# 2.9 Nonlinear Kalman Filter

The standard Kalman filter is designed mainly for use in linear systems, however, versions of this estimation process have been developed for nonlinear systems, including the extended Kalman filter and the unscented Kalman filter. Since many real-world systems cannot be described by linear models, these nonlinear estimation techniques play a large role in numerous real-world applications.

## Extended Kalman Filter

While the standard Kalman filter is a powerful estimation tool, its algorithms begin to break down when the system being estimated is nonlinear. Fortunately, a version of the standard Kalman filter, known as the extended Kalman filter (EKF), has been extended to nonlinear systems and relies on linearization in estimating these nonlinear systems. Linearization operates on the principle that at a small section around a selected operating point, a nonlinear function can be approximated as a linear function. This linearized function can be derived from the nonlinear function using the first-order terms in a Taylor series expansion shown in Equation \ref{nfk:le}.

\begin{equation}\label{nfk:le} \boldsymbol{g}(x) \approx \boldsymbol{g}(a) + \frac{\partial\boldsymbol{g(x)}}{\partial\boldsymbol{x}}\Bigg|_{x = a}(x-a)\end{equation}

Using this method of linearization, an EKF will follow the same propagate and update process as the standard Kalman filter, but with a few modifications to the standard equations. During the propagate step, rather than using Equation 5 in Section 2.8, the state vector is instead estimated by evaluating the nonlinear system model equations at the most recent state estimate as shown in Equation \ref{nkf:sp}. Additionally, in the state covariance matrix propagation, the state transition matrix, $\Phi$, is replaced with a matrix $F$, which is a Jacobian matrix containing the first-order partial derivatives of the nonlinear system model equations.

In the update step, the expected measurement vector is derived using the nonlinear measurement model equations, evaluated at the most recent state estimate as provided in Equation \ref{nkf:mu}. The measurement model matrix in each of the update equations is also replaced with the $H$ Jacobian matrix containing the first-order partial derivatives of the nonlinear measurement model equations.