Advertisement

Circuits, Systems, and Signal Processing

, Volume 38, Issue 11, pp 5323–5341 | Cite as

Finite-Time Stability of Homogeneous Impulsive Positive Systems of Degree One

  • Huitao Yang
  • Yu ZhangEmail author
Short Paper
First Online:
  • 105 Downloads

Abstract

This paper investigates the finite-time stability (FTS) of a special class of hybrid systems, namely homogeneous impulsive positive systems of degree one. By using max-separable Lyapunov functions together with average impulsive interval method, a sufficient FTS criterion is obtained for homogeneous impulsive positive systems of degree one. It should be noted that it’s the first time that the FTS result for homogeneous impulsive positive systems of degree one is given. Finally, some numerical examples are provided to demonstrate the effectiveness of the presented theoretical results.

Keywords

Finite-time stability Homogeneous systems Impulsive positive systems Max-separable Lyapunov functions 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions which improved the quality of this paper. This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 0800219386).

References

  1. 1.
    F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, G.D. Tommasi, Finite-Time Stability and Control (Springer, London, 2014)zbMATHGoogle Scholar
  2. 2.
    F. Amato, R. Ambrosino, C. Cosentino, G.D. Tommasi, Finite-time stabilization of impulsive dynamical linear systems. Nonlinear Anal. Hybrid Syst. 5(1), 89–101 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Amato, M. Ariola, C. Cosentino, Finite-time stabilization via dynamic output feedback. Automatica 42(2), 337–342 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Amato, G.D. Tommasi, A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems. Automatica 49(8), 2546–2550 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    R. Ambrosino, F. Calabrese, C. Cosentino, G.D. Tommasi, Sufficient conditions for finite-time stability of impulsive dynamical systems. IEEE Trans. Autom. Control 54(4), 861–865 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    M.U. Akhmet, M. Beklioglu, T. Ergenc, V.I. Tkachenko, An impulsive ratio-dependent predator-prey system with diffusion. Nonlinear Anal. Real World Appl. 7(5), 1255–1267 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    G. Ballinger, X. Liu, Existence and uniqueness results for impulsive delay differential equations. Dyn. Contin. Discrete Impuls. Syst. 5, 579–591 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    L. Benvenuti, L. Farina, Positive and compartmental systems. IEEE Trans. Autom. Control 47(2), 370–373 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    C. Briat, Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems. Nonlinear Anal. Hybrid Syst. 24, 198–226 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    W. Chen, W. Zheng, Exponential stability of nonlinear time-delay systems with delayed impulse effects. Automatica 47(5), 1075–1083 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    J.J. Distefano III, K.C. Wilson, M. Jang, P.H. Mak, Identification of the dynamics of thyroid hormone metabolism. Automatica 11(2), 149–159 (1975)zbMATHGoogle Scholar
  12. 12.
    W. Elloumi, D. Mehdi, M. Chaabane, G. Hashim, Exponential stability criteria for positive systems with time-varying delay: a delay decomposition technique. Circuits Syst. Signal Process. 35(5), 1545–1561 (2016)zbMATHGoogle Scholar
  13. 13.
    L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)zbMATHGoogle Scholar
  14. 14.
    H.R. Feyzmahdavian, T. Charalambous, M. Johansson, Exponential stability of homogeneous positive systems of degree one with time-varying delays. IEEE Trans. Autom. Control 59(6), 1594–1599 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    W.M. Haddada, V. Chellaboinab, Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst. Control Lett. 51, 355–361 (2004)MathSciNetGoogle Scholar
  16. 16.
    M. Hu, Y. Wang, J. Xiao, On finite-time stability and stabilization of positive systems with impulses. Nonlinear Anal. Hybrid Syst. 31, 275–291 (2019)MathSciNetzbMATHGoogle Scholar
  17. 17.
    M. Hu, J. Xiao, R. Xiao, W. Chen, Impulsive effects on the stability and stabilization of positive systems with delays. J. Franklin Inst. 354(10), 4034–4054 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    L. Lee, Y. Liu, J. Liang, X. Cai, Finite time stability of nonlinear impulsive systems and its applications in sampled-data systems. ISA Trans. 57, 172–178 (2015)Google Scholar
  19. 19.
    P.D. Leenheer, D. Aeyels, Stabilization of positive linear systems. Syst. Control Lett. 44(4), 259–271 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    X. Li, D.W.C. Ho, J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems. Automatica 99, 361–368 (2019)MathSciNetzbMATHGoogle Scholar
