Nonlinear Filters

5Kalman Filter

5.1 Introduction

Chapter 4 presented the general formulation of the optimal Bayesian filter. Furthermore, it was mentioned that except for special cases, a closed‐form solution for this problem does not exist. Perhaps the most famous special case is a linear dynamic system perturbed by additive white Gaussian noise, for which a recursive estimator that is optimal in the sense of minimizing the mean‐square of estimation error has an analytic solution [56]. Such an optimal estimator is the celebrated Kalman filter (KF), which is considered as the most useful and widely applied result of the state‐space approach of modern control theory [57]. Since the publication of Rudolf Emil Kalman's seminal paper in 1960 [58], KF found immediate applications in almost all practical control systems, and it has become a necessary tool in any control engineer's toolbox [3, 47, 56, 59–63].

However, almost all real systems are nonlinear in nature, and therefore, do not satisfy the basic assumptions for derivation of KF. In order to extend application of KF to nonlinear estimation problems, a number of filtering algorithms have been proposed over decades. These algorithms can be categorized into two main groups:

• Filters that use power series to approximate the nonlinear functions in the state and/or measurement equations. Extended Kalman filter (EKF), which uses Taylor series expansion [3], and divided‐difference filter (DDF), which uses Stirling's interpolation formula [64], belong ...