Accurate cubature and extended Kalman filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements

Applied Numerical Mathematics, 111 (2017), 260-275
http://dx.doi.org/10.1016/j.apnum.2016.09.015

This paper further advances the idea of accurate Gaussian filtering towards efficient cubature Kalman filters for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the moment differential equations describing evolution of the predicted mean and covariance of the propagated Gaussian density in time are solved accurately, i.e. with negligible error. The latter allows the total error of the cubature Kalman filtering to be reduced significantly and results in a new accurate continuous–discrete cubature Kalman filtering method. At the same time, we revise the earlier developed version of the accurate continuous–discrete extended Kalman filter by amending the involved iteration and relaxing the utilized global error control mechanism. In addition, we build a mixed-type method, which unifies the best features of the accurate continuous–discrete extended and cubature Kalman filters. More precisely, the time updates are done in this state estimator as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All these are examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, and compared to the state-of-the-art cubature Kalman filters.