How to use LSQNONLIN or LSQCURVEFIT when the function has a derivative form ?

I would like to use the least squares method to determine specific parameters. Knowing the time step, dt, time,t(i), and displacement, u(i), for every step i, the equation is the following:

F0(i)=k1*u(i)^k2

F1(i)=(F1(i-1)+k3*u(i)-k3*u(i-1))*(k4/(k4+k3*dt))

Y(i)=F0(i)+F1(i)=(k1*u(i)^k2)+(F1(i-1)+k3*u(i)-k3*u(i-1))*(k4/(k4+k3*dt))

or if it is more convenient for anyone (assuming that dF1(i)/dt=(F1(i)-F1(i-1))/dt) :

Y(i)=(k1*u(i)^k2)+(k4*u(i)-(k4/k3)*dF1(i)/dt)

Y_exp is the expected output, and the parameters I want to determine are the k1,k2,k3,k4. The thing that worries me the most, is the dependence of time step i on i-1 time step.

ANY IDEAS?

Answer 1

[Star Strider]

It’s difficult for me to follow what you’re doing. If you want to fit a (perhaps kinetic) differential equation to data, you can use the techniques in Monod kinetics and curve fitting.

[Christos]

Thank you for your reply. Just imagine that I have a system with a non linear spring with a stiffness parameter k1, and the displacement at the power of k2 (so its behaviour is described by F0(i)=k1*u(i)^k2 for every time step), and linear spring in series with a linear dashpot with a stiffness and dashpot parameters k3, and k4 respectively

(so its behaviour is described by: F1(i)=k4*u(i)-(k4/k3)*dF1(i)/dt).

The displacement of the non linear spring and the linear spring in series with the dashpot is the same, u). So, the total force can be determined as Y(i)=Fo(i)+F1(i).

I am trying to use the lsqnonlin command in order to optimize the k1,k2,k3,k4 parameters and I can't find any way to do that. So any help will be good. Thank you again