To be clear, you want to fit experimental data to a system of differential equations with some number of unknown parameters, correct? If so, your workflow will basically be performing a numerical optimization, with the ODE solution nested inside the optimization, with the following general step. Note that you don't need to have explicitly solved the ODEs to run this approach (and in fact, it is not necessary to even have an analytical solution to them).
1. Set up the system of differential equations in the dx = f(t,x) format required by Matlab's ODE solvers. (I assume since you mention Runge-kutta you have ODEs). See the Documentation for 'ode45' (for example) to see how to do this:
doc ode45
Let's call that function 'myodefcn' and have it take three inputs: your unknown parameters p, the time vector t and state vector x. This function could have the form:
function dx= myodefcn(p, t, x)
%your equations are here, in the form dx= f(t,x)
dx = p(1)*x-2*p(2); %for example
2. Create an objective function that solves the ODE equations and outputs a state variable that can be compared to your data. Calculate the difference between the model data and your experimental data, and sum the squared error (to perform a least-squares fit). Suppose you are modeling a 2-parameter first-order system, and you have experimental data time td and state xd. Such a function might look like:
function SSE = myobjective(td,xd, p, x0)
f = @(t,x) myodefcn(p,t,x); %function to call ode45
[ts,xs] = ode45(f, td, x0); %xs is the state variable predicted by the model
err = xd - xs;
SSE = sum(err.^2); %sum squared-error.
with inputs td (experimental time), xd (experimental data), p (model parameters), and x0 (initial conditions).
3. Use one of Matlab's optimization functions to solve for the parameters that minimize the SSE. If you have the Optimization Toolbox, you have lots of flexibility here, to implement constraints, bounds, etc. on the parameters. If you have basic Matlab, you can still use the fminsearch function:
psolve = fminsearch(@(p) myobjective(td,xd,p,x0), p0);
where p0 is your initial guesses for the unknown parameters p.
This is the general approach. Unfortunately, without more details, I cannot give anything more specific. Let me know if you have any questions.
