Fminsearch results shows shift from measured curve. How should I solve this?

I am working with several measurement data which I am reading through matlab and then using Fminsearch to estimate through different model equations to fit with the measurement data. But fminsearch is giving a shift from the measured data as shown in picture. Blue one is the measured curve and orange one is the fminsearch estimation. How should I solve this? I am a beginner so would be helpful with some guidance. Thank you.

My sample of fminsearch:

[John D'Errico]

When you post only a picture of your code, you make it diffiult to get help. That forces others to rewrite your code (with no typos made) themselves. And that makes it less likely you will get help in a timely manner. This is especially true when you could just have pasted in the text itself, rather than a screenshot of text.

If you want to make it easier yet to get help, then you would also attach data that someone can use as an example.

Why would you want to make it more difficult for someone to help you?

[Amit Das]

I am getting the position and torque values from measurement and then trying to fit the curve with the models. The fminsearch:

sigma_2  = 6.0672e-4;

Fc = 0.6339;
Fs = 2.235;
vs = 87.1746;

Delta=abs(-0.6732);
%             Delta=2.567;

F=torque;

Unrecognized function or variable 'torque'.

ts=1e-3;
%%speed calculation
speed=zeros(length(position)-1,1);
for j=2:1:length(position-1)
    speed(j)=(position(j)-position(j-1))/(time(j)-time(j-1));
end
%%plot
figure(3);plot(time, speed);hold on;
grid on;
xlabel('time in s'); 
ylabel('speed in 1/min'); 
plot(time, position);hold on;
grid on;
xlabel('time in s'); 
ylabel('position in m'); 

v = speed;

%% 2. Identification of LuGre Parameters
%% 2.1 Set starting values for optimization
x_Start=[0.4;0.4];
%% 2.2. Define Fitness-Function
erf=@(x)FitFunction_LuGre(x,Fc, Fs,vs,Delta,sigma_2,time,v, F);
%% 2.3. Perform Optimization
%% 2.3.1 Set Options for NM-Optimizer 
options = optimset('TolX',1e-50,'TolFun',1e-50,'MaxIter', 200000, 'MaxFunEvals', 200000,'PlotFcns',@optimplotfval);
%% 2.3.2 Optimize
[x,fiterr,exitflag,info]=fminsearch(erf,x_Start,options);
sigma_0_est=x(1)
sigma_1_est=x(2)
%% 3. Calculate and plot estimation
%% 3.1 Calculate LuGre
z=0;
for j = 1 : length(time)
    [F_est(j), z] = EQLugre(z, v(j), Fc, Fs, vs, Delta, sigma_0_est, sigma_1_est, sigma_2, ts);
end

figure(2);
plot(time,F_est');hold on;
grid on;
xlabel('time in s'); 
ylabel('Friction Force in Nm'); 
figure(1);
plot(position,F_est');hold on;
grid on;
xlabel('position in m'); 
ylabel('Friction Force in Nm'); 


Fit Function:
function F=FitFunction_LuGre(x,Fs,Fc,vs,Delta,sigma_2,time,speed,torque)

z=0; % Starting value z
ts=time(2)-time(1);
for j = 1 : length(time)
    [M(j), z] = EQLugre(z, speed(j), Fc, Fs, vs, Delta, x(1), x(2), sigma_2, ts);
end

[row_torque,col_torque]=size(torque);
[row_M,col_M]=size(M);
if(row_torque==row_M)&&(col_torque==col_M)
    M=M;
else
    M=M';
end

F=sum(abs(torque-M)).^2;

end

Eq:

function [F, z] = EQLugre(z, v, Fc, Fs, vs, Delta, sigma_0, sigma_1, sigma_2, ts)
r = -(v/vs).^Delta;
g_v = (Fc + (Fs - Fc) * exp(r)) ./ sigma_0;          % eq. 4
z_dot = v - abs(v) .* z ./ g_v;                      % eq. 1 

% note, this is forward euler integration. Your results depend heavily on the size of ts. - Jason Nicholson
z = z + z_dot.* ts;
F = sigma_0 * z + sigma_1 * z_dot + sigma_2.*v;    % eq. 3
end

[Matt J]

As you can see above, the code doesn't run. SOme variables have not been specified.

[Amit Das]

Hi the values needed to run the codes can be found in this excel file. Just run and plot should help.

% Read Measurements

[daten, header]=xlsread("your path");

time =daten(:,1);

position = daten(:,2);

torque = daten(:,3);

Answer 1

[Matt J]

The x_Start=[1,1] selection looks very arbitrary. If one of the parameters is the oscillation amplitude, it should be closer to 0.4.

Other than that, though, there is too little information to diagnose. We would need to see the full code including the FitnessFunction (and as text, not a screenshot).

[Matt J]

I notice that the least squares prediction F can be written as a linear combination of known things.

Q= abs(v) .* z ./ (Fc + (Fs - Fc) * exp(r));

zdot = v - sigma0*Q;                 


F = sigma0 * (z+zdot.*ts) + sigma1 * z_dot + sigma2.*v;    % eq. 3

  = sigma0*(z  + v) - sigma0^2.*Q.*ts   - sigma0*sigma1.*Q + (sigma1+sigma2)*v
  
  = C1*(z  + v) - C2.*Q.*ts   - C3.*Q + C4*v

So, you can solve for C1 .. C4 by solving a set of linear equations.

Then, from the C coefficients, you can easily solve for sigma0 and sigma1.

Then, you can use that as an initial guess for the original optimization.