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[MUSIC PLAYING]
PROFESSOR: During the course,
we've developed a number of
very powerful and
useful tools.
And we've seen how these can
be used in designing and
analyzing systems.
For example, for filtering,
for modulation, et cetera.
I'd like to conclude this series
of lectures with an
introduction to one more
important topic.
Namely, the analysis of
feedback systems.
And one of the principle reasons
that we have left this
discussion to the last part of
the course is so that we can
exploit some of the ideas that
we've just developed in some
of the previous lectures.
Namely, the tools afforded us
by Laplace and z-transforms.
Now, as I had indicated in one
of the very first lectures, a
common example of a feedback
system is the problem of,
let's say balancing a broom, or
in the case of that lecture
balancing my son's horse in
the palm of your hand.
And kind of the idea there is
that what that relies on, in
order to make that a stable
system, is feedback.
In that particular case,
visual feedback.
That specific problem, the one
of balancing something, let's
say in the palm of your hand,
is an example of a problem,
which is commonly referred to
as the inverted pendulum.
And it's one that we will
actually be analyzing in a
fair amount of detail
not in this lecture,
but in the next lecture.
But let me just kind of indicate
what some of the
issues are.
Let me describe this in the
context not of balancing a
broom on your hand, but let's
say that we have a mechanical
system which consists
of a cart.
And the cart can move, let's say
in one dimension, and it
has mounted on it a bar, a rod
with a weight on the top, and
it pivots around the base.
So that essentially represents
the inverted pendulum.
So that system can be, more
or less, depicted as I've
indicated here.
And this is the cart that can
move along the x-axis.
And here we have a pivot
point, a rod, a
weight at the top.
And then, of course, there are
several forces acting on this.
There is an acceleration that
can be applied to the cart,
and that will be thought of
as the external input.
And then on the pendulum itself,
on the weight, there
is the force of gravity.
And then typically, a set of
external disturbances that
might represent, for example,
air currents, or wind, or
whatever, that will attempt
to destabilize the system.
Specifically, to have the
pendulum fall down.
Now, if we look at this system
in, more or less, a
straightforward way, what we
have then are the system
dynamics and several inputs, one
of which is the external
disturbances and a second is the
acceleration, which is the
external acceleration
that's applied.
And the output of the system
can be thought of as the
angular displacement
of the pendulum.
Which if we want it balanced,
we would like that angular
displacement to be equal to 0.
Now, if we know exactly what the
system dynamics are and if
we knew exactly what the
external disturbances are,
then in principle, we could
design an acceleration.
Namely, an input.
That would exactly generate
0 output.
In other words, the angle
would be equal to 0.
But as you can imagine, just as
it's basically impossible
to balance a broom in the palm
of your hand with you eyes
closed, what is very hard to
ascertain in advance are what
the various dynamics and
disturbances are.
And so more typically what you
would think of doing is
measuring the output angle,
and then using that
measurement to somehow
influence the applied
acceleration or force.
And that, then, is an example
of a feedback system.
So we would measure the output
angle and generate an input
acceleration, which is some
function of what that
output angle is.
And if we choose the feedback
dynamics correctly, then in
fact, we can drive
this output to 0.
This is one example of a system
which is inherently
unstable because if we left
it to its own devices, the
pendulum would simply
fall down.
And essentially, by applying
feedback, what we're trying to
do is stabilize this inherently
unstable system.
And we'll talk a little bit
more about that specific
application of feedback
shortly.
Another common example of
feedback is in positioning or
tracking systems, and I indicate
one here which
corresponds to the problem of
positioning a telescope, which
is mounted on a rotating
platform.
So in a system of that type, for
example, I indicate here
the rotating platform
and the telescope.
It's driven by a motor.
And again, we could imagine, in
principle, the possibility
of driving this to the desired
angle by choosing an
appropriate applied
input voltage.
And as long as we know such
things as what the
disturbances are that influence
the telescope mount
and what the characteristics of
the motor are, in principle
we could in fact carry this
out in a form which is
referred to as open loop.
Namely, we can choose an
appropriate input voltage to
drive the motor to set the
platform angle at the desired
angular position.
However, again, there are enough
unknowns in a problem
like that, so that one is
motivated to employ feedback.
Namely, to make a measurement
of the output angle and use
that in a feedback loop to
influence the drive for the
motor, so that the telescope
platform is positioned
appropriately.
