% Select a demo number: 12

%     CASE STUDIES IN SYSTEM IDENTIFICATION
%     *************************************
%  
%     In this selection of case studies we illustrate typical techniques
%     and useful tricks when dealing with various system identification
%     problems.
%     
%     Case studies:
%  
%     1) A glass tube manufacturing process
%     2) Energizing a transformer
 
  
% Select a case study number: 2

% CASE STUDY # 2 ANALYSING A SIGNAL FROM A TRANSFORMER
%
% In this case study we shall consider the current signal from the
% R-phase when a  400 kV three-phase transformer is energized.
% The measurements were performed by Sydkraft AB in Sweden.

load current.mat

% Form the IDDATA object:
i4r = iddata(i4r,[],0.001);  % Second argument empty for no input
% The sampling interval is 1 ms.
% Let's take a look at the  data.
figure, plot(i4r)
                              

% A close up:
figure,plot(i4r(201:250))

% Let us first compute the periodogram:
ge = etfe(i4r);
 
figure,bode(ge)

% A smoothed periodogram is obtained by
ges=etfe(i4r,size(i4r,1)/4); 

% A plot with linear frequency scale  is given by
figure,ffplot(ges),grid

% We clearly see the dominant frequency component of 50 Hz, and its harmonics.
% Comparing with the pure periodogram gives:
figure, ffplot(ges,ge)

% The standard spectral analysis estimate gives (with the default window 
% size, which is not adjusted to resonant spectra)
gs =  spa(i4r);
figure,ffplot(gs,ges)

% We see that a very large lag window will be required to see all the fine
% resonances of the signal. Standard spectral analysis does not work well.
% Let us instead compute spectra by parametric AR-methods. Models of 2nd
% 4th and 8th order are obtained by
t2=ar(i4r,2); 
t4=ar(i4r,4); 
t8=ar(i4r,8); 
 
% Let us take a look at the spectra:
figure,ffplot(t2,t4,t8,ges)

% We see that the parametric spectra are not capable of picking up the
% harmonics. The reason is that the AR-models attach too much attention to
% the higher frequencies, which are difficult to  model. (See Ljung (1999)
% Example 8.5)
% We will have to go to high order models before the harmonics are picked
% up.
% What will a useful order be?
V=arxstruc(i4r(1:301),i4r(302:601),[1:30]'); % Checking all order up to 30
nn=selstruc(V,'log');

% We see a dramatic drop for n=20, so let's pick that order
t20=ar(i4r,20); 
figure,ffplot(ges,t20)

% All the harmonics are now picked up, but why has the level dropped?
% The reason is that t20 contains very thin but high peaks. With the
% crude gitter of frequency points in g20 we simply don't see the 
% true levels of the peaks. We can illustrate this as follows:
g20c = idfrd(t20,[551:650]/600*150*2*pi); % A frequency region around
                                             % 150 Hz
figure,ffplot(ges,t20,g20c)

% If we are primarily interested in the lower harmonics, and want to
% use lower order models we will have to apply prefiltering of
% the data. We select a fifth order Butterworth filter with cut-off
% frequency at 200 Hz. (This should cover the 50, 100 and 150 Hz modes):
i4rf = idfilt(i4r,5,200/500);% 500 Hz is the Nyquist frequency

t8f=ar(i4rf,8);  

% Let us now compare the spectrum obtained from the filtered data (8th
% order model) with that for unfiltered data (8th order) and with the
% periodogram:
figure,ffplot(t8f,t8, ges)

% We see that with the filtered data we pick up the three first peaks in
% the spectrum quite well. 
% We can compute the numerical values of the resonances as follows:
% The roots of a sampled sinusoid of frequency om are located on
% the unit circle at exp(i*om*T), T being the sampling interval. We
% thus proceed as follows:
 a = t8f.a % The AR-polynomial

omT=angle(roots(a))'


freqs=omT/0.001/2/pi'  % In Hz


% We thus find the harmonics quite well.  We could also test how well
% the model t8f is capable of predicting the signal, say 50 ms ahead:
figure,compare(i4rf,t8f,50,201:500);

% If we were interested in the fourth and fifth harmonics (around
% 200 and 250 Hz) we would proceed as follows:
i4rff=idfilt(i4r,5,[150 300]/500);
t8f=ar(i4rff,8);  
figure,ffplot(ges,t8f)

% We thus see that with proper prefiltering, low order parametric models
% can be built that give good descriptions of the signal over desired
% frequency ranges.
% What would the 20th order model give in terms of estimated harmonics:
a = t20.a  % The AR-polynomial

omT=angle(roots(a))'


freqs=omT/0.001/2/pi'  % In Hz


% We see that this model picks  up the harmonics very well.
% This model will also predict as follows:
figure,compare(i4r,t20,50,201:500);

% For a complete model of the signal, t20 thus is the natural choice,
% both in terms of finding the harmonics and in prediction capabilities.
% For models in certain frequency ranges we can however do very well
% with lower order models, but we then have to prefilter the data
% accordingly.