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y.0/
D
L
f
.h/.x
0
/
D
@L
f
.h/
.x
0
/f.x
0
/
@x
:
:
:
y
.k
1/
.0/
D
L
k
f
.h/.x
0
/
D
@L
k
f
.h/
@x
.x
0
/f.x
0
/
h;L
f
.h/;L
f
.h/;
distin-
guishes points then the system is observable. For a real analytic system this is a
necessary and sufficient condition for observability. For simplicity of exposition,
suppose
for some integer
k>0
. If the mapping from
x
0
to
p
D
1
. Consider the matrix
2
4
3
5
@h
@x
.x
0
/
@L
f
.h/
@x
.x
0
/
:
:
:
@L
n
f
.h/
@x
.x
0
/
If this matrix is invertible, then the system is locally observable at
x
0
.This
observability matrix is a topic addressed in almost all textbooks of linear and
nonlinear control theory, for instance
Kailath
(
1980
) for linear systems and
Isidori
(
1995
) for nonlinear systems.
For high dimensional systems, it is important to quantitatively define observabil-
ity. The observability Gramian is a widely used concept for this purpose (
Kailath
1980
). Consider a linear system (
1.1
), an arbitrary initial state
x
0
of a trajectory
x.t/
D
e
At
x
0
can be uniquely determined from the known function
y.t/
D
Cx.t/
if and only if
the columns in the matrix
Ce
At
are linearly independent over
Œt
0
;t
1
. This is equivalent to say that
Z
t
1
t
0
e
A
T
t
C
T
Ce
At
dt
G
D
L
2
-norm
is nonsingular. This matrix is called the observability Gramian. In fact, the
of the output satisfies
Z
t
1
jj
y.t/
jj
2
dt
D
x
0
Gx
0
t
0
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