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:

$AW_{X}+W_{X}A=-BC\,$

This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.

The triple $(A,B,C)$ is controllable and observable, and hence minimal, 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 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