MARTHA: In the last Screencast,
we went through the general concept of ratio control,
which is a type of feed-forward control.
Specifically, we measured the disturbance variable,
W1, and then adjusted W2, the manipulated variable,
in order to hold the two mass flow rates at a fixed ratio.
And based on the steady state values
that we've been using in previous Screencasts,
we decided to keep them at a ratio of three.
But in order to actually calculate
what the response would be,
we need some transfer functions for our general expression.
So previously, we derived a mass balance equation
for the tank for the blending process.
We have mass flow rates of two streams coming in
and one stream leaving.
And in this particular example,
we'll just assume that we're filling the tank,
and we're just trying to keep
the two inlet streams at a fixed ratio.
That doesn't really change the analysis,
but it just gives us one less term
to consider in our example.
Now, we can take the Laplace transform
of that time domain equation.
And again, we can just assume
that the tank is initially empty,
which won't really change our analysis at all.
So finally, we can get some transfer functions
for this process by dividing three by s.
This tank filling process is an integrated process,
so that's why we have integrators,
or one over s terms here in our transfer functions.
So here is our disturbance transfer function, Gd.
Here is our process transfer function, Gp.
In the last Screencast, we derived a general expression
for how the volume in the tank, V of s,
would change under ratio control.
So we had this general expression,
and one thing to notice here
is that there's only really one input
to this process under feed forward control.
The only input now is W1, because W2 is calculated
based on the measure of W1.
We have Gd and Gp for the process,
but we still need Gf.
Typically, one might use a PI controller
in order to get rid of steady state error,
so that W2 can track its set point.
But here, for simplicity of analysis and derivation,
we'll consider a proportional controller with Kp.
So we can go ahead and plug in our expressions
for the blending process.
And then finally, we would need to pick a value for Kp
for our proportional gain in our control law.
So we could try some different values
and see how the performance works out.
Let's go ahead here and just try a gain of 1000
and see what happens.
So we can pull out the one over s, the integrator term,
from both values, and then we get
ten to the minus three
plus three.
Now our Kp and our ten to the minus three
are gonna cancel there, and then we will just divide that
by 1001, one plus Kp.
And that all multiplies W1 of s.
So this actually gives us a value of 0.00398,
and if you think about what is going on here,
if we have a step change in W1,
then that's going to give us a change in V,
in the rate of change of V.
If we actually were able to keep our one to three ratio,
our flow rate going in would be four times
and not 3.98 times the amount of W1.
So we will have a little bit of steady state error here.
If we increase the gain further,
we could reduce that steady state error even further,
although we might actually be better off
using a PI controller
to be able to reduce our controller gain
and also get rid of any steady state error.