Approximate linear minimum variance filters for continuous-discrete state space models: convergence and practical adaptive algorithms

In this article, approximate linear minimum variance (LMV) filters for continuous-discrete state space models are introduced. The filters are derived from a wide class of recursive approximations to the predictions for the first two conditional moments of the state equation between each pair of consecutive observations. The convergence of the approximate filters to the exact LMV filter is proved when the error between the predictions and their approximations decreases no matter the time distance between observations. As particular instance, the order-beta local linearization filters are presented and expounded in detail. Practical adaptive algorithms are also provided and their performance in simulation is illustrated with various examples. The proposed filters are intended for the recurrent practical situation where a stochastic dynamical system should be identified from a reduced number of partial and noisy observations distant in time.