Geoscience Reference
In-Depth Information
An operator
A
is said to satisfy the
spectrum decomposition assumption at
˛
if
˛
˛
˛
˛
have no
intersection. Under this assumption, we can define the following spectral projection
is bounded and separated from
, i.e. the boundaries of
and
Z
1
2j
˛
.I
A/
1
xd;
P
˛
x
D
j
2
where
˛
is a curve traversed once in the positive direction
(counterclockwise) to enclose an open set containing
D
1
and
˛
˛
in its interior and
in its
exterior. The projection induces a decomposition of the state space
X
:
X
D
X
˛
˚
X
˛
;
X
˛
D
P
˛
X
X
˛
D
.I
P
˛
/X:
where
and
Next let us denote
A
˛
D
P
˛
A;
A
˛
D
.I
P
˛
/A;
(1.29)
C
˛
D
CP
˛
;
C
˛
D
C.I
P
˛
/:
Assume
C
has finite rank. We say that
.A;C/
is detectable with stability margin
greater than or equal to
˛
if there exists
L
2
L
.Y;X/
such that the
C
0
-semigroup
e
.A
C
LC/t
generated by
A
C
LC
satisfies
k
e
.A
C
LC/t
k
Me
ˇt
; M
1
(1.30)
holds for any
ˇ<˛
. The pair
.A;C/
is detectable if and only if
•
A
satisfies spectrum decomposition assumption at
˛
;
X
˛
•
is finite dimensional;
.A
˛
;C
˛
/
•
is observable;
e
A
˛
t
is exponentially stable with a stability margin that is least
˛
•
.
Sufficient conditions for the exponential detectability were obtained in 1975 by
Triggiani
(
1975
) (also see surveys by
Pritchard and Zabczyk
(
1981
) and by
Russell
(
1978
)). In 1985
Desch and Schappacher
(
1985
) show that these conditions are also
necessary for finite-rank inputs. These conditions can be simplified for systems of
either the Riesz-spectral type or the retarded delay type (
Bhat 1986
;
Curtain and
Pritchard 1974
;
Curtain and Zwart 1995
).
The observer for (
1.24
) has been studied by many authors (see
Orner and Foster
1971
;
Kitamura et al. 1972
;
Sakawa and Matsushita 1975
;
Balas 1980
;
Gressang
and Lamont 1975
;
Fuji 1980
) under the framework of distributed parameter
systems. However, due to its infinite-dimensional feature, in general, it is not
implementable in applications. Thus designs of finite dimensional observers (in
the context of compensators) were proposed based on eigenfunction projections
or direct state space projection (
Bernstein and Hyland 1986
;
Curtain 1982
,
1993
;
Kaman et al. 1985
;
Sakawa 1984
;
Schumacher 1983
;
Xiao and Ba¸ar 1999
). These
projection approximations usually lead to high dimensional observers in order to
achieve accurate estimation. For extensions to systems with unbounded input and
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