In control theory, the cross Gramian ( W X {\displaystyle W_{X}} , also referred to by W C O {\displaystyle W_{CO}} ) 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 {\displaystyle {\dot {x}}=Ax+Bu\,}
y = C x {\displaystyle y=Cx\,}
the cross Gramian is defined as:
W X := ∫ 0 ∞ e A t B C e A t d t {\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}
and thus also given by the solution to the Sylvester equation:
A W X + W X A = − B C {\displaystyle AW_{X}+W_{X}A=-BC\,}
This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric. The triple ( A , B , C ) {\displaystyle (A,B,C)} is controllable and observable, and hence minimal, if and only if the matrix W X {\displaystyle W_{X}} is nonsingular, (i.e. W X {\displaystyle W_{X}} has full rank, for any t > 0 {\displaystyle t>0} ). If the associated system ( A , B , C ) {\displaystyle (A,B,C)} is furthermore symmetric, such that there exists a transformation J {\displaystyle J} with
A J = J A T {\displaystyle AJ=JA^{T}\,}
B = J C T {\displaystyle B=JC^{T}\,}
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:
| λ ( W X ) | = λ ( W C W O ) . {\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}
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.
See also Controllability Gramian Observability Gramian
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