So if we look at this in a
feedback context, we would
then take the measured output
angle and the measured output
angle would be fed
back and compared
with the desired angle.
And the difference between
those, which essentially is
the error between the platform
positioning and the desired
position would be put perhaps
through an appropriate gain or
attenuation and used as the
excitation to the motor.
So in the mechanical or physical
system, that would
correspond to measuring the
angle, let's say with a
potentiometer.
So here we're measuring the
angle and we have an output,
which is proportional to
that measured angle.
And then we would use feedback,
comparing the
measured angle to some
proportionality factor
multiplying the desired angle.
So here we have the desired
angle, again through some type
of potentiometer.
The two are compared.
Out of the comparator, we
basically have an indication
of what the difference is, and
that represents an error
between the desired and
the true angle.
And then that is used through
perhaps an amplifier to
control the motor.
And in that case, of course,
when the error goes to 0, that
means that the actual
angle and the
desired angle are equal.
And in fact, in that case also
with this system, the input to
the motor is, likewise,
equal to 0.
Now, as I've illustrated it
here, it tends to be in the
context of a continuous
time or analog system.
And in fact, another very
common way of doing
positioning or tracking is to
instead implement the feedback
using a discrete-time
or digital system.
And so in that case, we would
basically take the position
output as it's measured, sample
it, essentially convert
that to a digital discrete-time
signal.
And then that is used in
conjunction with the desired
angle, which both form inputs
to this processor.
And the output of that is
converted, let's say, back to
an analog or continuous-time
voltage and used
to drive the motor.
Now, you could ask, why would
you go to a digital or
discrete-time measurement rather
than doing it the way I
showed on the previous overlay
which seemed relatively
straightforward?
And the reason, principally,
is that in the context of a
digital implementation of the
feedback process, often you
can implement a better
controlled and often also,
more sophisticated algorithm
for the feedback dynamics.
So that you can take a count,
perhaps not only of the angle
itself, but also of the rate
of change of angle.
And in fact, the rate of
change of the rate
of change of angle.
So the system, as it's shown
there then, basically has a
discrete-time or digital
feedback loop around a
continuous time system.
Now, this is an example of, in
fact, a more general way in
which discrete-time
feedback is used
with continuous systems.
And let me indicate, in general,
what the character or
block diagram of such
a system might be.
Typically, if we abstract away
from the telescope positioning
system, we might have
a more general
continuous-time system.
And around which we want to
apply some feedback, which we
could do with a continuous-time
system or with
a discrete-time system by first
converting these signals
to discrete-time signals.
Then, processing that with
a discrete-time system.
And then, through an appropriate
interpolation
algorithm, we would then
convert that back to a
continuous-time signal.
And the difference between
the input signal and this
continuous-time signal which
is fed back, then forms the
excitation to the system
that essentially
we're trying to control.
And in many systems of this
type, the advantage is that
this system can be implemented
in a very reproducible way,
either with a digital computer
or with a microprocessor.
And although we're not going to
go into this in any detail
in this lecture, there
is some discussion
of this in the text.
Essentially, if we make certain
assumptions about this
particular feedback system, we
can move the continuous to
discrete-time converter up to
this point and to this point,
and we can move the
interpolating system outside
the summer.
And what happens in that case
is that we end up with what
looks like an inherently
discrete-time feedback system.
So, in fact, if we take those
steps, then what we'll end up
with for a feedback system is a
system that essentially can
be analyzed as a discrete-time
system.
Here we have what is, in the
forward path, is basically the
continuous-time system with the
interpolator at one end
and the continuous to
discrete-time converter
at the other end.
And then we have whatever system
it was in the feedback
loop-- discrete-time--
that shows up in this
feedback loop.
Well, I show this mainly to
emphasize the fact, although
there are some steps there that
we obviously left out.
I show that mainly to emphasize
the fact that
feedback arises not just in the
context of continuous-time
systems, but also the analysis
of discrete-time feedback
systems becomes important.
Perhaps because we have used
discrete-time feedback around
a continuous-time system.
But also perhaps because the
feedback system is inherently
discrete-time.
And let me just illustrate one,
or indicate one example
in which that might arise.
This is an example which is
also discussed in somewhat
more detail in the text.
But basically, population
studies, for example,
represent examples of
discrete-time feedback systems.
Where let's say that we have
some type of model for
population growth.
