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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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