Extended Kalman Filter

The extended Kalman filter Extended Kalman Filter (EKF) is a nonlinear version of the Kalman filter used when the evolution or observation of the system state is nonlinear.

A discrete-time nonlinear system, consisting of state evolution and state observation, can be written in generalized form as

$$\left\{ \begin{array}{rl} \mathbf{x}_{k+1} & = f (\mathbf{x}... ..._k & = h (\mathbf{x}_{k}, \mathbf{v}_k) \\ \end{array}\right.$$ (3.30)

where, in addition to the state $\mathbf{x}_{k}$ and inputs $\mathbf{u}_{k}$, the process errors $\mathbf{w}_{k}$ and observation errors $\mathbf{v}_k$ may also affect the evolution of the state $f$ and the observation $h$ nonlinearly, thereby generalizing the concept of additive noise used earlier.

To be applied, the EKF requires computing the Jacobians of both $f$ and $h$. Applying the theory presented in Section 2.6 on uncertainty propagation through nonlinear functions, the same mathematical formulations used for the linear Kalman case can be applied to nonlinear functions through derivative matrices, using as matrices

$$\begin{array}{ll} \mathbf{A}_k = \left. \frac{\partial f(\ma... ...\partial \mathbf{v}}\right\vert _{\bar{\mathbf{x}}} \end{array}$$ (3.31)

and using the following update equation:

$$\hat{\mathbf{x}}_{k} = \hat{\mathbf{x}}^{-}_{k} + \mathbf{K}_k( \mathbf{z}_k - h(\hat{\mathbf{x}}^{-}_{k}) )$$ (3.32)

It should nevertheless be noted that the residual $\mathbf{z}_k - h(\hat{\mathbf{x}}^{-}_{k})$ may also be a nonlinear function, for example when comparing angles and the error is periodic.

Compared with the linear Kalman filter, the EKF is a suboptimal estimator, but it is nevertheless widely accepted and used in practical applications. By construction, the extended Kalman filter achieves only first-order accuracy, but it can still provide near-optimal results when the filter operates at points where the second derivatives are zero.