Cross Gramian

In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is.

For the stable time-invariant linear system

the cross Gramian is defined as:

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

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

then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:

Thus the direct truncation of the singular value decomposition of the cross Gramian allows model order reduction (see [1]) without a balancing procedure as opposed to balanced truncation.