Complex non-linear least squares fitting of impedance data

Hi,

I am trying to fit measured impedance complex data to an equation using fminsearch. I am using the following function to which locate the minimum:

function impedance_function=impedance_model(x,omega,ReData,ImData)

Zcpe=(1./(x(2))).*((1i.*omega).^(-x(3)));

Z=Zcpe+x(1);

obj=((ReData-real(Z)).^2)./abs(ReData)+((ImData-imag(Z)).^2)./abs(ImData);

impedance_function=sum(obj);

end

The problem that I am having is that the fitting is not robust and depends too much on the initial guess. I am not sure if there is something wrong with my function, I believe the equation to be minimised is correct for the case of complex data (but I am not fully certain). I am also not sure about how can I assess how good the fitting is.

Thanks!

[Torsten]

And "omega" is the vector of the independent variable and is of the same length as "ReData" and "ImData" ?

[Victor Calero]

Yes, that is it. Omega is the independent variable for both datasets, the real and the imaginary part.

[Star Strider]

If the System Identification Toolbox is available, this is likely a trivial problem. Begin with idfrd and follow the documentation to identify it. Then use the compare function to see how well the model fits the data.

Answer 1

[Matt J]

Because your model function is linear with respect to x(1) and 1/x(2), it would be beneficial to download fminspleas,

https://www.mathworks.com/matlabcentral/fileexchange/10093-fminspleas

Then, you only need a good initial guess x3Initial for the x(3) parameter.

xdata=omeaga(:);
ydata=[ReData(:);ImData(:)];


funlist={1, @(x3,omega) [real(  (1i.*omega).^(-x3) ) ; imag( (1i.*omega).^(-x3) )  ] };

[ILP,x3]=fminspleas(funlist, x3Initial, xdata, ydata);

x=[ILP(1), 1./ILP(2), x3];

[Victor Calero]

Thanks! Although I am not sure if this solves my problem, I am looking to minimise the sumatory function:

obj=((ReData-real(Z)).^2)./abs(ReData)+((ImData-imag(Z)).^2)./abs(ImData);

impedance_function=sum(obj);

Not only the impedance equation.

[Matt J]

I think it's just a weighted least squares.

xdata=omeaga(:);
ydata=[ReData(:);ImData(:)]; ydata=ydata./sqrt(abs(ydata));

absRe=sqrt( abs(ReData) ); absIm=sqrt(abs(ImData));

funlist={1, @(x3,omega) [real(  (1i.*omega).^(-x3) )./absRe ;...
                         imag(  (1i.*omega).^(-x3) )./absIm  ] };

[ILP,x3]=fminspleas(funlist, x3Initial, xdata, ydata);

x=[ILP(1), 1./ILP(2), x3];

[Torsten]

The weighting for small data becomes too strong when dividing by absRe or absIm.

Either divide only the data for which absRe, absIm > 1 by this factor or let away this weighting in total.