Circuits, Systems, and Signal Processing

, Volume 37, Issue 9, pp 3739–3755 | Cite as

Finite-Time Control of Uncertain Fractional-Order Positive Impulsive Switched Systems with Mode-Dependent Average Dwell Time

  • Leipo LiuEmail author
  • Xiangyang Cao
  • Zhumu Fu
  • Shuzhong Song
  • Hao Xing


This paper is concerned with the problem of finite-time control of uncertain fractional-order positive impulsive switched systems (UFOPISS) via mode-dependent average dwell time (MDADT). The uncertainties refer to interval and polytopic uncertainties. Firstly, the proof of the positivity of UFOPISS is given. By constructing linear copositive Lyapunov functions, the finite-time stability (FTS) of autonomous system with MDADT is studied. Then, state feedback controllers are designed to guarantee the FTS of the resulting closed-loop system with interval and polytopic uncertainties, respectively. All presented conditions can be easily solved by linear programming. Finally, a fractional-order circuit model is employed to illustrate the effectiveness of the proposed method.


Fractional-order positive impulsive switched systems Interval uncertainty Polytopic uncertainty Finite-time stability Linear programming 



The authors are grateful for the supports of the National Natural Science Foundation of China under Grants U1404610, 61473115 and 61374077, young key teachers plan of Henan province (2016GGJS-056).


