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l 2–l ∞ Filtering Design for Piecewise Discrete-Time Linear Systems

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Abstract

This paper is concerned with the l 2–l ∞ filtering problem for a class of piecewise discrete-time linear systems. Attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the l 2–l ∞ sense. By using the piecewise Lyapunov function, a sufficient condition for the solvability of this problem is obtained in terms of linear matrix inequalities (LMIs). It has been shown that the l 2–l ∞ filtering problem can be solved as an LMI optimization problem. Two numerical examples are presented to demonstrate the validity of the proposed design method.

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References

  1. G. Feng, Stability analysis of piecewise linear discrete time systems. IEEE Trans. Automat. Contr. 47(7), 1108–1112 (2002)

    Article  Google Scholar 

  2. G. Feng, Observer-based output feedback controller design of piecewise discrete-time linear systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(3), 448–451 (2003)

    Article  Google Scholar 

  3. G. Feng, Robust filtering design of piecewise discrete time linear systems. IEEE Trans. Signal Process. 53(2), 599–605 (2005)

    Article  MathSciNet  Google Scholar 

  4. G. Ferrari, R. Cuzzola, D. Mignone, M. Morari, Analysis and control with performance of piecewise affine and hybrid systems, in Proceedings of the American Control Conference, Arlington, VA (2001), pp. 200–205

  5. G. Ferrari, D. Mignone, M. Morari, Moving horizon estimation for hybrid systems. IEEE Trans. Automat. Contr. 47(10), 1663–1676 (2002)

    Article  Google Scholar 

  6. H. Gao, J. Lam, L. Xie, C. Wang, New approach to mixed H 2/H ∞ filtering for polytopic discrete time systems. IEEE Trans. Signal Process. 53(8), 3183–3192 (2005)

    Article  MathSciNet  Google Scholar 

  7. H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems. Syst. Contr. Lett. 55(2), 101–111 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Hassibi, S. Boyd, Quadratic stabilization and control of piecewise-linear systems, in Proceedings of the 1998 American Control Conference, Boston, MA (1998), pp. 3659–3664

  9. M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automat. Contr. 43(4), 555–559 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Johansson, A. Rantzer, Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Syst. 7(6), 713–722 (1999)

    Article  Google Scholar 

  11. V. Kulkarni, M. Jun, J. Hespanha, Piecewise quadratic Lyapunov functions for piecewise affine time-delay systems, in Proceedings of the 2004 American Control Conference, Boston, MA (2004), pp. 3885–3889

  12. H. Li, M. Fu, A linear matrix inequality approach to robust filtering. IEEE Trans. Signal Process. 45(9), 2338–2350 (1997)

    Article  Google Scholar 

  13. D. Mignone, G. Ferrari, M. Maorari, Stability and stabilization of piecewise affine and hybrid systems: an LMI approach, in Proceedings of 39th IEEE Conf. Decision and Control, Sydney, Australia (2000), pp. 1864–1876

  14. L. Rodrigues, J. How, Output feedback controller synthesis for piecewise-affine systems with multiple equilibria, in Proceedings of the American Control Conference, Chicago, IL (2000), pp. 1784–1789

  15. L. Rodrigues, J. How, Observer-based control of piecewise-affine systems. Int. J. Control 76(5), 459–477 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. O. Slupphaug, B. Foss, Constrained quadratic stabilization of discrete time uncertain nonlinear multi-model systems using piecewise affine state feedback. Int. J. Control 72(7), 686–701 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. X. Song, S. Zhou, H. Zhang, H. Zhao, Robust H ∞ controller design for continuous-time piecewise time-delay systems. Asian J. Control 10(5), 603–607 (2008)

    Article  MathSciNet  Google Scholar 

  18. K. Sun, A. Packard, Robust H ∞ filters for uncertain LFT systems. IEEE Trans. Automat. Contr. 50(5), 715–720 (2005)

    Article  MathSciNet  Google Scholar 

  19. D. Xie, L. Wang, F. Hao, G. Xie, An LMI approach to L 2 gain analysis and control synthesis of uncertain switched systems. IEE Proc. Control Theory Appl. 151(1), 21–28 (2004)

    Article  Google Scholar 

  20. S. Xu, T. Chen, Reduced-order H ∞ filtering for stochastic systems. IEEE Trans. Signal Process. 50(12), 2998–3007 (2002)

    Article  MathSciNet  Google Scholar 

  21. B. Zhang, S. Xu, Robust H ∞ filtering for uncertain discrete piecewise time-delay systems. Int. J. Control 80(4), 636–645 (2007)

    Article  MATH  Google Scholar 

  22. B. Zhang, S. Xu, Robust L 2–L ∞ filtering for uncertain nonlinearly parameterized stochastic systems with time-varying delays. Circuits Syst. Signal Process. 26(5), 751–772 (2007)

    Article  MATH  Google Scholar 

  23. S. Zhou, J. Lam, A. Xue, H ∞ filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach. Fuzzy Sets Syst. 158(2), 180–193 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, P.R. China

    Xinmin Song & Huanshui Zhang

  2. School of Computer and Telecommunication Engineering, Weifang University, Weifang, 261061, Shandong, P.R. China

    Haihui Liu

Authors
  1. Xinmin Song
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  2. Huanshui Zhang
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  3. Haihui Liu
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Correspondence to Xinmin Song.

Additional information

This work was supported in part by the National Science Foundation for Distinguished Young Scholars of P.R. China under grant 60825304, the Taishan Scholar Program of Shandong Province and the Foundation of Binzhou University of P.R. China under grant Bzxykj0802.

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Song, X., Zhang, H. & Liu, H. l 2–l ∞ Filtering Design for Piecewise Discrete-Time Linear Systems. Circuits Syst Signal Process 28, 883–898 (2009). https://doi.org/10.1007/s00034-009-9127-2

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  • DOI: https://doi.org/10.1007/s00034-009-9127-2

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