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For the stable time-invariant linear system
the cross Gramian is defined as:
and thus also given by the solution to the Sylvester equation:
If the associated system is furthermore symmetric, such that there exists a transformation with
The cross Gramian is also referred to by .
- Fortuna, Luigi; Fransca, 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. ISBN 9780898715293.
- Cross Gramian On the structure of balanced and other principal representations of SISO systems by K.V. Fernando and H. Nicholson; IEEE Transactions on Automatic Control; 1983