Novel Approaches to Stability and Stabilization of Positive Switched Systems with Unstable Subsystems

  • Yue Wang
  • Hongwei Wang
  • Jie LianEmail author
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)


This paper investigates the globally uniformly exponential stability of positive switched linear system (PSLS) in both continuous-time and discrete-time contexts. By using the multiple piecewise-continuous linear copositive Lyapunov function (MPLCLF) and exploring mode-dependent average dwell time (MDADT) switching, several stability criteria are developed with a switching strategy where slow switching and fast switching are applied to stable and unstable subsystems respectively. The proposed methods are also used to stabilize PSLS with controllable and uncontrolled subsystems. The obtained results provide lower bounds on MDADT of stable subsystems and higher bounds on MDADT of unstable subsystems and reduce the conservatism compared with the existing results. Finally, two numerical examples are provided to validate the advantages of the obtained results.


Positive systems Mode-dependent average dwell time Multiple copositive Lyapunov function Switched systems Linear programming 



This work was supported by the National Science Foundation of China under Grants 61773089, 61374070, 61473055, the Fundamental Research Funds for the Central Universities under Grants DUT17JC14, DUT17ZD227, and Youth Star of Dalian Science and Technology under 2016RQ014, 2015R052.


  1. 1.
    Farina L., Rinaldi S.: Positive Linear Systems: Theory and Applications. Wiley, New Jercy (2011)Google Scholar
  2. 2.
    Jadbabaie A., Lin J., Morse A.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. In: Proceedings 41st IEEE Conference on Decision and Control (2002)Google Scholar
  3. 3.
    Shorten, R., Wirth, F., Leith, D.: A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. (TON) 14(3), 616–629 (2006)CrossRefGoogle Scholar
  4. 4.
    Deaecto, G., Geromel, J.: Stability analysis and control design of discrete-time switched affine systems. IEEE Trans. Autom. Control 62(8), 4058–4065 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fei, Z., Shi, S., Wang, Z., Wu, L.: Quasi-time-dependent controller and filter design for discrete-time switched system with mode-dependent average dwell time. IEEE Trans. Autom. Control PP(99), 1–1 (2017)Google Scholar
  6. 6.
    Jungers, R., Ahmadi, A., Parrilo, P., Roozbehani, M.: A characterization of Lyapunov inequalities for stability of switched systems. IEEE Trans. Autom. Control 62(6), 3062–3067 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rami, M.: Solvability of static output-feedback stabilization for LTI positive systems. Syst. Control Lett. 60(9), 704–708 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rami M., Tadeo F.: Positive observation problem for linear discrete positive systems. In: Proceedings 45th IEEE Conference on Decision and Control, pp. 4729–4733 (2006)Google Scholar
  9. 9.
    Zhang, J., Han, Z., Zhu, F., Huang, J.: Stability and stabilization of positive switched systems with mode-dependent average dwell time. Nonlinear Anal. Hybrid Syst. 9(1), 42–55 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Xiang, W., Xiao, J.: Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50(3), 940–945 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cheng, J., Zhu, H., Zhong, S., Zheng, F., Zeng, Y.: Finite-time filtering for switched linear systems with a mode-dependent average dwell time. Nonlinear Anal. Hybrid Syst. 15, 145–156 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lian, J., Liu, J.: New results on stability of switched positive systems: an average dwell-time approach. IET Control Theory Appl. 7(12), 1651–1658 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xie, D., Zhang, H., Zhang, H., Wang, B.: Exponential stability of switched systems with unstable subsystems: a mode-dependent average dwell time approach. Circuits Syst. Signal Proc. 32(6), 3093–3105 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yin, Y., Zhao, X., Zheng, X.: New stability and stabilization conditions of switched systems with mode-dependent average dwell time. Circuits Syst. Signal Proc. 36(1), 82–98 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Liu, X., Dang, C.: Stability analysis of positive switched linear systems with delays. IEEE Trans. Autom. Control 56(7), 1684–1690 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)CrossRefGoogle Scholar
  19. 19.
    Li, Q., Zhao, J., Dimirovski, G.: Robust tracking control for switched linear systems with time-varying delays. IET Control Theory Appl. 2(6), 449–457 (2008)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Electronic Information and Electrical EngineeringDalian University of TechnologyDalianPeople’s Republic of China

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