Searching for Transferfunction between two functions

Hello,

the problem is as follows:

I have two functions (F1 and F2 both dependent on the time) from a data fit and I am searching for a transfer function (TF) that fulfills the following equation.

F2=F1*TF

TF should be given by an equation where two variables needs to be changed. The script has to change the variables to minimize the sum of squares errors between the original function F2 and the analytical result of F1*TF (which should be as close to F2 as possible).

R^2=(F2-F1*TF)^2 =!= min

I am open for any suggestions.

Thanks in advance

Answer 1

[Star Strider]

Use the System Identification Toolbox.

The actual transfer function is probably a simple delay term or that includes one, for example or something similar, if both functions are essentially the same otherwise.

To illustrate —

syms k s t 
f(t) = t*exp(-k*t)

f(t) = 

F(s) = laplace(f,t,s)

F(s) = 

clear k s t

s = tf('s');
HT = exp(-60*s);

figure
stepplot(HT)
grid

k1 = 0.1;

H1 = 0.01/(k1+s)^2

H1 =
 
         0.01
  ------------------
  s^2 + 0.2 s + 0.01
 
Continuous-time transfer function.

H2 = H1*HT

H2 =
 
                      0.01
  exp(-60*s) * ------------------
               s^2 + 0.2 s + 0.01
 
Continuous-time transfer function.

figure
impulseplot(H1)
hold on
impulseplot(H2)
hold off
grid
xlim([0 300])

This approximates your posted plot.

.

[User0815]

Hi, thanks for your input.

It is actually not a Laplace transformation. The roots lay in the residence time of a system. Those curves are measured with a Dirac impulse and the graphs are just there to illustrate the problem. The original curve is basicly the intensity of the tracer over the time.

In some case it is close to a time delay, in most cases the functions are not identical to each other (different peaks, integrals and so on).

Thats why we try to implement a model equation with two variables (Axial Dispersion Model) which should be analytically solved to that problem. It its not a "transfer function" which is used in the design of controllers.

[Star Strider]

It actually is, if you consider it to be such. You don’t have to specifically model it as a Laplace transformation. The System Identification Toolbox will model it that way.

If you use the System Identification Toolbox (available here even if you don’t have it licensed and installed on your computer), you can model each of the exponentioal impulse responses (it works best if you also have the inputs to the system) and then divide the later one by the earlier one to get the delay term, that will likely be the sort of exponential I described in my example. The exponential delay term seems to be what you’re looking for.