And since people come in
integer amounts that
represents essentially the
output of any population
model, essentially
or inherently
represents a sequence.
Namely, it's indexed on
an integer variable.
And typically, models for
population growth
are unstable systems.
You can kind of imagine that
because if you take these
simple models of population,
what happens is that in any
generation, the number of
people, or animals, or
whatever it is that this is
modeling, grows essentially
exponentially with the size of
the previous generation.
Now, where does the
feedback come in?
Well, the feedback typically
comes in, in incorporating in
the overall model various
retarding factors.
For example, as the population
increases, the food supply
becomes more limited.
And that essentially is a
feedback process that acts to
retard the population growth.
And so an overall model--
somewhat simplified--
for a population system is the
open loop model in the absence
of retarding factors.
And then, very often the
retarding factors can be
described as being related to
the size of the population.
And those essentially act to
reduce the overall input to
the population model.
And so population studies are
one very common example of
discrete-time feedback
systems.
Well, what we want to look at
and understand are the basic
properties of feedback
systems.
And to do that, let's look at
the basic block diagram and
equations for feedback
systems, either
continuous-time or
discrete-time.
Let's begin with the
continuous-time case.
And now what we've done is
simply abstract out any of the
applications to a fairly general
system, in which we
have a system H of s in what's
referred to as the forward
path, and a system G of s
in the feedback path.
The input to the system H of s
is the difference between the
input to the overall system
and the output of
the feedback loop.
And I draw your attention
to the fact that what we
illustrate here and what we're
analyzing is negative feedback.
Namely, this output is
subtracted from the input.
And that's done more for reasons
of convention then for
any other reasons.
It's typical to do that and
appropriate certainly in some
feedback systems, but not all.
And the output of the adder is
commonly referred to as the
error signal, indicating that
it's the difference between
the signal fed back and the
input to the overall system.
Now, if we want to analyze the
feedback system, we would do
that essentially by writing
the appropriate equations.
In generating the equivalent
system function for the
overall system, it's best done
in the frequency or Laplace
transform domain rather than
in the time domain.
And let me just indicate
what the steps
are that are involved.
And there are a few steps of
algebra that I'll leave in
your hands.
But basically, if we look at
this feedback system, we can
label, of course-- since the
output is y of t, we can label
the Laplace transform of
the output as Y of s.
And we also have Y of
s as the input here.
Because this is the system
function, the Laplace
transform of r of t is simply
the Laplace transform of this
input, which is Y of
s times G of s.
So here we have Y of
s times G of s.
At the adder, the input
here is x of s.
And so the Laplace transform of
the error signal is simply
x of s minus r of s, which
is Y of s G of s.
So this is minus
Y of s G of s.
That's the Laplace transform
of the error signal.
The Laplace transform of the
output of this system is
simply this expression
times H of s.
So that's what we have here.
But what we have here we
already called Y of s.
So in fact, we can simply say
that these two expressions
have to be equal.
And so we've essentially done
the analysis, saying that
those two expressions
are equal.
Let's solve for Y of s over x
of s, which is the overall
system function.
And if we do that, what we end
up with for the overall system
function is the algebraic
expression
that I indicate here.
It's H of s divided by
1 plus G of s H of s.
Said another way, it's the
system function in the open
loop forward path divided by 1
plus, what's referred to as
the loop gain, G of
s times H of s.
Let's just look back up
at the block diagram.
G of s times H of s is simply
the gain around the entire
loop from this point around
to this point.
So the overall system function
is the gain in the forward
path divided by 1 plus the loop
gain, which is H of s
times G of s.
Now, none of the equations
that we wrote had relied
specifically on this being
continuous-time.
We just did some algebra and
we used the system function
property of the systems.
And so, pretty obviously, the
same kind of algebraic
procedure would work
in discrete-time.
And so, in fact, if we carried
out a discrete-time analysis
rather than a continuous-time
analysis, we would simply end
up with exactly the same system
and exactly the same
equation for the overall
system function.
The only difference being that
here things are a function of
z, whereas if I just
flip back the other
overlay, we simply have--
previously everything is
function of t and in the
frequency domain s.
In the discrete-time case, we've
simply replaced in the
time domain, the independent
variable by n.
And in the frequency domain, the
independent variable by z.
So what we see is that we have a
basic feedback equation, and
that feedback equation is
exactly the same for
continuous-time and
discrete-time.
