Cross Gramian

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In control theory, the cross Gramian (, also referred to by ) is a Gramian matrix used to determine how controllable and observable a linear system is.[1][2]

For the stable time-invariant linear system

the cross Gramian is defined as:

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

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

The triple is controllable and observable, and hence minimal, if and only if the matrix is nonsingular, (i.e. has full rank, for any ).

If the associated system is furthermore symmetric, such that there exists a transformation with

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

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]


  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.