Geoscience Reference
In-Depth Information
n
X
n
X
0
D
@Q
@Q
@x
i
.x;t/f
i
.x/
C
1
@Q
@x
i
.x;t/a
ij
.x/
@Q
@t
.x;t/
C
@x
j
.x;t/
2
i
D
1
i;j
D
1
1
2
jj
y.t/
h.x/
jj
2
Similar to the
H
1
filter, this equation is very difficult to solve, if not impossi-
ble, either analytically or numerically. For linear systems, the partial differential
equation is reduced to a linear Riccati equation, which is numerically solvable for
systems with a moderate dimension. For systems with extremely high dimensions,
such as the models used for numerical weather forecast, special treatment must be
applied in the optimization process. In principle, 4D-Var is a discrete minimum
energy filter using a weighted norm. Some matrices of extremely large size exceed
the capacity of computational facility. The way to get around these difficulties
is to use tangent linear model and adjoint model in the computation (Liang,
some references here). More information on the general idea of minimum energy
estimation methods is referred to
Hijab
(
1980
),
Krener
(
2003a
), and
Mortensen
(
1968
) and references therein.
1.5
Observer Construction for PDE Systems
1.5.1
Linear Case
We consider a PDE system written in an abstract form
x.t/
D
Ax.t/; x.0/
D
x
0
; t
0
y.t/
D
Cx.t/; t
0;
(1.24)
on a Hilbert space
X
,where
A
is the infinitesimal generator of the strongly
e
At
on
continuous semigroup
X
and
C
is a bounded operator from
X
to a second
Hilbert space
(
Curtain and Zwart 1995
).
Similar to the finite dimensional case, the observability map of (
1.24
)on
Y
Œ0;T
is a bounded linear operator
C
T
W
X
!
L
2
.Œ0;T
I
Y/
defined as follows
C
T
.x/.t/
D
Ce
A.T
t/
x:
(1.25)
A widely adopted definition of observability is based on the property that the knowl-
edge about the output
y
over a finite time interval uniquely determines the initial
state. The following definition is essentially the same as the one following (
1.3
)for
finite dimensional systems:
Œ0;T
T>0
Definition 1.2.
System (
1.24
) is exactly observable on
(for some
)if
C
T
is injective and its inverse is bounded on the range of
C
T
.
Search WWH ::
Custom Search