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PROFESSOR: I wanted
to give a lecture,

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because as I told you in
presentations, I love feedback.

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In fact, I love
it so much, that I

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think the examples
we're going to do,

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00:00:40,890 --> 00:00:45,900
we can do analytically
are not maybe

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sufficiently compelling for you
to believe how exciting control

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can be.

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So let me just start by giving
a few minutes of a research

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00:00:54,090 --> 00:00:57,181
result we had a few years ago.

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That's why you've got color
images-- everybody noticed

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00:00:59,430 --> 00:01:02,670
the color images on the slides.

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I brought this robot in
to the first recitation

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00:01:06,180 --> 00:01:07,610
of [INAUDIBLE].

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It's a big 2 meter wing span
ornithopter, and in my case,

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we've been trying to design
control systems to make

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airplanes move more like birds.

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Here's one example of that.

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One thing birds do that
planes don't do very well

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is land on a perch.

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So we asked the question,
can we take a simple airplane

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with fixed wings,
no flapping allowed,

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and make it land on a perch--

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UAV stands for unmanned
aerial vehicle--

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00:01:33,880 --> 00:01:36,730
land on a perch like a bird.

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The story I want to tell you
is that with feedback design,

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you can.

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Here's why it's
interesting and hard.

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The reason airplanes
don't land on a perch

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is because, when your wing
is at a low angle, which

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a plane normally is
when it's flying,

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the air flow around the wing
is relatively easy to model.

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00:01:56,350 --> 00:01:59,730
It's easy to write a block
diagram description of.

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But if you go to moderate
angles, that's still true.

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The air stays attached to
the wings and everything--

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00:02:05,210 --> 00:02:12,050
[INAUDIBLE]

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At a low angle, the air is
still attached to the wing,

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and so we have very good models
of the airflow around here,

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and the lift and drag
forces that you get when

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you're flying in these regimes.

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But if you go up too far, like
this, then everything changes.

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So you think, the air can't
quite stay around the wing.

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You get separation.

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You get big vortices
pulling off the back.

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It's very nonlinear,
very unsteady.

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Very complicated flight regime.

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And as such, our best
aircrafts to date

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mostly stay down in this regime
where we have good models

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and we can design conventional
control systems for.

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But birds don't do that at all.

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So birds are, often, when
they're landing on a perch,

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they're up even far beyond this.

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Maybe even 90 degrees.

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"Angle of attack", it's called.

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They're way into this
deep stall regime--

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that's called stalling your
wings when that happens.

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And it seems like they're doing
a lot better landing on a perch

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because we don't see
our airplanes do it.

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But you can actually
try to quantify that.

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So if you want to compare
the performance of a bird

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landing versus a plane landing,
the first thing you have to do

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is you have to take out the
differences in mass and wing

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area and all these things.

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But you can do that.

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And a fair comparison
is the distance

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averaged drag coefficient,
which is just a way

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to scale out the effects
that you'd expect from having

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a bigger wing or a bigger mass.

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You can plot this
drag coefficient

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for a few different vehicles.

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This is a standard
runway landing of a 747,

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would get the drag
coefficient, just

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so you have some
calibration, of about a 0.16.

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The X-31 was a super
maneuverable research vehicle.

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It was designed to do super
short runway landings,

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and it got a drag coefficient
during those landings

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of about 0.3.

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There's a few other
projects out there

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about trying to make
perching planes that

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were getting similar numbers.

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But then we went out
and looked at nature.

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I have some
collaborators at Harvard

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that work with real birds.

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It turns out-- I
wanted him to tell me

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the numbers from some
really elite bird,

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like at least a hawk
or something like this.

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But they work on pigeons.

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So that was the only number
we could get, was a pigeon.

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They actually
convinced me by the end

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that pigeons are really good.

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The way they can sort of
dive through the fence

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and get your lunch
by the food trucks?

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I mean, that's actually--

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they're seriously skilled birds.

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We even had-- one
of the fun things

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about doing this kind of
research is you get visitors.

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We had will.i.am from the Black
Eyed Peas come to the lab.

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And he said-- after
we told him the story,

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he said pigeons
are ghetto birds.

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They got mad stop.

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That sort of summarizes it.

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You already know the answer.

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But if you look at
the drag coefficient

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that a bird gets when it's
landing, they get a 10.

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So they're doing
orders of magnitude

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better than our best planes
in terms of stopping.

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If you just want
to be fun about it,

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you could say, what
would it take for a 747

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to get a 10 to
impress will.i.am.

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And it turns out
that 747 would have

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to go-- they fly at about 450
miles an hour cruise speed.

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It would have to
stop in 40 meters

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to do what a bird's doing.

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The wings would
probably pop off.

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There's problems with that.

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But the dynamics are
impressive, of the birds.

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And what's more, when
they're doing that,

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they're getting
incredible accuracy.

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This is one of my favorite
videos of a perching owl.

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If you watch really closely,
you can see the airflow

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get complicated on his wings.

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You can see his leading edge
feathers start to flip over.

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Boom.

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The way you do that is you
put food by the camera.

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We tried to build
these simple planes

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and ask if could they do
what the owls are doing,

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what the pigeons are doing.

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Very simple planes.

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We did them in an
indoor environment.

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We made it so it was
basically a boring plane,

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but it can do very interesting
things in pitch, up and down.

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Then we spent some time trying
to build a dynamical systems

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representation of the plane.

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And how do you do that?

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Well, you shoot the
plane a bunch of times

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into a fishing net.

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And you collect a bunch of data.

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And like we talked
about in recitation,

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you can potentially
get from that data

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a model of the dynamics.

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And turns out, we had to
do a nonlinear dynamical

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model for the
vehicle, but we could

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do a linear actuator model
plus delays and everything.

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We built a pretty good model of
the plane, which is sort of--

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for an aerodynamics
crowd, this would

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tell you that this is
a surprisingly good fit

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to the lift and drag forces over
a very large range of angles.

