If 
is not linear or a linear relationship between
and 
cannot be written down, the so-called
Extended Kalman Filter (EKF for abbreviation) can be
applied
.
The EKF approach is to apply the standard Kalman filter (for linear systems) to nonlinear systems with additive white noise by continually updating a linearization around the previous state estimate, starting with an initial guess. In other words, we only consider a linear Taylor approximation of the system function at the previous state estimate and that of the observation function at the corresponding predicted position. This approach gives a simple and efficient algorithm to handle a nonlinear model. However, convergence to a reasonable estimate may not be obtained if the initial guess is poor or if the disturbances are so large that the linearization is inadequate to describe the system.
We expand 
into a Taylor series about
:
By ignoring the second order terms, we get a linearized measurement equation:
where 
is the new measurement vector, 
is the
noise vector of the new measurement, and 
is the linearized
transformation matrix. They are given by
Clearly, we have
, and
.
The extended Kalman filter equations are given in
the following algorithm, where the derivative 
is computed at 
.
