INSTRUCTOR: In this video,
we'll use the final value theorem for analysis
of our Laplace transform functions.
In particular, we will use our Laplace transform function
to calculate the final value of our variable.
So the final value theorem says
that we can calculate the limit of Y of T,
as T goes to infinity,
as the limit,
as S goes to zero of S, Y of S.
Laplace transform version.
So this means we don't always have to do
the inverse Laplace transform in order
to learn anything about our system.
One limitation of this method is that
it only applies for bounded Y.
And this could be a problem because
we don't always know ahead of time
if our variable, Y, is going to be bounded.
Lets apply this method to the blending process,
which is shown in the figure.
We have two streams that flow into a well-mixed tank,
with one outlet stream leaving
and we're most interested in controlling
the mass fraction, X, leaving the tank.
So the Laplace transform version for X
and deviation variables is 7.5,
times 10 to the minus six,
over S plus four times 10 to the minus four.
And then we multiply that by W one prime of S,
when we have a change in that disturbance variable.
So lets consider the same step input,
W one prime of S that we have considered
in previous examples.
Step input of height point one,
and then the Laplace transform of the step
is one over S.
We can go ahead and calculate the limit of X prime of T,
T goes to infinity as the limit as S goes to zero of S,
X prime of S.
Let's go ahead and just plug in,
we can go ahead and simplify this expression
by canceling out these S terms,
and get seven point five times 10 to the minus seven
over four times 10 to the minus four, SS goes to zero.
And we can also rewrite this as 1.875
times 10 to the minus three.
And this is the same result that we saw in earlier videos,
by actually calculating the time dependence of S prime.
Now we can apply a similar analysis
to the volume in the tank.
The problem is that we know, from previous videos,
that the volume in the tank is not gonna be bounded
under the step input.
If we increase the inflow
but we don't change the outlet flow,
then we will have a linear increase
in the volume in the tank.
So the assumptions of the final value theorem
do not actually apply.
In the blending process, we want to control X,
which is the mass fraction of species A,
leaving the tank.
We also have dynamics for the volume.
So we have previously computed that as
10 to the minus three over S,
times W one prime of S,
where W one prime of S was a step input
of height point one.
And so we could go ahead and calculate that term
SV prime of S,
is 10 to the minus three over S
times 0.1 over S,
times that S,
and so that gives us 10 to the minus four, over S.
Now, the limit of that term, as S goes to zero,
or into the final value theorem would be infinity.
Now, it turns out, that in fact,
the volume in the tank is unbounded.
It grows linearly as we increase the flow rate in
but don't decrease the flow rate out.
But really the final value theorem
does not apply in this case.
So really any result that we get here is really unreliable.
The value at long time is not bounded.
But what we can do is to define the rate,
as a new quantity,
so RT is the rate of change of volume.
DV prime DT, and that will be bounded
under a step input in W one.
So we can take the Laplace transform
to get R of S equals SV prime of S,
and this V prime of zero,
which we've been assuming is zero here.
Now we can apply the final value theorem to the rate, R.
So limit, this T goes to infinity,
R of T is equal to the limit
as S goes to zero of S, R of S.
So that is equal to the limit as S goes to zero,
S times R of S which is S times V prime of S,
10 to the minus four, over S squared,
and that's equal to 10 to the minus four,
meters cubed, per second.
So this now is a bounded value and it tells us
that the longtime value is 10 to the minus four.
Now we really knew that based on our inlet flow rate
but we are able to define this new variable,
R of S, for which we can apply the final value theorem.
And that will be helpful later,
when we look at multi-variable control.