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Open AccessArticle

Reduced Order Controller Design for Symmetric, Non-Symmetric and Unstable Systems Using Extended Cross-Gramian

Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan
School of Electrical, Electronic and Computer Engineering, The University of Western Australia, Perth 6009, Australia
Author to whom correspondence should be addressed.
Machines 2019, 7(3), 48;
Received: 28 February 2019 / Revised: 8 May 2019 / Accepted: 10 May 2019 / Published: 29 June 2019
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. View Full-Text
Keywords: cross-gramian; lyapunov Equation; plant reduction; schur-decomposition cross-gramian; lyapunov Equation; plant reduction; schur-decomposition
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Azhar, M.R.F.; Zulfiqar, U.; Liaquat, M.; Kumar, D. Reduced Order Controller Design for Symmetric, Non-Symmetric and Unstable Systems Using Extended Cross-Gramian. Machines 2019, 7, 48.

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