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?