Simultanous curve fitting to multiple datasets

I want to fit a nonlinear model simultaneously to multiple experimental datasets from different publications.

Besides the dependency of the model equation on the curve fitting parameters(a,b,c), my model also depends on an experimental variable, which defines the loading velocity of the experiment.

The loading velocity is different for each experiment and directly influences the model response. It is predefined and shall not be used for curve fitting.

The following sample data and model function is considered:

x1 = 0:0.1:1;
x2 = 0.05:0.1:0.75;

fun = @(x,a,b,c,velocity) a+b*x+velocity*exp(c.*x);

a_hat=1; b_hat=1; c_hat=1;
y1 = fun(x1, a_hat, b_hat, c_hat, 1.1)+(0.5-rand(1,length(x1)));
y2 = fun(x2, a_hat, b_hat, c_hat, 0.9)+(0.5-rand(1,length(x2)));

What is the best way to get one set of parameters, which fits both experiments?

Answer 1

[Torsten]

x1 = 0:0.1:1;
x2 = 0.05:0.1:0.75;
fun = @(x,a,b,c,velocity) a+b*x+velocity.*exp(c.*x);
a_hat=1; b_hat=1; c_hat=1;
y1 = fun(x1, a_hat, b_hat, c_hat, 1.1)+(0.5-rand(1,length(x1)));
y2 = fun(x2, a_hat, b_hat, c_hat, 0.9)+(0.5-rand(1,length(x2)));

x = [x1,x2];
y = [y1,y2];
fun_optim = @(p) fun(x,p(1),p(2),p(3),[1.1*ones(size(x1)),0.9*ones(size(x2))]) - y;
sol = lsqnonlin(fun_optim,[1;1;1])

Note that I changed your original fun from

fun = @(x,a,b,c,velocity) a+b*x+velocity*exp(c.*x);

to

fun = @(x,a,b,c,velocity) a+b*x+velocity.*exp(c.*x);

[lilythefirst]

Thanks a lot!