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Design of robust \(H_\infty \) filtering controller for discrete-time-varying delay systems with missing measurements

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Abstract

This work addresses the robust \(H_\infty \) filtering problem for a class of discrete-time-varying delay systems with missing measurements, when the uncertain parameters are inside a given convex polytope. The first step in applying a model transformation is to use a three-term approximation for delayed state variables, which leads to lower approximation error than other methods. By utilizing scaled small-gain theorem and Lyapunov–Krasovskii functional methods, sufficient conditions for the robust stability of the filtering error system with a prescribed \(H_\infty \) performance level are obtained. The use of the scaled small-gain theorem for the robust stability of discrete-time-varying systems with missing measurements has not been studied before in the literature. This represents the main novelty of this paper. Some previous results based on LMIs are found to be particular cases of that obtained in this paper. Finally, numerical examples are given to illustrate the efficiency of the proposed methodology.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Ahmida F, Tissir EH (2015) A new approach to \(H_\infty \) filtering for switched TS fuzzy systems with interval time varying delays and parameter uncertainties. Br J Math Comput Sci 6(3):215. https://doi.org/10.9734/BJMCS/2015/13916

    Article  Google Scholar 

  2. El-Amrani A, El Hajjaji A, Boumhidi I, Hmamed A (2020) Improved finite frequency \(H_\infty \) filtering for Takagi-Sugeno fuzzy systems. Int J Syst Control Commun 11(1):1–24. https://doi.org/10.1504/IJSCC.2020.105391

    Article  Google Scholar 

  3. Naamane K, Tissir EH (2019) \(H_\infty \) filter design for T-S fuzzy nonlinear quadratic systems with time-varying delay. Circuits Syst Signal Process 38(11):4999–5017. https://doi.org/10.1007/s00034-019-01128-8

    Article  Google Scholar 

  4. Xie X-P, Zhu X-L, Gong D-W (2013) Relaxed filtering designs for continuous-time nonlinear systems via novel fuzzy \(H_\infty \) filters. Signal Process 93(5):1251–1258. https://doi.org/10.1016/j.sigpro.2012.10.013

    Article  Google Scholar 

  5. Charqi M, Chaibi N, Tissir EH (2020) \(H_\infty \) filtering of discrete-time switched singular systems with time-varying delays. Int J Adapt Control Signal Process 34(4):444–468. https://doi.org/10.1002/acs.3093

    Article  MATH  Google Scholar 

  6. Lin J, Fei S, Wu Q (2012) Reliable \(H_\infty \) filtering for discrete-time switched singular systems with time-varying delay. Circuits Syst Signal Process 31(3):1191–1214. https://doi.org/10.1007/s00034-011-9361-2

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang D, Cai W, Wang Q-G (2014) Energy-efficient \(H_\infty \) filtering for networked systems with stochastic signal transmissions. Signal Process 101:134–141. https://doi.org/10.1016/j.sigpro.2014.01.032

    Article  Google Scholar 

  8. Hmamed A, El Kasri C, Tissir EH, Alvarez T, Tadeo F (2013) Robust \(H_\infty \) filtering for uncertain 2-D continuous systems with delays. Int J Innov Comput Inf Control IJICIC 9(5):2167–2183

    MATH  Google Scholar 

  9. Yang R, Gao H, Shi P (2011) Delay-dependent \({L_{2}}-{L_{\infty }}\) filter design for stochastic time-delay systems. IET Control Theory Appl 5(1):1–8. https://doi.org/10.1049/iet-cta.2009.0347

    Article  MathSciNet  Google Scholar 

  10. Tissir EH (2007) Robust control design for retarded discrete-time systems. J Dyn Syst Meas Control Trans ASME 129(1):72–76

    Article  Google Scholar 

  11. El Fezazi N, El Haoussi F, Tissir EH, Alvarez T (2017) Design of robust \(H_\infty \) controllers for congestion control in data networks. J Franklin Inst 354(17):7828–7845. https://doi.org/10.1016/j.jfranklin.2017.09.026

    Article  MathSciNet  MATH  Google Scholar 

  12. Pan Y, Li Q, Liang H, Lam H-K (2021) A novel mixed control approach for fuzzy systems via membership functions online learning policy. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2021.3130201

