Cross Gramian

In control theory, the cross Gramian ([math]\displaystyle{ W_X }[/math], also referred to by [math]\displaystyle{ W_{CO} }[/math]) is a Gramian matrix used to determine how controllable and observable a linear system is.^[1][2] For the stable time-invariant linear system

[math]\displaystyle{ \dot{x} = A x + B u \, }[/math]

[math]\displaystyle{ y = C x \, }[/math]

the cross Gramian is defined as:

[math]\displaystyle{ W_X := \int_0^\infty e^{At} BC e^{At} dt \, }[/math]

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

[math]\displaystyle{ A W_X + W_X A = -BC \, }[/math]

This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.

The triple [math]\displaystyle{ (A,B,C) }[/math] is controllable and observable, and hence minimal, if and only if the matrix [math]\displaystyle{ W_X }[/math] is nonsingular, (i.e. [math]\displaystyle{ W_X }[/math] has full rank, for any [math]\displaystyle{ t \gt 0 }[/math]).

If the associated system [math]\displaystyle{ (A,B,C) }[/math] is furthermore symmetric, such that there exists a transformation [math]\displaystyle{ J }[/math] with

[math]\displaystyle{ AJ = JA^T \, }[/math]

[math]\displaystyle{ B = JC^T \, }[/math]

then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:^[3]

[math]\displaystyle{ |\lambda(W_X)| = \sqrt{\lambda(W_C W_O)}. \, }[/math]

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

The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.^[4][5]

See also

• Controllability Gramian
• Observability Gramian

References

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