Skip to main content
SpringerLink
Log in

Exponential Stability of Nonlinear Impulsive and Switched Time-Delay Systems with Delayed Impulse Effects

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The exponential stability problem is considered for a class of nonlinear impulsive and switched time-delay systems with delayed impulse effects by using the method of multiple Lyapunov–Krasovskii functionals. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for stabilizing delayed impulses and destabilizing delayed impulses. It is shown that even if all the subsystems governing the continuous dynamics without impulse input delays are not exponential stable, if impulsive and switching signal satisfy a dwell-time upper bound condition, stabilizing delayed impulses can stabilize the systems in the exponential stability sense. Moreover, it is also shown that if the magnitude of the delayed impulses is sufficiently small, the exponential stability properties can be derived irrespective of the size of the impulse input delays under some conditions. The opposite situation is also developed. The efficiency of the proposed results is illustrated by two numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. D. Bainov, P.S. Simeonov, Systems with Impulsive Effect: Stability Theory and Applications (Ellis Horwood Limited, Chichester, 1989)

    Google Scholar 

  2. G. Ballinger, X.Z. Liu, Existence and uniqueness results for impulsive delay differential equations. Dyn. Contin. Impuls. Syst. 5, 579–591 (1999)

    MATH  MathSciNet  Google Scholar 

  3. W.H. Chen, J.G. Wang, Y.J. Tang, X. Lu, Robust \({H_\infty }\) control of uncertain linear impulsive stochastic systems. Int. J. Robust Nonlinear Control 18, 1348–1371 (2008)

    Article  MathSciNet  Google Scholar 

  4. W.H. Chen, W.X. Zheng, Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays. Automatica 45, 1481–1488 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. W.H. Chen, W.X. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects. Automatica 47, 1075–1083 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. H.L. Dong, Z.D. Wang, H.J. Gao, On design of quantized fault detection filters with randomly occurring nonlinearities and mixed time-delays. Signal Process. 4, 1117–1125 (2012)

    Article  Google Scholar 

  7. H.L. Dong, Z.D. Wang, H.J. Gao, Distributed \({H_\infty }\) filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. IEEE Trans. Ind. Electron. 60, 4665–4672 (2013)

    Article  Google Scholar 

  8. H.L. Dong, Z.D. Wang, H.J. Gao, Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts. IEEE Trans. Signal Process. 60, 3164–3173 (2013)

    Article  MathSciNet  Google Scholar 

  9. Y.G. Fang, K.A. Loparo, X.B. Feng, Almost sure and delta-moment stability of jump linear systems. Int. J. Control 59, 1281–1307 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)

    Book  MATH  Google Scholar 

  11. J.P. Hespanha, Uniform stability of switched linear systems: extension of Lasalle’s invariance principle. IEEE Trans. Autom. Control 49, 470–482 (2004)

    Article  MathSciNet  Google Scholar 

  12. J.P. Hespanha, D. Liberzon, A.R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems. Automatica 44, 2735–2744 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. D.W.C. Ho, J. Liang, J. Lam, Global exponential stability of impulsive high order BAM neural networks with time-varying delays. Neural Netw. 19, 1581–1590 (2006)

    Article  MATH  Google Scholar 

  14. Z.P. Jiang, Y. Wang, Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)

    Article  MATH  Google Scholar 

  15. A. Khadra, X. Liu, X.S. Shen, Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses. IEEE Trans. Autom. Control 4, 923–928 (2009)

    Article  MathSciNet  Google Scholar 

  16. S. Kim, S.A. Campbell, X.Z. Liu, Stability of a class of linear switching systems with time delay. IEEE Trans. Circuits Syst. 53, 384–393 (1999)

    MathSciNet  Google Scholar 

  17. V. Kolmanovskii, A. Myshkis, Introduction to The Theory and Applications of Functional Differential Equations (Kluwer, Dordrecht, 1999)

    Book  MATH  Google Scholar 

  18. D. Liberzon, Switching in Systems and Control (Birkhauser, Boston, 1999)

    Google Scholar 

  19. C. Li, F. Ma, G. Feng, Hybrid impulsive and switching time-delay systems, shape. IET Control Theory Appl. 3, 1487–98 (2009)

    Article  MathSciNet  Google Scholar 

  20. B. Liu, Stability of solutions for stochastic impulsive systems via comparison approach. IEEE Trans. Autom. Control 53, 2128–2133 (2008)

    Article  Google Scholar 

  21. B. Liu, X.Z. Liu, G.R. Chen, H.Y. Wang, Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans. Circuits Syst. I 52, 1431–1441 (2005)

    Article  MathSciNet  Google Scholar 

  22. J. Liu, X.Z. Liu, Input-to-state stability of impulsive and switching hybrid systems with time-delay. IEEE Trans. Circuits Syst. I 47, 899–908 (2011)

    MATH  Google Scholar 

  23. L. Liu, J.T. Sun, Finite-time stabilization of linear systems via impulsive control. Int. J. Control 81, 905–909 (2008)

    Article  MATH  Google Scholar 

  24. X. Liu, X.M. Zhong, X.Y. Ding, Robust exponential stability of impulsive switched systems with switching delays: a Razumikhin approach. Commun. Nonlinear Sci. Numer. Simul. 17, 1805–1812 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. X. Liu, X.S. Shen, Y. Zhang, Q. Wang, Stability criteria for impulsive systems with time delay and unstable system matrices. IEEE Trans. Circuits Syst. I. Regul. Pap. 54, 2288–2298 (2007)

