Fundamentals of Kalman Filtering and Applications to GNSS

Discrete Linear Kalman Estimator
System model:
n1
x
k
nn
k1
n1
k1
n1
k1
k
nn
x w , w ~ N 0,
Q
k
Measurement model:
Initial conditions:
1
n
n1
1
z H x v , v ~ N 0,
R
k
k
k
k
k
T
E x xˆ
~ ~
0 0 , E x0
x0
P0
k
nn
(white)
Other assumptions:
E
w k v T j
0
for all K, j
State estimate extrapolation:
Error covariance extrapolation:
State estimate update:
Error covariance update:
Kalman gain matrix:
xˆ
k
x ˆ k x
P
k
P
xˆ
K z H xˆ
k
k
ˆ k1 k1
T
k 1 P k 1 k 1
Qk
1
k
I
Kk
H k Pk
T
T
K P
H H
P H
R 1
k
k
k
k
k
k
k
k
k
k
Q
R
Kalman Filtering Consultant Associates © 2011
k
k
cov. of
cov. of
process noise w
process noise v
k
k
P
P
k
k
error cov. of states (
error cov. of states ( a priori)
a posteriori)
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