Model order reduction based on low-rank decomposition of the cross Gramian

Abstract

This paper considers model order reduction based on low-rank decomposition of the cross Gramian for linear systems. The proposed approach uses the low-rank factors of the cross Gramian to generate approximate balanced model for the large-scale system. Then, the reduced-order model is obtained by truncating the states corresponding to the small approximate Hankel singular values. The low-rank factors are directly constructed from the expansion coefficients of impulse responses in the space spanned by Legendre polynomials by solving block tridiagonal linear systems. In contrast to balanced truncation related approaches which require to solve Sylvester equation to compute the full cross Gramian, the proposed method needs to solve sparse linear equations, and only one singular value decomposition is applied to a low-dimensional matrix. The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, the reduction procedure is modified to alleviate the shortcoming, which may unexpectedly lead to unstable systems even though the original one is stable. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods.

Keywords:

• Model order reduction
• cross Gramian
• Legendre polynomials
• balanced truncation
• stability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 61873336 and 61803046]. Science and Technology Commission of Shanghai Municipality [Nr. 21190780300]; State Administration of Foreign Experts Affairs [Nr. QN2022013008L,G2021013008L, and G2022013050L]; National Key Research and Development Program [Nr. 2020YFB1708200].