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Then we had a 6003
problem, basically.

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The nonlinear terms make
it a little different.

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And what we did
as a group was we

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innovated a way to design
a feedback controller that

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could make this thing
very accurate during very

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high performance maneuvers.

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And a long story short, we
can now shoot an airplane

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into a motion capture arena.

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This is slowed down
about 11 times.

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It's this airplane right here.

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And it goes into
a very deep stall.

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And we can land with an
accuracy, enough accuracy

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to land on a perch.

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It turns out if you can build a
good enough control system that

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can handle the complexity
of the dynamics,

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then you can make
these things happen.

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This is just-- to convince
you that the dynamics

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are complicated, this is
the flow visualization.

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We built a wind tunnel releasing
smoke from the leading edge

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and took a picture
of it to show you how

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complicated the dynamics were.

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So control is potentially
an incredibly powerful idea.

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You can make-- we try to make
robots that run like ostriches.

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We try to make all
kinds of things happen.

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You could imagine
improving wind energy.

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You could imagine
all kinds of things

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where control is an
integral part of it.

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The feedback in there
was absolutely essential.

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If I took the same plane and
thought about the model a lot

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and then designed a controller,
just a set of commands

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to the elevator, to try to
make it land on the perch,

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it misses the perch
every single time.

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And with feedback,
we can hit-- it's

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only with feedback
that we can hit

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the perch every single time.

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And there's a reason for that.

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You guys probably heard
the idea that fighter jets

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are unstable without the
control system, right?

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Fighter jets are-- a lot
of times, the systems that

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have peak
maneuverability, that's

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often at odds with stability.

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In fact, when you try to make
a very high performance fighter

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jet, and you want to be
able to turn, you actually--

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if the control system is off,
this thing should be unstable.

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It's not a complete pathology,
but that's pretty true,

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because what you want, you
can elicit a very fast turn

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if you basically let
the system go unstable.

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Then you can let the unstable
dynamics of the system

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make a very rapid turn.

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In fact, it's common to build
a system that is unstable,

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and it's only stable when
there's feedback in the loop

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so that you can have very
high maneuverability when

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you need it.

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You've already done feedback
if you took 601, right?

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So what I want to do today is--

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00:09:28,190 --> 00:09:30,620
I taught 601 when some
of you were in there.

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I want to go through the example
that you already worked through

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in the 601 lab for those
of you that took it.

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00:09:36,050 --> 00:09:39,110
I think it's sufficiently
complete here

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00:09:39,110 --> 00:09:40,430
if you didn't take it.

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But we want to use that--

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I'm going to use that
as an example to build

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00:09:42,950 --> 00:09:45,366
on what you already know and
to show us with the new tools

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00:09:45,366 --> 00:09:48,050
how far you can get in thinking
about what that robot did

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00:09:48,050 --> 00:09:49,240
for you last year.

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00:09:49,240 --> 00:09:52,490
Do you remember this example of
the little pioneer robot that

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00:09:52,490 --> 00:09:56,240
had to go to the
wall, the foam wall,

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using the noisy sonar sensor?

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00:09:58,130 --> 00:10:00,500
And we wanted to get a desired
distance from the wall?

223
00:10:00,500 --> 00:10:02,450
And by the end, you guys
were moving the board around

224
00:10:02,450 --> 00:10:03,950
and it was trying
to track the wall?

225
00:10:06,860 --> 00:10:09,590
In that exercise,
in the 601 lab,

226
00:10:09,590 --> 00:10:11,930
you guys tried a bunch
of different gains.

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00:10:11,930 --> 00:10:14,810
For some gains, we saw responses
that sort of looked like this.

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00:10:14,810 --> 00:10:16,630
If you gave a desired
distance to a wall,

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00:10:16,630 --> 00:10:18,491
it would go up to
that desired distance.

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00:10:18,491 --> 00:10:20,240
For other gains, you
saw some oscillation.

231
00:10:20,240 --> 00:10:22,760
You saw some big
oscillations if you tried

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all the gains we recommended.

233
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Now you guys have a
deep understanding

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00:10:27,114 --> 00:10:29,405
of what kind of things can
cause that type of response.

235
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The way we told you to
think about it in 601

236
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is actually the
way that we often

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00:10:37,250 --> 00:10:39,590
think about control systems.

238
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Typically, we have something
called the plant, which

239
00:10:42,470 --> 00:10:44,480
describes the dynamics
of the robot or the thing

240
00:10:44,480 --> 00:10:46,160
we're trying to care about.

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00:10:46,160 --> 00:10:49,690
You guys know why it's
called the plant often?

242
00:10:49,690 --> 00:10:52,580
What would be the history
of calling it "plant"?

243
00:10:52,580 --> 00:10:55,634
It's a weird name for it, right?

244
00:10:55,634 --> 00:10:57,800
Actually, some of this stuff
grew up in the chemical

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00:10:57,800 --> 00:10:59,424
industry, in chemical plants.

246
00:10:59,424 --> 00:11:00,340
Is that what you said?

247
00:11:00,340 --> 00:11:01,730
No?

248
00:11:01,730 --> 00:11:04,560
It's actually-- it has a
history in chemical plants.

249
00:11:04,560 --> 00:11:09,020
But now everybody calls
their robot plants.

250
00:11:09,020 --> 00:11:12,140
The plant is the dynamics
of our robot, for instance.

251
00:11:12,140 --> 00:11:13,490
You've got some output of that.

252
00:11:13,490 --> 00:11:15,830
In this case, it's the
position to the wall.

253
00:11:15,830 --> 00:11:17,330
A sensor that reads it.

254
00:11:17,330 --> 00:11:19,820
It may have its own dynamics.

