Parameter estimation using fmincon - how to set up the problem in MATLAB?

I've an objective function of the form,

Note: M,D arematrices.

The second equation is the objective function that has to be minimized using fmincont to estimate diagonal entries of D tilde.

I'm using fmincon for the first time. I'd like to know if the objective function has to be approximated uisng trapzoidal quadrature .

Next, I want to provide the third equation mentioned above as dynamical constraints . Should this (eq 3) also be approximated by trapezoidal rule in a separate function?

I suppose the dynamical constraints must be specified as non-linear equality constraints. Is it also reuired to specify upper and lower bounds for variables in vctor phi?

Right now I have dphi/dt as a set of odes.

I am not sure how to set up this problem in MATLAB.

There are no linear inquality and linear equality constraints.

Any suggestions on how to proceed will be really helpful.

[Matt J]

How many unknowns are there in total? I tend to think you should use ga() instead of fmincon(). The method of integration that works best would probably depend on W(t). You can experiment with different methods, of course.

[Deepa Maheshvare]

Hi Matt,

I want to stick to fmincon. The numer of unknowns is 6 for the toy model that I am trying. The problem is in setting up the system.