Английская Википедия:Cross Gramian
In control theory, the cross Gramian (<math>W_X</math>, also referred to by <math>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>\dot{x} = A x + B u \, </math>
- <math>y = C x \, </math>
the cross Gramian is defined as:
- <math>W_X := \int_0^\infty e^{At} BC e^{At} dt \,</math>
and thus also given by the solution to the Sylvester equation:
- <math>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>(A,B,C)</math> is controllable and observable, and hence minimal, if and only if the matrix <math>W_X</math> is nonsingular, (i.e. <math>W_X</math> has full rank, for any <math>t > 0</math>).
If the associated system <math>(A,B,C)</math> is furthermore symmetric, such that there exists a transformation <math>J</math> with
- <math>AJ = JA^T \, </math>
- <math>B = JC^T \, </math>
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:[3]
- <math>|\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
References
Шаблон:Matrix-stub
Шаблон:Systemstheory-stub
