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Good afternoon, so having seen that, so now
we have to take care of these and design a
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controller appropriately. Now let us look
at performance, so what is the first requirement
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of performance.
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The first requirement of performance is that,
performance can be divided into two types
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in control, one is called steady-state performance
and other is called transient performance,
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in general steady-state performance is much
more important than transient performance,
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simply because of the fact that, that performance
holds over a much longer interval, generally
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in industrial automation the set point change
is somewhat infrequently.
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For example if you take a power station boiler,
then it set point over a day will typically
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be changed 7, 8, 9 times maybe less. So when
you will have load coming in the morning it
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set point will be changed, when lighting load
in the evening will start going down, after
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let us say 10 o'clock or 11o'clock load will
fall at that time you have to reduce the set
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point. So there are infrequent set point changes.
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And in between these set points are generally
maintained, this happens for a lot of process
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equipment. So if you have performance regulations
which are persisting during that phase, when
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this is, when the set point is held then,
that is generally considered much more serious
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than errors that can occur when the set point
is changing or immediately after that time
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for a short duration. So we would first like
to ensure steady state performance.
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And the major consideration for that is steady
state error that is we want that r should
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be equal to Y, at least in the steady state
obviously if r suddenly changes y cannot suddenly
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change. So y will have to, y will take some
time to come to the level of r but once it
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comes we want that this error will be 0, we
want zero steady state error, this is our
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wish. So, how do we obtain that right, so
for that, so we want, that this is the steady
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state error that is limit of T can be typically
we take a unit step response.
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If the reference input suddenly changes then
how is, as time passes does the error go to
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0 that is limit of T can be infinity e(t).
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We can also express it in frequency domain
form which says that, which is the final value
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theorem of Laplace transforms and which comes
down to the fact that e steady state is limit
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of s tending to zero, 1 divided by G(S) K(S)
right. So this is the steady state error and
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we have to ensure one of the prime requirements
is to ensure by control that this goes to
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zero. So how do we do that, so we have to
control for zero state, steady state, let
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us take simplest case of proportional control.
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The problem of proportional control is that,
if you want to proportional control is just
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you have a simple gain and if you get an error
of 1 volt you generate a maybe an output of
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100 volts if you get a 2 volts you generate
output of 200. So just simple multiplication.
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Now obviously we want to, what we want we
want that this r be equal to y, so we want
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to, r is a certain values. So we want to maintain
a certain value of y, now naturally in, it
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happens, so happens that for maintaining a
particular value of y, we need a particular
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value of u, so for the time being assume that,
this is not there. So now how are we going
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to get this u, if you want to maintain this
u.
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Then we have to maintain a particular value
of e, so unless we have a certain amount of
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steady state error, we cannot generate U and
therefore which you cannot generate y so we
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cannot make steady state error 0 ever, using
proportional control, that is what it turns
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out to be. So what happens is that the, there
are two things that could happen.
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Either you could artificially increase this
r, that is, if you want let us say an r of
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really one that the that the output stays
at 1 volt you give an r of maybe 1.1 volt,
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just what volt you have to give that if you
calculate but so you artificially increase
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1.1volts so that here you get 1 volt which
is a real output you want, right. Or what
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you could do is apart from what the controller
is doing you can give a you can give what
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is known as this manual bias that is you apply
some additional input, right directly to the
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plant.
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So you do one of these two things, in that
case you can maintain whatever output you
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want but what is the problem, the main problem
is that who is going to give this input, how
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do we know, by how much for example if we
give 1.1 volt for 1 volt how do we know let
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us say if you want an output of 3 volts what
output we have to give, by how much, so naturally
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it should not require a manual, you know,
manipulation. It should be done automatically
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right.
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So that brings us to the question that how
can we automatically generate this bias input
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without any manual intervention.
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So the question is that how do we, how to
create bias in put for 0 error, right. So
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that is the situation described here what
should be this.
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So we see, we see that what is the device
which for 0 error gives and gives an output,
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so that device is actually an integrator,
so somewhere in the loop there should be an
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integrator then even if the error is 0, we
will be able to give a finite output.
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So there are actually two cases in which an
integrator is required, so the first case
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is this one, so if you have an integrator,
imagine that, we have written Ki / S in Laplace
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domain notation which means that this is actually
an integrator that is U is equal to integral
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Ki e(t) d(t), so this is the integrator, so
this is equal to u. So now you see that even
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if, if after sometime this, at this point
if you even if we get e.
