Reduced order controller design for symmetric, non-symmetric and unstable systems using extended cross-gramian

Muhammad Raees Furquan Azhar, Umair Zulfiqar, Muwahida Liaquat, Deepak Kumar

Research output: Contribution to journalArticle

1 Citation (Scopus)


In model order reduction and system theory, the cross-gramian is widely applicable. The cross-gramian based model order reduction techniques have the advantage over conventional balanced truncation that it is computationally less complex, while providing a unique relationship with the Hankel singular values of the original system at the same time. This basic property of cross-gramian holds true for all symmetric systems. However, for non-square and non-symmetric dynamical systems, the standard cross-gramian does not satisfy this property. Hence, alternate approaches need to be developed for its evaluation. In this paper, a generalized frequency-weighted cross-gramian-based controller reduction algorithm is presented, which is applicable to both symmetric and non-symmetric systems. The proposed algorithm is also applicable to unstable systems even if they have poles of opposite polarities and equal magnitudes. The proposed technique produces an accurate approximation of the reduced order model in the desired frequency region with a reduced computational effort. A lower order controller can be designed using the proposed technique, which ensures closed-loop stability and performance with the original full order plant. Numerical examples provide evidence of the efficacy of the proposed technique.

Original languageEnglish
Article number48
Issue number3
Publication statusPublished - 29 Jun 2019

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