255
00:11:19,820 --> 00:11:23,090
And your goal is to take some
reference command, a position

256
00:11:23,090 --> 00:11:24,784
you want to go to,
let's say, compare it

257
00:11:24,784 --> 00:11:27,200
with what the sensor is saying
and build a controller that

258
00:11:27,200 --> 00:11:31,371
takes that error and makes the
robot or control system do what

259
00:11:31,371 --> 00:11:32,120
you want it to do.

260
00:11:35,560 --> 00:11:39,370
In this example, the
one you studied in 601,

261
00:11:39,370 --> 00:11:41,470
the plant was a
very simple model

262
00:11:41,470 --> 00:11:43,690
of the dynamics of the robot.

263
00:11:43,690 --> 00:11:45,700
It was just a first-order model.

264
00:11:45,700 --> 00:11:48,010
We said that the
output, the distance

265
00:11:48,010 --> 00:11:51,970
from the wall, the distance in
front of the wall, at time n

266
00:11:51,970 --> 00:11:56,530
was just the distance in
front of the wall time minus--

267
00:11:56,530 --> 00:11:58,210
because, you remember
the frustration

268
00:11:58,210 --> 00:11:59,590
of the flipped sign, too?

269
00:11:59,590 --> 00:12:01,730
But I kept it consistent here.

270
00:12:01,730 --> 00:12:03,399
So it's T times
the velocity, just

271
00:12:03,399 --> 00:12:05,440
happens that since the
velocity is going this way

272
00:12:05,440 --> 00:12:06,970
and the distance
is getting smaller,

273
00:12:06,970 --> 00:12:08,890
you get a minus T.
This looks almost

274
00:12:08,890 --> 00:12:13,590
like the first-order
approximation of a CT system,

275
00:12:13,590 --> 00:12:15,130
right?

276
00:12:15,130 --> 00:12:17,131
So that's the dynamics
of the vehicle.

277
00:12:17,131 --> 00:12:18,880
The sensor, we're going
to assume for now,

278
00:12:18,880 --> 00:12:21,070
is just perfect, that
it just immediately

279
00:12:21,070 --> 00:12:23,650
tells me the
distance to the wall

280
00:12:23,650 --> 00:12:25,570
and gives the perfect feedback.

281
00:12:25,570 --> 00:12:27,917
And then we designed
a controller, which

282
00:12:27,917 --> 00:12:30,250
just takes the error, the
difference between the desired

283
00:12:30,250 --> 00:12:32,560
position and the
actual position,

284
00:12:32,560 --> 00:12:35,800
multiplies it by a
constant K and comes

285
00:12:35,800 --> 00:12:38,170
up with a velocity command
that goes to the motors.

286
00:12:41,420 --> 00:12:45,810
You can visualize that as
a block diagram, of course.

287
00:12:45,810 --> 00:12:49,880
You get that this
plant model here--

288
00:12:49,880 --> 00:12:51,890
it's got a minus one here
and a minus one here,

289
00:12:51,890 --> 00:12:54,740
so that's equivalent having
a delay in the forward path.

290
00:12:54,740 --> 00:12:57,210
And then it's got feedback,
because the previous signal

291
00:12:57,210 --> 00:12:58,880
was directly put
around to the feedback.

292
00:12:58,880 --> 00:13:02,570
So this part here is
the model of the plant.

293
00:13:05,190 --> 00:13:07,130
Then we put the
T in, with a gain

294
00:13:07,130 --> 00:13:10,100
here, so I guess this
part here is the plant.

295
00:13:10,100 --> 00:13:14,334
Here's our simple controller
and our sensor is perfect.

296
00:13:14,334 --> 00:13:15,500
So that's our block diagram.

297
00:13:15,500 --> 00:13:17,541
We've just turned our
robot into a block diagram.

298
00:13:17,541 --> 00:13:21,740
And we know everything about
how to analyze those things.

299
00:13:21,740 --> 00:13:23,390
You can use the
operator notation.

300
00:13:23,390 --> 00:13:28,070
You could think of it as
a system function, too.

301
00:13:28,070 --> 00:13:29,900
This guy here you know is--

302
00:13:33,032 --> 00:13:34,990
you guys can do this in
your sleep now, almost?

303
00:13:34,990 --> 00:13:38,126
But if I just take
that by itself,

304
00:13:38,126 --> 00:13:39,250
what does that come out to?

305
00:13:41,800 --> 00:13:42,850
It's a plus here.

306
00:13:42,850 --> 00:13:44,920
This is x and y.

307
00:13:44,920 --> 00:13:52,630
So I get y is R x plus y.

308
00:13:52,630 --> 00:14:01,720
Or y over x is R
over 1 minus R. Then

309
00:14:01,720 --> 00:14:06,144
I multiply that by
K and a negative t.

310
00:14:06,144 --> 00:14:07,810
And then I do the
exact same computation

311
00:14:07,810 --> 00:14:09,550
again to get this loop around.

312
00:14:09,550 --> 00:14:14,060
And this is our input
output system function.

313
00:14:14,060 --> 00:14:16,510
Simplify it a little
bit, I get this.

314
00:14:16,510 --> 00:14:20,080
Simplify it a little bit
more to identify the pole,

315
00:14:20,080 --> 00:14:22,540
and you can see that the pole
now looks like 1 plus KT.

316
00:14:25,100 --> 00:14:29,200
T represents the time
step between updates,

317
00:14:29,200 --> 00:14:30,300
K is our feedback gain.

318
00:14:33,679 --> 00:14:35,220
Now, we can start
thinking about what

319
00:14:35,220 --> 00:14:37,050
happens if we choose
different-- let's just

320
00:14:37,050 --> 00:14:39,360
lump K and T together,
because the K you choose

321
00:14:39,360 --> 00:14:41,664
is going to be intimately
connected to time.

322
00:14:41,664 --> 00:14:43,830
In this system, there's no
point in separating them.

323
00:14:43,830 --> 00:14:46,980
So we just talk
about KT as a system.

