Cross Gramian

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In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is.[1] For the time-invariant linear system

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

the cross Gramian is given by the solution to the Sylvester equation:

AW_{X}+W_{X}A=-BC\,

The triple (A,B,C) is controllable and observable 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:[2]

|\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,

Thus the direct truncation of the singular value decomposition of the cross Gramian allows model reduction (see ) without a balancing procedure as opposed to balanced truncation.

Note

The cross Gramian is also referred to by W_{{CO}}.

See also

References

  1. ↑ Fortuna, Luigi; Fransca, Mattia (2012). Optimal and Robust Control: Advanced Topics with MATLAB́. CRC Press. pp. 83–. ISBN 9781466501911. Retrieved 29 April 2013. 
  2. ↑ 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
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