Mean-Square State Filtering and Parameter Identification for Stochastic Systems with Poisson Noises

First, this talk presents the mean-square filtering problem for incompletely measured polynomial system states, confused with white Poisson noises, over linear observations. Designing the mean-square filter for polynomial systems with white Poisson noises presents a significant advantage in the filtering theory and practice, since it enables one to address the mean-square estimation problems for nonlinear system states confused with other than Gaussian white noises. The procedure for obtaining a closed system of the filtering equations for any polynomial state with white Poisson noises over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular case of a third-order state equation. Next, the mean-square state and parameter estimation problem is addressed for stochastic linear systems with unknown multiplicative and additive parameters over linear observations, where unknown parameters are considered Poisson processes. The obtained optimal filter for the extended state vector also serves as the optimal identifier for the unknown parameters. Performance of the designed optimal state filter and parameter identifier is verified for both, stable and unstable, stochastic linear systems and compared against the mean-square estimator designed for polynomial systems with white Gaussian noises.