A closer look on the evolution of the Kalman gain
unveils a rather unexpected behavior: unlike the innovation term, which displays no sharp transitions during stationary load regimes, the elements of the Kalman gain
matrix exhibit a peculiar oscillation, with no apparent link to the load transients:
As indicated in Figure 1, the Kalman gain
is initially derived.
where P(n/n) is the error covariance matrix at time n; K(n) is the Kalman gain
matrix, responsible for minimizing P(n/n); and [??](n/n) is the state estimate at time n, according to the previous observations of y(n).
Then, we obtain the Kalman gain
in 12 via prior and posterior covariance calculated in 10 and 11.
Define the Kalman gain
matrix as [mathematical expression not reproducible] Step 5.
In the ROMS implementation of 4DVAR (Moore et al., 2004, 2011), nonlinearity of the underlying physics is dealt with by incrementally minimizing a linearized cost function via the Kalman gain
. 4DVAR ROMS remains computationally demanding, even for moderately sized problems.
L(k) is the Kalman gain
and [??](t | k) and P(t | k) are the state estimation and covariance matrix of estimation error at time tick t when the estimation is performed using measurements taken at time tick k.
Based on the above statistics, [P.sub.e] is used to calculate the Kalman gain
of discrete systems; this process has been shown by (34).
Let the processing times taken by error covariance matrix "[[PHI].sub.k]," Kalman gain
"[g.sub.k]," received signal estimation rfc, estimation error "[e.sub.k]," and update filter coefficient matrix "[[??].sub.k]" be [T.sub.[PHI]], [T.sub.g], [T.sub.r], [T.sub.e], and [T.sub.[??]], respectively.
The Kalman gain
K(k) is calculated as the blending factor that minimizes the a posteriori error covariance.
Here, the KF model computes the estimate state [x.sub.k|k] at time k which is evolved from the previous estimate state [x.sub.k-1|k-1]; the [F.sub.k] is the state transition model which is applied to the previous state; the [B.sub.k] is the control-input model which is applied to the control vector [u.sub.k]; the [P.sub.k|k] is the estimate of error covariance matrix; the [K.sub.k] is Kalman gain
; [z.sub.k] is the observation value at the current time k.