Cross Gramian

In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is.^{[1] [2]}

For the stable time-invariant linear system

${\displaystyle {\dot {x}}=Ax+Bu\,}$

${\displaystyle y=Cx\,}$

the cross Gramian is defined as:

${\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}$

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

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

The triple ${\displaystyle (A,B,C)}$ 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 ${\displaystyle (A,B,C)}$ is furthermore symmetric, such that there exists a transformation $J$ with

${\displaystyle AJ=JA^{T}\,}$

${\displaystyle B=JC^{T}\,}$

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

${\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}$

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

Note

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

See also

• Controllability Gramian
• Observability Gramian
• Gramian matrix

References

1. Fortuna, Luigi; Fransca, Mattia (2012). Optimal and Robust Control: Advanced Topics with MATLAB. CRC Press. pp. 83–. ISBN 9781466501911. Retrieved 29 April 2013.
2. Antoulas, Athanasios C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM. ISBN 9780898715293.
3. Cross Gramian On the structure of balanced and other principal representations of SISO systems by K.V. Fernando and H. Nicholson; IEEE Transactions on Automatic Control; 1983