Obatin s domain and transfer function from the data

I have two set of graph data

Frequency / phase , angle data - Freq_phase_angle.txt

Frequency / real , imaginary data - Freq_complex.txt

Could any one please help with procedure to obatin S-domain/ Transfer function from data(.txt) using system idenrtification tool in MATLAB?

To obtain this circuit. using cauer/foster method

Answer 1

[Star Strider]

Start with the idfrd function and go from there. The documentation explains everything.

Actually though, to analyse the circuit using the Foster-Cauer approach, plot the imaginary part of the complex frequency vector and go from there —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/828710/Freq_complex.txt', 'VariableNamingRule','preserve')

T1 = 5479×2 table
    Freq.                         V(i_s)                     
    ______    _______________________________________________

       100    {'5.25923694595431e-001,6.34606935959376e+000'}
    100.23    {'5.26471741717689e-001,6.36067773821260e+000'}
    100.46    {'5.27022310375074e-001,6.37531964864880e+000'}
    100.69    {'5.27575412141180e-001,6.38999516719951e+000'}
    100.93    {'5.28131058638060e-001,6.40470437033997e+000'}
    101.16    {'5.28689261538566e-001,6.41944733471207e+000'}
    101.39    {'5.29250032542678e-001,6.43422413714084e+000'}
    101.62    {'5.29813383450318e-001,6.44903485459486e+000'}
    101.86    {'5.30379326076855e-001,6.46387956423504e+000'}
    102.09    {'5.30947872306298e-001,6.47875834338622e+000'}
    102.33    {'5.31519034080622e-001,6.49367126954284e+000'}
    102.57    {'5.32092823401404e-001,6.50861842036846e+000'}
     102.8    {'5.32669252306374e-001,6.52359987370368e+000'}
    103.04    {'5.33248332892240e-001,6.53861570756800e+000'}
    103.28    {'5.33830077315765e-001,6.55366600013723e+000'}
    103.51    {'5.34414497792562e-001,6.56875082976168e+000'}

T1{:,2} = cellfun(@(x)sscanf(x, '%f, %f'), T1{:,2}, 'Unif',0);
RI = cell2mat(table2cell(T1(:,2)).').';
RI = complex(RI(:,1),RI(:,2));
T2 = [T1(:,1) table(RI)];
T2.Properties.VariableNames = {'Freq','V(i_s)'}

T2 = 5479×2 table
     Freq         V(i_s)     
    ______    _______________

       100    0.52592+6.3461i
    100.23    0.52647+6.3607i
    100.46    0.52702+6.3753i
    100.69    0.52758+6.39i  
    100.93    0.52813+6.4047i
    101.16    0.52869+6.4194i
    101.39    0.52925+6.4342i
    101.62    0.52981+6.449i 
    101.86    0.53038+6.4639i
    102.09    0.53095+6.4788i
    102.33    0.53152+6.4937i
    102.57    0.53209+6.5086i
     102.8    0.53267+6.5236i
    103.04    0.53325+6.5386i
    103.28    0.53383+6.5537i
    103.51    0.53441+6.5688i

figure
semilogx(T2{:,1}, imag(T2{:,2}))
grid

There is one obvious pole, however there may be two others with real values far from the imaginary axis. There is a zero at the origin, and likely one at infinity.

.

[Venkatkumar M]

@Star Strider Could you please let me know how to use vetcors in system identification tool to find the poles and zeros?

[Star Strider]

Using Cauer-Foster is not difficult, however I choose not to do it for you, since the assignment wants you to do it.

I will instead demonstrate how to identify it using the System Identification Toolbox —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/828710/Freq_complex.txt', 'VariableNamingRule','preserve')

T1 = 5479×2 table
    Freq.                         V(i_s)                     
    ______    _______________________________________________

       100    {'5.25923694595431e-001,6.34606935959376e+000'}
    100.23    {'5.26471741717689e-001,6.36067773821260e+000'}
    100.46    {'5.27022310375074e-001,6.37531964864880e+000'}
    100.69    {'5.27575412141180e-001,6.38999516719951e+000'}
    100.93    {'5.28131058638060e-001,6.40470437033997e+000'}
    101.16    {'5.28689261538566e-001,6.41944733471207e+000'}
    101.39    {'5.29250032542678e-001,6.43422413714084e+000'}
    101.62    {'5.29813383450318e-001,6.44903485459486e+000'}
    101.86    {'5.30379326076855e-001,6.46387956423504e+000'}
    102.09    {'5.30947872306298e-001,6.47875834338622e+000'}
    102.33    {'5.31519034080622e-001,6.49367126954284e+000'}
    102.57    {'5.32092823401404e-001,6.50861842036846e+000'}
     102.8    {'5.32669252306374e-001,6.52359987370368e+000'}
    103.04    {'5.33248332892240e-001,6.53861570756800e+000'}
    103.28    {'5.33830077315765e-001,6.55366600013723e+000'}
    103.51    {'5.34414497792562e-001,6.56875082976168e+000'}

