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A frequency-partitioning approach to stability analysis of two-dimensional discrete systems

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Abstract

Although there exist necessary and sufficient conditions for two-dimensional (2-D) discrete systems, their application scope is almost limited to system analysis only and no work has demonstrated their possible extension to system synthesis. In this paper, we propose a novel frequency-partitioning approach to analyzing stability of 2-D discrete state-space systems. A piecewise constant matrix function is introduced to approximate the solution to the frequency-dependent Lyapunov inequality, whose solvability is known to be equivalent to stability of a 2-D state-space model. Then by the generalized Kalman–Yakubovich–Popov Lemma, new stability conditions are derived for the Roesser model and the Fornasini–Marchesini (FM) first and second models, respectively. Stability criteria in the paper simultaneously overcome the drawbacks of the simple 2-D Lyapunov inequality approach and the existing necessary and sufficient conditions: (1) They are expressed in terms of linear matrix inequalities (LMIs), which are generally less conservative than the existing simple 2-D Lyapunov inequality-based results, and could be improved by increasing the partitioning number; (2) Since each of the LMIs corresponds to a simple 2-D Lyapunov inequality, they are more suitable for further development for system synthesis than the existing necessary and sufficient conditions, which is demonstrated by an illustrative application to state-feedback control of an uncertain FM second model.

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Acknowledgments

This work was supported in part by the 973 Project under Grant (2009CB320600), in part by National Natural Science Foundation of China under Grants (61273201, 61203035, 61021002) and in part by the Key Laboratory of Integrated Automation for the Process Industry (Northeast University). The authors sincerely thank the anonymous reviewers for their insightful comments that have helped improve the paper.

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Authors and Affiliations

  1. Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150080, Heilongjiang Province, People’s Republic of China

    Xianwei Li & Huijun Gao

  2. Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, People’s Republic of China

    James Lam

  3. Research Institute of Mechatronics and Automation, Bohai University, Jinzhou, 121000, Liaoning Province, People’s Republic of China

    Huijun Gao

  4. Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai Institute of Spaceflight Control Technology, Shanghai, 200233, People’s Republic of China

    Yue Gu

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Li, X., Lam, J., Gao, H. et al. A frequency-partitioning approach to stability analysis of two-dimensional discrete systems. Multidim Syst Sign Process 26, 67–93 (2015). https://doi.org/10.1007/s11045-013-0237-4

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