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Delay-Dependent Stability and \(H_{\infty }\) Performance of 2-D Continuous Systems with Delays

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Abstract

The analysis of stability and \(H_{\infty }\) performance of two-dimensional (2-D) Roesser-like continuous systems with delayed states is solved here. Firstly, based on the delay partitioning method, and on the use of an auxiliary function-based integral inequality, a new delay-dependent sufficient condition for asymptotical stability of these systems is developed. Then, the obtained result is extended to \(H_{\infty }\) performance analysis, with all conditions formulated as linear matrix inequalities. Finally, some numerical examples are provided to demonstrate the effectiveness and benefits of the proposed methodology.

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Change history

  • 03 July 2018

    The original version of the article unfortunately contained typographical errors in unnumbered equations before (12).

References

  1. M. Alfidi, A. Hmamed, Robust stability analysis for 2-D continuous systems via parameter-dependent Lyapunov functions. WSEAS Trans. Syst. Control 2(11), 497–503 (2007)

    Google Scholar 

  2. M. Benhayoun, F. Mesquine, A. Benzaouia, Delay-dependent stabilizability of 2D delayed continuous systems with saturating control. Circuits Syst. Signal Process. 32(6), 2723–2743 (2013)

    Article  MathSciNet  Google Scholar 

  3. A. Benzaouia, M. Benhayoun, F. Tadeo, State-feedback stabilization of 2D continuous systems with delays. Int. J. Innov. Comput. Inf. Control 7(2), 977–988 (2011)

    Google Scholar 

  4. C. Du, L. Xie, C. Zhang, \(H_{\infty }\) control and robust stabilization of two-dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001)

    Article  MathSciNet  Google Scholar 

  5. M. Dymkov, E. Rogers, S. Dymkou, K. Galkowski, Constrained optimal control theory for differential linear repetitive processes. SIAM J. Control Optim. 47(1), 396–420 (2008)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Z.Y. Feng, L. Xu, M. Wu, Y. He, Delay-dependent robust stability and stabilisation of uncertain two-dimensional discrete systems with time-varying delays. IET Control Theory Appl. 4(10), 1959–1971 (2010)

    Article  MathSciNet  Google Scholar 

  8. Z. Feng, W.X. Zheng, Improved stability condition for Takagi-Sugeno fuzzy systems with time-varying delay. IEEE Trans. Cybern. 47(3), 661–670 (2017)

    Article  Google Scholar 

  9. K. Galkowski, LMI based stability analysis for 2D continuous systems, in International Conference on Electronics, Circuits and Systems, 3, pp. 923–926 (2002)

  10. M. Ghamgui, N. Yeganefar, O. Bachelier, D. Mehdi, \(H_ {\infty }\) performance analysis of 2D continuous time-varying delay systems. Circuits Syst. Signal Process. 34(11), 3489–3504 (2015)

    Article  MathSciNet  Google Scholar 

  11. M. Ghamgui, N. Yeganefar, O. Bachelier, D. Mehdi, Stability and stabilization of 2D continuous state-delayed systems, in Proceedings of 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 1878–1883 (2011)

  12. I. Ghous, Z. Xiang, Robust state feedback \(H_{\infty }\) control for uncertain 2-D continuous state delayed systems in the Roesser model. Multidimens. Syst. Signal Process. 27(2), 297–319 (2016)

    Article  MathSciNet  Google Scholar 

  13. K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings of the 39th IEEE Conference on Decision and Control, 3, pp. 2805–2810 (2000)

  14. A. Hmamed, M. Alfidi, A. Benzaouia, F. Tadeo, LMI conditions for robust stability of 2D linear discrete-time systems. Mathematical Problems in Engineering 2008, 356124, 11 p. (2008)

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

    MATH  Google Scholar 

  16. A. Hmamed, S. Kririm, A. Benzaouia, F. Tadeo, Delay-dependent stability and stabilisation of continuous 2D delayed systems with saturating control. Int. J. Syst. Sci. 47(12), 3004–3015 (2016)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  18. S. Kririm, A. Hmamed, F. Tadeo, Analysis and design of \( H_ {\infty } \) controllers for 2D singular systems with delays. Circuits Syst. Signal Process. 35(5), 1579–1592 (2016)

    Article  Google Scholar 

  19. E.B. Lee, W.S. Lu, Stabilization of two-dimensional systems. IEEE Trans. Autom. Control AC 30, 409–411 (1985)

    Article  MathSciNet  Google Scholar 

  20. W.S. Lu, Some new results on stability robustness of two-dimensional discrete systems. Multidimens. Syst. Signal Process. 5, 345–361 (1994)

    Article  MathSciNet  Google Scholar 

  21. P. Park, W.I. Lee, S.Y. Lee, Auxiliary function-based integral/summation inequalities: application to continuous/discrete time-delay systems. Int. J. Control Autom. Syst. 14(1), 3 (2016)

    Article  Google Scholar 

  22. W. Paszke, J. Lam, K. Galkowski, S. Xu, Z. Lin, Robust stability and stabilisation of 2D discrete state-delayed systems. Syst. Control Lett. 51(3), 277–291 (2004)

    Article  MathSciNet  Google Scholar 

  23. D. Peng, C. Hua, Improved approach to delay-dependent stability and stabilisation of two-dimensional discrete-time systems with interval time-varying delays. IET Control Theory Appl. 9(12), 1839–1845 (2015)

    Article  MathSciNet  Google Scholar 

  24. R. Roesser, A discrete state-space model for linear image processing. IEEE Trans. Autom. Control AC–20, 1–10 (1975)

    Article  MathSciNet  Google Scholar 

  25. S. Shi, Z. Fei, X. Sun, X. Yang, Stabilization of 2-D switched systems with all modes unstable via switching signal regulation. IEEE Trans. Autom. Control 63(3), 857–863 (2018)

    Article  MathSciNet  Google Scholar 

  26. J. Xu, Y. Nan, G. Zhang, G. Ou, H. Ni, Delay-dependent \(H_{\infty }\) control for uncertain 2-D discrete systems with state delay in the Roesser model. Circuits Syst. Signal Process. 32(3), 1097–1112 (2013)

    Article  Google Scholar 

  27. J. Yao, W. Wang, Y. Zou, The delay-range-dependent robust stability analysis for 2-D state-delayed systems with uncertainty. Multidimens. Syst. Signal Process. 24(1), 87–103 (2013)

    Article  MathSciNet  Google Scholar 

  28. E. Yaz, On state-feedback stabilization of two-dimensional digital systems. IEEE Trans. Circuits Syst. CAS–32, 1069–1070 (1985)

    Article  Google Scholar 

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Authors and Affiliations

  1. Engineering, Systems and Applications Laboratory, National School of Applied Sciences, BP 72, My Abdallah Avenue Km. 5 Imouzzer Road, Fez, Morocco

    Khalid Badie, Mohammed Alfidi & Zakaria Chalh

  2. Industrial Engineering School, University of Valladolid, 47005, Valladolid, Spain

    Fernando Tadeo

Authors
  1. Khalid Badie
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  2. Mohammed Alfidi
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  3. Fernando Tadeo
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  4. Zakaria Chalh
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Correspondence to Fernando Tadeo.

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Badie, K., Alfidi, M., Tadeo, F. et al. Delay-Dependent Stability and \(H_{\infty }\) Performance of 2-D Continuous Systems with Delays. Circuits Syst Signal Process 37, 5333–5350 (2018). https://doi.org/10.1007/s00034-018-0839-z

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