324
00:14:46,980 --> 00:14:48,150
If we chose KT--

325
00:14:48,150 --> 00:14:50,100
KT is almost always
going to be negative.

326
00:14:50,100 --> 00:14:52,410
You want negative
feedback, in general,

327
00:14:52,410 --> 00:14:54,960
for making things stable.

328
00:14:54,960 --> 00:14:59,280
If we choose KT equals 0.5,
then with that system function,

329
00:14:59,280 --> 00:15:00,780
there's a delay in
the forward path,

330
00:15:00,780 --> 00:15:02,154
so it's going to
be offset by one

331
00:15:02,154 --> 00:15:06,000
and then it's just got the
simple single pole response.

332
00:15:06,000 --> 00:15:09,222
The unit step response
similarly looks like this.

333
00:15:09,222 --> 00:15:10,680
So the question
is, what determines

334
00:15:10,680 --> 00:15:14,030
the speed of that response?

335
00:15:14,030 --> 00:15:15,780
Here you go.

336
00:15:15,780 --> 00:15:17,530
I have to get you to
talk to your neighbor

337
00:15:17,530 --> 00:15:18,904
the way Denny
always seems to get

338
00:15:18,904 --> 00:15:20,587
you to talk to your neighbor.

339
00:15:20,587 --> 00:15:21,170
Take a minute.

340
00:15:21,170 --> 00:15:23,570
Figure out which of these
or none of them do you think

341
00:15:23,570 --> 00:15:26,420
would give you the best response
for this simple block diagram

342
00:15:26,420 --> 00:15:26,920
system.

343
00:15:31,889 --> 00:15:33,180
Talk out loud to your neighbor.

344
00:16:48,501 --> 00:16:49,000
OK.

345
00:16:49,000 --> 00:16:50,770
Show of fingers, what
do you think it is?

346
00:16:54,760 --> 00:16:55,260
OK.

347
00:16:55,260 --> 00:16:56,520
A lot of right answers.

348
00:16:56,520 --> 00:16:57,885
Let's just do it real quick.

349
00:17:01,100 --> 00:17:03,830
The fastest possible
convergence is

350
00:17:03,830 --> 00:17:06,470
going to be at the pole zero.

351
00:17:06,470 --> 00:17:08,730
Pole zero, what's it
going to look like?

352
00:17:08,730 --> 00:17:11,240
It's going to give you--

353
00:17:11,240 --> 00:17:13,280
the best you can do
if you zero this out,

354
00:17:13,280 --> 00:17:15,391
you get the impulse response.

355
00:17:15,391 --> 00:17:16,849
The unit sample
response would just

356
00:17:16,849 --> 00:17:21,780
be R. You're going to have a
delay of 1, that's inevitable.

357
00:17:21,780 --> 00:17:27,340
You get one non-zero entry and
then zeroes everywhere else.

358
00:17:27,340 --> 00:17:29,090
In discrete time
systems, you can actually

359
00:17:29,090 --> 00:17:30,560
kill things in a single step.

360
00:17:30,560 --> 00:17:35,560
You can set a pole at 0 that's
the fastest possible response.

361
00:17:35,560 --> 00:17:37,550
But if you think about
the different responses

362
00:17:37,550 --> 00:17:40,700
for different values of K, you
can use the pole zero diagrams

363
00:17:40,700 --> 00:17:42,230
to pretty much
understand everything

364
00:17:42,230 --> 00:17:45,020
there is to know about it.

365
00:17:45,020 --> 00:17:49,220
For KT less than 0 to
negative 1-- like I said,

366
00:17:49,220 --> 00:17:51,140
we want to think about
negative feedback--

367
00:17:51,140 --> 00:17:54,860
that's going to take us from
out here on the unit circle back

368
00:17:54,860 --> 00:17:55,760
towards the origin.

369
00:17:55,760 --> 00:17:58,400
If I keep making
KT more negative,

370
00:17:58,400 --> 00:17:59,620
it's going to go out here.

371
00:17:59,620 --> 00:18:01,697
If I make KT too
negative, bad things

372
00:18:01,697 --> 00:18:02,780
are going to happen again.

373
00:18:02,780 --> 00:18:04,820
The poles are going to go
outside the unit circle

374
00:18:04,820 --> 00:18:06,236
and will actually
have alternating

375
00:18:06,236 --> 00:18:07,340
diverging responses.

376
00:18:12,650 --> 00:18:14,570
The answer is negative 1.

377
00:18:14,570 --> 00:18:16,880
Negative 1 puts
you at a pole of 0.

378
00:18:21,810 --> 00:18:24,540
Here's to think
about it physically.

379
00:18:24,540 --> 00:18:30,690
If I choose K correctly,
since KT was negative 1,

380
00:18:30,690 --> 00:18:33,130
that means K's going
to be negative 10.

381
00:18:33,130 --> 00:18:36,660
That's going to be commanding
exactly the right speed so

382
00:18:36,660 --> 00:18:41,580
that the robot, after the one
tenth of a second, let's say,

383
00:18:41,580 --> 00:18:43,610
it gets you exactly
to the right position.

384
00:18:43,610 --> 00:18:48,090
Unit sample response in
this case would be saying,

385
00:18:48,090 --> 00:18:50,257
the robot's at zero,
I want it to be at 1,

386
00:18:50,257 --> 00:18:51,840
and then I want it
to be back at zero.

387
00:18:51,840 --> 00:18:54,660
The command is going 0, 1, 0.

388
00:18:54,660 --> 00:18:57,130
And the robot is going to be
almost doing exactly that.

389
00:18:57,130 --> 00:19:00,300
It's going to go from 0
to 1 in a single step.

390
00:19:00,300 --> 00:19:03,180
And then back to 0
in a single step.

391
00:19:03,180 --> 00:19:04,830
It's just going to
be delayed by one.

