over 0,

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which will give an amplitude
response that goes to infinity,

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so the largest response.

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So we want to choose
our frequency, i*omega.

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Basically, we want to
look at this diagram

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and look at the poles that are
the closest to the values that

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are going to give us
p of s equals to 0.

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And this is basically
diagram that gives us

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the pole of this 1 over p.

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And so we want to choose
a value for i*omega,

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which will be on
the imaginary axis,

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that is the closest to
one of these two poles.

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Given that we work with positive
frequencies, we would want,

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for example, to choose
i*omega equals i*4.

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So it would be here, or
just omega equals to 4.

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And this value of
omega would get

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us very close to this
pole, which would mean then

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that we are 1 over a
very small number, which

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means that the amplitude
response will be the biggest.

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So that's the idea--
look behind the resonance

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and looking at,
again, the phenomenon

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of the resonance from the
pole diagram perspective

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and with the idea of the
use of the Laplace transform

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behind the whole technique.

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So that ends this session.

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And the goal was really
to learn to interpret

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the meaning of the poles
in the pole diagram

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and how do the polls
give us an idea

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and insight on the behavior of
the system in the long term.