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e goes to 0, so for example suppose the error
goes to 0 this is, this is y, this level is
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r, this is the level so here error is going
to 0, but what will be the value of the integral,
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what will be the output of this block, the
output of this block is going to be the area
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under the curve because this is the error,
this is y and this is r, so r – y is this
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vertical distance, so this is the area, right.
So this area, even if the error remains 0,
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this area remains finite.
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So you can generate a finite u even when the
error is 0, you can keep generating, so having
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an integrator helps.
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There is another case in which you can have
an, you can have an integrator that is the
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integrator is actually part of the plant itself
which means that to be able to sustain an
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output, the plant may not need an input all
the time, so the plant itself is an integrator,
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what is the typical example of this, a typical
example of this is a tank, so suppose you
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have a system, you know, like we have in our
toilets.
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So you can have a flow in and the, this flow
in is actually this flow is actually proportional
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to the level, actually there is a, if you
might have noticed if you have looked into
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the system, that there is a, there is a ball
cock, floating ball, so when the, when there
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is no water then the ball cock is hanging
like this and water flowing, so as water rises
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so the ball cock goes up, up, up and at a
certain level the valve through which the
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water is flowing will actually close.
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So at that level it will be maintained, so
you see now so this tank is a plant which
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this tank is a plant,
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which, which has an integrator with respect
to flow because the level is nothing but an
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integral of flow, so at some point to be able
to maintain a level, this is a level it does
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not need any flow so the flow can go to zero,
so still, so we are having a simple proportional
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controller so the error going to zero flow
goes to zero but still level is maintained,
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so this is a another situation where you can
say have 0 steady state error.
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Now, so exactly this is what happens, so now
you see, if you, if you put the integrator
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what happens, is very interesting, so what
happens is that, this bias input which was
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previously coming now comes automatically,
that is the integral output which I was talking
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about now increases, increases, increases,
still it can, then it goes to zero and then
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if you, if you change the set point again
from here to here again some error will be
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created.
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And again integrate, integrate, integrate,
and it will generate it just enough output
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such that it can be, it can be sustained without
the error, so you see that the integrator
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is actually, the integrator actually works
as.
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The integrator actually works as a very interesting
thing it actually gives a bias input but which
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is not manual, so the bias input, integrator
is actually, exactly like the, like the, like
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the bias input but it generates, it automatically,
you do not have to give it manual, you do
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not have to give it manually, and it will
adjust itself depending on, if you, depending
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on the set point, so the integrator will automatically
build up and give enough additional input
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such that at zero error it can be sustained.
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So that is the principle by which 0 steady
state error is obtained.
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Now, only thing is that, now this has certain,
there are certain drawbacks too, for example
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let us see that if you give a step response,
if we give a step response then how does it,
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how does it work how does the loop work, so
you see that, suppose the process starts from
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here, okay. So the process starts from here,
it starts, so as you have given, so here there
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is a lot of error so the proportional controller
now generates a positive input,
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which drives the plant, so the plant goes
up and typical, we are likely to get a step
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response like this, so what happens during
this phase, during this phase, during this
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phase you have proportional is positive, error
is positive, so output is positive but since
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the error is decreasing, so the output is
going down, so it is positive but going down
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that is the output of the proportional controller,
what does the integral controller do?
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Integral controller is also positive because
it is integrating positive error and it is
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increasing because the, because the area,
as it is going with time this area is continuously
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increasing, so it is positive and going up,
increasing, right. So what happens at this
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point, at this point error is zero, so the
proportional controller output is zero but
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because the integral controller output is
still positive, so the plant continues on
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this journey, in this path, right.
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Now at when it is here let us say, when it
is here the proportional controller around
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this point P is negative, P is negative but
I is still positive because of the fact that
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there is already a large positive integral
accumulated here, so here integral is negative,
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this part of the integral is negative but
still there is a large positive integral,
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so it so that overall net output is, maybe
still positive, so it continues on this journey.
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But eventually the, this integral value also
reduces and the proportional controller value
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also becomes enough negative, so the overall
input turns negative and the plant turns to
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move, now again the same thing happens here,
once it crosses this line, now that, now,
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again it will, it will oscillate, so you see
that typically because of integral control
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there tends to be an oscillation, so that
tends to be a high overshoot
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and an oscillation, so this is a drawback
of integral control, that is to gain steady
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state error, to gain, to gain zero steady
state error, this is the price that you are
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paying that in your transient response you
are likely to get some
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Overshoot so.
