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Decentralized \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) Filtering for Interconnected Markovian Jump Systems with Delays

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Abstract

This paper deals with the problem of decentralized \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) filtering for a class of interconnected (or large-scale) Markovian jump systems with constant time delays. The purpose is to present delay-dependent conditions for the existence of mode-dependent decentralized filters, which guarantees that the filtering error system is stochastically stable with a prescribed \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) disturbance attenuation level. Such a purpose is achieved by using a mode-dependent centralized Lyapunov functional together with the so-called Jensen’s inequality. The obtained synthesis conditions are expressed in terms of linear matrix inequalities (LMIs), which leads to a convex design method for the concerned filters. An example including numerical and simulation results is provided finally to illustrate the effectiveness of the proposed design method.

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References

  1. Y.Y. Cao, Y.X. Sun, W.J. Mao, Output feedback decentralized stabilization: ILMI approach. Syst. Control Lett. 35(3), 183–194 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Z. Deng, P. Shi, H. Yang, Y. Xia, Robust H ∞ filtering for nonlinear systems with interval time-varying delays. Int. J. Innov. Comput. Inf. Control 6(12), 5527–5538 (2010)

    Google Scholar 

  3. Q. Ding, M. Zhong, On designing H ∞ fault detection filter for Markovian jump linear systems with polytopic uncertainties. Int. J. Innov. Comput., Inf. Control (A) 6(3), 995–1004 (2010)

    MathSciNet  Google Scholar 

  4. Z. Hu, Decentralized stabilization of large-scale interconnected systems with delays. IEEE Trans. Autom. Control 39(1), 180–182 (1994)

    Article  MATH  Google Scholar 

  5. M. Ikeda, G. Zhai, Y. Fujisaki, Decentralized H ∞ controller design for large-scale systems: a matrix inequality approach using a homotopy method, in Proceedings of 35th IEEE Conference on Decision and Control, Kobe (1996), pp. 1–6

    Google Scholar 

  6. H. Gao, T. Chen, H ∞ estimation for uncertain systems with limited communication capacity. IEEE Trans. Autom. Control 52(11), 2070–2084 (2007)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. T.N. Lee, U.L. Radovic, Decentralized stabilization of linear continuous and discrete time systems with delays in interconnections. IEEE Trans. Autom. Control 33(8), 757–761 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Li, V.A. Ugrinovskii, R. Orsi, Decentralized robust control of uncertain Markov jump parameter systems via output feedback. Automatica 43(11), 1932–1944 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Li, Y. Chu, S. Xu, Delay-dependent nonfragile robust H ∞ filtering of T-S fuzzy time-delay systems. Circuits Syst. Signal Process. 29(3), 361–375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Lin, Q.G. Wang, T.H. Lee, Less conservative stability conditions for fuzzy large-scale systems with time delays. Chaos Solitons Fractals 29(5), 1147–1154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Liu, D.C.W. Ho, Y. Niu, Stabilization of Markovian jump linear system over networks with random communication delay. Automatica 45(2), 416–421 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Y. Liu, W. Wang, Fuzzy H ∞ filtering for nonlinear stochastic systems with missing measurements. ICIC Express Lett. (B) 3(3), 739–744 (2009)

    Google Scholar 

  14. M.S. Mahmoud, S. Bingulac, Robust design of stabilizing controllers for interconnected time-delay systems. Automatica 34(6), 795–800 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. M.S. Mahmoud, P. Shi, A. Ismail, Control of interconnected jumping systems: an H ∞ approach. Asian J. Control 6(1), 1–8 (2004)

    Article  MathSciNet  Google Scholar 

  16. X. Meng, H. Gao, S. Mou, A new parameter-dependent approach to robust energy-to-peak filter design. Circuits Syst. Signal Process. 26(4), 451–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Nakura, Stochastic optimal tracking with preview by state feedback for linear discrete-time Markovian jump systems. Int. J. Innov. Comput. Inf. Control 6(1), 15–28 (2010)

    Google Scholar 

  18. Y. Niu, D.C.W. Ho, X. Wang, Sliding mode control for Itô stochastic systems with Markovian switching. Automatica 43(10), 1784–1790 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. J.H. Park, On design of dynamic output feedback controller for GCS of large-scale systems with delays in interconnections: LMI optimization approach. Appl. Math. Comput. 161(2), 423–432 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Qiu, K. Lu, New robust passive stability criteria for uncertain singularly Markov jump systems with time delays. ICIC Express Lett. (B) 3(3), 651–656 (2009)

