Advertisement

Finite-Time Stability and Stabilization of Fractional-Order Switched Singular Continuous-Time Systems

  • Tian Feng
  • Baowei WuEmail author
  • Lili Liu
  • Yue-E Wang
Article
  • 16 Downloads

Abstract

The finite-time stability and stabilization of a class of fractional-order switched singular continuous-time systems with order \(0<\alpha <1\) are investigated in this paper. First, by employing the average dwell time switching technique, together with the introduction of multiple Lyapunov functions, some sufficient conditions of the finite-time stability and finite-time boundedness are derived for the considered system. Second, based on the obtained conditions, suitable state feedback controllers can be designed if a set of linear matrix inequalities are feasible. Finally, an illustrative example is presented to show the effectiveness of the proposed results.

Keywords

Switched singular system Fractional-order system Finite-time stability Average dwell time switching 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61403241), by the Fundamental Research Funds for the Central Universities (Nos. GK201703009, GK201903004, GK201905001) and also by the China Scholarship Council (No. 201806870032).

References

  1. 1.
    N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. 19(9), 2951–2957 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 1(4), 304–311 (1989)Google Scholar
  3. 3.
    G.P. Chen, Y. Yang, Finite-time stability of switched positive linear systems. Int. J. Robust Nonlinear Control 24(1), 179–190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J.P. Clerc, A.M.S. Tremblay, G. Albinet, C. Mitescu, AC response of fractal networks. J. de Physique Lett. 45(19), 913–924 (1984)CrossRefGoogle Scholar
  5. 5.
    H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems. Nonlinear Dyn. 67(4), 2433–2439 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Ding, Q.L. Han, X.M. Zhang, Distributed secondary control for active power sharing and frequency regulation in islanded microgrids using an event-triggered communication mechanism. IEEE Trans. Ind. Inform. to be published.  https://doi.org/10.1109/TII.2018.2884494
  7. 7.
    X. Gao, J.B. Yu, Synchronization of two coupled fractional-order chaotic oscillators. Chaos Solitons Fract. 26(1), 141–145 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    L. Gaul, P. Klein, S. Kemple, Damping description involving fractional operators. Mech. Syst. Signal Process. 5(2), 81–88 (1991)CrossRefGoogle Scholar
  9. 9.
    X. Ge, Q.L. Han, X.M. Zhang, Achieving cluster formation of multi-agent systems under aperiodic sampling and communication delays. IEEE Trans. Ind. Electron. 65(4), 3417–3426 (2018)CrossRefGoogle Scholar
  10. 10.
    M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. 33(2), 253–265 (1971)CrossRefGoogle Scholar
  11. 11.
    T. Kaczorek, Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T. Kaczorek, Stability of positive fractional switched continuous-time linear systems. B. Pol. Acad. Sci-Tech. 61(2), 349–352 (2013)Google Scholar
  13. 13.
    S.T. Li, X.M. Liu, Y.Y. Tan, Optimal switching time control of discrete-time switched autonomous systems. Int. J. Innov. Comput. I. 11(6), 2043–2050 (2015)Google Scholar
  14. 14.
    Y. Li, Y.Q. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    C. Lin, B. Chen, P. Shi, J.P. Yu, Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Syst. Control Lett. 112, 31–35 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Liu, X. Wu, X.F. Zhou, Asymptotical stability of Riemann–Liouville fractional nonlinear systems. Nonlinear Dyn. 86(1), 65–71 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order \(0<\alpha <1\) case. IEEE Trans. Autom. Control 55(1), 152–158 (2015)MathSciNetGoogle Scholar
  18. 18.
    Y.J. Ma, B.W. Wu, Y.E. Wang, Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing 173(3), 2076–2082 (2016)CrossRefGoogle Scholar
  19. 19.
    S. Marir, M. Chadli, D. Bouagada, New admissibility conditions for singular linear continuous-time fractional-order systems. J. Frankl. Inst. 354, 752–766 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    E.T. McAdams, A. Lackermeier, J.A. McLaughlin, D. Macken, J. Jossinet, The linear and non-linear electrical properties of the electrode–electrolyte interface. Biosens. Bioelectron. 10(1), 67–74 (1995)CrossRefGoogle Scholar
