I have seen some literature where the covariance is updated first, like $(P_k)^{-1} = (P_k^-)^{-1} + H^T R^{-1} H$, where $P^-$ is the a priori estimate of the state covariance $P$. Then, the updated covariance is used to calculate the Kalman gain $K$.

And I have seen other literature where the Kalman gain is first calculated using the a priori estimate of $P$, then $P$ is updated as $P_k^- - K_k H P_k^-$.

Could anybody provide any unification on this? Are the two methods mathematically equivalent?

  • $\begingroup$ Hi: there are two different ways that the kalman filter can be written-setup ( as far as how the observation equation is lagged with the system equation ) so, as long as the time subscripting of the setup is done correctly, the two calculation methods are equivalent. But you need to be careful when doing the time subscripting. The equivalence of the two setups might be in (anderson and moore) or jazwinsky. I'll look around and see if I can track it down. $\endgroup$
    – mark leeds
    May 10 '19 at 9:39

If you look at Gary Bishop, Greg Welch - An Introduction to the Kalman Filter (SIGGRAPH 2001, Course 8) then, on page 17, there are equations 3.3 and 3.4:

enter image description here

This is probably not the KF formulation you normally see because the system equation, 3.3, is kind of lagged one behind the observation equation, 3.4, time-wise ( with respect to their right hand sides ). This is less common than the one where the RHS's of both equations has the same time subscript. So, without going into all the gory details, if you derive the KF based on 3.3 and 3.4, you get totally different recursions but they are equivalent to those in the KF formulation where the RHS's of the observation and system equations are contemporaneous rather than lagged by one. Hopefully this gives the general idea. I would have put it in comments since it doesn't provide the details but it wouldn't fit.

  • $\begingroup$ Thanks Peter for editing. I should learn how to do the inclusion of the paper and I will. $\endgroup$
    – mark leeds
    May 11 '19 at 1:12
  • 1
    $\begingroup$ I advise you no to use "This Paper". Always assume one day the link won't be available hence it is better to leave text the readers can search for and find. $\endgroup$
    – Royi
    Jun 9 '19 at 20:30
  • $\begingroup$ good point. thanks. $\endgroup$
    – mark leeds
    Jun 10 '19 at 0:35

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