Cross Gramian

In control theory, the cross Gramian (${\displaystyle W_{X}}$, also referred to by ${\displaystyle W_{CO}}$) is a Gramian matrix used to determine how controllable and observable a linear system is.^[1][2]

For the stable time-invariant linear system

${\displaystyle {\dot {x}}=Ax+Bu\,}$

${\displaystyle y=Cx\,}$

the cross Gramian is defined as:

${\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}$

and thus also given by the solution to the Sylvester equation:

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

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

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

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

${\displaystyle AJ=JA^{T}\,}$

${\displaystyle B=JC^{T}\,}$

then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:^[3]

${\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}$

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.^[4][5]

• Controllability Gramian
• Observability Gramian