Although we have to be careful
about what implications we
draw, depending on whether
we're talking about
continuous-time or
discrete-time.
Now, to illustrate the
importance of feedback, let's
look at a number of common
applications.
And also, as we talk about these
applications, what will
see is that while these
applications and context in
which feedback is used are
extremely useful and powerful,
they fall out in an almost
straightforward way from this
very simple feedback equation
that we've just derrived.
Well, the examples that I want
to just talk about are, first
of all, the use of feedback
in amplifier design.
And we're not going to design
amplifiers in detail, but what
I'd like to illustrate is the
basic principle behind why
feedback is useful in designing
amplifiers.
In particular, how it plays a
role in compensating for a
non-constant frequency
response.
So that's one context that
we'll talk about.
A second that I'll indicate
is the use of feedback for
implementing inverse systems.
And the third, which we
indicated in the case of the
inverted pendulum, is an
important context in which
feedback is used is in
stabilizing unstable systems.
And what we want to see is why
or how a feedback system or
the basic feedback equation, in
fact, let's us do each of
these various things.
Well, let's begin with
amplifier design.
And let's suppose that we've
built somehow without
feedback, an amplifier that is
terrific in terms of its gain,
but has the problem that whereas
we might like the
amplifier to have a very flat
frequency response, in fact
the frequency response of this
amplifier is not constant.
And what we'd like to do
is compensate for that.
Well, it turns out,
interestingly, that if we
embed the amplifier in a
feedback loop where in the
feedback path we incorporate an
attenuator, then in fact,
we can compensate for that
non-constant frequency response.
Well, let's see how that
works out from
the feedback equation.
We have the basic feedback
equation that we derived.
And we want to look at frequency
response, so we'll
look specifically at the
Fourier transform.
And of course, the frequency
response of the overall system
is the frequency response of the
Fourier transform of the
output divided by the input.
Using the feedback equation that
we had just arrived, that
has, in the numerator, the
frequency response in the
forward path divided by 1 plus
the loop gain, which is H of j
omega times k.
And this is the key.
Because here, if we choose k
times H of j omega to be very
large, much larger than 1,
then what happens is that
these two cancel out.
H of j omega here and in the
denominator will cancel out as
long as this term dominates.
And in that case, under that
assumption, the overall system
function is approximately 1/k.
Well, if k is constant as a
function of frequency, then we
somehow magically have ended up
with an amplifier that has
a flat frequency response.
Well, it seems like we're
getting something for nothing.
And actually, we're not.
There's a price that
we pay for that.
Because notice the fact that
in order to get gain out of
the overall system, k
must be less than 1.
So this has to correspond
to attenuator.
And we also require that k,
which is less than 1, times
the gain of the original
amplifier, that that product
be greater than 1.
And the implication of this,
without tracking it in detail
right now, the implication in
this is that whereas we
flatten the frequency response,
we have in fact paid
a price for that.
The price that we've paid is
that the gain is somewhat
reduced from the gain that we
had before the feedback.
Because k times h must be much
larger than 1, but the gain is
proportional to 1/k.
Now, one last point to
make related to that.
One could ask, well, why is it
any easier to make k flat with
frequency than to build an
amplifier with a flat
frequency response?
The reason is that the gain
in the feedback path is an
attenuator, not an amplifier.
And generally, attenuation with
a flat frequency response
is much easier to get
than gain is.
For example, a resistor, which
attenuates, would generally
have a flatter frequency
response than a
very high-gain amplifier.
So that's one common example
of feedback.
And feedback, in fact, is very
often used in high-quality
amplifier systems.
Another very common example in
which feedback is used is in
implementing inverse systems.
Now, what I mean by that is,
suppose that we have a system,
which I indicate
here, P of s--
input and output.
And what we would like to do is
implement a system which is
the inverse of this system.
Namely, has a Laplace transform
or system function
which is 1 over P of s.
For example, we may have
measured a particular system
and what we would like
to design is a
compensator for it.
And the question is, by putting
this in a feedback
loop, can we, in fact,
implement the
inverse of this system?
The answer to that is yes.
And the feedback system,
in that case, is
as I indicate here.
So here what we choose to do
is to put the system whose
inverse we're trying to generate
in the feedback loop.
And in this case, a high-gain
in the forward path.
Now for this situation, k
is, again, a constant.
But in fact, it's a high-gain
constant.