    Article  MathSciNet  Google Scholar 

  26. Y. Liu, W. Feng, Razumikhin–Lyapunov functional method for the stability of impulsive switched systems with time delay. Math. Comput. Model. 54, 2288–2298 (2009)

    MathSciNet  Google Scholar 

  27. P. Naghshtabrizi, J.P. Hespanha, A.R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems. Syst. Control Lett. 57, 378–385 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Nesic, A.R. Teel, Input-to-state stability of networked control systems. Automatica 40, 2121–2128 (2004a)

    MATH  MathSciNet  Google Scholar 

  29. D. Nesic, A.R. Teel, Input–output stability properties of networked control systems. IEEE Trans. Autom. Control 49, 1650–1667 (2004b)

    Article  MathSciNet  Google Scholar 

  30. H. Shen, S.Y. Xu, X.N. Sonh, G.D. Shi, Passivity-based control for Markovian jump systems via retarded output feedback. Circuits Syst. Signal Process. 31, 189–202 (2012)

    Article  MATH  Google Scholar 

  31. Y.G. Sun, Stabilization of switched systems with nonlinear impulse effects and disturbances. IEEE Trans. Autom. Control 56, 2739–2743 (2011)

    Article  Google Scholar 

  32. N. Van de Wouw, P. Naghshtabrizi, M.B.G. Cloosterman, J.P. Hespanha, Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays. Int. J. Robust Nonlinear Control 20, 387–411 (2010)

    Google Scholar 

  33. Y.J. Wang, X.M. Shi, Z.Q. Zuo, M.Z.Q. Chen, Y.T. Shao, On finite-time stability for nonlinear impulsive switched systems. Nonlinear Anal. Real World Appl. 14, 807–814 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  34. Y.J. Wang, G. Wang, X.M. Shi, Z.Q. Zuo, Finite-time stability analysis of impulsive switched discrete-time linear systems: the average dwell time approach. Circuits Syst. Signal Process. 31, 1877–1886 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. H.L. Xu, X.Z. Liu, K.L. Teo, Delay independent stability criteria of impulsive switched systems with time-invariant delays. Math. Comput. Model. 47, 372–379 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. L.G. Xu, D.H. He, Mean square exponential stability analysis of impulsive stochastic switched systems with mixed delays. Comput. Math. Appl. 62, 109–117 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. T. Yang, Impulsive Systems and Control: Theory and Application (Nova Science, New York, 2001)

    Google Scholar 

  38. Y. Yang, C. Xiang, T.H. Lee, Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching. Int. J. Control 85, 1977–1995 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Z. Yang, D. Xu, Stability analysis of delay neural networks with impulsive effects. IEEE Trans. Circuits Syst. II 52, 517–521 (2005)

    Article  Google Scholar 

  40. G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int. J. Syst. Sci. 32, 1055–1061 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  41. G. Zhai, H. Lin, P.J. Antsaklis, Quadratic stabilizability of switched linear systems with polytopic uncertainties. Int. J. Control 76, 747–753 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  42. L.X. Zhang, H.J. Gao, Asynchronously switched control of switched linear systems with average dwell time. Int. J. Syst. Sci. 46, 953–958 (2010)

    MATH  Google Scholar 

  43. W. Zhang, A. Abate, J.H. Hu, M.P. Vitus, Exponential stabilization of discrete-time switched linear systems. Automatica 45, 2526–2536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  44. P. Zhao, Y. Kang, D.H. Zhai, On input-to-state stability of stochastic nonlinear systems with Markovian jumping parameters. Int. J. Control 85, 343–349 (2009)

    Article  MathSciNet  Google Scholar 

  45. G.D. Zong, J. Li, Robust \({l_2}-{l_\infty }\) guaranteed cost filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Circuits Syst. Signal Process. 30, 17–33 (2011)

    Article  MATH  Google Scholar 

  46. G.D. Zong, R.H. Wang, W.X. Zheng, L.L. Hou, Finite-time \(H_\infty \) control for discrete-time switched nonlinear systems with time delay. Int. J. Robust Nonlinear Control (2011). doi:10.1002/rnc.3121

  47. G.D. Zong, L.L. Hou, J.F. Li, A descriptor system approach to filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Int. J. Innov. Comput. Inf. Control 7(5: A), 2213–2224 (2011)

    Google Scholar 

  48. G.D. Zong, R.H. Wang, W.X. Zheng, L.L. Hou, Finite time stabilization for a class of switched time-delay systems under asynchronous switching. Appl. Math. Comput. 219(11), 5757–5771 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the Editor and the reviewers for their valuable comments to improve the quality of the manuscript. This work is supported by NNSF of China under Grants 61104007, 61273091, 61273123, 61304066, Natural Science Foundation of Shandong province under Grant ZR2011FM033, Shandong Provincial Scientific Research Reward Foundation for Excellent Young and Middle-aged Scientists of China under grant BS2011DX013, BS2012SF008, and Taishan Scholar Project of Shandong Province of China.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Automation, Qufu Normal University, Rizhao, 276826, China

    Lijun Gao & Yuqiang Wu

  2. School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan , 243002, China

    Hao Shen

  3. Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan , 712-749, Republic of Korea

    Hao Shen

Authors
  1. Lijun Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuqiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lijun Gao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gao, L., Wu, Y. & Shen, H. Exponential Stability of Nonlinear Impulsive and Switched Time-Delay Systems with Delayed Impulse Effects. Circuits Syst Signal Process 33, 2107–2129 (2014). https://doi.org/10.1007/s00034-014-9743-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9743-3

Keywords

Search

Navigation