7. The Extended Kalman Filter
Society for Industrial and Applied Mathematics eBooks · 2008
The solution of the estimation problem for a nonlinear system requires the construction of the conditional probability density function. Based on the conditional probability density function, state estimates, such as the conditional mean estimates, are not implementable for real-time application. Therefore, approximate filters are presented, called the extended Kalman filter. Other nonlinear filters, such as the particle filter [1] and its simplification, the unscented Kalman filter [24], are important new innovations but are beyond the scope of this book. 7.1 Linearized Kalman Filtering Real problems are nonlinear; practical solutions, however, tend to fall out from linear theory. This describes Kalman filtering in the real world. The theory is linear; the applications are not. Yet, there are countless applications that demonstrate that one can effectively use the Kalman filter on nonlinear problems (though this success is by no means universal, nor uniform). In this chapter, we will examine nonlinear Kalman filtering and some variations on these methods to solve particular problems. Our study is by no means comprehensive, since the applications of Kalman filtering are too numerous and varied to fit into a reasonably sized volume. 7.1.1 Continuous-Time Theory Nonlinear Kalman filtering is perhaps a bit of a misleading title. What we are really doing is adapting the linear Kalman filter so that we can apply it to nonlinear problems. Ultimately, this requires us to linearize the problem in some way.