Journal of Global Optimization

, Volume 65, Issue 1, pp 109–118 | Cite as

Practical exponential set stabilization for switched nonlinear systems with multiple subsystem equilibria

  • Honglei XuEmail author
  • Yi Zhang
  • Jin Yang
  • Guanglu Zhou
  • Louis Caccetta


This paper studies the practical exponential set stabilization problem for switched nonlinear systems via a \(\tau \)-persistent approach. In these kinds of switched systems, every autonomous subsystem has one unique equilibrium point and these subsystems’ equilibria are different each other. Based on previous stability results of switched systems and a set of Gronwall–Bellman inequalities, we prove that the switched nonlinear system will reach the neighborhood of the corresponding subsystem equilibrium at every switching time. In addition, we constructively design a suitable \(\tau \)-persistent switching law to practically exponentially set stabilize the switched system. Finally, a numerical example is presented to illustrate the obtained results.


\(\varepsilon \)-Practical set stability Switched nonlinear systems \(\tau \)-Persistent switching law 



This work was partially supported by the NSFC under Grants 11171079 and 11371371, the ARC Discovery Projects, Natural Science Foundation of Hubei Province of China (2014CFB141), HUST Independent Innovation Research Fund (GF and Natural Science).


  1. 1.
    Alpcan, T., Basar, T.: A hybrid systems model for power control in multicell wireless data networks. Perform. Eval. 57, 477–495 (2004)CrossRefGoogle Scholar
  2. 2.
    Alpcan, T., Basar, T.: A stability result for switched systems with multiple equilibria. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17, 949–958 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chen, Y., Xu, H.: Exponential stability analysis and impulsive tracking control of uncertain time-delayed systems. J. Global Optim. 52(2), 323–334 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Liu, B., Hill, D.J.: Uniform stability and ISS of discrete-time impulsive hybrid system. Nonlinear Anal. Hybrid Syst. 217, 2067–2083 (2010)MathSciNetGoogle Scholar
  6. 6.
    Shi, P., Boukas, E.K., Agarwal, R.K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Autom. Control 44, 2139–2144 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Xu, X., Zhai, G.: Practical stability and stabilization of hybrid and switched systems. IEEE Trans. Autom. Control 50, 1897–1903 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhai, G., Michel, A.N.: On practical stability of switched systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, pp. 3488–3493, Las Vegas, Nevada, USA (2002)Google Scholar
  9. 9.
    Zhang, G., Han, C., Guan, Y., Wu, L.: Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays. Int. J. Innov. Comput. Inf. Control 8, 1973–1986 (2012)Google Scholar
  10. 10.
    Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability of switched positive linear systems with average dwell time switching. Automatica 48, 1132–1137 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Honglei Xu
    • 1
    • 2
    Email author
  • Yi Zhang
    • 3
  • Jin Yang
    • 3
  • Guanglu Zhou
    • 2
  • Louis Caccetta
    • 2
  1. 1.School of Energy and Power EngineeringHuazhong University of Science and TechnologyWuhanChina
  2. 2.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia
  3. 3.Department of MathematicsChina University of Petroleum (Beijing)BeijingChina

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