FOPDT Model Tutorial

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FOPDT
Modelling Tutorial

Modelling FOPDT - Excel

FOPDT

If the input function to a FOPDT is a step function, having a step of height=A
occur at time t=t*, is m(t)

then the time response of the FOPDT system is c(t) [Step output function]

where K is the gain, to is the dead-time and t is the first-order time constant,
or characteristic time, of the FOPDT system.
The variables m(t) and c(t) in these equations are deviation variables.
We want to have Excel draw a graph of these equations so we can see the effect of the various parameters.

TUTORIAL

Open Excel by double clicking on the Excel icon.

You'll see a spreadsheet open, named Worksheet 1. We're going to enter in three columns
of numbers to have Excel plot for us. The columns are "Time," "Input" and "Output."
Type each of these names in the top row of the spreadsheet in columns A, B and C, respectively.
Move to the right in the spreadsheet by touching the "tab" key or using the -> key or by clicking where you want to move to.

In cell A2, enter a 0 (zero). In cell A3, enter this:
=0.1+A2
and then touch "return." This has put a formula in cell A3 that adds 0.1 to the value of cell A2.

We're going to put formulas in cells B2 and C2 for the functions m(t) and c(t).
For now, we're going to use values for t*, A, K, to and t.
Let's use 2, 3, 4, 5 and 6, respsectively.

Click on B2 and put in this formula:
=if(A2>2,3,0)
This says, "if the value of cell A2 is greater than 2, then this cell takes the value 3,
else it takes the value 0."

Click on C2 and put in this formula:
=if(A2>2+5,3*4*(1-exp(-(A2-2-5)/6)),0)

Now click on cell A3 and notice in the lower right corner of the cell there is a small square dot.
Move your cursor to that square (your cursor will change shape into a + ), click-and-drag your mouse
down to near the bottom of your screen. Release the mouse and the formula will have been copied
into all the cells you dragged over and the values of time will be in that column.

Now, do a similar thing in column B, dragging down to the same cell you dragged to in column A.

Now, do a similar thing in column C, dragging down to the same cell you dragged to in column A.

Now do a related thing. Click in the last time-value in column A and drag across to column C
and release the mouse. All three cells should be selected. Now click on the square at the right bottom
of that selection and drag it down (off the screen and it will scroll down) to a ways, maybe, say, to row 400.

Now you should have 3 columns with the values in them that we want.
The values in row 400 should be 39.8, 3 and about 11.9.

USING PARAMETERS

Close the graph window and let's go back to the spreadsheet and enter a
table of the parameters that we are interested in. Lets make a table.
[Excel Table]

Now go back and click on cell B2. Click next to the "2" that represents the t*,
delete it and then click on cell E1, the cell that has the value of t* (2) in it. Now, fix that reference
to that cell by pulling down the Formula menu to Reference.
(On Macs the key combination is Command-L; for Excel on Windows it is the F4 key.)

Do this also for the "3" in the formula. Replace with E2 and choose Formula/Reference.

Repeat in the formula in C2. Replacing the 2, 3, 4, 5, and 6;
remember to always choose Formula/Reference.

Now, again highlite cells B2 and C2, then drag them down to row 400, as you did before.

Look at the chart. It should be the same.

Now go back to the parameter cells and change one of them and watch the graph change.

To download this entire Excel SpreadSheet model for step response:

Step Response Model on Excel SpreadSheet

GRAPHING

Scroll back to the top of the spreadsheet. Highlight the three columns we want to plot by
clicking on the A at the top of column A and dragging over to the C above column C.
Now click on the graphing button.

Move the cursor over the spreeadsheet and click-and-drag a rectangle as large as you want your graph.
Then answer questions posed by the ChartWizard. ALWAYS CHOOSE SCATTER PLOT!

When the graph is drawn, double click on it to make it a separate window,
then click on the upper right "grow"-box to enlarge the graph.

**** CONTINUOUS IMPROVEMENT ***

For suggestions and feedback,

contact Jim Henry - Lab-Master Send E-Mail to Jim Henry

Last revised 11 February 1998

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