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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:

$AW_{X}+W_{X}A=-BC\,$

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

$AJ=JA^{T}\,$

$B=JC^{T}\,$

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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