392
00:19:04,830 --> 00:19:07,814
So you give it a command saying,
go here, go here, go back.

393
00:19:07,814 --> 00:19:10,230
And it's going to do exactly
the right thing-- boom, boom,

394
00:19:10,230 --> 00:19:12,480
except for one step delayed,
because you can't get rid

395
00:19:12,480 --> 00:19:14,490
of that R.

396
00:19:14,490 --> 00:19:18,002
So if I plot it-- let
me draw a stem diagram,

397
00:19:18,002 --> 00:19:20,460
but coming down in time so that
I can line up with the axes

398
00:19:20,460 --> 00:19:21,440
up here.

399
00:19:21,440 --> 00:19:24,570
If I start with an
initial position here,

400
00:19:24,570 --> 00:19:28,020
and I command it to go to
the desired front position,

401
00:19:28,020 --> 00:19:30,720
it's going to go boom right
to that front position

402
00:19:30,720 --> 00:19:32,460
with one unit of delay.

403
00:19:32,460 --> 00:19:37,880
And then it's going to stay
there for the rest of the time.

404
00:19:37,880 --> 00:19:40,260
It would be a unit
step response.

405
00:19:44,220 --> 00:19:47,890
But that's not what we got
to see on the robots in 601,

406
00:19:47,890 --> 00:19:48,390
right?

407
00:19:48,390 --> 00:19:51,670
The real robot didn't
work like that.

408
00:19:51,670 --> 00:19:55,420
And the way we made the robot
model more realistic was we

409
00:19:55,420 --> 00:19:58,240
said, OK, but your
sensor's got some delay.

410
00:19:58,240 --> 00:20:00,310
And actually, if you
knew what was going on

411
00:20:00,310 --> 00:20:02,850
behind those 601 robots, it's
actually had a lot of delay.

412
00:20:02,850 --> 00:20:05,200
There's Python running
serial interfaces

413
00:20:05,200 --> 00:20:10,480
to over the serial link to
the fairly old controller

414
00:20:10,480 --> 00:20:11,370
in the pioneer robot.

415
00:20:11,370 --> 00:20:13,440
So the delay is real.

416
00:20:13,440 --> 00:20:18,670
There's a delay of about 1/10
of a second in the sensor.

417
00:20:18,670 --> 00:20:22,180
If I take that exact same
controller, exact same gain

418
00:20:22,180 --> 00:20:24,850
that I already did, now put
it in this new system that

419
00:20:24,850 --> 00:20:27,320
has an extra delay,
then what happens?

420
00:20:27,320 --> 00:20:30,640
I get a velocity of 10
after one step, looks good.

421
00:20:30,640 --> 00:20:32,260
But then, uh-oh,
there was some delay

422
00:20:32,260 --> 00:20:34,245
in the sensor I didn't
realize was here,

423
00:20:34,245 --> 00:20:36,370
so I'm still going to take
corrective action trying

424
00:20:36,370 --> 00:20:37,060
to get me there.

425
00:20:37,060 --> 00:20:38,976
It's going to move me
all the way to the wall,

426
00:20:38,976 --> 00:20:40,660
smash into the
wall, and then it's

427
00:20:40,660 --> 00:20:41,920
going to realize it was zero.

428
00:20:41,920 --> 00:20:44,003
It's going to-- there's
some delay in seeing that.

429
00:20:44,003 --> 00:20:45,460
It's going to move me back.

430
00:20:45,460 --> 00:20:49,620
And you're going to
get oscillations.

431
00:20:49,620 --> 00:20:52,650
You can see that now by just
adding the model to our block

432
00:20:52,650 --> 00:20:54,790
diagram.

433
00:20:54,790 --> 00:20:59,550
If we put the delay now
in the feedback path,

434
00:20:59,550 --> 00:21:03,630
otherwise, keep the block
diagram exactly the same.

435
00:21:03,630 --> 00:21:07,020
Now, you can write
the system function.

436
00:21:07,020 --> 00:21:08,524
What's the resulting
system function

437
00:21:08,524 --> 00:21:10,440
given that this thing
is in the feedback path?

438
00:22:48,317 --> 00:22:50,900
Go ahead and put your fingers
up when you think you've got it.

439
00:22:54,600 --> 00:22:55,170
All right.

440
00:22:55,170 --> 00:22:55,770
Fantastic.

441
00:23:02,060 --> 00:23:03,810
Just like I did here,
you can just quickly

442
00:23:03,810 --> 00:23:07,350
replace the accumulator there
with the equivalent block

443
00:23:07,350 --> 00:23:08,790
diagram.

444
00:23:08,790 --> 00:23:09,750
Do the loop again.

445
00:23:09,750 --> 00:23:11,610
It's going to be exactly
what we did before,

446
00:23:11,610 --> 00:23:15,910
but it's got a new R
in the feedback path.

447
00:23:15,910 --> 00:23:19,720
Giving us the R-squared here,
just on the feedback path.

448
00:23:19,720 --> 00:23:21,920
It's exactly the same
otherwise except for that R,

449
00:23:21,920 --> 00:23:24,750
and that simplifies out to this.

450
00:23:24,750 --> 00:23:27,130
So the answer was 4.

451
00:23:27,130 --> 00:23:29,640
That's just operator
notation, polynomial algebra.

452
00:23:34,720 --> 00:23:37,750
If we want to find the
poles of that system,

453
00:23:37,750 --> 00:23:43,270
we can just go ahead and
factor the quadratic form

454
00:23:43,270 --> 00:23:45,289
in the denominator.

455
00:23:45,289 --> 00:23:47,080
The roots of the
denominator look something

456
00:23:47,080 --> 00:23:50,290
like this, which is a
little ugly to think about.

457
00:23:50,290 --> 00:23:53,980
But we can think it through.