    Article  Google Scholar 

  13. Pan Y, Wu Y, Lam H-K (2022) Security-based fuzzy control for nonlinear networked control systems with dos attacks via a resilient event-triggered scheme. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2022.3148875

    Article  Google Scholar 

  14. Ech-charqy A, Ouahi M, Tissir EH (2018) Delay-dependent robust stability criteria for singular time-delay systems by delay-partitioning approach. Int J Syst Sci 49(14):2957–2967. https://doi.org/10.1080/00207721.2018.1529255

    Article  MathSciNet  MATH  Google Scholar 

  15. Rachid H, Lamrabet O, Tissir EH (2019) Robust stability for delta operator systems via Wirtinger inequality. In: 2019 international conference of computer science and renewable energies (ICCSRE). 2019 International conference of computer science and renewable energies (ICCSRE), Agadir, Morocco: IEEE, pp 1–4 . https://doi.org/10.1109/ICCSRE.2019.8807471

  16. Lian Z, He Y, Zhang C-K, Wu M (2016) Stability analysis for T-S fuzzy systems with time-varying delay via free-matrix-based integral inequality. Int J Control Autom Syst 14(1):21–28. https://doi.org/10.1007/s12555-015-2001-z

    Article  Google Scholar 

  17. Liao D, Zhong S, Luo J, Zhang X, Yu Y, Zhong Q (2019) Improved delay-dependent stability criteria for networked control system with two additive input delays. Int J Control Autom Syst 17(9):2174–2182. https://doi.org/10.1007/s12555-018-0481-3

    Article  Google Scholar 

  18. Zoulagh T, El Aiss H, Hmamed A, El Hajjaji A (2018) \(H_\infty \) filter design for discrete time-varying delay systems: three-term approximation approach. IET Control Theory Appl 12(2):254–262. https://doi.org/10.1049/iet-cta.2017.0325

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin Z, Huijun G, Karimi HR (2013) Robust stability and stabilization of uncertain T-S fuzzy systems with time-varying delay: an Input–Output approach. IEEE Trans Fuzzy Syst 21(5):883–897. https://doi.org/10.1109/TFUZZ.2012.2235840

    Article  Google Scholar 

  20. Hmamed A, El Aiss H, EL Hajjaji A (2015) Stability analysis of linear systems with time varying delay: an input output approach. In: 2015 54th IEEE conference on decision and control (CDC). 2015 54th IEEE conference on decision and control (CDC), Osaka: IEEE, pp 1756–1761. https://doi.org/10.1109/CDC.2015.7402464

  21. Fridman E, Shaked U (2006) Input-output approach to stability and \(L_2\)-gain analysis of systems with time-varying delays. Syst Control Lett 55(12):1041–1053. https://doi.org/10.1016/j.sysconle.2006.07.002

    Article  MATH  Google Scholar 

  22. Li Z, Gao H, Agarwal RK (2013) Stability analysis and controller synthesis for discrete-time delayed fuzzy systems via small gain theorem. Inf Sci 226:93–104. https://doi.org/10.1016/j.ins.2012.11.008

    Article  MathSciNet  MATH  Google Scholar 

  23. El Aiss H, Hmamed A, El Adel EM (2017) Delay-dependent stability criteria and stabilization for discrete-time systems via three terms approximation. J Control Eng Appl Inform 19(4):3–12

    Google Scholar 

  24. El Aiss H, Rachid H, Hmamed A, El Hajjaji A (2018) Approach to delay-dependent robust stability and stabilisation of delta operator systems with time-varying delays. IET Control Theory Appl 12(6):786–792. https://doi.org/10.1049/iet-cta.2017.0006

    Article  MathSciNet  Google Scholar 

  25. Duan Z, Shen J (2019) Synchronization problem of 2-D coupled dynamical networks with communication delays and missing measurements. Multidimens Syst Signal Process 30(1):39–67. https://doi.org/10.1007/s11045-017-0545-1

    Article  MATH  Google Scholar 

  26. Li W, Wei G, Wang L (2012) Probability-dependent static output feedback control for discrete-time nonlinear stochastic systems with missing measurements. Math Probl Eng. https://doi.org/10.1155/2012/696742

    Article  MathSciNet  MATH  Google Scholar 

  27. Li X, Karimi HR, Xiang Z (2013) Output feedback control of discrete impulsive switched systems with state delays and missing measurements. Math Probl Eng. https://doi.org/10.1155/2013/283426

