The cross-gramian is a well-known matrix with
embedded controllability and observability information. The
cross-gramian is related to the Hankel operator and the Hankel
singular values of a linear square system and it has several
interesting properties. These properties make the cross-gramian
popular in several applications including model reduction, control
configuration selection and sensitivity analysis. The ordinary
cross-gramian which has been defined in the literature is the
solution of a Sylvester equation. This Sylvester equation is not
always solvable and therefore for some linear square symmetric
systems, the ordinary cross-gramian does not exist. To cope with
this problem, a new generalized cross-gramian is introduced
in this paper. In contrast to the ordinary cross-gramian, the
generalized cross-gramian can be easily obtained for general
linear systems and therefore can be used in the applications
instead of the ordinary cross-gramian.