Tobias,
Your Kalman post triggered me to study this "dummy" level introduction
to Kalman filters: https://www.kalmanfilter.net/
and now I'm trying to write the Kalman filter.
An example of a Kalman filter described in the link above uses distance,
speed and acceleration as predictors for a one dimensional Kalman filter
So I wondered if it would be possible to do the same for the phase, eg
use phase, frequency and drift in the prediction model and still have a
one dimensional model?
The Kalman filter thus uses the normalized phase as input (the measured
time difference between the 1 PPS and the 1 second pulse derived from
the VCO output) to predict the next phase, frequency and drift
p = phase
f = frequency (d Phase/d T)
d = drift (d Frequency / d T)
m = measured phase
with a 1 second interval between observations the state extrapolation
(or prediction) is
p[k+1] = p[k] + f[k] + 0.5*d[k]
f[k+1] = f[k] + d[k]
d[k+1] = d[k]
and the state update is
p[k] = p[k-1] + alpha*( m - p[k-1])
f[k] = f[k-1] + beta*(m-p[k-1])
d[k] = d[k-1] + gamma*(m-p[k-1])/0.5)
In a small simulation (in excel) I used a VCO that initially is on the
correct frequency, after some seconds Vtune has a jump and after some
more seconds the Vtune starts to drift.
The Vtune of the VCO can have noise(system error) and the m can have
noise (measurement error).
The p,f and m nicely track the input values even when measurement noise
is added but after this I'm stuck as I do not yet understand how to
calculate the alpha, beta and gamma values from the observed measurement
error as the above website does not yet explain how to do this.
And I do not understand how to close the loop to either keep the phase
error or frequency error as low as possible.
So much to learn.
Erik.
On 16-4-2022 22:44, Pluess, Tobias via time-nuts wrote:
Hallo all,
In the meantime I had to refresh my knowledge about state-space
representation and Kalman filters, since it was quite a while ago since I
had this topic.
So I looked at the equations of the Kalman filter. To my understanding,
we
can use it like an observer, and instead of using the phase error and
feeding it to the PI controller, we use the output of the Kalman filter
as
input to the PI controller. And the Kalman filter gets its input from the
phase error, but we also tell it how much variance this phase error has.
Luckily, the GPS module outputs an estimate of the timing accuracy, so I
believe one could use this (after squaring) as the estimate of the timing
variance, correct?
I believe depending on how we model the VCO, we can get away with a
scalar
Kalman filter and circumvent the matrix and vector operations.
I tried to simulate it in Matlab, and it kind of worked, but I noticed
some
strange effects.
a) I made a very simple VCO model, that simulates the phase error. It is
x[k+1] = x[k] + KVCO * u with u being the DAC code. If KVCO is chosen
correctly, this perfectly models the phase measurements. I assumed the
process noise is zero, and the 1PPS jitter contributes only to the
measurement noise.
b) from the above model, we have a very simple state space model, if you
want to call it like this. We have A = 1, B=KVCO, C=1, D=0.
c) in the "prediction phase" for the Kalman filter, the error covariance
(in this case, the error variance, actually) is P_new=APA' + Q, which
reduces in this case to P_new=P+Q with Q being the process noise
variance,
which I believe is negligible in this case.
d) in the "update phase" of the Kalman filter, we find the Kalman gain as
K=P*C*inv(C'*P*C + R), and this reduces, as everything is scalar and C=1,
to K=P/(P+R), with R being the measurement noise, which, I believe, is
equal to the timing accuracy estimate of the GPS module. Correct? we then
update the model xhat = A*xhat + b*u + K*(y-c*xhat), which simplifies to
xhat=xhat + Kvco*u + K*(y-xhat). Nothing special so far.
e) now comes my weird observation. I don't know whether this is correct.
The error covariance is now updated according to P=(I-K*C)*P, this breaks
down to P=P-K*P. I now observe that P behaves very odd, first we set
P=P+Q,
and then we set P=P-K*P. It does not really converge in my Matlab
Simulation, and I see that the noise is filtered somewhat, but not very
good. It could also be related to my variances not being correctly set, I
am not sure. Or I made some mistakes with the equations.
Any hints?
As soon as I see it sort of working in Matlab, I want to test it on my
GPSDO. Especially the fact that I have an estimate of the timing error
(the
GPS module announces this value via a special telegram!) I find very
amazing and hope I can make use of this.
best
Tobias
HB9FSX
On Tue, Apr 12, 2022 at 11:23 PM glen english LIST <
glenl...@cortexrf.com.au>
wrote:
For isolating noise, (for the purpose of off line analysis) , using ICA
(Independent Component Analysis) , a kind of blind source separation ,
can assist parting out the various noises to assist understanding the
system better . There are some Python primers around for it.
fantastic discussion going on here. love it.
glen
On 12/04/2022 6:42 pm, Markus Kleinhenz via time-nuts wrote:
Hi Tobias,
Am 11.04.2022 um 13:33 schrieb Pluess, Tobias via tim
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