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Robust Stabilization of Markovian Jump Linear Singular Systems with Wiener Process and Generally Incomplete Transition Rates

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Abstract

This paper is devoted to the investigation of robust stability and robust stabilizability for a class of uncertain Markovian jump continuous-time singular systems with Wiener process and generally uncertain transition rates (GUTRs). This new uncertain model is more general than the existing ones and can be applicable to more practical situations because of its uncertain parameters and transition rates. Each transition rate can be completely unknown, or only its estimate value is known. Our interests are focused on establishing sufficient criteria on robust stability and the design of a state feedback controller such that the robust stochastic stability is guaranteed, even if the singular system incorporates GUTRs, Wiener process disturbance and norm-bounded uncertainties. These sufficient criteria are derived in terms of linear matrix inequalities. Finally, a numerical example is presented to illustrate the effectiveness and applicability of the proposed method.

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References

  1. L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974)

    Google Scholar 

  2. E. Boukas, Z. Liu, Robust stability and stability of Markov jump linear uncertain systems with mode-dependent time delays. J. Opt. Thoery Appl. 209, 587–600 (2001)

    Article  MathSciNet  Google Scholar 

  3. E. Boukas, Control of Singular Systems with Random Abrupt Changes (Springer, Berlin, 2008)

    MATH  Google Scholar 

  4. L. Dai, Singular Control Systems (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  5. B. Du, J. Lam, Y. Zou, Z. Shu, Stability and stabilization for Markovian jump time-delay systems with partially unknown transition rates. IEEE Trans. Circuit Syst. 60(2), 341–351 (2013)

    Article  MathSciNet  Google Scholar 

  6. Y. Guo, Z. Wang, Stability of Markovian jump systems with generally uncertain transition rates. J. Franklin Inst. 350(9), 2826–2836 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Kao, C. Wang, L. Zhang, Delay-dependent exponential stability of impulsive Markovian jumping cohen-grossberg neural networks with reaction-diffusion and mixed delays. Neural Process. Lett. 38(3), 321–346 (2013)

    Article  Google Scholar 

  8. Y. Kao, J. Guo, C. Wang, X. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays. J. Franklin Inst. 349(6), 1972–1988 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Kao, C. Wang, H.R. Karimi, R. Bi, Global stability of coupled Markovian switching reaction-diffusion systems on networks. Nonlinear Anal. Hybrid Syst. 13, 61–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Y. Kao, J. Xie, C.H. Wang, Stabilisation of singular Markovian jump systems with generally uncertain transition rates. IEEE Trans. Autom. Control 59(9), 2604–2610 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Karan, P. Shi, C. Kaya, Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems. Automatica 42, 2159–2168 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Kumaresan, P. Balasubramaniam, Optimal control for stochastic nonlinear singular system using neural networks. Comput. Math. Appl. 56(9), 2145–2154 (2008)

    Article  MathSciNet  Google Scholar 

  13. Y. Li, Y. Kao, Stability of stochastic reaction-diffusion systems with Markovian switching and impulsive perturbations. Math. Probl. Eng. 1–13 (2012). Article ID 429568

  14. C. Lin, J. Wang, D. Wang, Robustness of uncertain descriptor systems. Syst. Control Lett. 31(3), 129–138 (1997)

    Article  MATH  Google Scholar 

  15. J. Lin, S. Fei, J. Shen, Delay-dependent \(H_{\infty }\) filtering for discrete-time singular Markovian jump systems with time-varying delay and partially unknown transition probabilities. Signal Process. 91(2), 277–289 (2011)

    Article  MATH  Google Scholar 

  16. S. Ma, E. Boukas, Robust \(H_{\infty }\) filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay. IET Control Theory Appl. 3(3), 351–361 (2009)

    Article  MathSciNet  Google Scholar 

  17. S. Ma, E. Boukas, Guaranteed cost control of uncertain discrete-time singular Markov jump systems with indefinite quadratic cost. Int. J. Robust Nonlinear 21, 1031–1045 (2011)

    Article  MathSciNet  Google Scholar 

  18. X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching (Imperial College Press, London, 2006)

    Book  MATH  Google Scholar 

  19. I.R. Peterson, A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett. 8, 351–357 (1987)

    Article  Google Scholar 

  20. J. Raouf, E.K. Boukas, Robust stabilization of Markovian jump linear singular systems with Wiener process. in Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30–July 2, 2004

