# Cross Gramian

In control theory, the cross Gramian ($W_{X}$ , also referred to by $W_{CO}$ ) is a Gramian matrix used to determine how controllable and observable a linear system is.

For the stable time-invariant linear system

${\dot {x}}=Ax+Bu\,$ $y=Cx\,$ the cross Gramian is defined as:

$W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,$ and thus also given by the solution to the Sylvester equation:

$AW_{X}+W_{X}A=-BC\,$ The triple $(A,B,C)$ is controllable and observable if and only if the matrix $W_{X}$ is nonsingular, (i.e. $W_{X}$ has full rank, for any $t>0$ ).

If the associated system $(A,B,C)$ is furthermore symmetric, such that there exists a transformation $J$ with

$AJ=JA^{T}\,$ $B=JC^{T}\,$ then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:

$|\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,$ Thus the direct truncation of the singular value decomposition of the cross Gramian allows model order reduction (see ) without a balancing procedure as opposed to balanced truncation.