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Therefore, the eigenvalues of
G
represent the gain from the initial state to the
output. If
has a zero eigenvalue, then its eigenvector results in a zero output. The
system is unobservable. If
G
has a very small eigenvalue, then the system is weakly
observable, i.e. a small noise in
G
y.t/
can cause a large estimation error. Therefore,
the smallest eigenvalue of
is used as a quantitative measure of observability.
For nonlinear systems, an empirical observability Gramian can be numerically
computed (
Krener and Ide 2009
). Consider (
1.2
) and a nominal trajectory
G
x.t/
with
initial state
x.0/
D
x
0
. Define a mapping
ıx
0
!
h.x.t//
h.x.t//
subject to
x.t/
D
f.x.t//
x.0/
D
x
0
C
ıx
0
(1.4)
R
n
.Let
Let
v
1
;
v
2
;
;
v
n
be an orthonormal basis in
>0
be a small number. In
the direction of
v
i
, the variation of the output can be estimated empirically by
h.x
C
.t//
h.x
.t//
;
i
.t/
D
1
2
(1.5)
where
x
˙
.t/
D
f.x
˙
.t//
x
˙
.0/
D
x
0
˙
v
i
;
The mapping, (
1.4
), from the initial state to the output space can be locally
approximated by a linear function
n
X
n
X
ıx
0
D
˛
i
v
i
!
˛
i
i
.t/
(1.6)
i
D
0
i
D
0
Therefore, the observability Gramian of the nonlinear system can be approximated
by the Gramian associated to (
1.6
)
G
D
.G
ij
/
i;j
D
1
G
ij
D
Z
t
1
t
0
i
.t/
j
.t/dt
(1.7)
Locally around the nominal trajectory, the eigenvalues of (
1.7
) measure the gain
from the variation of the initial state to the variation of the output. If
G
has a small
eigenvalue, then
x.t/
is weakly observable. A small noise in
y.t/
can result in a
large estimation error.
The Gramian or empirical Gramian in
Kailath
(
1980
)and
Krener and Ide
(
2009
)
measures the observability of full initial states. However, for systems with very high
dimensions, the problem of full observability is, in many cases, ill-posed. Some
discussions on the partial observability, or
Z
-observability, for complex systems
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