The Observability Gramian is a Gramian used in control theory to determine whether or not a linear system is observable.
For a linear time variant system described by
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
the observability Gramian is given by
W
o
(
t
0
,
t
1
)
=
∫
t
0
t
1
Φ
T
(
s
,
t
0
)
C
T
(
s
)
C
(
s
)
Φ
(
s
,
t
0
)
d
s
,
where
Φ
is the state transition matrix.
The system is observable on the interval
t
∈
[
t
0
,
t
1
]
if and only if
W
o
(
t
0
,
t
1
)
is nonsingular. In the case of a linear time invariant system, this can be simplified to finding the rank of the "observability matrix". If
x
(
t
)
is a
n
-dimensional real-valued vector, then the system is observable if and only if
rank
[
C
T
,
A
T
C
T
,
.
.
.
,
(
A
T
)
n
−
1
C
T
]
=
n