  21. P. Shi, E. Boukas, \(H_\infty \) control for Markovian jumping linear systems with parametric uncertainty. J. Opt. Thoery Appl. 95(1), 75–99 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Tian, D. Yue, G. Wei, Robust control for Markovian jump systems with partially known transition probabilities and nonlinearities. J. Franklin Inst. 350(8), 2069–2083 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Tian, Y. Li, J. Zhao, S. Zhong, Delay-dependent stochastic stability crieria for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates. Appl. Math. Comput. 218(9), 5769–5781 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Wang, H. Wang, A. Xue, R. Lu, Delay-dependent \(H_{\infty }\) control for singular Markovian jump systems with time delay. Nonlinear Anal. Hybrid Syst. 8, 1–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Wei, J. Qiu, H.R. Karimi, M. Wang, A new design \(H_{\infty }\) filtering for contrinuous-time Markovian jump systems with time-varying delay and partially accessible mode information. Signal Process. 93, 2392–2407 (2013)

    Article  Google Scholar 

  26. L. Wu, X. Su, P. Shi, Sliding mode control with bounded \(H_{2}\) gain performance of Markovian jump singular time-delay systems. Automatica 48(8), 1929–1933 (2010)

    Article  MathSciNet  Google Scholar 

  27. Z. Wu, J.H. Park, H. Su, J. Chu, Stochastic stability analysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilities. J. Franklin Inst. 349(9), 2889–2902 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Z. Wu, P. Shi, H. Su, J. Chu, Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data. IEEE Trans. Cybern. 43(6), 1796–1806 (2013)

    Article  Google Scholar 

  29. Z. Wu, P. Shi, H. Shi, J. Chu, Asynchronous \(l_2-l_\infty \) filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities. Automatica 50(1), 180–186 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Xia, E. Boukas, P. Shi, J. Zhang, Stability and stabilization of continuous-time singular hybrid systems. Automatica 45(6), 1504–1509 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Xiong, J. Lam, Robust \(H_{2}\) control of Markovian jump systems with uncertain switching probabilities. Int. J. Syst. Sci. 45(6), 255–265 (2009)

    Article  MathSciNet  Google Scholar 

  32. J. Xiong, J. Lam, H. Gao, D.W.C. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica 41(6), 897–903 (2005)

    Article  MathSciNet  Google Scholar 

  33. S. Xu, J. Lam, Robust Control and Filtering of Singular Systems (Springer, Berlin, 2006)

  34. D. Yang, Q. Zhang, B. Yao, Singular Systems (Science Press, Beijing, 2004)

  35. L. Zhang, \(H_{\infty }\) estimation for discrete-time piecewise homogeneous Markov jump linear systems. Automatica 45(8), 2570–2576 (2009)

    Article  Google Scholar 

  36. L. Zhang, E. Boukas, Mode-dependent \(H_{\infty }\) control for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Int. J. Robust Nonlinear 19(8), 868–883 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Zhang, E. Boukas, Mode-dependent \(H_{\infty }\) filtering for discrete-time Markovian jump linear systems with partly unknown transition probability. Automatica 45(6), 1462–1467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Zhang, E. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(2), 463–468 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. L. Zhang, E. Boukas, J. Lam, Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Trans. Autom. Control 53(10), 2458–2464 (2008)

    Article  MathSciNet  Google Scholar 

  40. L. Zhang, J. Lam, Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete tran sition descriptions. IEEE Trans. Autom. Control 55(7), 1695–1701 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the Editor-in-Chief, Dr. M.N.S. Swamy, for his helpful comments in improving the presentation of the paper. This research is supported by the National Natural Science Foundations of China (61473097), National 863 Plan Project (2008 AA04Z401, 2009 AA043404), the Natural Science Foundation of Shandong Province (No. ZR2012FM006).

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Authors and Affiliations

  1. School of Technology, Nanjing Audit University, Nanjing, 211815, People’s Republic of China

    Guowei Yang

  2. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, People’s Republic of China

    Yonggui Kao & Lei Shi

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  1. Guowei Yang
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  2. Yonggui Kao
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  3. Lei Shi
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Correspondence to Yonggui Kao.

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Yang, G., Kao, Y. & Shi, L. Robust Stabilization of Markovian Jump Linear Singular Systems with Wiener Process and Generally Incomplete Transition Rates. Circuits Syst Signal Process 34, 2475–2498 (2015). https://doi.org/10.1007/s00034-014-9964-5

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  • DOI: https://doi.org/10.1007/s00034-014-9964-5

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