Extended Kalman Filtering for Nonlinear Stochastic Modeling Tasks

This chapter presents a sound insight into the theory of nonlinear filtering for continuous–discrete stochastic systems whose process models are of a continuous-time fashion, whereas the measurement ones in use are discrete time. In particular, it gives precise definitions and explains all basic issues and notions of state estimation in nonlinear Gaussian systems of such sort. A special emphasis is placed on two extended Kalman filtering design approaches and on their practical implementation aspects since these can expose instabilities in solving real-world state estimation tasks because of the linearization, discretization and rounding operations implemented in computer-based simulations. Under some circumstances, such numerical integration and round-off errors committed may affect severely the calculation and result in non-symmetric and / or indefinite covariance matrices yielded, which compromise the theoretical rigor of the extended Kalman filtering and produce state estimates of poor accuracy. This chapter pays its particular attention to the issue of numerical stability and presents a remedy for treating such a covariance-matrix-symmetry-and-positivity-loss in the fashion of square-root filtering. Two specific square-rooting techniques grounded on the Cholesky factorization and SVD are considered and justified, here. The theoretical analysis of extended Kalman filters under exploration, which are summarized in the kind of concise pseudo-codes situated in Appendix of this chapter, is supported with illustrative calculations performed in MATLAB.