From: Nan Li (annli_at_cs.washington.edu)
Date: Fri Apr 25 2003 - 03:44:46 PDT
An Introduction to the Kalman Filter"/Greg Welch and Gary Bishop
The Kalman Filter is a recursive solution for state estimation in a
linear, discrete-timed domain.
The underlying idea of the Kalman Filter is like most closed cycle control
algorithms: estimate and correct from the feedback. It includes two sets
of equations to implement the estimation-correct iteration: time update
equations (estimate the current state based on the previous state), and
measurement update equations (correct the estimation based on feedback).
It's optimal in the sense that it minimizes the estimated error
covarinace, which is implemented buy choosing some parameter in the
measurement update equations.
Meanwhile, the Kalman Filter maintains an elegant style: the write-upis
neat, the computation is tractable. This advantage is earned from two
aspects. One is the recursive idea borrowed from closed cycle control
theories. The current state is estimated based on information merely in
the previous step, no need to store a whole history any more. The other
one is an important assumption: the process is linear. Thus, the
normal-distributed feature can be kept on through transformations. So the
algorithm only need to maintain the first two moments of the state
distribution.
Yet this assumption is a two-sided sword. For some non-linear processes,
EKF is useful by making local linear approximations. But all non-linear
processes cannot be done in this way. An example is when the state value
is defined on a discrete domain. Probably some work has been done
connecting KF and HMM.
One little complaint to the paper is, it would have been even clearer
if some details had been provided. For example, equation (4.7) is the core
of the KF, yet I can not understand from 4.1.3 where it comes from.
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