Geoscience Reference
In-Depth Information
x
D
f.x/
C
g.x/
w
y
D
h.x/
C
v
(1.21)
x.0/
D
x
0
C
x
0
where disturbances
w
and
v
are unknown
L
2
functions,
x
0
is an unknown error in
the initial condition. The total “energy” of the disturbances is formulated using
L
2
norms
Z
t
jj
x
0
jj
2
C
./
jj
2
Cjj
v
./
jj
2
d
jj
w
0
For some
>0
, if a filter satisfies
Z
t
jj
x./
x./
jj
2
d
2
Z
t
jj
x
0
jj
2
C
./
jj
2
Cjj
v
./
jj
2
d
jj
w
0
0
for arbitrary
x
0
,
w
,and
v
, then we say that the gain from the disturbance to
the estimation error is bounded by
. We seek an estimator based on worst case
scenarios. Define
2
2
Z
t
jj
x
0
jj
2
C
2
2
./
jj
2
Cjj
y./
h.x.//
jj
2
d
Q.x;t/
D
inf
x
0
;
w
.
/
jj
w
Z
t
0
1
2
jj
x./
x./
jj
2
d
0
where
, then it is guaranteed
that the gain from the disturbance to the estimation error is bounded by
x.
/
is subject to (
1.21
)and
x.t/
D
x
.If
Q.x;t/
0
.From
dynamic programming,
Q.x;t/
satisfies the following partial differential equation
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
2
i
D
1
i;j
D
1
2
2
jj
y.t/
h.x/
jj
2
C
1
2
jj
x
x
jj
2
(1.22)
If the equation has a solution, then the optimal estimate is given by
x.t/
D argmin
x
Q.x;t/
(1.23)
It is of Hamilton-Jacobi type, first order, nonlinear PDE driven by the observa-
tions. It is very difficult, if not impossible, to compute an accurate solution in real
time. Moreover it may not admit a smooth solution so the (
1.22
) must be interpreted
in the viscosity sense. This is an infinite dimensional observer with state
Q.
;t/
evolving according to (
1.22
) with state estimate given by (
1.23
). Hence it is of
limited practical use.
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