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That is the picture for the step response.
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So we want to improve transient response without
sacrificing the steady state error, so if
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you want to do that then what we have to do
is that, around this point only here, we have
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to, we have to, we have to keep breaking,
you know, we have to keep breaking and around
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this point, so that we can quickly turn, so
what happens is that here now we have to you
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have to slow down, so slowing down means during
this phase we have to create more negative
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input.
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Which will grow towards this point, as it
comes closer it should this negative input
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or increase similarly around this point this
negative input should also, this negative
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input should also keep increasing, so that
it quickly turns and then actually it will
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settle very fast, so you see, it will not
oscillate, if you, we want that it does not
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oscillate, so, many times but rather follows
this yellow curve maybe does a small overshoot
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and then immediately settles down.
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This is the kind of curve that we want, so
now it turns out that this kind of curve we
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can obtain if we add.
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A derivative term to error right, so we want
to reduce rise time what is rise time, we
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want to reduce rise time, rise time, we want
to reduce this time, we want to typically
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speaking, we want to reduce overshoot, that
is this height and we want to also reduce
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settling time, that is total time, but taken
for it to come to a steady state, so we want
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to reduce that all these now getting all this
is somewhat difficult and that is why you
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need to have a you need to have a non-trivial
tuning exercise. To, you know, come to a compromise
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between these we could.
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Typically what we do is, it turns out that
these things can be achieved, we have already
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discussed proportional and integral controller
and we have also seen that adding a derivative
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will you know try to break, so that it does
not go towards much overshoot.
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So we need a we need a proportional controller
because if you do not have a proportional
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controller then there will be tend to be too
much too much of oscillation we need an integral
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00:17:35,669 --> 00:17:39,730
controller to have zero steady state error
and we need to have a derivative controller
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to have low overshoot and fast settling time
and we need to tune this gains Kp, Ki and
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Kd nicely so that we get a good transient
response without sacrificing on the zero steady
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state error.
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One thing interesting to see is that, this,
so we now calculate input like this, now interestingly
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that you see that in the steady state, in
the steady state the total of e is zero, so
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therefore this term is zero, so the proportional
controller is zero and since he is not also
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changing, so there is a de/dt is also zero,
so that is also gone so we only have the whole
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output coming from the integral part. So the
zero steady state error concept that you have
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studied for the integral control holds.
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Now, so as I said that we need, we are, we
are looking for a step response like this,
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this would be a very good step response and
we if you can make it even sharper even better
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but generally if we want to make it sharper
then, if we can make it like this even better,
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so this is a good step response that we would
like to achieve.
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So, now come to the point of disturbance rejection,
so what is the disturbance response, the disturbance,
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so let us talk about there are there are there
are several types of disturbances and you
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can do the same kind of analysis but the most
predominant disturbance which occurs generally
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is the load disturbance in a process, occur
due to various reasons property variations
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of materials, variations in power sources,
voltages, variations in pressure sources all
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sorts of things.
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So, it turns out that the transfer function
we want to reduce the effect of the, we want
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00:19:45,789 --> 00:19:52,280
to reduce the effect of d0 on y, so what is
the transfer function between y and d0 that
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turns out to be this, so now again we see
that if you have so, you know, it is not possible
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to exactly neutralize all kinds of disturbances
but let us say one of the major kinds of disturbance
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is, step disturbance again, that disturbances
will change once or twice and then stay on
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okay.
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So if you have step disturbances then you
can again see that.
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If you have an, if you have an integral then
this term will actually go very high as s
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tends to zero, so the effect of, so for the
same reason exactly similar transfer function
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is coming, so the same reason why e goes to
zero with integral control, if you put integral
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control even the effect of step disturbances
will also go away because the integral value
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will rise and it will provide the additional
torque to actually take care of the disturbance.
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So it will produce an integral will rise and
it will, here, it will, instead of producing
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y it will automatically produce an output
y plus d0 or rather y minus d0, so that after
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plus d0 you will get the desired value of
y, so this is going to happen, so integral
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control is one of the major ways of reducing
disturbance response.
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So, this is, this is, now let us look at some
other issues for example, this, we have to,
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we have to actually remember these things,
they are very practical issues and we possibly
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did not learn about these in our, in our earlier
control course, where we treated things rather
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ideally, but we must remember here that some
other non idealness exits.
