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From: Vladimir Vassilevsky on 30 Jun 2008 13:50
Hello All,

There is a physical system which is described well by the 2nd order pole
only model. The poles can be complex (Q up to 10), or real, including
the degenerated cases with one pole or no poles at all. The impulse
response can be measured.

The problem is to find the system parameters from the impulse response
as accurate as possible. There are methods based on the Bode plots,
min/max peaks, slopes, zero crossing position estimations. However they
look pretty much ad-hoc, require the initial decision about the system
type, and prone to errors.

I made the AR estimator: compute the autocorrelation function, then
Levinson-Durbin to get the model coefficients, then solve for the poles.
However it appears that the result is dependent on the sample rate,
especially in the case of low Q poles. This is not a numeric issue. The
result could be completely wrong with the higher oversampling, and this
looks to be the problem of the approach.

What would be the right way to estimate the parameters?


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com



From: Rune Allnor on 30 Jun 2008 14:17
On 30 Jun, 19:50, Vladimir Vassilevsky <antispam_bo...(a)hotmail.com>
wrote:
> Hello All,
>
> There is a physical system which is described well by the 2nd order pole
>   only model. The poles can be complex (Q up to 10), or real, including
> the degenerated cases with one pole or no poles at all. The impulse
> response can be measured.
>
> The problem is to find the system parameters from the impulse response
> as accurate as possible. There are methods based on the Bode plots,
> min/max peaks, slopes, zero crossing position estimations. However they
> look pretty much ad-hoc, require the initial decision about the system
> type, and prone to errors.
>
> I made the AR estimator: compute the autocorrelation function, then
> Levinson-Durbin to get the model coefficients, then solve for the poles.
> However it appears that the result is dependent on the sample rate,
> especially in the case of low Q poles. This is not a numeric issue. The
> result could be completely wrong with the higher oversampling, and this
> looks to be the problem of the approach.
>
> What would be the right way to estimate the parameters?

I would pursue the AR line, with focus on numerical methods.
Use an as long observation record as you dare (you don't
mention whether this is stationary) and get a good estimate
for the signal autocovariance matrix. The matrix ought to
be dimension at least 6 x 6, provided you are right that
there is at most one pair of poles.

You could try the Levinson recursion + order estimators
with succesively longer frames and see if better statistics
help.

My second approach would be to use the SVD to compute the
parameters for the AR model. Set up a data matrix (details
in the 1982 paper by Tufts and Kumaresan) compute the
SVD and look at the singular values to determine the order.

These sorts of inquiries *might* work, but may also be
too computationally expensive for your application.

Is there any chance that you post example data somewhere?
Preferably both easy and tricky ones. Ah, and a control,
where you have modeled the data (both easy and tricky)
and know the answers. If you could post these data
somewhere and not disclose what data set is which, it
would be very interesting to see what might be done with
them.

Rune
From: Greg Berchin on 30 Jun 2008 14:28
On Jun 30, 1:50 pm, Vladimir Vassilevsky <antispam_bo...(a)hotmail.com>
wrote:

> The problem is to find the system parameters from the impulse response
> as accurate as possible.

How noisy is your impulse response characterization? Prony's Method
does exactly this, but becomes inconsistent in the presence of noise.

Greg
From: Vladimir Vassilevsky on 30 Jun 2008 15:09


Rune Allnor wrote:
> On 30 Jun, 19:50, Vladimir Vassilevsky <antispam_bo...(a)hotmail.com>
> wrote:
>
>>There is a physical system which is described well by the 2nd order pole
>> only model.
>>The problem is to find the system parameters from the impulse response
>>as accurate as possible.
> Is there any chance that you post example data somewhere?

> Preferably both easy and tricky ones. Ah, and a control,
> where you have modeled the data (both easy and tricky)
> and know the answers.

Here is the typical one:

http://www.abvolt.com/misc/data.cpp

> If you could post these data
> somewhere and not disclose what data set is which, it
> would be very interesting to see what might be done with
> them.

VLV
From: stanp on 30 Jun 2008 15:24
On Jun 30, 1:50 pm, Vladimir Vassilevsky <antispam_bo...(a)hotmail.com>
wrote:
> Hello All,
>
> There is a physical system which is described well by the 2nd order pole
> only model. The poles can be complex (Q up to 10), or real, including
> the degenerated cases with one pole or no poles at all. The impulse
> response can be measured.
>
> The problem is to find the system parameters from the impulse response
> as accurate as possible. There are methods based on the Bode plots,
> min/max peaks, slopes, zero crossing position estimations. However they
> look pretty much ad-hoc, require the initial decision about the system
> type, and prone to errors.
>
> I made the AR estimator: compute the autocorrelation function, then
> Levinson-Durbin to get the model coefficients, then solve for the poles.
> However it appears that the result is dependent on the sample rate,
> especially in the case of low Q poles. This is not a numeric issue. The
> result could be completely wrong with the higher oversampling, and this
> looks to be the problem of the approach.
>
> What would be the right way to estimate the parameters?
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultanthttp://www.abvolt.com

Sample rate dependency implies you might have some MA in the system.
Additive measurement noise also makes AR systems ARMA.
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