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H ∞ Filtering of Two-Dimensional T-S Fuzzy Systems

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Abstract

The H ∞ filtering problem for two-dimensional Takagi–Sugeno fuzzy systems described by the Fornasini–Marchesini (FM) model is studied. Attention is focused on the design of an H ∞ fuzzy filter such that the filter error system is asymptotically stable and preserves a guaranteed H ∞ performance. By using basis-dependent Lyapunov functions and adding slack matrix variables, the coupling between the Lyapunov matrix and the system matrices is eliminated. Then, a linear matrix inequality (LMI)-based approach is developed for designing the H ∞ fuzzy filter. Finally, an illustrative example is provided to show the effectiveness of the proposed approach and less conservatism.

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References

  1. J. An, G. Wen, C. Lin, R. Li, New results on a delay-derivative-dependent fuzzy H ∞ filter design for T-S fuzzy systems. IEEE Trans. Fuzzy Syst. 19(4), 770–779 (2011)

    Article  Google Scholar 

  2. M. Benhayoun, F. Mesquine, A. Benzaouia, Delay-dependent stabilizability of 2D delayed continuous systems with saturating control. Circuits Syst. Signal Process. (2013). doi:10.1007/s00034-013-9585-4

    MathSciNet  Google Scholar 

  3. C.W. Chen, J.S.H. Tsai, L.S. Shieh, Two-dimensional discrete-continuous model conversion. Circuits Syst. Signal Process. 18, 565–585 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. J.R. Cui, G.D. Hu, Q. Zhu, Stability and robust stability of 2D discrete stochastic systems. Discrete Dyn. Nat. Soc. 545361, 11 pp. (2011). doi:10.1155/2011/545361

    MathSciNet  Google Scholar 

  5. C. De Souza, L. Xie, D. Coutinho, Robust filtering for 2D discrete-time linear systems with convex-bounded parameter uncertainty. Automatica 46(4), 673–681 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Du, L. Xie, H ∞ Control and Filtering of Two-dimensional Systems (Springer, Berlin, 2002)

    MATH  Google Scholar 

  7. C. Du, L. Xie, Y. Soh, H ∞ filtering of 2D discrete systems. IEEE Trans. Signal Process. 48(6), 1760–1768 (2000)

    Article  MATH  Google Scholar 

  8. C. El-Kasri, A. Hmamed, T. Alvarez, F. Tadeo, Robust H ∞ filtering of 2D Roesser discrete systems: a polynomial approach. Math. Problems Eng. 521675, 15 pp. (2012)

  9. A. Hmamed, A. El Hajjaji, A. Benzaouia, Stabilization of discrete-time 2D T-S fuzzy systems by state feedback control, in 17th Mediterranean Conference on Control and Automation, MED’09, Thessaloniki (2009), pp. 1–6

    Chapter  Google Scholar 

  10. A. Hmamed, F. Mesquine, F. Tadeo, M. Benhayoun, A. Benzaouia, Stabilization of 2D saturated systems by state feedback control. Multidimens. Syst. Signal Process. 21, 277–292 (2010). doi:10.1007/s11045-010-0107-2

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, LMI conditions for robust stability of 2D linear discrete-time systems. Math. Probl. Eng. 2008, 356124 (2008)

    Article  MathSciNet  Google Scholar 

  12. A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, Robust stabilisation under linear fractional parametric uncertaines of two-dimensional system with Rosser models. Int. J. Sci. Tech. Autom. Control Comput. Eng. (2007). Special Issue. CSC

  13. A. Hmamed, C. El-Kasri, E.H. Tissir, T. Alvarez, F. Tadeo, Robust H ∞ filtering for uncertain 2-D continuous systems with delays. Int. J. Innov. Comput. Inf. Control 9(5), 2167–2183 (2013)

    Google Scholar 

  14. S. Huang, Z. Xiang, Delay-dependent robust H ∞ control for 2-D discrete nonlinear systems with state delays. Multidimens. Syst. Signal Process. (2013). doi:10.1007/s11045-013-0230-y

    Google Scholar 

  15. S. Huang, Z. Xiang, Delay-dependent stability for discrete 2D switched systems with state delays in the Roesser model. Circuits Syst. Signal Process. (2013). doi:10.1007/s00034-013-9600-9

    Google Scholar 

  16. A. Jadbabaie, A reduction in conservatism in stability and L 2 gain analysis of Takagi–Sugeno fuzzy systems via linear matrix inequalities, in Proceedings of the 14th IFAC Triennial World Congress, (1999), pp. 285–289

    Google Scholar 

  17. T. Kaczorek, Two-Dimensional Linear Systems (Springer, Berlin, 1985)

    MATH  Google Scholar 

  18. L. Li, W. Wang, X. Li, New approach to H ∞ filtering of two-dimensional T-S fuzzy systems. Int. J. Robust Nonlinear Control (2012). doi:10.1002/rnc.2866

    Google Scholar 

  19. S.K. Nguang, W. Assawinchaichote, H ∞ filtering for fuzzy dynamical systems with D stability constraints. IEEE Trans. Circuits Syst. 50(11), 1503–1508 (2003)

