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 is controllable and observable if and only if the matrix is nonsingular,.
If the associated system is furthermore symmetric, such that there exists a transformation 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 without a balancing procedure as opposed to balanced truncation.
