The Kalman filter and the extended Kalman filter

This chapter provides an overview of the main results of the linear Kalman filter (KF), and outlines some extensions for nonlinear systems. The linear KF is perhaps one of the most celebrated signal processing algorithms in practice. A complete solution to the filtering problem for discrete time state space systems with Gaussian noise was published in 1960 by Kalman. An extension of the KF for nonlinear systems was used for onboard guidance and navigation for the Apollo spacecraft. The Unscented KF and the associated deterministic sampling based filter is perhaps the newest family of filtering algorithms. A separate strand of research within the research area of filtering based on deterministic sampling derives from computing the moments of nonlinear functions by computationally efficient means of explicit integration. These filters are called cubature KF and cubature quadrature KF. Extended KF and its variants approach nonlinear filtering by linearizing the system dynamics and then using the KF.