Transfer function with non-zero initial value

[Dammi87]

Hi there

This is my first time trying this webpage

I have this equation: , with x(0) = E
Here x(t) is the output, D is the input

By taking Laplace transform, first with D = 0 and x(0) = E, then D = D and x(0) = 0 I get these two equations:

and So the final answer for X(S) would be the sum of these two or

Now I know that x(t) = G(s)u(t) .. how can I find G(s) ? (I'm assuming I haven't done anything wrong up till now, unless you can find a error somewhere?)

This is a part of an assignment of Parametric Signal Identification which I've done, but I can't verify my solution unless I can derive the transfer function (Or state space model) to put into SimuLink.
I at least know that the input is a step and the output looks like the output for a first-order system.

You would be of great help if you can help me with this, it's really embarrassing to go through really hard PSI methods, and then get stuck on the small matter of a simple Transfer Function!

 

[Dammi87]

Just to give you guys some more insight into what I'm doing.

The differential equation actually looks like this: , where p,c,C,W,T and Q are constants. What we are doing is heating tissue with the power Q and describing the heat in that tissue at time t --> x(t)
Now, the initial temperature of the tissue is of course 37°C.

What we are supposed to do is to estimate the parameter W according to a given dataset. I've done that and the DISCRETE results I get are the dots, on this figure. The actual data is the solid line



And this is an okay estimation for W I think. I use Backward Euler on x'(t) and convert the equation to discrete time and some more magic to be able to iterate my calculations until I get a good enough estimation of W.
Now to test this I wanted to make a simulink model with all these parameters and see if I'd get similar results.

Hopefully you have some more magic up your sleeve

 

[Dammi87]

Never mind, found out my error