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For example, there can be very often, there
can be actuated saturation, that is the characteristic
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of the, this is the control input CI and this
is the plant input PI, so this is the actuator
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characteristic, so as you increase the control
input the actuator will also proportionally
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include the plant input but only up to a certain
point, after which it will saturate, so if
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you give more and more control input it will
not give you proportionally high plant input.
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So, that should as typically will saturated,
now when the actuator saturates effectively
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the feedback loop is opened because the effect
of the error no longer transmits to the output
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or rather the plant input so the input does
not change, in response to the error but is
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held constant that is the case of an open
loop operation, so your control is gone.
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And not only that, these sometimes as, we
shall see later that the, that says persistently
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actuator saturation, persistent actuator saturation
has very bad effects on controllers especially
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with, because of the fact that controllers
have memory, so this is a particular phenomenon
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which occurs in PID controllers and we are
going to take a look at it in great detail
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in the next lecture.
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So next issue that we should look at is sensor
bias, so you know remember that if you have
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a sensor bias, sensor has errors then sensors
are the eyes of the controller, so whatever
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the sensor sees that the controller simply
works on that, so if you have bias you will
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think that is the controller will produce
an input.
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Which will have zero error but actually there
will be non zero error, so that is, so you
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have zero error but you have non zero error,
similarly we have to remember that, that you
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always have, actually you design controllers
based on some models but it always turns out
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that these models are actually inaccurate
so you are always going to have model errors
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and this model errors are typically dominant
in the high frequency band.
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And now remember that if you have inaccuracy
in the high frequency band it is the error
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in the high, actually typically instability
occurs in the high frequency band not in the
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DC band, low frequency band, so if you have
modeling error in the high frequency band
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then such modeling inaccuracies can also lead
to stability problems, so we have to remember
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these things.
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So to take care of these things, various kinds
of other architectures are possible then what
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we have that is the loop structure, what feedback
will use how we will use the controllers and
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we will see some of them for example feed-forward
configuration, cascade configuration, we will
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see all of them, so they are possible.
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Finally while we are going towards the conclusion,
let me mention some of my pet facts about
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control in colloquial terms, so I say that
what you feedback is what you control, what
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you, so the controller exactly tries to maintain
the feedback so if you feedback s erroneous
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then your control is erroneous, if you are
sitting in the middle of the room and if you
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have put your temperature sensor, at the,
at the roof of the room then you are controlling
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the temperature at the roof of the room not
in the of the room right .
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Then, when you, what you cannot actuate, you
cannot control, so you may be giving whatever
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in output from the controller you may be using
a fancy algorithm but if you cannot actuate
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it then you are not controlling it well.
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If you can, if you can measure or estimate
the disturbances then you can compensate them,
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so one of the, I mean, there are many advanced
algorithm which, which precisely try to do
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that. Stability is basically, stability is
not enough, stability is barely basic performance,
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you must ensure stability and then ensure
performance but while you ensure performance
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if you, as you try to go more and more, drive
more and more and improve performance eventually
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instability results.
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So instability is actually comes, it actually
decides the, generally decides the maximum
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performance that you can achieve.
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And models are always approximate, most systems
are actually nonlinear but that does not mean
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that we can work with approximate.
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So linear control can work very well for non
linear plans but sometimes nonlinear control
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may be working better but 95% of industrial
controllers are linear.
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So we have come to the end of the lesson and
let us review quickly, so we looked at the
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objectives of automatic control, maintain
stability, follow set point, reject, reject
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disturbance, we looked at stability and found
the causes of stability in because of instability
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in the process loop, we looked at steady state
error and the ways of reducing it and we also
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looked at the ways of reducing transient performance
keeping the steady state error at zero.
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So we saw that the PID control is a very effective
ways and simple and effective way of doing
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that and eventually we say, we saw that PID
control can also do some amount of very common
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disturbance rejections.
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So that is the end of the lesson, let us here
are some points for you to ponder, first is
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that state the three major objectives of automatic
control which I have just now said, state
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three major cause of instability in a control
loop and give, this is tricky try it given
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example for each case practical example and
is it possible for a proportional controller
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to achieve zero steady-state error I have
already explained an example.
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You try to explain it in your own language
and explain how a PID controller can achieve
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good transient performance as well as zero
steady state response and finally justify
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or contradict the statement that PID control
achieves zero steady-state error with step
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set points and disturbances, thank you very
much, we will see in the next lecture.