  21. 21.
    C.A. Monje, Y.Q. Chen, B.M. Vinagre, D.Y. Xue, V. Feliu, Fractional-Order Systems and Controls (Springer, London, 2010)CrossRefzbMATHGoogle Scholar
  22. 22.
    I. N’Doye, M. Darouach, M. Zasadzinski, Robust stabilization of uncertain descriptor fractional-order systems. Automatica 49(6), 1907–1913 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. Podlubny, Fractional differential equations. Int. J. Differ. Equ. 3, 553–563 (2010)Google Scholar
  24. 24.
    W.H. Qi, G.D. Zong, J. Cheng, T.C. Jiao, Robust finite-time stabilization for positive delayed semi-Markovian switching systems. Appl. Math. Comput. 351, 139–152 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Y.H. Wei, J.C. Wang, T.Y. Liu, Y. Wang, Fixed pole based modeling and simulation schemes for fractional order systems. ISA Trans. 84, 43–54 (2019)CrossRefGoogle Scholar
  26. 26.
    Y.H. Wei, J.C. Wang, T.Y. Liu, Y. Wang, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state. J. Frankl. Inst. 356(4), 1975–1990 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    T.B. Wu, F.B. Li, C.H. Yang, W.H. Gui, Event-based fault detection filtering for complex networked jump systems. IEEE-ASME T. Mech. 23(2), 497–505 (2018)CrossRefGoogle Scholar
  28. 28.
    Y. Yang, G.P. Chen, Finite-time stability of fractional order impulsive switched systems. Int. J. Robust Nonlinear Control 25, 2207–2222 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J.F. Zhang, X.D. Zhao, Y. Chen, Finite-time stability and stabilization of fractional order positive switched systems. Circuits Syst. Signal Process. 35(7), 2450–2470 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M. Zhang, P. Shi, L. Ma, Cai J, Su H, Network-based fuzzy control for nonlinear Markov jump systems subject to quantization and dropout compensation. Fuzzy Sets Syst. (2018). https://doi.org/10.1016/j.fss.2018.09.007
  31. 31.
    M. Zhang, P. Shi, L. Ma, Cai J, Su H, Quantized feedback control of fuzzy Markov jump systems. IEEE Trans. Cybern. 49(9), 3375–3384 (2018)CrossRefGoogle Scholar
  32. 32.
    X.F. Zhang, Y.Q. Chen, Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order \(\alpha \): The \(0<\alpha <1\) case. ISA Trans. 82, 42–50 (2018)CrossRefGoogle Scholar
  33. 33.
    X.M. Zhang, Q.L. Han, Network-based \(H_\infty \) filtering using a logic jumping-like trigger. Automatica 49(5), 1428–1435 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    X.M. Zhang, Q.L. Han, J. Wang, Admissible delay upper bounds for global asymptotic stability of neural networks with time-varying delays. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5319–5329 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    X.M. Zhang, Q.L. Han, A. Seuret, F. Gouaisbaut, Y. He, Overview of recent advances in stability of linear systems with time-varying delays. IET Control Theory Appl. 13(1), 1–16 (2019)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Y.L. Zhang, B.W. Wu, Y.E. Wang, Finite-time stability for switched singular systems. Acta Phys. Sinica. 63(17), 32–41 (2014)Google Scholar
  37. 37.
    L. Zhou, L. Cheng, J. She, Z. Zhang, Generalized extended state observer-based repetitive control for systems with mismatched disturbances. Int. J. Robust Nonlinear Control (to be published).  https://doi.org/10.1002/rnc.4582
  38. 38.
    L. Zhou, D.W.C. Ho, G. Zhai, Stability analysis of switched linear singular systems. Automatica 49(5), 1481–1487 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    L. Zhou, J. She, S. Zhou, Robust \(H_\infty \) control of an observer-based repetitive-control system. J. Frankl. Inst. 355(12), 4952–4969 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    L. Zhou, J. She, S. Zhou, C. Li, Compensation for state-dependent nonlinearity in a modified repetitive-control system. Int. J. Robust Nonlinear Control 28(1), 213–226 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Z. Zuo, Q.L. Han, B. Ning, X. Ge, X.M. Zhang, An overview of recent advances in fixed-time cooperative control of multi-agent systems. IEEE Trans. Ind. Inform. 14(6), 2322–2334 (2018)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

Personalised recommendations