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Integration drift with numerical simulations

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Hi Guys
I have solved a system of differential equations, whereby some of the variables are presenting integration drift in the results.
I have attached a spreadsheet with the results of one of the variables (theta) note that the time component was exluded to keep the file small. The time increments are 0.0001seconds with t starting at 0.
The plot of the results is presented below.
The inital conditions of the problem has theta = 0.1, which is correct when reviewing the data in the spreadsheet.
I have used the code below to process the data, however upon processing the data the inital condition is no longer true as the value is no longer 1. Note that the code does not present the data being read from the excel file as it already exists in a variable (theta).
Could someone please give me some advice on how one could accurately process the data?
T = detrend(theta,1);
figure(33)
plot(t,T)
Fs = 1/inc;
Fc = 1/(Fs/2);
[b a] = butter(4,Fc,'high');
T = filtfilt(b,a,T);
  3 Comments
Mathieu NOE
Mathieu NOE on 26 Oct 2021
hello Mishal
well doing filter can alter initial condition
if you use filter, this is only a forward time filter and you see the IC are kept the same so this is fine , but this way of doing filtering will cause a good amount of phase lag as will do any filter
if you use filtfilt , this applies the filtering both in forward and backward time axis, so this will cancel the phase lag and keep your signals time aligned , but the code cannot garantee that IC will be kept exact the same.
I will suggest you a solution in the answer section
see if it fills your needs and if you accept it

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Accepted Answer

Mathieu NOE
Mathieu NOE on 26 Oct 2021
here one possible solution
we want to have the signal not time distorted so we must keep with filtfilt even with the "bad" IC effects
a work around I suggest is to blend the unfiltered (raw) data and the filtered data within the first second, so that it will almost look like the raw data and then progressively match only the filtered data . So IC are kept , whatever the characteristics of your high pass filter
zoom on the start
overal behaviour
code :
theta = importdata('theta.txt');
theta = theta.data;
inc = 0.0001; % second
samples = length(theta);
t = inc*(0:samples-1);
Fs = 1/inc;
Fc = 1/(Fs/2);
[b, a] = butter(2,Fc,'high');
T = filtfilt(b,a,theta);
% blending the data on the first second
blend = ones(size(t))';
tmp = (0:inc:1);
blend(1:length(tmp)) = tmp;
T2 = blend.*T + (1-blend).*theta;
plot(t,theta,t,T,t,T2)
legend('raw','HP filtered','HP filtered and blended')
  2 Comments
Mathieu NOE
Mathieu NOE on 27 Oct 2021
hello Mishal
ok - tx for accepting my submission
I have tried many other options, but none could really give me the right output
finally went to this trick as best available short term solution....

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