Cross Gramian

In control theory, the cross Gramian ( W X {\displaystyle W_{X}} , also referred to by  W C O {\displaystyle 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 {\displaystyle {\dot {x}}=Ax+Bu\,}
y = C x {\displaystyle y=Cx\,}

the cross Gramian is defined as:

W X := ∫ 0 ∞ e A t B C e A t d t {\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}

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

A W X + W X A = − B C {\displaystyle 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 ) {\displaystyle (A,B,C)} is controllable and observable, and hence minimal, if and only if the matrix  W X {\displaystyle W_{X}} is nonsingular, (i.e.  W X {\displaystyle W_{X}} has full rank, for any  t > 0 {\displaystyle t>0} ).

If the associated system  ( A , B , C ) {\displaystyle (A,B,C)} is furthermore symmetric, such that there exists a transformation  J {\displaystyle J} with

A J = J A T {\displaystyle AJ=JA^{T}\,}
B = J C T {\displaystyle B=JC^{T}\,}

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

| λ ( W X ) | = λ ( W C W O ) . {\displaystyle |\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 [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

[edit]

* Controllability Gramian
* Observability Gramian

## References

[edit]

1. ^ Fortuna, Luigi; Frasca, 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. 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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