How can I fit experimental data to a system of implicit equations (using optimization toolbox?)

I constructed the following code to fit experimental data to one implicit equation using optimization toolbox.

y = (A3*(x/(x+A1))*exp(B1*(B2-B3+A4-y*V*A2)))

Here

• x is the independent variable
• y is the dependent variable.
• B1, B2 and B3 as well as V are known constants
• A1, A2, A3, and A4 are unknown parameters that I want to use the code to find.

The code I used is the following:

% define experimental data

x = ...

[240;220;195;170;145;120;95;75;50;35;25;12.5;6;0]*1e-6;

y = ...

[-1;-0.95;-0.94;-0.90;-0.87;-0.84;-0.77;-0.67;-0.56;-0.48;-0.37;-0.20;-0.13;0];

x = x';

y = y';

% define initial guess for A1-A4, and upper limit (ub) and lower limit (lb)

Fit_Params = [100e-6 200 10 0.1]; %insert initial guess here

lb = [0 0 0 0];

ub = [1e-3 300 1000 0.8];

% Plot y = f(x) using initial guess parameters

ezplot(@(x)myImplicitF(x,Fit_Params),[0e-6 250e-6]) %plot y = f(x)

% Find best fit value for parameters using lsqcurvefit

lsqcurvefit(@(params, xdata) myImplicitF(xdata,params),Fit_Params, x, y, lb, ub)

% Define implict function y = f(y,x)

% numerically computes the value of y by calling the FSOLVE function.

function yout = myImplicitF(x, Parameters)

B1= -40;

B2 = 0.1;

B3 = 0.5;

V = 20e-4;

A1=Parameters(1);

A2=Parameters(2);

A3=Parameters(3);

A4=Parameters(4);

yout = zeros(size(x));

opt = optimset('display','off');

for i=1:length(x)

yout(i) = fsolve(@(y) y+(A3*(x(i)/(x(i)+A1))*exp(B1*(B2-B3+A4-y*V*A2))), 1e-15, opt);

end

end

Question:

In the end, this gives a best fit for parameters for A1, A2, A3, and A4.

The challenge for me is to fit a system of equations. I have multiple datasets for variable y for different values of reaction volume (V = [20, 40, 60, 80]), while the independet variable x remains the same.

x = [240;220;195;170;145;120;95;75;50;35;25;12.5;6;0]*1e-6;

y1 = [-1;-0.95;-0.94;-0.90;-0.87;-0.84;-0.77;-0.67;-0.56;-0.48;-0.37;-0.20;-0.13;0]; %for v = 20.

y2 = [another array]; %for v = 40,

y3 = [another array]; %for v = 60,

y4 = [another array]; %for v = 80,

I want to find the best fit for A1, A2, A3 and A4 to minimze the sum of square error for all four equations at once, instead of the code I showed which fits to one equation at a time. Any suggestions on how to do this?

Thanks in advance