Circuits, Systems, and Signal Processing

, Volume 38, Issue 7, pp 2931–2950 | Cite as

Finite-Time Stability and Boundedness of Switched Systems with Finite-Time Unstable Subsystems

  • Jialin TanEmail author
  • Weiqun Wang
  • Juan Yao


In this paper, problems covering finite-time stability and boundedness of switched systems with finite-time unstable subsystems are researched through the method of multi-Lyapunov function. On basis of the mode-dependent average dwell time method, the systems are required to meet the standards of remaining finite-time stable and finite-time bounded through the practice of designing the switching signal for finite-time stable and unstable subsystems respectively. Finally, stabilization conditions for switched linear systems based on linear matrix inequalities are presented to guarantee the finite-time stability of the closed-loop system. Numerical examples are put forward attempting to verify the efficiency through different methodologies.


Switched nonlinear system Finite-time stability Finite-time boundedness Mode-dependent average dwell time 



This work was supported by the National Natural Science Foundation of China (Nos. 61603188, 61573007).


  1. 1.
    F. Amato, M. Ariola, Finite-time control of discrete-time linear system. IEEE Trans. Autom. Control 50(5), 724–729 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, Finite-time stability of linear time-varying systems with jumps. Automatica 45(5), 1354–1358 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    F. Amato, M. Ariola, P. Dorato, Finite-time stabilization via dynamic output feedback. Automatica 42(2), 337–342 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Czornik, M. Niezabitowski, Controllability and stability of switched systems. Bull. Polish Acad. Sci. Tech. Sci. 61(3), 16–21 (2013)zbMATHGoogle Scholar
  8. 8.
    H. Du, X. Lin, S. Li, Finite-time boundedness and stabilization of switched linear systems. Kybernetika 5(5), 1365–1372 (2010)MathSciNetGoogle Scholar
  9. 9.
    P. Dorato, Short time stability in linear time-varying systems, in Proceeding of the IRE International Convention Record Part, vol. 4, pp. 83–7 (1961)Google Scholar
  10. 10.
    J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time. IEEE Trans. Autom. Control 2655–2660 (1999)Google Scholar
  11. 11.
    X. Li, X. Lin, S. Li, Y. Zou, Finite-time stability of switched nonlinear systems with finite-time unstable subsystems. J. Frankl. Inst. 352(3), 1192–1214 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19(5), 59–70 (2001)zbMATHGoogle Scholar
  13. 13.
    J.L. Mancilla-Aguilar, R.A. Garcia, A converse Lyapunov theorem for nonlinear switched systems. Syst. Control Lett. 41(1), 67–71 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A.S. Morse, Supervisory control of families of linear set-point controllers—part 1: exact matching. IEEE Trans. Autom. Control 41(10), 1413–1431 (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    E. Moulay, M. Dambrine, N. Yeganefar, W. Perruquetti, Finite-time stability and stabilization of time-delay systems. Syst. Control Lett. 57(7), 561–566 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    E. Moulay, W. Perruquetti, Finite time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323(2), 1430–1443 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Y. Orlov, Finite-time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43(4), 1253–1271 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Y.G. Sun, L. Wang, G. Xie, Stability of switched systems with time-varying delays: delay-dependent common Lyapunov functional approach, in American Control Conference (2006)Google Scholar
  19. 19.
    J. Tan, W. Wang, J. Yao, A study on finite-time stability of switched linear systems with finite-time unstable subsystems, in China Control Conference (2017)Google Scholar
  20. 20.
    V. Valdivia, R. Todd, F.J. Bryan, Behavioral modeling of a switched reluctance generator for aircraft power systems. IEEE Trans. Ind. Electron. 61(6), 2690–2699 (2014)CrossRefGoogle Scholar
  21. 21.
    L. Vu, D. Chatterjee, D. Liberzon, Input-to-state stability of switched systems and switching adaptive control. Automatica 43(4), 639–646 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    H. Wang, W. Sun, P.X. Liu, Adaptive intelligent control for a class of nonaffine nonlinear time-delay systems with dynamic uncertainties. IEEE Trans. Syst. Man Cybern. Syst. 47(7), 1474–1485 (2017)CrossRefGoogle Scholar
  23. 23.
    L. Weiss, E.F. Infante, Finite time stability under perturbing forces and on product spaces. IEEE Trans Autom. Control 12, 54C9 (1967)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Y. Wu, R. Lu, Event-based control for network systems via integral quadratic constraints. IEEE Trans. Circuits Syst. I Reg. Pap. PP(99), 1–9 (2017)Google Scholar
  25. 25.
    L. Wu, R. Yang, P. Shi, X. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings. Automatica 59(C), 206–215 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    W. Xiang, J. Xiao, \(H_{\infty }\) finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance. J. Frankl. Inst. 348(2), 331–352 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    D. Xie, H. Zhang, H. Zhang, B. Wang, Exponential stability of switched systems with unstable subsystems: a mode-dependent average dwell time approach. Circuits Syst. Signal Process. 32(6), 3093–3105 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Y. Yin, X. Zhao, X. Zheng, New stability and stabilization conditions of switched systems with mode-dependent average dwell time. Circuits Syst. Signal Process. 36(1), 1–17 (2016)MathSciNetGoogle Scholar
  29. 29.
    J. Zhang, Z. Han, F. Zhu, J. Huang, Stability and stabilization of positive switched systems with mode-dependent average dwell time. Nonlinear Anal. Hubrid Syst. 9(1), 42–55 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    J. Zhao, D.J. Hill, On stability, L2-gain and \(H_{\infty }\) control for switched systems. Automatic 44(5), 1220–1232 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    L. Zhou, D.W.C. Ho, G. Zhai, Stability analysis of switched linear singular systems. Automatica 49(5), 1481–1487 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    X. Zhao, P. Shi, Y. Yin, S.K. Nguang, New results on stability of slowly switched systems: a multiple discontinuous Lyapunov function approach. IEEE Trans. Autom. Control PP(99), 1 (2016)zbMATHGoogle Scholar
  33. 33.
    X. Zhao, P. Shi, X. Zheng, L. Zhang, Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone. Automatic 60(C), 193–200 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    S. Zhao, Jitao Sun, Li Liu, Finite-time stability of linear time-varying singular systems with impulsive effects. Int. J. Control 81(11), 1824–1829 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    X. Zhao, S. Yin, H. Li, B. Niu, Switching stabilization for a class of slowly switched systems. IEEE Trans. Autom. Control 60(1), 221–226 (2015)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Nanjing University of Science and TechnologyNanjingChina

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