Stabilizing the Fast Kalman Algorithms

Fast Kalman algorithms are algorithms that solve, in a very efficient way, the recursive least-squares estimation problem. Unfortunately they are known to exhibit a very unstable behavior, due basically to the accumulation of roundoff errors. It is the structure of the algorithms that favors this accumulation, which is present even when the data are well behaved. In this paper, by introducing a redundant equation, that is, by computing a specific quantity of the algorithms in two different w ays, we use the difference of these two ways as a measure of the accumulation of the roundoff errors. This difference is consequently used to correct the variables of the algorithm at every time step in order to stabilize it. The correction is defined as the solution of a specific minimization problem. The resulting algorithm still has the nice complexity properties of the original algorithm (linear in the number of parameters to be estimated), but has a much more stable behavior.