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then it can be verified that
e.1/M
T
H
T
C
e.2/.M
2
/
T
H
T
r
x.0/
f.x.0//
D
1
2
(2.76)
0
a
and
0
a
2
,weget
M
T
H
T
D
.M
2
/
T
H
T
D
But since
0
ae.1/
C
a
2
e.2/
r
x.0/
f
D
1
2
(2.77)
this in turn implies that
f.x.0//
is a constant with respect to the first component of
x.0/
. Hence, the initial condition
x.0/
cannot be recovered from
z
.1/
and
z
.2/
2.4.2
Observer-Based Nudging: Linear Dynamics
Let
x.k
C
1/
D
M x.k/
(2.78)
where
x.k/
is the true linear dynamical state with true initial condition
x.0/
and
z
.k/
D
H x.k/
C
V.k/
(2.79)
.M;H/
the observations. It is assumed that the pair
is observable (See Sect.
2.4.1
).
Let the observer be given by (
Luenberger 1964
,
1971
) the dynamics
x.k
C
1/
D
Mx.k/
C
G.
.k/
Hx.k//
z
(2.80)
G
2
R
n
m
. The idea of the observer is that the observer state
x.k/
where
is an
x.k/
estimate of the true state
. In the parlance of meteorology this observer is
called the nudged dynamics (
Anthes
(
1974
)) and the matrix
G
is called the nudging
coefficient.
To analyze the behavior of (
2.80
), using (
2.79
) and simplifying, we obtain
e.k/
D
x.k/
x.k/
(2.81)
Subtracting (
2.78
) from (
2.80
), using (
2.81
) and simplifying, we obtain
e.k
C
1/
D
.M
GH/e.k/
C
GV.k/
(2.82)
Since it is given that the pair
.M:H/
is observable, by the fact 2.1 in Sect.
2.4.1
,
G
2
R
n
m
such that
there exists a matrix
.M
GH/
is a Hurwitz matrix. Then
setting
A
D
.M
GH/
, from (
2.82
) we obtain
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