How can I minimize an objective function using lsqnonlin? I am minimizing the difference between theoretical S parameter and measured S-parameter to extract the dielectric constant?

The function has complex values and every time I change the initial value example: x0 = 1-1i the value of dielectric constant changes. I think something is wrong with the optimization process here.

 F = Reflx_tot_thy-refl_cmplx_measrd;
 x = lsqnonlin(F,x0)
er = real(x)-i*imag(x)

Answer 1

[Alan Weiss]

Perhaps your function has multiple local minima. Perhaps your function F is not specified correctly (is it a function handle?). I do not understand why you take er as real(x) - i*imag(x) when I would think that you should add the imaginary part, and maybe should take er = F(x) instead of er = x.

It is difficult to know without seeing more details, and a formula of what the dielectric constant is as a function of x.

Alan Weiss

MATLAB mathematical toolbox documentation

[Ankit Regmi]

% I am calculating the dielectric constant as a function of angle of incidence. Currently I am calculating at a single frequency with diffrenent incidence angle data for S21(oblique incidence) and will compare the results of this data at single angle with the measured frequency range. Thus, I am having the same problem and you are right about the multiple local minimum. My code is attached below, dont know where the problem is coming from.

function F = constant(x,t,d,w,c,S21)

er = x;

 gama = (1i*w*sqrt(er))/c;
reflx_thy_n = zeros(1,length(t));
 reflx_thy_d = zeros(1,length(t));
 reflx_thy = zeros(1,length(t));
 Reflx_totn = zeros(1,length(t));
 Reflx_totd = zeros(1,length(t));
 Reflx_tot = zeros(1,length(t));

   for i = 1:length(t)

     reflx_thy_n(1,i) = cos(t(i))-sqrt(er-sin(t(i)).^2);
     reflx_thy_d(1,i) = cos(t(i))+sqrt(er-sin(t(i).^2));
     reflx_thy(1,i) = reflx_thy_n(i)./reflx_thy_d(i);

    %  Total Reflection Coefficient Calculation

 Reflx_totn(1,i) = 1-exp(-2*gama*d/cos(t(i)))*(reflx_thy(i));
 Reflx_totd (1,i)= 1-reflx_thy(i)^2*exp(-2*gama*d/cos(t(i))); 
   Reflx_tot(1,i) = Reflx_totn(i)/Reflx_totd(i);

   end
   F = (Reflx_tot-refl_cmplx);
end

% Main Code

t = [0.83,0.97,1.01];

 fre = 1.03075e+09;
 e_v = 8.854e-12;
 mu_v = 4*pi*10^-7;
 c = 1/sqrt(e_v*mu_v);
 lamda = c/fre;
 w = 2*pi*fre;
S21 = ......(measured complex data)
F = @(x)constant(x,t,d,w,c,S21);

opts = optimoptions(@lsqnonlin,'Algorithm','levenberg-marquardt','Display','iter');

x0 = 3-0.025i;

[x1,resnorm,residuals,exitflag,output] = lsqnonlin(F,x0,[],[],opts); x1,resnorm,exitflag,output.iterations

 er = x1;

Answer 2

[Alan Weiss]

I am surprised that this code works at all, because I do not see where refl_cmplx is defined.

Even if that is not a real issue, I suggest that you take a wide variety of initial values x0. And it seems to me that you might have better luck if you scale some of your parameters. e_v*mu_v is of order 1e-20, which might be giving you some trouble.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Ankit Regmi]

sorry I forgot to replace refl_complx here with S21. The problem here seems the same as before multiple local minima. How do I search for the global minimum and select that value?

[Alan Weiss]

That is a hard question in general. See what the documentation has to say about it. And, since you seem to have a low-dimensional problem, you might be able to take a set of x0 that cover the space pretty well.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Ankit Regmi]

Thankyou for your advise. I'll do my best.