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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 time-invariant linear system

$\dot{x}=Ax+Bu$

$y=Cx$

the cross Gramian is given by the solution to the Sylvester equation:

$A{W}_X+W_{X}A=-BC$

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

$AJ=J{A}^T$

$B=J{C}^T$

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

$|\lambda (W_X)|=\sqrt{\lambda (W_C{W}_O)}.$

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

Note

The cross Gramian is also referred to by $W_{CO}$.

See also

• Controllability Gramian
• Observability Gramian
• Gramian matrix

References

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