I was recently experimenting with UKF that compensates nonlinear effects of noises by forming appended state vector. The general way of dealing with such case in the literature is the following:
- Form appended state vector and expanded state covariance matrix.
- Do a Cholesky decomposition (or some other square root decomposition) and form sigma points.
- Propagate sigma points through state equation and calculate predicted state and state covariance matrix.
- Additionally propagate predicted sigma points through measurement equation as well and form estimated output vector and output covariance matrix
- Do standard UKF update calculations
This works fine, but I wondered why couldn't we do double Cholesky decomposition like we do in "vanilla" UKF with additive noises where we can do another sigma point generation with propagated state covariance matrix. So I tried to do first Cholesky decomposition where state vector was appended with state noises and expanded covariance matrix, and then do a second Cholesky decomposition with predicted state and covariance matrix which were appended by measurement noise effects. To my surprise the results of such filter were very bad. I cannot seem to explain to myself why would that be the case. Could anyone shed some light on why would such procedure not work very well.
Just as a note, I did make sure to adjust filter parameters taking into account sizes of appended state vectors (which were different in cases for first and second decompositions).