Robust \(H_{\infty }\) Filtering for Two-Dimensional Delayed Systems

  • Abdellah Benzaouia
  • Abdelaziz Hmamed
  • Fernando Tadeo
Chapter
First Online:
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 28)

Abstract

This chapter deals with robust \(H_{\infty }\) filtering for continuous two-dimensional systems with delays, considering several situations: constant or time-varying delays and combined with polytopic or linear fractional uncertainties. Sufficient conditions to have an \(H_{\infty }\) noise attenuation are given in terms of linear matrix inequalities, so \(H_{\infty }\) filters can be obtained by solving a convex optimization problem. Examples are given to illustrate the effectiveness of the proposed results.

Keywords

Robust filtering Continuous 2-D systems State delays \(H_{\infty }\) Filtering LMIs 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    C. Du, L. Xie, \(H_{\infty }\) Control and Filtering of Two-dimensional Systems (Springer, Heidelberg, 2002)Google Scholar
  2. 2.
    H.D. Tuan, P. Apkarian, T.Q. Nguyen, T. Narikiys, Robust mixed \(H_{2}/H_{\infty }\) filtering of 2-D systems. IEEE Trans. Signal Process. 50(7), 1759–1771 (2002)CrossRefGoogle Scholar
  3. 3.
    L. Xie, C. Du, C. Zhang, Y.C. Soh, \(H_{2}/H_{\infty }\) deconvolution filtering of 2-D digital systems. IEEE Trans. Signal Process. 50(9), 2319–2332 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    N.E. Mastorakis, M. Swamy, A new method for computing the stability margin of two-dimensional continuous systems. IEEE Trans. Circuits Syst. I 49(6), 869–872 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    E. Rogers, K. Galkowski, D.H. Owens, Delay differential control theory applied to differential linear repetitive processes. In Proceedings of the 2002 American Control Conference, Anchorage, AK, 8–10 May 2002Google Scholar
  6. 6.
    J. Huang, G. Lu, X. Zou, Existence of traveling wave fronts of delayed lattice differential equations. J. Math. Anal. Appl. 298(2), 538–558 (2004)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    B.G. Zhang, C.J. Tian, Oscillation criteria of a class of partial difference equations with delays. Comput. Math. Appl. 48(1–2), 291–303 (2004)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    C.W. Chen, J.S.H. Tsai, L.S. Shieh, Modeling and solution of two-dimensional input time-delay systems. J. Frankl. Inst. 337(5), 569–578 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    W. Paszke, J. Lam, K. Galkowski, S. Xu, Z. Lin, Robust stability and stabilization of 2-D discrete state-delayed systems. Syst. Control Lett. 51(3–4), 277–291 (2004)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    L. Dugard, E.I. Verriest, Stability and Control of Time-delay Systems, Lecture Notes in Control and Information Sciences vol. 228, (Springer, London, 1998)Google Scholar
  11. 11.
    H. Gao, C. Wang, A delay-dependent approach to robust \(H_{\infty }\) filtering for uncertain discrete-time state-delayed systems. IEEE Trans. Signal Process. 52(6), 1631–1640 (2004)Google Scholar
  12. 12.
    M.S. Mahmoud, Robust Control and Filtering for Time-delay Systems (Marcel Dekker, New York, 2000)MATHGoogle Scholar
  13. 13.
    Y.S. Moon, P. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems. Int. J. Control 74(14), 1445–1447 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust \(H_{\infty }\) filtering for uncertain discrete-time state-delayed systems. IEEE Trans. Signal Process. 49(8), 1696–1703 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    C. De Souza, L. Xie, D. Coutinho, Robust filtering for 2-D discrete-time linear systems with convex-bounded parameter uncertainty. Automatica 46(4), 673–681 (2010)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    S.F. Chen, I.K. Fong, Robust filtering for 2-D state-delayed systems with NFT uncertainties. IEEE Trans. Signal Process. 54(1), 274–285 (2006)CrossRefGoogle Scholar
