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New Stability Criteria for Uncertain Nonlinear Stochastic Time-Delay Systems

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Abstract

This paper deals with the problem of robust stabilization and non-fragile robust control for a class of uncertain stochastic nonlinear time-delay systems that satisfy a one-sided Lipschitz condition. The parametric uncertainties are assumed to be real time-varying and norm bounded. Based on the one-sided Lipschitz condition including useful information of the nonlinear part, a new stability criterion for this class of nonlinear systems is provided. A memoryless non-fragile state-feedback controller is designed to guarantee robust stochastic stability of closed-loop systems. The approach of linear matrix inequalities is proposed to solve the robust stability for stochastic nonlinear systems with time-varying delay, and to obtain new delay-dependent sufficient conditions. Numerical examples are given to illustrate the validity and advantages of the proposed theoretical results.

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References

  1. Y. Cao, J. Lam, Computation of robust stability bounds for time-delay systems with nonlinear time-varying perturbation. Int. J. Syst. Sci. 31, 350–365 (2000)

    Google Scholar 

  2. W.H. Chen, Z.H. Guan, X.M. Lu, Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Syst. Control Lett. 54, 547–555 (2005)

    Article  MathSciNet  Google Scholar 

  3. Y. Chen, W.X. Zheng, A.K. Xue, A new result on stability analysis for stochastic neutral systems. Automatica 46, 2100–2104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Dekker, J.G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984)

    MATH  Google Scholar 

  5. T. Donchev, V. Rios, P. Wolenski, Strong invariance and one-sided Lipschitz multifunctions. Nonlinear Anal. Theory 60, 849–862 (2005)

    Article  MathSciNet  Google Scholar 

  6. Q.L. Han, Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations. Comput. Math. Appl. 47, 1201–1209 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. G.D. Hu, A note on observer for one-sided Lipschitz nonlinear systems. IMA J. Math. Control 1(25), 297–303 (2007)

    Article  Google Scholar 

  8. M.G. Hua, F.Q. Deng, X.Z. Liu, Y.J. Peng, Robust delay-dependent exponential stability of uncertain stochastic system with time-varying delay. Circ. Syst. Signal Process. 29, 515–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Jeetendra, J. Vernold Vivin, Delay range-dependent stability analysis for markovian jumping stochastic systems with nonlinear perturbations. Stoch. Anal. Appl. 30, 590–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. O.M. Kwon, Stability criteria for uncertain stochastic dynamic systems with time-varying delays. Int. J. Robust. Nonlinear Control 21, 338–350 (2011)

    Article  Google Scholar 

  11. W.Q. Li, X.J. Xie, Inverse optimal stabilization for stochastic nonlinear systems whose linearizations are not stabilizable. Automatica 45, 498–503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. X.R. Mao, Robustness of exponential stability of stochastic differential delay equations. IEEE Trans. Autom. Control 41(3), 442–447 (1996)

    Article  MATH  Google Scholar 

  13. X.R. Mao, Stochastic Differential Equations and Applications (Horwood, Chichester, 1997)

    MATH  Google Scholar 

  14. X.R. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125–142 (2002)

    Article  MathSciNet  Google Scholar 

  15. X.R. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosion in population dynamics. Stoch. Process. Appl. 97, 95–110 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Y.G. Niu, D.W.C. Ho, J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. Automatica 41, 873–880 (2005)

    Article  MathSciNet  Google Scholar 

  17. Y.G. Niu, D.W.C. Ho, Robust observer design for It\(\hat{o}\) stochastic time-delay systems via sliding mode control. Syst. Control Lett. 55, 781–793 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. C. Wang, Y. Shen, Delay-dependent non-fragile robust stabilization and \(H_{\infty }\) control of uncertain stochastic systems with time-varying delay and nonlinearity. J. Frankl. Inst. 348, 2174–2190 (2011)

    Article  MATH  Google Scholar 

  19. Y.T. Wang, X. Zhang, Y.M. Hu, Robust \(H_{\infty }\) control for a class of uncertain neutral stochastic systems with mixed delays: a CCL approach. Circ. Syst. Signal Process. 32(2), 631–646 (2013)

    Article  MathSciNet  Google Scholar 

  20. Y.T. Wang, A.H. Yu, X. Zhang, Robust stability of stochastic genetic regulatory networks with time-varying delays: a delay fractioning approach. Neural Comput. Appl. 23(5), 1217–1227 (2013)

    Article  Google Scholar 

  21. G.L. Wei, Z.D. Wang, H.S. Shu, J.A. Fang, Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances. Syst. Control Lett. 56, 623–633 (2007)

    Article  MathSciNet  Google Scholar 

  22. M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40, 1435–1439 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. S.Y. Xu, J. Lam, T.W. Chen, Robust \(H_{\infty }\) control for uncertain discrete stochastic time-delay systems. Syst. Control Lett. 51, 203–215 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. M.Y. Xu, G.D. Hu, Y.B. Zhao, Reduced-order observer design for one-sided Lipschitz non-linear systems. IMA J. Math. Control 1(26), 299–307 (2009)

    Article  MathSciNet  Google Scholar 

  25. V.A. Yakubovich, S-Procedure in Nonlinear Theory (Vestnik Leningradskogo Universiteta, Ser. Matematika, 1997)

    Google Scholar 

  26. J.H. Zhang, P. Shi, J.Q. Qiu, Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems. J. Frankl. Inst. 346, 676–690 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. J.H. Zhang, P. Shi, H.J. Yang, Non-fragile robust stabilization and \(H_{\infty }\) control for uncertain stochastic nonlinear time-delay systems. Chaos Solitons Fractals 42, 3187–3196 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. W. Zhang, X.S. Cai, Z.Z. Han, Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations. J. Comput. Appl. Math. 234, 174–180 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. W.B. Zhang, Y. Tang, Q.Y. Miao, W. Du, Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. 24, 1316–1326 (2013)

    Article  MATH  Google Scholar 

  30. W.B. Zhang, Y. Tang, X.T. Wu, J.A. Fang, Synchronization of nonlinear dynamical networks with heterogeneous impulses, IEEE Trans. Circ. Syst. I Reg. Papers (2013). doi:10.1109/TCSI.2013.2286027

  31. Y.B. Zhao, J. Tao, N.Z. Shi, A note on observer design for one-sided Lipschitz nonlinear systems. Syst. Control Lett. 59, 66–71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Q. Zhou, S.Y. Xu, B. Chen, Y.M. Chu, \(H_{\infty }\) filtering for stochastic systems with time-varying delay. Int. J. Syst. Sci. 42, 235–244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Z. Zuo, Y. Wang, New stability criterion for a class of linear systems with time-varying delay and nonlinear perturbations. IEE P Cont. Theory Appl. 153, 623–626 (2006)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Xiufeng Miao & Longsuo Li

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  1. Xiufeng Miao
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  2. Longsuo Li
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Correspondence to Longsuo Li.

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Miao, X., Li, L. New Stability Criteria for Uncertain Nonlinear Stochastic Time-Delay Systems. Circuits Syst Signal Process 34, 2441–2456 (2015). https://doi.org/10.1007/s00034-014-9943-x

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  • DOI: https://doi.org/10.1007/s00034-014-9943-x

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