Extended Kalman Filter - Discrete-time Extended Kalman Filter

Discrete-time Extended Kalman Filter

Most physical systems are represented as continuous-time models while discrete-time measurements are frequently taken for state estimation via a digital processor. Therefore, the system model and measurement model are given by

\begin{align} \dot{\mathbf{x}}(t) &= f\bigl(\mathbf{x}(t), \mathbf{u}(t)\bigr) + \mathbf{w}(t), &\mathbf{w}(t) &\sim N\bigl(\mathbf{0},\mathbf{Q}(t)\bigr) \\ \mathbf{z}_k &= h(\mathbf{x}_k) + \mathbf{v}_k, &\mathbf{v}_k &\sim N(\mathbf{0},\mathbf{R}_k) \end{align}

where .

Initialize

\hat{\mathbf{x}}_{0|0}=E\bigl, \mathbf{P}_{0|0}=Var\bigl

Predict

\begin{align} &\begin{cases} \dot{\hat{\mathbf{x}}}(t) = f\bigl(\hat{\mathbf{x}}(t), \mathbf{u}(t)\bigr), \\ \dot{\mathbf{P}}(t) = \mathbf{F}(t)\mathbf{P}(t)+\mathbf{P}(t)\mathbf{F}(t)^\top + \mathbf{Q}(t), \end{cases}\qquad \text{with } \begin{cases} \hat{\mathbf{x}}(t_{k-1}) = \hat{\mathbf{x}}_{k-1|k-1}, \\ \mathbf{P}(t_{k-1}) = \mathbf{P}_{k-1|k-1}, \end{cases} \\ \Rightarrow &\begin{cases} \hat{\mathbf{x}}_{k|k-1} = \hat{\mathbf{x}}(t_k) \\ \mathbf{P}_{k|k-1} = \mathbf{P}(t_k) \end{cases} \end{align}

where

Update

where

The update equations are identical to those of discrete-time extended Kalman filter.

Read more about this topic:  Extended Kalman Filter

Famous quotes containing the word extended:

    “Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. The earth is given as a common stock for man to labor and live on.... The small landowners are the most precious part of a state.”
    —Thomas Jefferson (1743–1826)