If you have Statistics Toolbox, you can use nlinfit, although there's no guarantee that k will be in the interval (0,1). Make a function handle to your function:
f = @(c,x) c(1)*x+c(2)*c(3)+c(2)*(c(3)^(1/c(4))+c(5)^2+(c(5)-x).^2).^c(4);
f is a function of the parameters ( c ) and x. Make an initial guess of the parameters:
c = rand(5,1);
Then call nlinfit with the data you have:
cfit = nlinfit(xdata,ydata,f,c)
If you don't have Stats TB, you can brute-force it in MATLAB by making an error function to minimize. Define f as above. Then define
g = @(c) norm(f(c,xdata)-ydata);
Now use fminsearch to find the coefficients, starting with an initial guess (as before)
cfit = fminsearch(g,c)
If you have Optimization Toolbox, you can use fmincon to constrain k.
