Nonlinear Filtering

Abstract

Some various approaches to state estimation for nonlinear stochastic systems are presented in this chapter. Typically, systems of the form

$$ \begin{array}{*{20}{c}} {x(t + 1) = f(x(t)) + v(t),{\text{ }}} \\ {y(t) = h(x(t)) + e(t),} \end{array}{\text{ }}$$

((9.1))

are treated, where v(t) and e(t) are mutually independent white noise sequences. Recall, from Chapter 5, that the optimal state estimate is given by the conditional mean, E[x(t)|Y ^t]. To compute the optimal state estimate, it is also necessary to compute recursively the conditional pdfs p(x(t)|Y ^t). As explained in Section 5.4, in most cases this will require a huge amount of computation, so suboptimal schemes are of practical interest. We also present a numerical scheme based on Monte Carlo simulations for numerically evaluating how the conditional pdfs are propagating. In another section we present a practically important suboptimal algorithm, called the interacting multiple model (IMM) approach. The basic idea is to let the dynamics jump between a fixed number of linear models. Some other nonlinear estimation problems, such as median filters and quantization effects, are also discussed in this chapter.