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

$\dot{x} = A x + B u \,$

$y = C x \,$

the cross Gramian is defined as:

$W_X := \int_0^\infty e^{At} BC e^{At} dt \,$

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

$A W_X + W_X A = -BC \,$

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:^[3]

$|\lambda(W_X)| = \sqrt{\lambda(W_C W_O)}. \,$

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 $W_{CO}$.

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