Bayesian Estimation and Tracking: A Practical Guide

7.2 Multidimensional Consideration

For multidimensional functions with multidimensional arguments, the state prediction integrals for are given by (5.51)–(5.57). In both the linear Kalman filter and the one-dimensional EKF it is obvious from (6.3) and (7.23) that the deterministic control function u_n is additive for the state prediction and does not appear in any of the remaining filter equations. So, for what follows, we will ignore u_n and add it back in if it is needed for the case studies.

The general multidimensional Taylor polynomial was presented in (2.63). For a second-order Taylor polynomial for expanded about the zero-mean value of c, using (2.69) in (2.63) results in

(7.30)

It follows immediately that (7.30) can be rewritten as

(7.31)

7.2.1 The State Prediction Equation

From (5.51), using the Taylor polynomial (7.31), we can express the predictive first moment as

(7.32)

Using ...