T1{:,2} = cellfun(@(x)sscanf(x, '%f, %f'), T1{:,2}, 'Unif',0);
RI = cell2mat(table2cell(T1(:,2)).').';
RI = complex(RI(:,1),RI(:,2));
T2 = [T1(:,1) table(RI)];
T2.Properties.VariableNames = {'Freq','V(i_s)'}

T2 = 5479×2 table
     Freq         V(i_s)     
    ______    _______________

       100    0.52592+6.3461i
    100.23    0.52647+6.3607i
    100.46    0.52702+6.3753i
    100.69    0.52758+6.39i  
    100.93    0.52813+6.4047i
    101.16    0.52869+6.4194i
    101.39    0.52925+6.4342i
    101.62    0.52981+6.449i 
    101.86    0.53038+6.4639i
    102.09    0.53095+6.4788i
    102.33    0.53152+6.4937i
    102.57    0.53209+6.5086i
     102.8    0.53267+6.5236i
    103.04    0.53325+6.5386i
    103.28    0.53383+6.5537i
    103.51    0.53441+6.5688i

figure
semilogx(T2{:,1}, imag(T2{:,2}))
grid

frd = idfrd(T2{:,2}, T2{:,1}, 1/(2*T2{end,1}))

frd =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s).
Response data is available at 5479 frequency points, ranging from 100 rad/s to 3e+07 rad/s.
 
Sample time: 1.6667e-08 seconds
Status:                                                         
Created by direct construction or transformation. Not estimated.

sys_tf = tfest(frd, 4, 4)                                       % Poles: 4, Zeros: 4

sys_tf =
 
  24.81 s^4 + 5.406e08 s^3 + 6.957e13 s^2 + 5.884e17 s + 2.879e19
  ---------------------------------------------------------------
     s^4 + 1.295e05 s^3 + 4.636e10 s^2 + 2.381e15 s + 1.011e19
 
Continuous-time identified transfer function.

Parameterization:
   Number of poles: 4   Number of zeros: 4
   Number of free coefficients: 9
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                
Estimated using TFEST on frequency response data "frd".
Fit to estimation data: 98.7%                          
FPE: 459.9, MSE: 458.4                                 

poles = pole(sys_tf)

poles = 
   1.0e+05 *

  -0.3661 + 2.0164i
  -0.3661 - 2.0164i
  -0.5160 + 0.0000i
  -0.0467 + 0.0000i

zeros = zero(sys_tf)

zeros = 4×1

1.0e+07 * -2.1663 -0.0120 -0.0009 -0.0000

figure
pzmap(sys_tf)

figure
pzmap(sys_tf)
axis([-100000  1000    -500 500])                                   % Detail Near Origin

sys_ss = ss(sys_tf)

sys_ss =
 
  A = 
               x1          x2          x3          x4
   x1  -1.295e+05  -1.769e+05  -6.929e+04  -1.796e+04
   x2   2.621e+05           0           0           0
   x3           0   1.311e+05           0           0
   x4           0           0   1.638e+04           0
 
  B = 
              u1
   x1  3.277e+04
   x2          0
   x3          0
   x4          0
 
  C = 
             x1        x2        x3        x4
   y1  1.64e+04      7965     470.1    -12.04
 
  D = 
          u1
   y1  24.81
 
Continuous-time state-space model.

figure
compare(frd,sys_ss)
grid

There is a constant phase difference from the original spectrum of about 360° (by observation, not calculation), and the approsimation appears to be good. I cannot determine where the output is taken from that network (assuming it is a two-port), so I assume it is simply a one-port with 4 obvious poles and 4 obvious zeros. The Cauer-Foster synthesis requires the transfer function, and this identification provides it.

Since that is the point of the assignment, I leave that to you. It would be easier to do it symbolically using polynomial division, the other option being to use the deconv function to get a numeric polynomial continued-fraction decomposition of the transfer function.

.