  17. 17.
    C. El-Kasri, A. Hmamed, T. Alvarez, F. Tadeo, Robust \(H_{\infty }\) filtering of 2-D Roesser discrete systems: a polynomial approach. Math. Probl. Eng., Article ID 521675, pp. 15 (2012)Google Scholar
  18. 18.
    C. El-Kasri, A. Hmamed, E.H. Tissir, F. Tadeo, Robust \(H_{\infty }\) filtering for uncertain two-dimensional continuous systems with time-varying delays. Multidimens. Syst. Signal Process. 24(4), 685–706 (2013)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    C. El-Kasri, A. Hmamed, F. Tadeo, Reduced-order \(H_{\infty }\) filters for uncertain 2-D continuous systems, via LMIs and polynomial matrices. Circuits Syst. Signal Process. 33(4), 1189–1214 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    C. El-Kasri, A. Hmamed, T. Alvarez, F. Tadeo, Robust \(H_{\infty }\) filtering for uncertain 2-D continuous systems, based on a polynomially parameter-dependent Lyapunov function, In 7th International Workshop on Multidimensional (nD) Systems (nDs), Poitiers, France, 5–7 September 2011Google Scholar
  21. 21.
    C. El-Kasri, A. Hmamed, T. Alvarez, F. Tadeo, Uncertain 2-D continuous systems with state delay: filter design using an \(H_{\infty }\) polynomial approach. Int. J. Comput. Appl. 44(18), 13–21 (2012)Google Scholar
  22. 22.
    C. El-Kasri, Filtrage \(H_{\infty }\) robuste des systèmes linéaires bidimensionnels. Ph.D. thesis of University Mohammed Ben Abdallah, Fès, Morocco (2013)Google Scholar
  23. 23.
    S. Xu, J. Lam, Y. Zou, Z. Lin, W. Paszke, Robust \(H_{\infty }\) filtering for uncertain 2-D continuous systems. IEEE Trans. Signal Process. 53(5), 1731–1738 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    El-K Boukas, Z.K. Liu, Deterministic and Stochastic Time Delay Systems, 1st edn. (Birkhauser, Boston, 2002)MATHCrossRefGoogle Scholar
  25. 25.
    S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics (SIAM Philadephia, 1994)Google Scholar
  26. 26.
    B. Chen, J. Lam, S. Xu, Memory state feedback guaranteed cost control for neutral delay systems. Int. J. Innov. Comput. Inf. Control 2(2), 293–303 (2006)Google Scholar
  27. 27.
    S.F. Chen, Delay-dependent stability for 2-D systems with delays in Roesser model, In American Conference Control, Baltimore, June 30–July 2, pp. 3470–3474 (2010)Google Scholar
  28. 28.
    K. Sun, J. Chen, G.P. Liu, D. Rees, Delay-dependent robust \(H_{\infty }\) filter design for uncertain linear systems with time-varying delay. Circuits Syst. Signal Process. 28(5), 763–779 (2009)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    W. Marszalek, Two-dimensional state-space discrete models for hyperbolic partial differential equations. Appl. Math. Model. 8(1), 11–14 (1984)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Benzaouia, M. Benhayoun, F. Tadeo, State-feedback stabilization of 2-D continuous systems with delays. Int. J. Innov. Comput. Inf. Control 7(2), 977–988 (2011)Google Scholar
  31. 31.
    A. Hmamed, F. Mesquine, M. Benhayoun, A. Benzaouia, F. Tadeo, Stabilization of 2-D saturated systems by state feedback control. Multidimens. Syst. Signal Process. 21(3), 277–292 (2010)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Abdellah Benzaouia
    • 1
  • Abdelaziz Hmamed
    • 2
  • Fernando Tadeo
    • 3
  1. 1.Department of PhysicsUniversity of Cadi AyyadMarrakechMorocco
  2. 2.Ecole Supérieure de TechnologieUniversity of Sidi Mohammed Ben AbdellahFèsMorocco
  3. 3.Departamento de Ingeniería de Sistemas y AutomáticaUniversity of ValladolidValladolidSpain

Personalised recommendations