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:

$W_X:=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{At}BC{e}^{At}dt$

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

$A{W}_X+W_{X}A=-BC$

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

$AJ=J{A}^T$

$B=J{C}^T$

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