Cross Gramian

In control theory, the cross Gramian (WX, also referred to by WCO) 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˙=Ax+Bu
y=Cx

the cross Gramian is defined as:

WX:=∫0∞eAtBCeAtdt

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

AWX+WXA=−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 WX is nonsingular, (i.e. WX 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=JAT
B=JCT

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

|λ(WX)|=λ(WCWO).

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.[4][5]

## See also

* Controllability Gramian
* Observability Gramian

## References

1. ↑ Fortuna, Luigi; Frasca, Mattia (2012). Optimal and Robust Control: Advanced Topics with MATLAB. CRC Press. pp. 83–. ISBN 9781466501911. https://books.google.com/books?id=WM3OzyHKlD4C&pg=PA83. Retrieved 29 April 2013.
2. ↑ Antoulas, Athanasios C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM. doi:10.1137/1.9780898718713. ISBN 9780898715293.
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. doi:10.1109/tac.1983.1103195. ISSN 0018-9286.
4. ↑ Himpe, C. (2018). "emgr -- The Empirical Gramian Framework". Algorithms 11 (7): 91. doi:10.3390/a11070091.
5. ↑ Blower, G.; Newsham, S. (2021). "Tau functions for linear systems". Operator Theory Advances and Applications: IWOTA Lisbon 2019. https://eprints.lancs.ac.uk/id/eprint/142050/1/GBlowerIWOTAconferencepaperrevised.pdf.

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