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?

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

Jay Goldfarb

Reply to
Jay Goldfarb

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

Reply to
kyle

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

Reply to
Lars Imsland

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