A general approach to constructing parameter identification algorithms in the class of square root filters with orthogonal and $J$-orthogonal tranformations

We study modern implementations of the discrete Kalman filter, namely array square-root algorithms. An important feature of such algorithms is the use of orthogonal and $J$-orthogonal transformations on each filtering step. For the first time, we develop for this class of algorithms a simple universal approach that lets us generalize any numerically stable implementation of this type to the case of updates in sensitivity equations of the filter with respect to unknown system model parameters. An advantage of the resulting adaptive schemes is their numerical stability with respect to machine rounding errors. Estimation of the noisy state vector of the system and identification of unknown system parameters occur simultaneously. The proposed approach can be used for parameter identification problems, adaptive control problems, experiment planning, and others.