# Kalman Filter - Order of Update Step?

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?

• 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. – mark leeds May 10 '19 at 9:39