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Kalman filter with elliptical (quadratic) constraint

Started by Jay Goldfarb November 8, 2003
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