458
00:23:53,980 --> 00:24:00,190
For general KT-- when KT's
small, KT's about zero,

459
00:24:00,190 --> 00:24:03,300
then I'll go ahead and simplify
that a little bit to say,

460
00:24:03,300 --> 00:24:04,990
KT could sneak inside the--

461
00:24:04,990 --> 00:24:07,420
KT and KT-squared
aren't so different.

462
00:24:07,420 --> 00:24:09,340
We're going to sneak
it inside here.

463
00:24:09,340 --> 00:24:12,090
And then you can see that the
poles end up at-- around K

464
00:24:12,090 --> 00:24:17,130
equal to zero you get
a pole at the origin.

465
00:24:17,130 --> 00:24:20,860
But you also get a pole
up by the unit circle.

466
00:24:20,860 --> 00:24:24,620
Around 1 and around 0.

467
00:24:24,620 --> 00:24:30,560
Remember, when you've got
multiple poles in the system,

468
00:24:30,560 --> 00:24:34,220
the total response is
going to be dominated

469
00:24:34,220 --> 00:24:38,204
by whatever is the slower pole.

470
00:24:38,204 --> 00:24:39,620
The total system
response is going

471
00:24:39,620 --> 00:24:40,760
to be dominated by
the slower pole.

472
00:24:40,760 --> 00:24:43,250
In this case, the slower pole's
the one closer to the unit

473
00:24:43,250 --> 00:24:43,749
circle.

474
00:24:48,540 --> 00:24:52,410
What about if KT
equals negative 0.25?

475
00:24:52,410 --> 00:24:54,390
We can pop that in and solve it.

476
00:24:54,390 --> 00:24:57,510
Exactly-- math works out nicely.

477
00:24:57,510 --> 00:25:02,820
We get poles at a half,
two poles at a half.

478
00:25:02,820 --> 00:25:05,940
And in fact, there's a smooth
transition between-- if you

479
00:25:05,940 --> 00:25:10,080
look at the numbers between KT
equals zero and KT equals 0.25,

480
00:25:10,080 --> 00:25:12,630
you'll see as you
vary that gain,

481
00:25:12,630 --> 00:25:17,040
the poles move together
along the real axis

482
00:25:17,040 --> 00:25:18,990
until they come
together at a half.

483
00:25:21,860 --> 00:25:29,270
So here, you've got a purely
real response dominated

484
00:25:29,270 --> 00:25:31,130
by these poles at a half.

485
00:25:31,130 --> 00:25:33,080
System's stable.

486
00:25:33,080 --> 00:25:36,170
If you keep changing K though,
the poles came together

487
00:25:36,170 --> 00:25:40,310
and then they split off
and start going this way.

488
00:25:40,310 --> 00:25:42,260
And in fact, if you look
at negative 1, which

489
00:25:42,260 --> 00:25:44,960
is the one that was
the best response,

490
00:25:44,960 --> 00:25:50,300
it put a pole exactly at 0,
for the system with no delay.

491
00:25:50,300 --> 00:25:52,130
If you put it at the
system with delay,

492
00:25:52,130 --> 00:25:55,280
they land exactly
on the unit circle.

493
00:25:55,280 --> 00:25:57,260
Right there.

494
00:25:57,260 --> 00:26:00,230
Complex poles on
the unit circle.

495
00:26:00,230 --> 00:26:02,269
You're going to get
a stable oscillation.

496
00:26:07,720 --> 00:26:09,580
Which is exactly
what we saw there.

497
00:26:13,400 --> 00:26:16,490
Just a quick-- you know this
like the back of your hand now,

498
00:26:16,490 --> 00:26:19,955
but what's the period
of that oscillation?

499
00:26:26,260 --> 00:26:31,660
You've got two poles on the
unit circle right there.

500
00:26:31,660 --> 00:26:33,855
What's the period
of oscillation?

501
00:26:39,300 --> 00:26:41,370
Put your fingers up
when you think you know.

502
00:26:52,380 --> 00:26:53,120
Yep.

503
00:26:53,120 --> 00:26:54,290
Good.

504
00:26:54,290 --> 00:26:56,650
Most people got it.

505
00:26:56,650 --> 00:27:00,820
This thing was a half, so it's
1/2 square root of 3 over 2.

506
00:27:00,820 --> 00:27:03,497
So that thing had
to be at pi over 3.

507
00:27:03,497 --> 00:27:05,080
It's actually the
same pole that I was

508
00:27:05,080 --> 00:27:09,782
using in recitation yesterday.

509
00:27:09,782 --> 00:27:11,240
So if you have a
pole at pi over 3,

510
00:27:11,240 --> 00:27:15,140
it makes it around in six steps.

511
00:27:15,140 --> 00:27:17,340
The period of the
oscillation is six.

512
00:27:26,140 --> 00:27:28,840
This is generally
true, that if you

513
00:27:28,840 --> 00:27:32,110
put a controlled
gain, a feedback gain,

514
00:27:32,110 --> 00:27:36,700
into the closed loop dynamics,
then even a simple gain

515
00:27:36,700 --> 00:27:40,240
can allow you to really shift
around the poles of the system.

516
00:27:40,240 --> 00:27:42,850
And since we know the response
of the system, the zeros

517
00:27:42,850 --> 00:27:46,570
matter, but for convergence,
for the rate of convergence,

518
00:27:46,570 --> 00:27:49,410
it's the biggest
pole that dominates.

519
00:27:49,410 --> 00:27:53,436
It's very nice to understand
how those poles move with K.

520
00:27:53,436 --> 00:27:54,810
If you change the
system, the way

521
00:27:54,810 --> 00:27:56,880
they move with K is different.

522
00:27:56,880 --> 00:28:00,000
And you can just change
K to tune the response

523
00:28:00,000 --> 00:28:03,330
to be what you want, from
KT equals 0 to infinity,

524
00:28:03,330 --> 00:28:06,810
the poles go towards
here and then off in both

525
00:28:06,810 --> 00:28:08,608
directions, actually,
to infinity.