And now if we look at the
feedback equation, then what
we see is an equation
of this form.
And notice that if k times P
of s is large compared with
one, then this term dominates.
The gain in the forward
path cancels out.
And what we're left with is a
system function, which is just
1 over P of s.
And a system of this
type is used in a
whole variety of contexts.
One very common one is in
building what are called
logarithmic devices or
logarithmic amplifiers.
Ones in which the input-output
characteristic is logarithmic.
It's common to do that with a
diode that has an exponential
characteristic.
And using that with feedback--
as feedback around a high-gain
operational amplifier.
And by the way, the logarithmic
amplifier is nonlinear.
What I've said here
is linear, or the
analysis here was linear.
But that example, in fact,
suggests something which is
true, which is that same basic
idea, in fact, can be used
often in the context of
nonlinear feedback and
nonlinear feedback systems.
Well, as a final example, what
I'd like to analyze is the
context in which we
would consider
stabilizing unstable systems.
And I had indicated that one
context in which that arises
and which we will be analyzing
in the next lecture is the
inverted pendulum.
And in that situation, or in
a situation where we're
attempting to stabilize an
unstable system, we have now
in the forward path a system
which is unstable.
And in the feedback path,
we've put an appropriate
system so that the overall
system, in fact, is stable.
Now, how can stability arise
out of having an initially
unstable system?
Well, again, if we look at the
basic feedback equation, the
overall system function is the
system function for the
forward path divided by 1
plus the loop gain, G of
s times H of s.
And for stability what we want
to examine are the roots of 1
plus G of s times H of s.
And in particular, the poles are
the zeroes of that factor.
And as long as we choose
G of s, so that the
poles of this term--
I'm sorry, so that the zeroes
of this term are in the left
half of the s-plane,
then what we'll
end up with is stability.
So stability is dependent not
just on h of s for the
closed-loop system, but on 1
plus G of s times H of s.
And this kind of notion is used
in lots of situations.
I indicated the inverted
pendulum.
Another very common example is
in some very high-performance
aircraft where the basic
aircraft system
is an unstable system.
But in fact, it's stabilized by
putting the right kind of
feedback dynamics around it.
And those feedback dynamics
might, in fact, involve the
pilot as well.
Now, for the system that we
just talked about, the
stability was described
in terms of a
continuous-time system.
And the stability condition
that we end up with, of
course, relates to the zeroes
of this denominator term.
And we require for stability
that the real parts of the
associated roots be in the
left half of the s-plane.
Exactly the same kind of
analysis, in terms of
stability, applies
in discrete-time.
That is, in discrete-time, as
we saw previously, the basic
discrete-time feedback system
is exactly the same, except
that the independent variable
is now an integer variable
rather than a continuous
variable.
The feedback equation
is exactly the same.
So to analyze stability of the
feedback system, we would want
to look at the zeroes of 1
plus G of z times H of z.
So again, it's those zeroes
that affect stability.
And the principal difference
between the continuous-time
and discrete-time cases is the
fact that the stability
condition in discrete-time is
different than it is in
continuous-time.
Namely, in continuous-time, we
care for stability about poles
of the overall system being in
the left half of the s-plane
or the right half
of the s-plane.
In discrete-time, what we care
about is whether the poles are
inside or outside
the unit circle.
So in the discrete-time case,
what we would impose for
stability is that the zeroes
have a magnitude which
is less than 1.
So the basic analysis is the
same, but the details of the
stability condition, of
course, are different.
Now, what I've just indicated
is that feedback can be used
to stabilize an unstable
system.
And as you can imagine there's
the other side of the coin.
Namely, if you start with a
stable system and put feedback
around it, if you're not careful
what can happen, in
fact, is that you can
destabilize the system.
So there's always the potential
hazard, unless it's
something you want to have
happen, that feedback around
what used to be a stable system
now generates a system
which is unstable.
And there are lots of
examples of that.
One very common example
is in audio systems.
And this is probably an example
that you're somewhat
familiar with.
Basically, an audio system, if
you have the speaker and the
microphone in any kind of
proximity to each other is, in
fact, a feedback system.
Well, first of all, the audio
input to the microphone
consists of the external audio
inputs, and the external audio
inputs might, for example,
be my voice.
It might be the room noise.
And in fact, as we'll illustrate
shortly if I'm not
careful, might in fact be the
output from a speaker, which
represents feedback.
