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

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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.
  2 Comments
Matt J
Matt J on 17 Mar 2020
Edited: Matt J on 17 Mar 2020
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
Deepa Maheshvare on 18 Mar 2020
Edited: Deepa Maheshvare on 18 Mar 2020
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.

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