526
00:28:14,450 --> 00:28:19,740
So, KT equals negative 1 was
the fastest possible response

527
00:28:19,740 --> 00:28:22,020
for the system without
any delay in the sensor.

528
00:28:22,020 --> 00:28:24,700
What's the fastest possible
response for this one?

529
00:28:40,200 --> 00:28:41,250
Oscillations are allowed.

530
00:28:41,250 --> 00:28:43,050
I just want the fastest
possible response.

531
00:29:14,840 --> 00:29:15,340
Yeah.

532
00:29:15,340 --> 00:29:15,880
Looks good.

533
00:29:15,880 --> 00:29:19,431
So where do I want the poles
to be for the fastest possible

534
00:29:19,431 --> 00:29:19,930
response?

535
00:29:26,470 --> 00:29:31,180
These poles are still
stable and oscillations are

536
00:29:31,180 --> 00:29:33,370
fine, but the absolute--

537
00:29:33,370 --> 00:29:35,920
the magnitude of
those poles is larger

538
00:29:35,920 --> 00:29:38,260
when they're out
here in the complex.

539
00:29:38,260 --> 00:29:39,910
When I'm down here,
I have got a pole

540
00:29:39,910 --> 00:29:41,960
over by one, that's
going to dominate.

541
00:29:41,960 --> 00:29:45,400
So the best I can do is if
I put the poles at a half.

542
00:29:45,400 --> 00:29:48,130
That gives us the largest
pole, the smallest

543
00:29:48,130 --> 00:29:49,950
possible magnitude.

544
00:29:49,950 --> 00:29:54,640
And that happened if the poles--
the double poles at a half

545
00:29:54,640 --> 00:29:57,940
happened when KT
was negative 0.25.

546
00:29:57,940 --> 00:30:00,289
Most of you said,
the answer is 2.

547
00:30:08,560 --> 00:30:11,920
In general, delay
is a bad thing.

548
00:30:11,920 --> 00:30:14,530
In DT systems, we have good
representations of delay.

549
00:30:14,530 --> 00:30:17,290
It's even worse in
continuous time.

550
00:30:17,290 --> 00:30:18,250
Delay is a bad thing.

551
00:30:18,250 --> 00:30:23,770
It tends to make control
systems not work as well.

552
00:30:23,770 --> 00:30:27,460
If you just took the ideal
sensor, we had a K equals 1,

553
00:30:27,460 --> 00:30:30,580
we had a response that started
here, we could put it anywhere

554
00:30:30,580 --> 00:30:33,534
we wanted along the real axis.

555
00:30:33,534 --> 00:30:35,200
As far negative as
we wanted, of course,

556
00:30:35,200 --> 00:30:38,980
what we chose was to put
it right at the origin.

557
00:30:38,980 --> 00:30:41,380
But as soon as we just added
that one piece of delay,

558
00:30:41,380 --> 00:30:44,140
the things we could do
with proportional feedback

559
00:30:44,140 --> 00:30:46,630
changed completely,
and ultimately,

560
00:30:46,630 --> 00:30:51,320
got worse because I have
two poles now, first of all,

561
00:30:51,320 --> 00:30:54,130
and I can't simultaneously
get both of those poles

562
00:30:54,130 --> 00:30:55,450
to go to zero.

563
00:30:55,450 --> 00:30:58,949
In fact, as I change them,
they go off into complex.

564
00:30:58,949 --> 00:31:00,490
The fact that it's
complex isn't bad.

565
00:31:00,490 --> 00:31:03,156
But the fact that there's two of
them, and the best I can do now

566
00:31:03,156 --> 00:31:05,620
is get to a half, which is a
much slower response, the best

567
00:31:05,620 --> 00:31:06,480
possible response.

568
00:31:10,030 --> 00:31:13,600
If I added even more delay,
things get even worse.

569
00:31:13,600 --> 00:31:16,004
If I had two units
of delay and I

570
00:31:16,004 --> 00:31:17,920
went through the same
exercise, what you'd see

571
00:31:17,920 --> 00:31:21,780
is that the poles would
start in the same place.

572
00:31:21,780 --> 00:31:23,735
They'd come together
and go there,

573
00:31:23,735 --> 00:31:25,360
and there's another
pole that goes off.

574
00:31:25,360 --> 00:31:27,870
Just because there are
three poles in that system.

575
00:31:27,870 --> 00:31:32,230
Two poles come together
and split off this way.

576
00:31:32,230 --> 00:31:35,290
And this one goes off this way.

577
00:31:35,290 --> 00:31:37,104
And the place you
probably want, depending

578
00:31:37,104 --> 00:31:39,020
on how fast this one
splits off, but the place

579
00:31:39,020 --> 00:31:41,630
you probably want to put it is
when the two poles are together

580
00:31:41,630 --> 00:31:42,270
right there.

581
00:31:42,270 --> 00:31:44,370
That's going to give you
the fastest response.

582
00:31:44,370 --> 00:31:48,570
But that fastest response is
still slower than what we had--

583
00:31:48,570 --> 00:31:50,780
it's a bigger
number than a half.

584
00:31:50,780 --> 00:31:54,340
It's 0.682.

585
00:31:54,340 --> 00:31:57,890
It's going to be a
worse response than when

586
00:31:57,890 --> 00:32:00,421
I had a delay of one,
which is intuitive.

587
00:32:05,420 --> 00:32:08,460
That's a quick reminder
of something you already

588
00:32:08,460 --> 00:32:13,080
did in 601, of using feedback
and the tools we've already

589
00:32:13,080 --> 00:32:16,072
got, which is poles and zeros
and everything, to understand

590
00:32:16,072 --> 00:32:17,280
how to design feedback gains.

591
00:32:17,280 --> 00:32:20,020
We're going to get more into it
in the next couple of lectures.