    Article  MathSciNet  Google Scholar 

  20. T. Ooba, On stability analysis of 2D systems based on 2D Lyapunov matrix inequalities. IEEE Trans. Circuits Syst. I 47(8), 1263–1265 (2000)

    Article  MathSciNet  Google Scholar 

  21. D. Peng, X. Guan, H ∞ filtering of 2D discrete state-delayed systems. Multidimens. Syst. Signal Process. 20(3), 265–284 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust H ∞ filtering for continuous-time polytopic systems with time-varying delay. Int. J. Robust Nonlinear Control 20, 346–365 (2010). doi:10.1002/rnc.1439

    Article  MATH  MathSciNet  Google Scholar 

  23. T. Tanaka, T. Hori, H.O. Wang, A fuzzy Lyapunov approach to fuzzy control system design, in Proceedings of the American Control Conference, Arlington, V (2001), pp. 4790–4795

    Google Scholar 

  24. H. Tuan, P. Apkarian, T. Nguyen, T. Narikiyo, Robust mixed H 2–H ∞ filtering of 2D systems. IEEE Trans. Signal Process. 50(7), 1759–1771 (2002)

    Article  Google Scholar 

  25. L. Wu, Z. Wang, H. Gao, C. Wang, Filtering for uncertain 2D discrete systems with state delays. Signal Process. 87(9), 2213–2230 (2007)

    Article  MATH  Google Scholar 

  26. L. Wu, P. Shi, H. Gao, C. Wang, H ∞ filtering for 2D Markovian jump systems. Automatica 44(7), 1849–1858 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. L. Wu, Z. Wang, H. Gao, C. Wang, H ∞ and l 2−l 1 filtering for two-dimensional linear parameter-varying systems. Int. J. Robust Nonlinear Control 17(12), 1129–1154 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Z. Xiang, S. Huang, Stability analysis and stabilization of discrete-time 2D switched systems. Circuits Syst. Signal Process. 32(1), 401–414 (2013)

    Article  MathSciNet  Google Scholar 

  29. X. Xie, H. Zhang, Convergent stabilization conditions of discrete-time 2D T-S fuzzy systems via improved homogeneous polynomial techniques. Acta Autom. Sin. 36(9), 1305–1311 (2010)

    Article  MathSciNet  Google Scholar 

  30. X. Xie, H. Zhang, Stabilization of discrete-time 2D T-S fuzzy systems based on new relaxed conditions. Acta Autom. Sin. 36(2), 267–273 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Xu, J. Lam, Y. Zou, Z. Lin, W. Paszke, Robust H ∞ filtering for uncertain 2D continuous systems. IEEE Trans. Signal Process. 53(5), 1731–1738 (2005)

    Article  MathSciNet  Google Scholar 

  32. S. Xu, J. Lam, Exponential H ∞ filter design for uncertain Takagi–Sugeno fuzzy systems with time delay. J. Eng. Appl. Artif. Intell. 17(6), 645–659 (2004)

    Article  Google Scholar 

  33. H. Xu, Z. Lin, A. Makur, Non-fragile H 2 and H ∞ filter designs for polytopic two-dimensional systems in Roesser model. Multidimens. Syst. Signal Process. 21(3), 255–275 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Yang, Y. Li, Set-membership fuzzy filtering for nonlinear discrete-time systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(1), 116–124 (2010)

    Article  Google Scholar 

  35. R. Yang, L. Xie, C. Zhang, H 2 and mixed H 2/H ∞ control of two-dimensional systems in Roesser model. Automatica 42(9), 1507–1514 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. 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 

  37. S. Zhou, B. Zhang, W.X. Zheng, Gain-scheduled H ∞ filtering of parameter-varying systems. Int. J. Robust Nonlinear Control 16(8), 397–411 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  38. S. Zhou, T. Li, Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets Syst. 151(1), 139–153 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Faculty of Sciences Dhar El Mehraz, University of Sidi Mohamed Ben Abdellah, B.P. 1796, Fes-Atlas, Morocco

    Bensalem Boukili & Abdelaziz Hmamed

  2. Faculty of Science Semlalia, LAEPT URAC 28, University Cadi Ayyad, BP 2390, Marrakech, Morocco

    Abdellah Benzaouia

  3. MIS-University of Picardie Jules-Vernes, 7, Rue du Moulin Neuf, 80000, Amiens, France

    Ahmed El Hajjaji

Authors
  1. Bensalem Boukili
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  2. Abdelaziz Hmamed
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  3. Abdellah Benzaouia
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  4. Ahmed El Hajjaji
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Correspondence to Abdellah Benzaouia.

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Boukili, B., Hmamed, A., Benzaouia, A. et al. H ∞ Filtering of Two-Dimensional T-S Fuzzy Systems. Circuits Syst Signal Process 33, 1737–1761 (2014). https://doi.org/10.1007/s00034-013-9720-2

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  • DOI: https://doi.org/10.1007/s00034-013-9720-2

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