Kalman filter with elliptical (quadratic) constraint

I am struggling with a problem in which the states and measurements
are both implicit in a constraint of the form

(mx-bx)^2/(1+sx)^2 + (my-by)^2/(1+sy)^2 = 1

where mx, bx are measurements and bx, by sx, sy are states to be
estimated. The states are generally constant but occassionally exhibit
discontinuities, and it is these discontinuities which I would like to
track.

I have been treating the constraint equation as a
"pseudo-measurement".

I have tried a standard extended KF, an extended "Bayes" filter, a
Schmidt KF (estimating bx and by only) and several variations.
Everything I have tried has been unstable. The matrix H*Px*HT (H -
Jacobian of constraint, Px state covariance) is very ill-conditioned.

I have experimented with various a priori covariances and with both
constant and Markov process models (with varying correlation times)
for the states.

Does anyone have any suggestions as to how to proceed?

Note that I made a mistake in the previous post. The measured
quantities are mx and my.

Jay Goldfarb

jmg@gci.net (Jay Goldfarb) wrote in message news:<413ae280.0311081258.20af11f8@posting.google.com>...
> I am struggling with a problem in which the states and measurements
> are both implicit in a constraint of the form
> 
> (mx-bx)^2/(1+sx)^2 + (my-by)^2/(1+sy)^2 = 1
> 
> where mx, bx are measurements and bx, by sx, sy are states to be
> estimated. The states are generally constant but occassionally exhibit
> discontinuities, and it is these discontinuities which I would like to
> track.
> 
> I have been treating the constraint equation as a
> "pseudo-measurement".
> 
> I have tried a standard extended KF, an extended "Bayes" filter, a
> Schmidt KF (estimating bx and by only) and several variations.
> Everything I have tried has been unstable. The matrix H*Px*HT (H -
> Jacobian of constraint, Px state covariance) is very ill-conditioned.
> 
> I have experimented with various a priori covariances and with both
> constant and Markov process models (with varying correlation times)
> for the states.
> 
> Does anyone have any suggestions as to how to proceed?

Jay Goldfarb wrote:
> Note that I made a mistake in the previous post. The measured
> quantities are mx and my.
> 
> Jay Goldfarb
> 
> jmg@gci.net (Jay Goldfarb) wrote in message news:<413ae280.0311081258.20af11f8@posting.google.com>...
> 
>>I am struggling with a problem in which the states and measurements
>>are both implicit in a constraint of the form
>>
>>(mx-bx)^2/(1+sx)^2 + (my-by)^2/(1+sy)^2 = 1
>>
>>where mx, bx are measurements and bx, by sx, sy are states to be
>>estimated. The states are generally constant but occassionally exhibit
>>discontinuities, and it is these discontinuities which I would like to
>>track.
>>
>>I have been treating the constraint equation as a
>>"pseudo-measurement".
>>
>>I have tried a standard extended KF, an extended "Bayes" filter, a
>>Schmidt KF (estimating bx and by only) and several variations.
>>Everything I have tried has been unstable. The matrix H*Px*HT (H -
>>Jacobian of constraint, Px state covariance) is very ill-conditioned.
>>
>>I have experimented with various a priori covariances and with both
>>constant and Markov process models (with varying correlation times)
>>for the states.
>>
>>Does anyone have any suggestions as to how to proceed?

Have you tried to transform your state space to another coordinate 
system, like polar coordinates?

jmg@gci.net (Jay Goldfarb) writes:

> I am struggling with a problem in which the states and measurements
> are both implicit in a constraint of the form
> 
> (mx-bx)^2/(1+sx)^2 + (my-by)^2/(1+sy)^2 = 1
> 
> where mx, bx are measurements and bx, by sx, sy are states to be
> estimated. The states are generally constant but occassionally exhibit
> discontinuities, and it is these discontinuities which I would like to
> track.

It seems to me that the linearization (since A=0) is bound to be
unobservable, thus (I think) the Kalman Filter can not work?

Lars