    Article  MathSciNet  MATH  Google Scholar 

  28. Wei G, Wang Z, Shen B, Li M (2011) Probability-dependent gain-scheduled filtering for stochastic systems with missing measurements. IEEE Trans Circuits Syst II Express Briefs 58(11):753–757. https://doi.org/10.1109/TCSII.2011.2168018

    Article  Google Scholar 

  29. Zou Y, Niu Y (2012) Predictive control design subject to multiple missing measurements. In: Proceedings of the 10th world congress on intelligent control and automation. 2012 10th world congress on intelligent control and automation (WCICA 2012), Beijing, China: IEEE, pp 2701–2705 . https://doi.org/10.1109/WCICA.2012.6358330

  30. Zabari A, Tissir EH, Kririm S (2016) Delay dependent robust \(H_\infty \) filter design for discrete time-delay systems with missing measurements via homogeneous polynomial matrices. Int J Autom Smart Technol 6(3):163–175. https://doi.org/10.5875/AUSMT.V6I3.1112

    Article  Google Scholar 

  31. Zhang M, Shen C, Wu Z-G, Zhang D (2020) Dissipative filtering for switched fuzzy systems with missing measurements. IEEE Trans Cybern 50(5):1931–1940. https://doi.org/10.1109/TCYB.2019.2908430

    Article  Google Scholar 

  32. Tian E, Yue D, Wei G (2013) Robust \(H_\infty \) filter for discrete-time linear system with uncertain missing measurements and non-linearity. IET Signal Proc 7(3):239–248. https://doi.org/10.1049/iet-spr.2012.0029

    Article  MathSciNet  Google Scholar 

  33. You J, Gao H, Basin MV (2013) Further improved results on \(H_\infty \) filtering for discrete time-delay systems. Signal Process 93(7):1845–1852. https://doi.org/10.1016/j.sigpro.2013.01.021

    Article  Google Scholar 

  34. Huaicheng Y, Zhenzhen S, Hao Z, Hongbo S (2012) Quantized \(H_\infty \) filtering for discrete-time networked systems with mixed delays and missing measurements. In: Proceedings of the 31st Chinese control conference. Proceedings of the 31st Chinese Control Conference, pp 5966–5971

  35. Gu Z, Fei S, Yue D, Tian E (2014) \(H_\infty \) filtering for discrete-time systems subject to stochastic missing measurements: a decomposition approach. Int J Syst Sci 45(7):1356–1363. https://doi.org/10.1080/00207721.2013.871372

    Article  MathSciNet  MATH  Google Scholar 

  36. He X, Wang Z, Zhou D (2007) Robust \(H_\infty \) filtering for time-delay systems with missing measurements: a parameter-dependent approach. J Control Theory Appl 5(4):336–344. https://doi.org/10.1007/s11768-006-6095-y

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang Z, Yang F, Ho DWC, Liu X (2006) Robust \(H_\infty \) filtering for stochastic time-delay systems with missing measurements. IEEE Trans Signal Process 54(7):2579–2587. https://doi.org/10.1109/TSP.2006.874370

    Article  MATH  Google Scholar 

  38. Zabari A, Tissir EH, Tadeo F (2017) Delay-dependent robust \(H_\infty \) filtering with lossy measurements for discrete-time systems. Arab J Sci Eng 42(12):5263–5273. https://doi.org/10.1007/s13369-017-2608-x

    Article  MathSciNet  MATH  Google Scholar 

  39. Gao H, Meng X, Chen T (2009) \(H_\infty \) filter design for discrete delay systems: a new parameter-dependent approach. Int J Control 82(6):993–1005. https://doi.org/10.1080/00207170802277501

    Article  MathSciNet  MATH  Google Scholar 

  40. He Y, Liu G-P, Rees D, Wu M (2009) \(H_\infty \) filtering for discrete-time systems with time-varying delay. Signal Process 89(3):275–282. https://doi.org/10.1016/j.sigpro.2008.08.008

    Article  MATH  Google Scholar 

  41. Zhang X-M, Han Q-L (2006) Delay-dependent robust \(H_\infty \) filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality. IEEE Trans Circuits Syst II Express Briefs 53(12):1466–1470. https://doi.org/10.1109/TCSII.2006.884116

