Fitting multiple curves with multiple data sets, partial and globally shared parameters using lsqcurvefit

Hello everyone, I hope you can help me with the following problem. I have 3 measurement data sets, consisting of x-values (e.g. x1) and 2 corresponding y-data sets (e.g. A1 and B1). The functions of these parameters have partially shared parameters. A simultaneous fit over both curves works so far. Now I want to run a global fit over all 3 measurement datasets simultaneously, where all 3 measurement datasets share one parameter (beta(10)). How do I do this? Previously, I received a value for beta (10) for each fit, but now I want to change this. My data and functions (non-linear) are very extensive, so I have created a simplified example.

b1 = 1; b2 = 0.85; b3 = 2.5;
b4 = 1.1; b5 = 2.2; b6 = 4.5;
b7 = 1.3; b8 = 7.2; b9 = 9.5;
b10 = 0.5;
%x data
x1 = linspace(0, 10, 20).'; x2 = linspace(0, 10, 23).'; x3 = linspace(0, 10, 14).';
%constants
C1 = 3.7 ; C2 = 4.2; C3 = 20.2;
%measurement dataset 1
A1 = b1 + b2*x1 + C1 * b10 + rand(20,1);
B1 = b3 - b2*x1 + C1 * b10 + rand(20,1);
%measurement dataset 2
A2 = b4 + b5*x2 + C2 * b10 + rand(23,1);
B2 = b6 - b5*x2 + C2 * b10 + rand(23,1);
%measurement dataset 3
A3 = b7 + b8*x3 + C3 * b10 + rand(14,1);
B3 = b9 - b8*x3 + C3 * b10 + rand(14,1);

mdl1 = @(beta, x) [beta(1) + beta(2).*x + C1 .* beta(10) ,...
    beta(3) - beta(2).*x + C1 .* beta(10)];
mdl2 = @(beta, x) [beta(4) + beta(5).*x + C2 .* beta(10) ,...
    beta(6) - beta(5).*x + C2 .* beta(10)];
mdl3 = @(beta, x) [beta(7) + beta(8).*x + C3 .* beta(10) ,...
    beta(9) - beta(8).*x + C3 .* beta(10)];
beta0 = [0.92, 0.8, 2, 0.7, 2, 4, 1, 7, 9, 1];
lb = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
ub = [15, 15, 15, 15, 15, 15, 15, 15, 15, 15];
options = optimoptions(@lsqcurvefit,'Algorithm','levenberg-marquardt');
beta1 = lsqcurvefit(mdl1,beta0,x1,[A1, B1],lb,ub,options);
beta2 = lsqcurvefit(mdl2,beta0,x2,[A2, B2],lb,ub,options);
beta3 = lsqcurvefit(mdl3,beta0,x3,[A3, B3],lb,ub,options);
A1_fit = beta1(1) + beta1(2)*x1 + C1 * beta1(10);
B1_fit = beta1(3) - beta1(2)*x1 + C1 * beta1(10);
A2_fit = beta2(4) + beta2(5)*x2 + C2 * beta2(10);
B2_fit = beta2(6) - beta2(5)*x2 + C2 * beta2(10);
A3_fit = beta3(7) + beta3(8)*x3 + C3 * beta3(10);
B3_fit = beta3(9) - beta3(8)*x3 + C3 * beta3(10);
figure(1);
subplot(2,1,1); 
hold on;
plot(x1, A1,'s');
plot(x1, A1_fit);
plot(x2, A2,'d');
plot(x2, A2_fit);
plot(x3, A3,'p');
plot(x3, A3_fit);
subplot(2,1,2);
hold on;
plot(x1, B1,'s');
plot(x1, B1_fit);
plot(x2, B2,'d');
plot(x2, B2_fit);
plot(x3, B3,'p');
plot(x3, B3_fit);
hold off

Answer 1

[Torsten]

rng("default")
b1 = 1; b2 = 0.85; b3 = 2.5;
b4 = 1.1; b5 = 2.2; b6 = 4.5;
b7 = 1.3; b8 = 7.2; b9 = 9.5;
b10 = 0.5;
%x data
x1 = linspace(0, 10, 20).'; x2 = linspace(0, 10, 23).'; x3 = linspace(0, 10, 14).';
%constants
C1 = 3.7 ; C2 = 4.2; C3 = 20.2;
%measurement dataset 1
A1 = b1 + b2*x1 + C1 * b10 + rand(20,1);
B1 = b3 - b2*x1 + C1 * b10 + rand(20,1);
%measurement dataset 2
A2 = b4 + b5*x2 + C2 * b10 + rand(23,1);
B2 = b6 - b5*x2 + C2 * b10 + rand(23,1);
%measurement dataset 3
A3 = b7 + b8*x3 + C3 * b10 + rand(14,1);
B3 = b9 - b8*x3 + C3 * b10 + rand(14,1);
beta0 = [0.92, 0.8, 2, 0.7, 2, 4, 1, 7, 9, 1];
lb = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0];
ub = [15, 15, 15, 15, 15, 15, 15, 15, 15, 15];
fun = @(beta,x)[beta(1) + beta(2).*x1 + C1* beta(10) ;
                beta(3) - beta(2).*x1 + C1* beta(10) ;
                beta(4) + beta(5).*x2 + C2* beta(10) ;
                beta(6) - beta(5).*x2 + C2* beta(10) ;
                beta(7) + beta(8).*x3 + C3* beta(10) ;
                beta(9) - beta(8).*x3 + C3* beta(10) ];
beta = lsqcurvefit(fun,beta0,[x1;x1;x2;x2;x3;x3],[A1;B1;A2;B2;A3;B3],lb,ub)

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

beta = 1x10

1.2801 0.8703 2.8336 1.3172 2.1985 4.7244 0.4327 7.1798 8.4477 0.5704

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A1_fit = beta(1) + beta(2)*x1 + C1 * beta(10);
B1_fit = beta(3) - beta(2)*x1 + C1 * beta(10);
A2_fit = beta(4) + beta(5)*x2 + C2 * beta(10);
B2_fit = beta(6) - beta(5)*x2 + C2 * beta(10);
A3_fit = beta(7) + beta(8)*x3 + C3 * beta(10);
B3_fit = beta(9) - beta(8)*x3 + C3 * beta(10);
figure(1);
subplot(2,1,1); 
hold on;
plot(x1, A1,'s');
plot(x1, A1_fit);
plot(x2, A2,'d');
plot(x2, A2_fit);
plot(x3, A3,'p');
plot(x3, A3_fit);
subplot(2,1,2);
hold on;
plot(x1, B1,'s');
plot(x1, B1_fit);
plot(x2, B2,'d');
plot(x2, B2_fit);
plot(x3, B3,'p');
plot(x3, B3_fit);
hold off

[WR]

I have applied your suggestion to my non-linear problem and it works. Thank you very much. I tip my cap to you.