  21. 21.
    X. Li, S. Song, Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control 62(1), 406–411 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63–69 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    L. Liu, X. Cao, Z. Fu, S. Song, S. Xing, Finite-time control of uncertain fractional-order positive impulsive switched systems with mode-dependent average dwell time. Circuits Syst. Signal Process. 37(9), 3739–3755 (2018)MathSciNetGoogle Scholar
  24. 24.
    L. Liu, J. Sun, Finite-time stabilization of linear systems via impulsive control. Int. J. Control 81(6), 905–909 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    X. Liu, Stability results for impulsive differential systems with applications to population growth models. Dyn. Stab. Syst. 9(2), 163–174 (1994)MathSciNetzbMATHGoogle Scholar
  26. 26.
    X. Liu, W. Yu, L. Wang, Stability analysis of positive systems with bounded time-varying delays. IEEE Trans. Circuits Syst. II Exp. Briefs 56(7), 600–604 (2009)Google Scholar
  27. 27.
    J. Lu, D.W.C. Ho, J. Cao, A unified synchronization criterion for impulsive dynamical networks. Automatica 46(7), 1215–1221 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    X. Lv, X. Li, Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications. ISA Trans. 70, 30–36 (2017)Google Scholar
  29. 29.
    Y. Ma, B. Wu, Y. Wang, Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing 173, 2076–2082 (2016)Google Scholar
  30. 30.
    Y. Ma, B. Wu, Y. Wang, Input-output finite time stability of fractional order linear systems with \( 0 < \alpha < 1 \). Trans. Inst. Meas. Control 39(5), 653–659 (2017)Google Scholar
  31. 31.
    O. Mason, M. Verwoerd, Observations on the stability properties of cooperative systems. Syst. Control Lett. 58(6), 461–467 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    S.G. Nersesov, W.M. Haddadb, Finite-time stabilization of nonlinear impulsive dynamical systems. Nonlinear Anal. Hybrid Syst. 2, 832–845 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    P.H.A. Ngoc, Stability of positive differential systems with delay. IEEE Trans. Autom. Control 58(1), 203–209 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II Exp. Briefs 54(2), 151–155 (2007)Google Scholar
  35. 35.
    H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (American Mathematical Society, Rhode Island, 1995)zbMATHGoogle Scholar
  36. 36.
    R.J. Smith, L.M. Wahl, Drug resistance in an immunological model of HIV-1 infection with impulsive drug effects. Bull. Math. Biol. 67(4), 783–813 (2005)MathSciNetzbMATHGoogle Scholar
  37. 37.
    W. Tao, Y. Liu, J. Lu, Stability and \( {L}_{2}\)-gain analysis for switched singular linear systems with jumps. Math. Methods Appl. Sci. 40(3), 589–599 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Y. Wang, J. Zhang, M. Liu, Exponential stability of impulsive positive systems with mixed time-varying delays. IET Control Theory Appl. 8(15), 1537–1542 (2014)MathSciNetGoogle Scholar
  39. 39.
    X. Wu, Y. Zhang, Input-to-state stability of discrete-time delay systems with delayed impulses. Circuits Syst. Signal Process. 37(6), 2320–2356 (2018)MathSciNetzbMATHGoogle Scholar
  40. 40.
    H. Yang, Y. Zhang, Stability of positive delay systems with delayed impulses. IET Control Theory Appl. 12(2), 194–205 (2018)MathSciNetGoogle Scholar
  41. 41.
    H. Yang, Y. Zhang, Exponential stability of homogeneous impulsive positive delay systems of degree one. Int. J. Control (2019).  https://doi.org/10.1080/00207179.2019.1584335 CrossRefGoogle Scholar
  42. 42.
    T. Yang, Impulsive Control Theory (Springer, Berlin, 2001)zbMATHGoogle Scholar
  43. 43.
    J. Zhang, X. Zhao, Y. Chen, Finite-time stability and stabilization of fractional order positive switched systems. Circuits Syst. Signal Process. 35(7), 2450–2470 (2016)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Q. Zhu, Y. Liu, J. Lu, J. Cao, On the optimal control of Boolean control networks. SIAM J. Control Opti. 56(2), 1321–1341 (2018)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Q. Zhu, Y. Liu, J. Lu, J. Cao, Further results on the controllability of Boolean control networks. IEEE Trans. Autom. Control 64(1), 440–442 (2019)MathSciNetzbMATHGoogle Scholar
  46. 46.
    S. Zhu, J. Lou, Y. Liu, Y. Li, Z. Wang, Event-triggered control for the stabilization of probabilistic Boolean control networks. Complexity 2018, 9259348, 7p (2018)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsJinggangshan UniversityJi’anChina
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiChina

Personalised recommendations

Cite article

Buy options