592
00:32:20,020 --> 00:32:23,490
But let me just convince
you that this stuff is real.

593
00:32:23,490 --> 00:32:25,620
And I showed this
diagram once in 601,

594
00:32:25,620 --> 00:32:27,940
but it probably means
even more to you now.

595
00:32:33,170 --> 00:32:36,810
This is an F-14.

596
00:32:36,810 --> 00:32:41,900
It's one of the best
modern engineering control

597
00:32:41,900 --> 00:32:44,750
systems ever built. It
was built for this F-14.

598
00:32:44,750 --> 00:32:49,220
They got more research money
and modeling this vehicle,

599
00:32:49,220 --> 00:32:53,480
designing ultimate gains for it.

600
00:32:53,480 --> 00:32:56,420
It's such a success story
that you can actually,

601
00:32:56,420 --> 00:32:59,900
if you're in MATLAB, you
can open up the F-14 demo

602
00:32:59,900 --> 00:33:04,940
and see what a flight control
system for an F-14 looks like.

603
00:33:04,940 --> 00:33:07,190
MATLAB-- I don't know if
you've played with Simulink--

604
00:33:07,190 --> 00:33:11,000
MATLAB has a language called
Simulink, a graphical language

605
00:33:11,000 --> 00:33:13,460
that allows you to
draw the block diagrams

606
00:33:13,460 --> 00:33:17,430
and simulate them and even
design controllers for them.

607
00:33:17,430 --> 00:33:19,310
And it turns out you
can make a block diagram

608
00:33:19,310 --> 00:33:27,860
description of an F-14 using
only tools from 601 and 6003.

609
00:33:27,860 --> 00:33:31,520
You can see all the
same adders and gains,

610
00:33:31,520 --> 00:33:36,480
transfer functions, system
functions like this.

611
00:33:36,480 --> 00:33:39,710
The only essential difference
is that in some of the diagrams,

612
00:33:39,710 --> 00:33:41,810
you see multiple
inputs coming in.

613
00:33:41,810 --> 00:33:43,610
In this class, we've
restricted ourselves

614
00:33:43,610 --> 00:33:46,850
so far to thinking about single
input, single output systems,

615
00:33:46,850 --> 00:33:48,190
which keeps everything clean.

616
00:33:48,190 --> 00:33:51,410
All the intuition scales
to multiple inputs

617
00:33:51,410 --> 00:33:53,120
and multiple outputs.

618
00:33:53,120 --> 00:33:58,570
But that's sort of the only
big addition of complexity

619
00:33:58,570 --> 00:34:02,200
when you go to this model
from what we've done before.

620
00:34:02,200 --> 00:34:06,759
If you zoom in onto the
controller here, you can--

621
00:34:06,759 --> 00:34:08,300
these block diagrams
in this language

622
00:34:08,300 --> 00:34:10,777
allow you to abstract away
a bunch of hidden things

623
00:34:10,777 --> 00:34:12,110
with a single transfer function.

624
00:34:12,110 --> 00:34:16,670
If I zoom in, then you can
see things you recognize.

625
00:34:16,670 --> 00:34:18,770
It's got a low-pass
filter, which

626
00:34:18,770 --> 00:34:21,139
is just a system with
a pole at negative 1,

627
00:34:21,139 --> 00:34:25,150
exactly what we did in
recitation the other day.

628
00:34:25,150 --> 00:34:27,590
This is what people--

629
00:34:27,590 --> 00:34:29,510
this is what MATLAB
uses on an F-14.

630
00:34:29,510 --> 00:34:31,468
I guess it might not be
what the military uses.

631
00:34:31,468 --> 00:34:35,739
But it's modeled after what
the unclassified documents say

632
00:34:35,739 --> 00:34:36,739
is happening on an F-14.

633
00:34:39,949 --> 00:34:41,659
Everything here, you
understand, right?

634
00:34:41,659 --> 00:34:44,330
A proportional
controller in the end.

635
00:34:44,330 --> 00:34:47,300
One difference in
F-14s is that--

636
00:34:47,300 --> 00:34:50,060
and in general for these
more complicated systems--

637
00:34:50,060 --> 00:34:52,520
they'll design slightly
different controllers

638
00:34:52,520 --> 00:34:53,929
given the situation.

639
00:34:53,929 --> 00:34:59,746
So if you have an altimeter
on there and pressure sensors,

640
00:34:59,746 --> 00:35:03,720
an inclinometer will tell
you the angle of the plane--

641
00:35:03,720 --> 00:35:07,490
given those sensors,
they'll pick a different K

642
00:35:07,490 --> 00:35:09,230
for a proportional
controller out

643
00:35:09,230 --> 00:35:12,500
of a pot of a library
of controllers

644
00:35:12,500 --> 00:35:14,300
they've already designed.

645
00:35:14,300 --> 00:35:16,610
So it's called gain
scheduled control.

646
00:35:16,610 --> 00:35:19,340
But the analysis
and design of each K

647
00:35:19,340 --> 00:35:23,310
is a linear systems
design and analysis.

648
00:35:23,310 --> 00:35:24,900
This is super powerful stuff.

649
00:35:24,900 --> 00:35:28,310
I think-- signal
processing is good, too,

650
00:35:28,310 --> 00:35:30,170
but control is where it's at.

651
00:35:32,900 --> 00:35:35,810
I guess I went a
little fast, but that's

652
00:35:35,810 --> 00:35:39,260
your introduction to feedback.

653
00:35:39,260 --> 00:35:42,410
We'll do DT and CT feedback in
the next couple of lectures.

654
00:35:42,410 --> 00:35:44,450
And if anybody hasn't
picked up their exams,

655
00:35:44,450 --> 00:35:45,930
we have them over here.

656
00:35:45,930 --> 00:35:48,670
And we'll see you in
recitation tomorrow.