Geoscience Reference
In-Depth Information
2.4
Observability and Observer-Based Nudging
We start by reviewing some of the fundamental concepts related to observability
that are key to the analysis of observer-based nudging. Loosely stated, observability
relates to the goal of reconstruction of a past state, say
x.q/
at time
q
, from
a finite collection of
of a
system. The basic theory of controllability/reachability and its dual observability
of linear deterministic dynamical systems was first introduced by
Kalman
(
1960a
).
The notion of observer was introduced by
Luenberger
(
1964
,
1971
). If the given
dynamical system is observable, then the observer is a derived dynamical system
that estimates the state of the original system. In this sense, observers are the
deterministic counterpart of the well known Kalman filters which provide the “best”
estimate of the state of a stochastic dynamical system. The notion of observability
and the design of observers have been extended to nonlinear deterministic systems.
Refer to the topic by
Isidori
(
1995
) for a thorough treatment of this topic.
N
future observations
z
.k/
for
q
k
.N
C
q/
2.4.1
Conditions for Observability
x.0/
2
R
n
be the initial state of a linear dynamical system
Let
x.k
C
1/
D
Mx.k/
(2.63)
M
2
R
n
n
is a nonsingular matrix. Iterating (
2.63
), it can be verified that
where
x.k
C
q/
D
M
k
x.q/
(2.64)
.k/
2
R
m
be the observation at time
for any integer
q
0
and
k
0
.Let
z
k
given
by
z
.k/
D
Hx.k/
C
V.k/
(2.65)
H
2
R
m
n
and
R
2
R
m
n
,
where
V.k/
N.0;R/
is a white Gaussian noise with
a known symmetric and positive definite matrix.
Assume that we are given a set f
z
.k/
W
q
k
N
C
q
1
g of
N
observations.
Substituting (
2.64
)in(
2.65
), this set of
N
observations can be collectively repre-
sented by
2
3
2
3
2
3
.q/
H
HM
HM
2
HM
N
1
V.q/
V.q
C
1/
V.q
C
2/
V.q
C
N
1/
z
4
5
4
5
4
5
.q
C
1/
z
z
.q
C
2/
D
x.q/
C
(2.66)
z
.q
C
N
1/
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