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Generalized \({\varvec{H}_{\infty }}\) model reduction for stable two-dimensional discrete systems

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Abstract

For model reduction, the approximation performance sometimes needs to be enhanced over a specific frequency range. Motivated by this fact, the paper investigates generalized \(H_{\infty }\) model reduction for stable two-dimensional (2-D) discrete systems represented by the Roesser model and the Fornasini–Machesini local state-space model, respectively. The generalized \(H_{\infty }\) norm of 2-D systems is introduced to evaluate the approximation error over a specific finite frequency (FF) domain. In light of the 2-D generalized Kalman–Yakubovich–Popov lemmas, sufficient conditions in terms of linear matrix inequalities are derived for the existence of a stable reduced-order model satisfying a specified generalized \(H_{\infty }\) level. Several examples are provided to illustrate the effectiveness and advantages of the proposed method. Compared with most of the existing 2-D model reduction results, the proposed method has the following merits: (1) The generalized \(H_\infty \) model reduction problems are considered for both important types of 2-D models, and no structural assumption is made for the plant model, so that our method has a broader applicable scope. (2) The reduced-order model is guaranteed to be stable, and an upper bound on the generalized \(H_{\infty }\) error is provided. Moreover, no frequency weighting function is needed. (3) The proposed method is applicable for 2-D model reduction with multiple FF specifications.

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Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61273201, in part by GRF HKU 7140/11E and in part by HKU CRCG 201309176107.

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong

    Xianwei Li, James Lam & Kie Chung Cheung

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  1. Xianwei Li
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Correspondence to Xianwei Li.

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Li, X., Lam, J. & Cheung, K.C. Generalized \({\varvec{H}_{\infty }}\) model reduction for stable two-dimensional discrete systems. Multidim Syst Sign Process 27, 359–382 (2016). https://doi.org/10.1007/s11045-014-0306-3

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