Approximate finite-dimensional filtering for polynomial states over polynomial observations

Abstract

In this article, the mean-square filtering problem for polynomial system states over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, this article deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third-order sensor filtering problem for a quadratic state, assuming a conditionally Gaussian initial condition for the extended third-order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.

Keywords:

• filtering
• nonlinear systems
• stochastic systems

Acknowledgements

The authors thank the UK Engineering and Physical Sciences Research Council (EPSRC) and the Mexican National Science and Technology Council (CONACyT) for financial support under EPSRC Grant EP/F029195/1 and CONACyT Grants 55584 and 52953.