. 2016 Jul 19;16(7):1109. doi: 10.3390/s16071109

Table 2.

Kalman Filter Update Equations ( ${\hat{x}}_{k}^{-}$ —A priori state estimate at step $k$ , $P_{k}^{-}$ —A priori estimate error covariance at step $k$ , ${\hat{x}}_{k}$ —A posteriori state estimate at step $k$ , $P_{k}$ —A posteriori estimate error covariance at step $k$ , $K_{k}$ —Kalman Gain that is deduced by minimizing $P_{k}$ ).

Time Update Equations		Measurement Update Equations
${\hat{x}}_{k}^{-} = A {\hat{x}}_{k - 1} + B u_{k - 1}$	(12)	$K_{k} = P_{k} H^{T} R_{k}^{- 1}$	(14)
$P_{k}^{-} = A P_{k - 1} A^{T} + Q_{k - 1}$	(13)	${\hat{x}}_{k} = {\hat{x}}_{k}^{-} + K_{k} (z_{k} - H {\hat{x}}_{k}^{-})$	(15)
		$P_{k} = {(P_{k}^{-}^{- 1} + H^{T} R_{k}^{- 1} H)}^{- 1}$	(16)