That audio, of course, after
appropriate amplification
drives a speaker.
And if, in fact, the speaker
is, let's say has any
proximity to the microphone,
then there can be a certain
amount of the output of the
speaker that feeds back around
and is fed back into
the microphone.
Now, the system function
associated with the feedback I
indicate here as a constant
times e to the minus s times
capital T. The e to the minus
s times capital T represents
the fact that there is, in
general, some delay between
the time delay between the
speaker output and the input
that it generates to
the microphone.
The reason for that delay of
course, being that there may
be some distance between the
speaker and the microphone.
And then the constant K2 that
I have in the feedback path
represents the fact that between
the speaker and the
microphone, there may
be some attenuation.
So if I have, for example, a
speaker as I happen to have
here, and I were to have that
speaker putting out what in
fact I'm putting into the
microphone, or the output of
the microphone, then what we
have is a feedback path.
And the feedback path is from
the microphone, through the
speaker, out of the speaker,
back into the microphone.
And the feedback path is from
here to the microphone.
And the characteristics or
frequency response or system
function is associated with
the characteristics of
propagation or transmission.
If I were to move closer to the
speaker and I, by the way,
don't have the speaker
on right now.
And I'm sure you all
understand why.
If I move closer, then the
constant K2 gets what?
Gets larger.
And if I move further away the
constant K2 gets smaller.
Well, let's look at an analysis
of this and see what
it is, or why it is, that
in fact we get an
instability in terms--
or that an instability is
predicted by the basic
feedback equation.
Now, notice first of all, that
we're talking about positive
feedback here.
And just simply substituting
the appropriate system
functions into our basic
feedback equation, we have an
equation that says that the
overall system function is
given by the forward gain,
which is the gain of the
amplifier between the microphone
and the speaker,
divided by 1 minus--
and the minus because we have
positive feedback--
the overall loop gain, which
is K1, K2, e to the minus s
capital T. And these two gains,
K1 and K2 are assumed
to be positive, and generally
are positive.
So in order for us to--
well, if we want to look at the
poles of the system, then
we want to look at the zeroes
of this denominator.
And the zeroes of this
denominator occur at values of
s such that e to the minus s
capital T is equal to 1 over
K1 times K2.
And equivalently that says that
the poles of the closed
loop system occur at 1 over
capital T, and capital T is
related to the time delay.
1 over capital T times the log
to the base e of K1 times K2.
Well, for stability we want
these poles to all be in the
left half of the s-plane.
And what that means then is
that for stability what we
require is that K1 times
K2 be less than 1.
In other words, we require that
the overall loop gain be
less-- the magnitude of the
loop gain be less than 1.
If it's not, then what we
generate is an instability.
And just to illustrate that,
let's turn the speaker on.
And what we'll demonstrate
is feedback.
Right now the system
is stable.
And I'm being careful to keep my
distance from the speaker.
As I get closer, K2
will increase.
And as K2 increases, eventually
the poles will move
into the right half of the
s-plane, or they'll try to.
What will happen is that the
system will start to oscillate
and go into nonlinear
distortion.
So as I get closer, you can hear
that we get feedback, we
get oscillation.
And I guess neither you nor I
can take too much of that.
But you can see that
what's happening--
if we can just turn the
speaker off now.
You can see that what's
happening is that as K2
increases, the poles are moving
on to the j omega axis,
the system starts
to oscillate.
They won't actually move into
the right half plane because
there are nonlinearities that
inherently control the system.
OK, so what we've seen in
today's lecture is the basic
analysis equation and a few
of the applications.
And one application, or one both
application and hazard
that we've talked about, is the
application in which we
may stabilize unstable
systems.
Or if we're not careful,
destabilize stable systems.
As I've indicated at several
times during the lecture, one
common example of an unstable
system which feedback can be
used to stabilize is the
inverted pendulum, which I've
referred to several times.
And in the next lecture, what
I'd like to do is focus in on
a more detailed analysis
of this.
And what we'll see, in fact, is
that the feedback dynamics,
the form of the feedback
dynamics are important with
regard to whether you can and
can't stabilize the system.
Interestingly enough, for this
particular system, as we'll
see in the next lecture, if you
simply try to measure the
angle and feed that back that,
in fact, you can't stabilize
the system.
What it requires is not only
the angle, but some
information about the rate
of change of angle.
But we'll see that in much more
detail in the next lecture.
Thank you.