  1. 1.
    M.P. Aghababa, M. Borjkhani, Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme. Complexity 20(2), 37–46 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Babiarz, A. Legowski, M. Niezabitowski, Controllability of positive discrete-time switched fractional order systems for fixed switching sequence. Lect. Notes Artif. Intell. 9875, 303–312 (2016)Google Scholar
  3. 3.
    D. Baleanu, Z.B. Guvenc, J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications (Springer, Netherlands, 2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    N. Bigdeli, H.A. Ziazi, Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems. J. Frankl. I 354(1), 160–183 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Chen, Y. Yang, Stability of a class of nonlinear fractional order impulsive switched systems. Trans. Inst. Meas. Control 39(5), 1–10 (2016)Google Scholar
  6. 6.
    W. Chen, W. Zheng, Robust stability and \(H_\infty \)-control of uncertain impulsive systems with time-delay. Automatica 45, 109–117 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Y. Chen, Y. Wei, H. Zhong et al., Sliding mode control with a second-order switching law for a class of nonlinear fractional order systems. Nonlinear Dyn. 85(1), 1–11 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Dong, Stability of switched positive nonlinear systems. Int. J. Robust Nonlinear 26(14), 3118–3129 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    X. Dong, Y. Zhou, Z. Ren et al., Time-varying formation control for unmanned aerial vehicles with switching interaction topologies. Control Eng. Pract. 46, 26–36 (2016)CrossRefGoogle Scholar
  10. 10.
    R. Elkhazali, Fractional-order (PID mu)-D-lambda controller design. Comput. Math. Appl. 66(5), 639–646 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Erenturk, Fractional-order (pid mu)-d-lambda and active disturbance rejection control of nonlinear two-mass drive system. IEEE Trans. Ind. Electron. 60(9), 3806–3813 (2013)CrossRefGoogle Scholar
  12. 12.
    V. Filipovic, N. Nedic, V. Stojanovic, Robust identification of pneumatic servo actuators in the real situations. Forsch. Ingenieurwes. 75(4), 183–196 (2011)CrossRefGoogle Scholar
  13. 13.
    T.T. Hartley, C.F. Lorenzo, H.K. Qammer, Chaos in a fractional-order Chuas system. IEEE Trans. Circ. Syst. I(42), 485–490 (1995)CrossRefGoogle Scholar
  14. 14.
    S.H. Hosseinnia, I. Tejado, B.M. Vinagre, Stability of fractional order switching systems. Comput. Math. Appl. 66(5), 585–596 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Huang, J. Zhang, S. Shi, Circuit simulation on control and synchronization of fractional order switching chaotic system. Math. Comput. Simulat. 113(C), 28–39 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    H. Jia, Z. Chen, W. Xue, Analysis and circuit implementation for the fractional-order Lorenz system. Physics 62(14), 31–37 (2013)Google Scholar
  17. 17.
    S. Li, Z. Xiang, Stability and \(L_\infty \)-gain analysis for positive switched systems with time-varying delay under state-dependent switching. Circ. Syst. Signal Process. 35(3), 1045–1062 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    H. Liu, S. Li, J. Cao et al., Adaptive fuzzy prescribed performance controller design for a class of uncertain fractional-order nonlinear systems with external disturbances. Neurocomputing 219(C), 422–430 (2017)CrossRefGoogle Scholar
  19. 19.
    L. Liu, X. Cao, Z. Fu et al., Guaranteed cost finite-time control of fractional-order positive switched systems. Adv. Math. Phys. 3, 1–11 (2017)MathSciNetGoogle Scholar
  20. 20.
    X. Liu, C. Dang, Stability analysis of positive switched linear systems with delays. IEEE Trans. Autom. Control 56(7), 1684–1690 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J.E. Mazur, G.M. Mason, J.R. Dwyer, The mixing of interplanetary magnetic field lines: a significant transport effect in studies of the energy spectra of impulsive flares. Acceleration and Transport of Energetic Particle, pp. 47–54 (2000)Google Scholar
  22. 22.
    S. Shao, M. Chen, Q. Wu, Stabilization control of continuous-time fractional positive systems based on disturbance observer. IEEE Access 4, 3054–3064 (2016)CrossRefGoogle Scholar
  23. 23.
    J. Shen, J. Lam, Stability and performance analysis for positive fractional-order systems with time-varying delays. IEEE Trans. Autom. Control 61(9), 2676–2681 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    H. Sira-Ramrez, V.F. Batlle, On the gpi-pwm control of a class of switched fractional order systems. IFAC Proc. Vol. 39(11), 161–166 (2006)CrossRefGoogle Scholar
  25. 25.
    V. Stojanovic, V. Filipovic, Adaptive input design for identification of output error model with constrained output. Circ. Syst. Signal Process. 33(1), 97–113 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    V. Stojanovic, N. Nedic, Identification of time-varying OE models in presence of non-Gaussian noise: application to pneumatic servo drives. Int. J. Robust. Nonlinear 26(18), 3974–3995 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    V. Stojanovic, N. Nedic, D. Prsic et al., Optimal experiment design for identification of ARX models with constrained output in non-Gaussian noise. Appl. Math. Model. 40(13–14), 6676–6689 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    S. Tang, L. Chen, The periodic predator-prey lotka-volterra model with impulsive effect. J. Mech. Med. Biol. 2(03–04), 267–296 (2011)Google Scholar
  29. 29.
    M. Tozzi, A. Cavallini, G.C. Montanari, Monitoring off-line and on-line PD under impulsive voltage on induction motors-Part 2: testing. IEEE Electr. Insul. Mag. 27(1), 14–21 (2011)CrossRefGoogle Scholar
  30. 30.
    M. Xiang, Z. Xiang, H.R. Karimi, Asynchronous \(L_1\) control of delayed switched positive systems with mode-dependent average dwell time. Inf. Sci. 278(10), 703–714 (2014)CrossRefzbMATHGoogle Scholar
  31. 31.
    H. Yang, B. Jiang, On stability of fractional order switched nonlinear systems. IET Control Theory A 10(8), 965–970 (2016)CrossRefGoogle Scholar
  32. 32.
    Y. Yang, G. Chen, Finite-time stability of fractional order impulsive switched systems. Int. J. Robust Nonlinear 25(13), 2207–2222 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    E. Zambrano-Serrano, E. Campos-Cantn, J.M. Munoz-Pacheco, Strange attractors generated by a fractional order switching system and its topological horseshoe. Nonlinear Dyn. 83(3), 1629–1641 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    J. Zhang, X. Zhao, Y. Chen, Finite-time stability and stabilization of fractional order positive switched systems. Circ. Syst. Signal Process. 35(7), 2450–2470 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    X. Zhao, Y. Yin, X. Zheng, State-dependent switching control of switched positive fractional-order systems. ISA Trans. 62, 103–108 (2016)CrossRefGoogle Scholar
  36. 36.
    X. Zhao, L. Zhang, P. Shi, Stability of a class of switched positive linear time-delay systems. Int. J. Robust Nonlinear 23(5), 578–589 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Leipo Liu
    • 1
    Email author
  • Xiangyang Cao
    • 1
  • Zhumu Fu
    • 1
  • Shuzhong Song
    • 1
  • Hao Xing
    • 1
  1. 1.School of Information EngineeringHenan University of Science and TechnologyLuoyangChina

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