Minimization of a definite integral

I want to find out the unknown parameter (x) by minimizing the error norm of the equations attached. The objective is to fined value of sai (greek letter).

• Syy, S1y and S2y are column vectors (power spectral density)
• H1_bar and H2_bar have trigonometric equation defined based on sai-->which have an integer values *L is length-->have an integer values*0-1st resonant frequency is the range of integration

I tried using fminbnd but I cannot define the integral when I'm creating an anonymous function. Also, I did define sai as syms but when I ran fminbnd I got a few errors. I need some guidance on how may I get the minimum value of sai. Should I define integration separately or can be embedded in the optimization command? Any suggestions is warmly appreciated.

[Shz713]

Here is the link to equation: Integral minimization equation

Answer 1

[Torsten]

Create an .m file to define the function to be minimized and call MATLAB's "integral" within this function to evaluate the integral depending on zeta.

Best wishes

Torsten.

[Shz713]

Dear Torsten, Thanks for your response. Well, I did define integral within minimization command fminbnd but I got error. The error was that the output should be scalar. Would you provide a schematic example to grasp the idea? I can post the script for you to have a look as well. I really appreciate it as I'm stuck in this part for almost several days :(

[Torsten]

Please post the code you use together with the exact error message.

Best wishes

Torsten.

[Shz713]

if true
  D_1 = load ('d_1.txt');                 %vertical acceleration @ initial
D_3 = load ('d_3.txt');                 %vertical acceleration @ end
Fi_1 = load ('fi_1.txt');               %rotational acceleration @ initial
Fi_3 = load ('fi_3.txt');               %rotational acceleration @ end
D_2 = load ('d2.txt');                  % vertical acceleration at middle
u_1 = D_1 + D_3;
u_2 = Fi_1 - Fi_3;              
sub_length = 39.3;              %length of internal substructure in meter-> 25sec measurement used
fs = 20;                                 %sampling frequency Hz
ts= 1/fs;                                %intervals between data points/distance between data points
N = length (u_1);                        %number of data points 
tmax= (N-1)*ts;
t = 0:ts:tmax;                           %time-step size
YY = abs (fft (D_2));                    %FFT of signal
Y_1 = abs (fft (u_1));                   %abs gives magnitude spectrum, otherwise shows complex numbers
Y_2 = abs (fft (u_2));
bin_num = 0:N-1;                        %by default bins are added by one in Matlab->(N-1)bins are started from zero not one
Syy= YY.* conj(YY);                       % power spectral density (PSD)  
S1y = Y_1.*conj (YY);                     % cross spectral density (CSD)
S2y = Y_2.*conj (YY);
fbins = bin_num*fs/N;                    %shows frequencies in Hz
%syms sai;                               %creates a symbolic variable in workspace with the value sai assigned to sai
                               %y=sym('y') is equivalent to above
% J= max(fbins);
subplot (3,1,1);
% plot(bin_num,Syy,'r');
plot(fbins,Syy,'m');
xlabel('Frequency (Hz)');
ylabel('Power/Frequency (dB/Hz)');
title ('Syy (\Omega)');
subplot (3,1,2);
plot(fbins,S1y,'r');
xlabel('Frequency (Hz)');
ylabel('Power/Frequency (dB/Hz)');
title ('S1y (\Omega)');
subplot (3,1,3);
plot(fbins,S2y,'g');
xlabel('Frequency (Hz)');
ylabel('Power/Frequency (dB/Hz)');
title ('S2y (\Omega)');

syms sai

% error_norm = @ (sai) (Syy - ((h1_bar * S1y) + (h2_bar * sub_length * S2y))) ^ 2; error_norm = @ (sai) (Syy - ((0.5*(((sin (sai)+ sinh(sai))/((sin(sai)*cosh(sai))+(cos(sai)*sinh(sai))))) .* S1y) + ... (((-1 / (4 * sai)) (((cos (sai)- cosh(sai))/((sin(sai)*cosh(sai))+(cos(sai)*sinh(sai))))) * sub_length) . S2y))) .^ 2;

% h1_bar = 0.5*(((sin (sai)+ sinh(sai))/((sin(sai)*cosh(sai))+(cos(sai)*sinh(sai))))); % h2_bar = (-1 / (4 * sai)) *(((cos (sai)- cosh(sai))/((sin(sai)*cosh(sai))+(cos(sai)*sinh(sai)))));

min_sai = fminbnd (error_norm,0,3); end I tried first to define sai (zeta) as symbolic but I cannot use it in optimization. So I created anonymous function handle for zeta. Now I am not sure where to define optimization and integration. end

[Shz713]

The input files are here:

https://drive.google.com/file/d/0B6y79ILk_COETHA3S0RGNXFqZ1E/view?usp=sharing