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 .

Thu Feb 8 11:42:20 MET 1996