# Cross Gramian

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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

x ˙ = A x + B u y = C x

the cross Gramian is defined as:

W X := 0 e A t B C e A t d t

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

A W X + W X A = B C

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

A J = J A T B = J C T

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

| λ ( W X ) | = λ ( W C W O ) .

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.

## Note

The cross Gramian is also referred to by W C O .

## References

Cross Gramian Wikipedia

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