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Extended Kalman Filter


  If tex2html_wrap_inline3178 is not linear or a linear relationship between tex2html_wrap_inline2849 and tex2html_wrap_inline3258 cannot be written down, the so-called Extended Kalman Filter (EKF for abbreviation) can be appliedgif.

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 tex2html_wrap_inline3324 into a Taylor series about tex2html_wrap_inline3330 :


By ignoring the second order terms, we get a linearized measurement equation:


where tex2html_wrap_inline3332 is the new measurement vector, tex2html_wrap_inline3334 is the noise vector of the new measurement, and tex2html_wrap_inline3336 is the linearized transformation matrix. They are given by


Clearly, we have tex2html_wrap_inline3338 , and tex2html_wrap_inline3340 .

The extended Kalman filter equations are given in the following algorithm, where the derivative tex2html_wrap_inline3342 is computed at tex2html_wrap_inline3344 .


Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996