Before revision: For the time-invariant linear system the cross Gramian is given by the solution to the Sylvester equation: The triple is controllable and observable if and only if the matrix is nonsingular, (i.e.
After revision: For the stabledefined as:and thus also time-invariant linear system the cross Gramian is given by the solution to the Sylvester equation: The triple is controllable and observable if and only if the matrix is nonsingular, (i.e.