    Google Scholar 

  21. N.R. Sandell, P. Varaiya, M. Athans, M.G. Safonov, Survey of decentralized control methods for large scale systems. IEEE Trans. Autom. Control 23(2), 108–128 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. H.E.E. Shabaik, M.S. Mahmoud, Y. Shi, H 2 and H ∞ filtering for linear systems with unknown inputs and polytopic uncertainty. Int. J. Innov. Comput. Inf. Control 6(10), 2211–2220 (2010)

    Google Scholar 

  23. D.D. Siljak, Large-Scale Dynamic Systems: Stability and Structure (Elsevier/North-Holland, Amsterdam, 1978), pp. 11–24

    MATH  Google Scholar 

  24. Y. Su, M. Chen, B. Chen, C. Lin, Fuzzy H ∞ filter design for nonlinear discrete-time systems with time-varying delays. Int. J. Innov. Comput. Inf. Control 6(10), 4335–4346 (2010)

    Google Scholar 

  25. V.A. Ugrinovskii, H.R. Pota, Decentralized control of power systems via robust control of uncertain Markov jump parameter systems. Int. J. Control 78(9), 662–677 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Wu, P. Shi, H. Gao, C. Wang, A new approach to robust H ∞ filtering for uncertain systems with both discrete and distributed delays. Circuits Syst. Signal Process. 26(2), 229–248 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Xia, Z. Zhu, M.S. Mahmoud, H 2 control for networked control systems with Markovian data losses and delays. ICIC Express Lett. (A) 3(3), 271–276 (2009)

    Google Scholar 

  28. J. Xiong, V.A. Ugrinovskii, I.R. Petersen, Local mode dependent decentralized control of uncertain Markovian jump large-scale systems, in Proceedings of 47th IEEE Conference on Decision and Control, Cancun (2008), pp. 4345–4350

    Chapter  Google Scholar 

  29. J. Xiong, V.A. Ugrinovskii, I.R. Petersen, Local mode dependent decentralized stabilization of uncertain Markovian jump large-scale systems. IEEE Trans. Autom. Control 54(11), 2632–2637 (2009)

    Article  MathSciNet  Google Scholar 

  30. S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 12(39), 1095–1113 (2008)

    Article  MathSciNet  Google Scholar 

  31. S. Xu, J. Lam, H. Gao, Y. Zou, Robust H ∞ filtering for uncertain discrete stochastic systems with time delays. Circuits Syst. Signal Process. 24(6), 753–770 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Xu, J. Lam, X. Mao, Delay-dependent H ∞ control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans. Circuits Syst. I, Regul. Pap. 54(9), 2070–2077 (2007)

    Article  MathSciNet  Google Scholar 

  33. H.B. Zhang, C.Y. Dang, Piecewise H ∞ controller design of uncertain discrete-time fuzzy systems with time delays. IEEE Trans. Fuzzy Syst. 16(6), 1649–1655 (2008)

    Article  Google Scholar 

  34. H.B. Zhang, G. Feng, Stability analysis and H ∞ controller design of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. IEEE Trans. Syst. Man Cybern., B Cybern. 38(5), 1390–1401 (2008)

    Article  MathSciNet  Google Scholar 

  35. 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 

  36. B. Zhang, S. Xu, Y. Zou, Output feedback stabilization for delayed large-scale stochastic systems with Markovian jumping parameters. Asian J. Control 11(4), 457–460 (2009)

    Article  MathSciNet  Google Scholar 

  37. X. Zhang, Y. Zheng, Nonlinear H ∞ filtering for interconnected Markovian jump systems. Journal of Systems Engineering and. Electronics 17(1), 138–146 (2006)

    MATH  Google Scholar 

  38. P. Zhao, Practical stability, controllability and optimal control of stochastic Markovian jump systems with time-delays. Automatica 44(12), 3120–3125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Yongmin Li

    Present address: School of Science, Huzhou Teachers College, Huzhou, 313000, Zhejiang, China

Authors and Affiliations

  1. Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, School of Automation, Southeast University, Nanjing, 210096, Jiangsu, China

    Yongmin Li & Shumin Fei

  2. School of Science, Huzhou Teachers College, Huzhou, 313000, Zhejiang, China

    Yuming Chu

  3. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China

    Baoyong Zhang

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  1. Yongmin Li
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  2. Shumin Fei
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  4. Yuming Chu
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Correspondence to Yongmin Li.

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Li, Y., Fei, S., Zhang, B. et al. Decentralized \(\mathcal{L}_{2}\)–\(\mathcal{L}_{\infty}\) Filtering for Interconnected Markovian Jump Systems with Delays. Circuits Syst Signal Process 31, 889–909 (2012). https://doi.org/10.1007/s00034-011-9350-5

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  • DOI: https://doi.org/10.1007/s00034-011-9350-5

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