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must be of full rank, that is
Rank.H.0
W
N
1//
D
n
. Consequently, if
H.0
W
N
1/
satisfies this condition, the pair
.M;H/
is said to be observable. By Cayley-
M
n
can be expressed as a linear combination of
M
k
for
Hamilton theorem since
0
k
n
1
, it follows that
N
D
n
observations would suffice. Hence, the pair
.M;H/
.
We now quote a mathematical fact that we need in the analysis of observer-based
nudging considered in Sect.
2.4.2
.
Fact 2.1
: If the system (
2.63
)and(
2.65
) is such that the pair
is observable if
H.0
W
n
1/
is of rank
n
.M;H/
is observable
G
2
RH
n
m
such that
then there exists a matrix
.M
GH/
is a Hurwitz matrix
(2.73)
That is, the eigenvalues
i
,
1
i
n
,of
.M
GH/
are such that j
i
j
<1
for all
1
i
n
where j
a
j denotes the absolute value of the complex number
a
.Thatis,
the eigenvalues of
lie within the unit circle in the complex plane. Refer
to Chap.
12
in
Bernstein
(
2009
) for a proof of this fact.
.M
GH/
x.k/
D
.x
1
.k/;x
2
.k//
T
;
Example 2.3.
Let
n
D
2
and
m
D
1
.Then
z
.k/
2
R:
Let
11
0a
.Then
H
D
Œ0;1
and
M
D
x.k
C
1/
D
Mx.k/
in component form is given
by
x
1
.k
C
1/
D
x
1
.k/
C
x
2
.k/
x
2
.k
C
1/
D
ax
2
.k/
and
z
.k/
x
2
.k/
C
V.k/
V.k/
N.0;
2
/
Where
. It can be verified that
H
HM
01
0a
HŒ0
W
1
D
D
is of rank 1 and hence the pair
is not observable.
The import of the above example can be interpreted from another angle by
using the standard data assimilation point of view. Let
z
.M;H/
.1/
and
z
.2/
be the two
observations and let
.k/
HM
k
x.0/
e.k/
D
z
.k/
Hx.k/
D
z
(2.74)
be the residuals for
k
D
1
and
2
. Consider the sum of the squared residuals
e
2
.1/
C
e
2
.2/
1
2
2
f.x.0//
D
(2.75)
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