    Article  Google Scholar 

  42. Zhang B, Xu S, Zou Y (2008) Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica 44(11):2963–2967. https://doi.org/10.1016/j.automatica.2008.04.017

    Article  MathSciNet  MATH  Google Scholar 

  43. Xiao S, Xu L, Zeng H-B, Teo KL (2018) Improved stability criteria for discrete-time delay systems via novel summation inequalities. Int J Control Autom Syst 16(4):1592–1602. https://doi.org/10.1007/s12555-017-0279-8

    Article  Google Scholar 

  44. Gao H, Chen T (2007) New results on stability of discrete-time systems with time-varying state delay. IEEE Trans Autom Control 52(2):328–334. https://doi.org/10.1109/TAC.2006.890320

    Article  MathSciNet  MATH  Google Scholar 

  45. Huang H, Feng G (2010) Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay. IET Control Theory Appl 4(10):2152–2159. https://doi.org/10.1049/iet-cta.2009.0225

    Article  MathSciNet  Google Scholar 

  46. Kwon O-M, Park M-J, Park JH, Lee S-M, Cha E-J (2013) Improved delay-dependent stability criteria for discrete-time systems with time-varying delays. Circuits Systems Signal Process 32(4):1949–1962. https://doi.org/10.1007/s00034-012-9543-6

    Article  MathSciNet  Google Scholar 

  47. Charqi M, Tissir EH, Boukili B (2021) Asymptotic stability and stabilization of discrete-time switched systems with time-varying delay. Opt Control Appl Methods 42(2):486–502. https://doi.org/10.1002/oca.2684

    Article  MathSciNet  MATH  Google Scholar 

  48. Phat VN, Ratchagit K (2011) Stability and stabilization of switched linear discrete-time systems with interval time-varying delay. Nonlinear Anal Hybrid Syst 5(4):605–612. https://doi.org/10.1016/j.nahs.2011.05.006

    Article  MathSciNet  MATH  Google Scholar 

  49. Du D, Jiang B, Zhou S (2008) Delay-dependent robust stabilisation of uncertain discrete-time switched systems with time-varying state delay. Int J Syst Sci 39(3):305–313. https://doi.org/10.1080/00207720701805982

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang L, Shi P, Basin M (2008) Robust stability and stabilisation of uncertain switched linear discrete time-delay systems. IET Control Theory Appl 2(7):606–614. https://doi.org/10.1049/iet-cta:20070327

    Article  MathSciNet  Google Scholar 

  51. Du D, Zhang J (2010) Novel robust stabilization and \(H_\infty \) control results for discrete-time switched systems with time-varying delays. In: 2010 international conference on computer and communication technologies in agriculture engineering vol 1, pp 93–96 . https://doi.org/10.1109/CCTAE.2010.5543187

  52. Chen J-D, Wu I-T, Lien C-H, Lee C-T, Chen R-S, Yu K-W (2014) Robust exponential stability for uncertain discrete-time switched systems with interval time-varying delay through a switching signal. J Appl Res Technol 12(6):1187–1197

    Article  Google Scholar 

  53. Lien C-H, Yu K-W, Yang J-H, Chou I-C, Chen J-D, Chung L-Y (2016) Novel delay-partitioning approach on stability of uncertain discrete switched time-delay systems via switching signal design. IMA J Math Control Inf 33(3):843–857. https://doi.org/10.1093/imamci/dnv011

    Article  MathSciNet  MATH  Google Scholar 

  54. Park M, Kwon O, Choi S (2017) Stability analysis of discrete-time switched systems with time-varying delays via a new summation inequality. Nonlinear Anal Hybrid Syst 23:76–90. https://doi.org/10.1016/j.nahs.2016.08.001

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions that improved the quality of this paper.

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

    Fatima Zahra Darouiche & E L Houssaine Tissir

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  1. Fatima Zahra Darouiche
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  2. E L Houssaine Tissir
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TEH contributed to supervision and review and editing; DFZ contributed to writing—original draft.

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Correspondence to Fatima Zahra Darouiche or E L Houssaine Tissir.

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Darouiche, F.Z., Tissir, E.L.H. Design of robust \(H_\infty \) filtering controller for discrete-time-varying delay systems with missing measurements. Int. J. Dynam. Control 11, 1671–1689 (2023). https://doi.org/10.1007/s40435-022-01084-3

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