Scaling Distillation model

Dear Alexander Schwarm,

Atle forwarded your email to me.

> On p.495 the book says that the
>model is scaled to magnitude of 1. But if I give an input of 1 to the
>first two inputs (L and V) I do not get the correct output of 1, even at
>steady state.

Maybe there is a misunderstanding. The "model is scaled" means
that the input and output variables are scaled,
but the model gain is not one. Actually, the steady-state gain is
about 100 between these scaled variables, see (12.17).

Scaling of the linear model is generally recommended before you do
controllability analysis, model reduction, etc. The scaling should
make all inputs/disturbances of about equal importance, and all outputs
of about equal importance. Usually, this is done by dividing each
variable by it maximum change, i.e.

         u = U / Umax ; y = Y / Ymax; d = D / Dmax

where U is the deviation in original units, Umax is the maximum
allowed or expected deviation, and u is the scaled variable. For
more details see p. 5 in Skogestad and Postlethwaite (1997).

For example, for the outputs, Y is in mole fraction units and Ymax = 0.01.
So, a scaled output y= 1 corresponds to Y=0.01 mole fractions units,
and an output y=100 corresponds to 1 mole fraction unit.
We want the output magnitude |y| to be less than 1, i.e. we want
Y to change less than +- 0.01 mole fraction units.

If you increase u1=L by 0.01, then at steady-state, see (12.17),
y1=yD will increase by 0.878 (or 0.00878 mole fraction units).
Note that (12.17) is a linear model which applies to small changes.

I hope this makes things clearer.

-Sigurd Skogestad

>From: Alexander Timothy Schwarm <ats5026@unix.tamu.edu>
>Date: Sat, 26 Apr 1997 16:07:24 -0500 (CDT)
>Subject: Distillation model
>
>To Atle,
>Sorry to bother you again, but....
>I emailed you quite a while ago about looking for a distillation
>model. I got the one from Dr. Skogestad's web site and I am using it as
>a basis for a project in a class using Dr. Skogestad's book. I am
>having trouble with the linear model. On p.495 the book says that the
>model is scaled to magnitude of 1. But if I give an input of 1 to the
>first two inputs (L and V) I do not get the correct output of 1, even at
>steady state. I checked and rechecked my typing of the A, B, C matrices
>and they are correct. I have tried many permutations of the other 82
>state linear models, G4 and G4u, and have had no success with them either.
>What am I doing wrong? What are the initial conditions for these linear
>>models? What is the input and output scaling for these models? I am
>using simulink in MATLAB could there be something wrong with the
>solution? I am most interested in the reduced 5 state model from the book,
>because I will be using the model for a model based compensator. Any
>help is appreciated.
> -ATS
>
>*******************************************************************************
>Alexander Schwarm * God is dead. -Nietzche(1886)
>Texas A & M University * Nietzche is dead. -God(1900)
>Department of Chemical Engineering ------------------------------------------
>>alexander@tamu.edu * Vote Cthulhu!
>ats5026@unix.tamu.edu * Why settle for the lesser evil?
>*******************************************************************************
Received on Mon Apr 28 13:31:50 1997