Cross Gramian

Last updated January 03, 2024

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

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

y=Cx\,

the cross Gramian is defined as:

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

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

AW_{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:^[3]

|\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.^[4]^[5]

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References

↑ Fortuna, Luigi; Frasca, Mattia (2012). Optimal and Robust Control: Advanced Topics with MATLAB. CRC Press. pp. 83–. ISBN 9781466501911 . Retrieved 29 April 2013.
↑ Antoulas, Athanasios C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM. doi:10.1137/1.9780898718713. ISBN 9780898715293. S2CID 117896525.
↑ 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.
↑ Himpe, C. (2018). "emgr -- The Empirical Gramian Framework". Algorithms. 11 (7): 91. arXiv: 1611.00675 . doi: 10.3390/a11070091 .
↑ Blower, G.; Newsham, S. (2021). "Tau functions for linear systems" (PDF). Operator Theory Advances and Applications: IWOTA Lisbon 2019.

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[FortunaFrasca2012-1] Fortuna, Luigi; Frasca, Mattia (2012). Optimal and Robust Control: Advanced Topics with MATLAB. CRC Press. pp. 83–. ISBN 9781466501911 . Retrieved 29 April 2013.

[antoulas05-2] Antoulas, Athanasios C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM. doi:10.1137/1.9780898718713. ISBN 9780898715293. S2CID 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. doi:10.1109/tac.1983.1103195. ISSN 0018-9286.

[4] Himpe, C. (2018). "emgr -- The Empirical Gramian Framework". Algorithms. 11 (7): 91. arXiv: 1611.00675 . doi: 10.3390/a11070091 .

[5] Blower, G.; Newsham, S. (2021). "Tau functions for linear systems" (PDF). Operator Theory Advances and Applications: IWOTA Lisbon 2019.

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