Estimate differential equation parameters

Dear all,

I need to estimate a parameter of a second order differential equation. This equation is a law of the displacement of a floating body. I know a solution in time domain coming from the experimental investigation but the roblem are the values of the parameters. Is there any function to estimate a differetial equation parameters?

Thanking in advance

Alessandro Antonini

Answer 1

[Shashank Prasanna]

You can set up an optimization problem to 'fit' your ode to the experimental data for certain parameters that minimize the error between the fitted and real data.

Here is a link in the documentation that explains how to go about this: http://www.mathworks.com/help/optim/ug/optimizing-a-simulation-or-ordinary-differential-equation.html

[Alessandro Antonini]

Can you suggest an example,I did not understand too much about that link.

[Shashank Prasanna]

Ok, at this point i am assuming you know how to set up an ode and solve it using ODE45, if you are not sure then please look through some examples in: http://www.mathworks.com/help/matlab/ref/ode45.html

I am going to assume you ode function is 'odefun' as in some examples in the link, but with an extra parameter p passed by the optimizer.

Create your objective function such that it minimizes the error between your fitted ode result for a given parameter(s) p.

function err = odefit(exp_t,exp_y,p)
[t,y] = ode45(@(t,y)odefun(t,y,p),exp_t,0) % i am using y0=0 you can choose whatever.
err = sum((y-exp_y).^2); % compute error between experimental y and fitted y
end

p_estimate = fminsearch(@(p)odefit(exp_t,exp_y,p),p0);

Choice of optimization function is left to you and some dangers are listed in the link i gave in the previous post.

[Alessandro Antonini]

Yes I know how set up ODE. Thanks

Answer 2

[Matt J]

If it's a linear differential equation with constant unknown coefficients, just evaluate both sides of your differential equation at lots of time points. This will result in a linear system of equalities in the unknown parameters x(i), representable in matrix/vector form as

 A*x=b

Then solve by doing x=A\b.