Cross Gramian Explained

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.^{[1] [2]}

For the stable time-invariant linear system

• x = A x + B u y = C x

the cross Gramian is defined as:

W _{X} : = \int ^{inf ty} _{0} 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 withAJ = JA ^{T} B = JC ^{T}

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

| λ (W _{X)|} = \sqrt{ λ (W _{C} W _{O)}.}

Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see http://modelreduction.org) 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

• Controllability Gramian
• Observability Gramian

Notes and References

1. Book: Fortuna. Luigi. Frasca. Mattia. Optimal and Robust Control: Advanced Topics with MATLAB. 29 April 2013. 2012. CRC Press. 9781466501911. 83–.
2. Book: Athanasios C. Antoulas. Antoulas. Athanasios C.. Approximation of Large-Scale Dynamical Systems. 2005. SIAM. 9780898715293. 10.1137/1.9780898718713. 117896525 .
3. Fernando. K.. Nicholson. H.. February 1983. On the structure of balanced and other principal representations of SISO systems. IEEE Transactions on Automatic Control. 28. 2. 228–231. 10.1109/tac.1983.1103195. 0018-9286.
4. Himpe. C.. 2018. emgr -- The Empirical Gramian Framework. Algorithms. 11. 7. 91. 10.3390/a11070091. free. 1611.00675.
5. Blower. G.. Newsham. S.. 2021. Tau functions for linear systems. Operator Theory Advances and Applications: IWOTA Lisbon 2019.