Alright folks, welcome to Simulink for ChemE's, where we go through all of the ways that
Simulink can ease your process control pain. Today we're dealing with systems of
higher order transfer functions, but we're also going to make first-order plus dead time
approximations to make the math easier. Higher order systems can be approximated with first-order
plus dead time functions, or FOPDTs. These can be expressed in the general form K times
e to the negative theta s over tau s plus one. Notice that it looks a lot like a regular
first-order transfer function, but it has this extra exponential term added. This is
the dead time. And there are several different methods you can use, such as the Taylor or
the Skogestad methods to develop these FOPDTs. Now our goal for today is to use the Taylor
and the Skogestad methods to approximate this given series of transfer functions. How do
the two approximations differ from one another and from the original model? Now Simulink
is a great tool for answering graphical, qualitative questions like this one. So we're already
presented with the system on the screen. It consists of a step input, three first-order
transfer functions linked in series, a mux, and a scope. Placing Simulink blocks in series
like this is equivalent to multiplying the blocks together. If we wanted to we could
write these three first-order transfer functions as one third-order transfer function. So we're
going to develop these first-order plus dead time models, and then compare them to the
system, as well as to one another. And, there is no magical FOPDT block that we can drag
and drop in Simulink, but we can make a FOPDT by multiplying two different blocks together.
First we put down a transfer function block, then we multiply that signal by a transport
delay block, which gives us our dead time. Alright, so let's hook up the step input to
the transfer function, and attach the transport delay to the mux. This will enable us to compare
the real transfer function to the approximation. We're also going to make a second FOPDT. So
we can highlight the blocks by shift clicking them, press control C to copy, and then press
control V to paste. We'll need an extra port on the mux block for this new input, and then
we have to connect it to the step. So let's make this top FOPDT a Taylor approximation,
the bottom one a Skogestad approximation. Let's do the Taylor one first. For a Taylor
FOPDT the gain of the process is still the gain of the FOPDT, therefore K equals six.
Tau, the time constant, is equal to the largest time constant in the process. So in the process
we have a time constant of two, of four, and a six, so the tau for the Taylor approximation
is six. The dead time is equal to the other time constants in the denominator that aren't
tau, plus any dead time that already exists in the system. So we already used the six
as our tau, but we still have time constants of two and four, and there aren't any transport
delay blocks in the original process to signify dead time, so the dead time for our Taylor
FOPDT is six, it's these two extra transfer functions here. Therefore, our Taylor approximation
is six times e to the negative six s over six s plus one. What this means for the Simulink
model is that the transfer function will equal six divided by six s plus one. The e to the
negative six s term goes in the transport delay as a time lag of six minutes. The FOPDT
using Skogestad's half rule will be a little bit different. The process gain is again the
gain of the FOPDT, K equals six. Now for a Skogestad approximation tau equals the largest
time constant plus half of the second largest time constant. So this would be six plus one
half of four, which is two, so our tau is eight, because six plus two is eight. The
dead time is equal to the other half of the second largest time constant plus all the
others. So we have the half of the four, which is two, plus this two, which equals four,
so our dead time is four. Therefore, our Skogestad approximation is six times e to the negative
four s over eight s plus one. So in our transfer function block we put in six over eight s
plus one, and then our transport delay is four minutes. We'll save our progress just
in case. Set our run time to fifty minutes. Save again, and press run, and click on scope.
So what do we see? The original third-order transfer function is in yellow. You can tell
this because it doesn't have any dead time, it just starts increasing at time zero. The
magenta curve is from the Taylor FOPDT, it has the longest dead time. It underestimates
the values for the original system until about nine minutes in, then it overestimates the
values. The cyan line comes from the FOPDT derived from Skogestad's half rule. It tends
to stay closer to the yellow line, and because they intersect twice, the Skogestad FOPDT
serves as a better approximation of the original system, both at earlier and at later times.
Cool. I hope you enjoyed doing some first-order plus dead time approximations this afternoon.
I know I did. If you want to learn more about using Simulink to solve process control problems,
you can subscribe to the Lafayette CHBE YouTube channel.