The following content is provided under a Creative
Commons license.
Your support will help MIT OpenCourseWare continue to
offer high-quality educational resources for free.
To make a donation, or view additional materials from
hundreds of MIT courses, visit MIT OpenCourseWare at
ocw.mit.edu.
[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.