4.2 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Cross Gramian | 1/1 | https://en.wikipedia.org/wiki/Cross_Gramian | reference | science, encyclopedia | 2026-05-05T12:27:58.997582+00:00 | kb-cron |
In control theory, the cross Gramian (
W
X
{\displaystyle W_{X}}
, also referred to by
W
C
O
{\displaystyle W_{CO}}
) is a Gramian matrix used to determine how controllable and observable a linear system is. For the stable time-invariant linear system
x
˙
=
A
x
+
B
u
{\displaystyle {\dot {x}}=Ax+Bu\,}
y
=
C
x
{\displaystyle y=Cx\,}
the cross Gramian is defined as:
W
X
:=
∫
0
∞
e
A
t
B
C
e
A
t
d
t
{\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}
and thus also given by the solution to the Sylvester equation:
A
W
X
+
W
X
A
=
−
B
C
{\displaystyle AW_{X}+W_{X}A=-BC\,}
This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric. The triple
(
A
,
B
,
C
)
{\displaystyle (A,B,C)}
is controllable and observable, and hence minimal, if and only if the matrix
W
X
{\displaystyle W_{X}}
is nonsingular, (i.e.
W
X
{\displaystyle W_{X}}
has full rank, for any
t
>
0
{\displaystyle t>0}
). If the associated system
(
A
,
B
,
C
)
{\displaystyle (A,B,C)}
is furthermore symmetric, such that there exists a transformation
J
{\displaystyle J}
with
A
J
=
J
A
T
{\displaystyle AJ=JA^{T}\,}
B
=
J
C
T
{\displaystyle B=JC^{T}\,}
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:
|
λ
(
W
X
)
|
=
λ
(
W
C
W
O
)
.
{\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}
Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see [1]) without a balancing procedure as opposed to balanced truncation. The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.
== See also == Controllability Gramian Observability Gramian
== References ==