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From: "Bruno Luong" <b.luong@fogale.findmycountry>
Newsgroups: comp.soft-sys.matlab
Subject: Re: simple parameter estimation
Date: Fri, 10 Oct 2008 21:25:03 +0000 (UTC)
Organization: FOGALE nanotech
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"Marcio " <marciobarbalho@yahoo.com> wrote in message <gcog9f$219$1@fred.mathworks.com>...

> > 
> > Do you know the cauchy data? Or is it a free parameters?
> 
> t  = [t0 t1 t2];
> y1 = [x1 x2 x3];
> y2 = [p1 p2 p3];
> y3 = [w1 w2 w3];
> 

OK, so you have state parameters (y) at three times. So your problem is estimation parameters from state data. This problem is well know.

I'll repeat myself again: See papers from Jacques Louis Lions on Control Optimal theory.

What you need is compute the derivative of y (in particular at {t0,t1,t2}) with respect to {a,b,c}. This can be accomplished in two steps:
1. Solving the linearization of the direct ODE
2. Solving an adjoint equation of the linearized ODE, with a forcing term as the deviation to data.

More details on the above reference.

After the derivative is computed, plug it in an optimizer and let it does the works.

Bruno

> I've been trying to transform that system into a simple non-linear equation system by transforming the derivatives into new functions, for example, dy1/dt could be called f1(t) once y1 is known. I'm working on it. I just had this insight, whether it works out, my day will be made!