Extended Kalman Filter - Continuous-time Extended Kalman Filter

Continuous-time Extended Kalman Filter

Model

\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}(t) &= h\bigl(\mathbf{x}(t)\bigr) + \mathbf{v}(t), &\mathbf{v}(t) &\sim N\bigl(\mathbf{0},\mathbf{R}(t)\bigr) \end{align}

Initialize

\hat{\mathbf{x}}(t_0)=E\bigl \text{, } \mathbf{P}(t_0)=Var\bigl

Predict-Update

\begin{align} \dot{\hat{\mathbf{x}}}(t) &= f\bigl(\hat{\mathbf{x}}(t),\mathbf{u}(t)\bigr)+\mathbf{K}(t)\Bigl(\mathbf{z}(t)-h\bigl(\hat{\mathbf{x}}(t)\bigr)\Bigr)\\ \dot{\mathbf{P}}(t) &= \mathbf{F}(t)\mathbf{P}(t)+\mathbf{P}(t)\mathbf{F}(t)^{\top}-\mathbf{K}(t)\mathbf{H}(t)\mathbf{P}(t)+\mathbf{Q}(t)\\ \mathbf{K}(t) &= \mathbf{P}(t)\mathbf{H}(t)^{\top}\mathbf{R}(t)^{-1}\\ \mathbf{F}(t) &= \left . \frac{\partial f}{\partial \mathbf{x} } \right \vert _{\hat{\mathbf{x}}(t),\mathbf{u}(t)}\\ \mathbf{H}(t) &= \left . \frac{\partial h}{\partial \mathbf{x} } \right \vert _{\hat{\mathbf{x}}(t)} \end{align}

Unlike discrete-time extended Kalman filter, the prediction and update steps are coupled in continuous-time extended Kalman filter.

Read more about this topic:  Extended Kalman Filter

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