This problem is clearly difficult. It has multiple minima, and a strong dependence on initial conditions. Instead of NLINFIT, I tried two different approaches which both give me good fits for values of Vg > 18.
In both cases, since it seems this equation can't really model the Vg < 18 region, I weighted that region to be zero. Also, just to scale things better, I am working with the log10's of K and R.
Option 1: Pattern Search Algorithm from the Global Optimization Toolbox
This gives a solution that is almost identical to the one that you found from your other software.
G = @(X) (((10.^(X(1)).*(Vg-X(2)).^(X(3)+1)).^(-1))+10.^(X(4))).^(-1);
E = @(X) ( sum(((real(G(X)) - Id)).^2 .* (Vg >= 18)) );
opts = psoptimset;
opts.Display = 'iter';
opts.MaxIter = realmax;
opts.MaxFunEvals = realmax;
X0 = [log10(8.6E-10);1.7;0.8;log10(5E6)];
X = patternsearch(E,X0,[],[],[],[],[],[],[],opts);
K = 8.6E-10; V = 17.3; b = 0.618; R = 2.3E6;
F0 = (((K.*(Vg-V).^(b+1)).^(-1))+R).^(-1);
plot(Vg, Id);hold all; plot(Vg, G(X)); plot(Vg,F0);
Option 2: Simply using FMINSEARCH from various starting points
Again, identical to the previous results.
G = @(X) (((10.^(X(1)).*(Vg-X(2)).^(X(3)+1)).^(-1))+10.^(X(4))).^(-1);
E = @(X) ( sum(((real(G(X)) - Id)).^2 .* (Vg >= 18)) );
opts = optimset('fminsearch');
opts.MaxIter = realmax;
opts.MaxFunEvals = realmax;
options.TolFun = realmin;
options.TolX = realmin;
X0 = [log10(8.6E-10);1.7;0.8;log10(5E6)];
Emin = Inf;
for n = 1:100
disp(n)
[Xn, err(n)] = fminsearch(E,X0.*(1+0.01*randn(size(X0))),opts);
if err(n) < Emin
Emin = err(n);
X = Xn;
end
end
K = 8.6E-10; V = 17.3; b = 0.618; R = 2.3E6;
F0 = (((K.*(Vg-V).^(b+1)).^(-1))+R).^(-1);
plot(Vg, Id);hold all; plot(Vg, G(X)); plot(Vg,F0);
