In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system
x
˙
=
A
x
+
B
u
if all eigenvalues of
A
have negative real part, then the unique solution
W
c
of the Lyapunov equation
A
W
c
+
W
c
A
T
=
−
B
B
T
is positive definite if and only if the pair
(
A
,
B
)
is controllable.
W
c
is known as the controllability Gramian and can also be expressed as
W
c
=
∫
0
∞
e
A
τ
B
B
T
e
A
T
τ
d
τ
A related matrix used for determining controllability is
W
c
(
t
0
,
t
1
)
=
∫
t
0
t
1
e
A
(
t
0
−
τ
)
B
B
T
e
A
T
(
t
0
−
τ
)
d
τ
The pair
(
A
,
B
)
is controllable if and only if the matrix
W
c
(
t
0
,
t
1
)
is nonsingular, for any
t
1
>
t
0
. A physical interpretation of the controllability Gramian is that if the input to the system is white gaussian noise, then
W
c
is the covariance of the state.
Linear time-variant state space models of form
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
,
y
(
t
)
=
C
(
t
)
x
(
t
)
+
D
(
t
)
u
(
t
)
are controllable in an interval
[
t
0
,
t
1
]
if and only if the Gramian matrix
W
c
(
t
0
,
t
1
)
is nonsingular, where
W
c
(
t
0
,
t
1
)
=
∫
t
0
t
1
Φ
(
t
0
,
τ
)
B
(
τ
)
B
T
(
τ
)
Φ
T
(
t
0
,
τ
)
d
τ
