Cross Gramian

In control theory, the cross Gramian ($$W_X$$, also referred to by $$W_{CO}$$) is a Gramian matrix used to determine how controllable and observable a linear system is.

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:


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

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

The triple $$(A,B,C)$$ is controllable and observable, and hence minimal, 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:


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

Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see ) 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.