Synchronization for fractional-order neural networks with full/under-actuation using fractional-order sliding mode control

Original Article

Abstract

This paper considers synchronization between two fractional-order neural networks (FONNs). To handle the case of full/under-actuation, i.e. the dimension of the synchronization controller is equal to or less than that of the FONNs, a novel fractional-order integral sliding surface is designed, and the feasibility of the proposed approach is shown by solving two linear matrix inequalities. Then, based on the fractional Lyapunov stability criterion, a fractional-order sliding mode controller equipped with fractional-order adaptation laws is constructed to guarantee the synchronization error converging to an arbitrary small region of the origin. The effectiveness of the proposed control method is verified by two simulation examples.

Keywords

Fractional-order neural network Fractional-order sliding mode control Fractional-order adaptation law 

Notes

Acknowledgements

The authors are indebted to the anonymous reviewers’ valuable comments, which improved the presentation and quality of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11401243, 61403157), the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (Grant No. KJ2015A256), the Fundamental Research Funds for the Central Universities of China (Grant No. GK201504002), the Foundation for Distinguished Young Talents in Higher Education of Anhui, China (Grant No. GXYQZD2016257), and the Innovation Funds of Graduate Programs of Shaanxi Normal University (Grant No. 2015CXB008).

References

  1. 1.
    Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Academic pressGoogle Scholar
  2. 2.
    Shen J, Lam J (2014) Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50(2):547–551MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19(9):2951–2957MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bouzeriba A, Boulkroune A, Bouden T (2016) Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. Int J Mach Learn Cybern 5(7):893–908CrossRefMATHGoogle Scholar
  5. 5.
    Chen L, Wu R, He Y, Yin L (2015) Robust stability and stabilization of fractional-order linear systems with polytopic uncertainties. Appl Math Comput 257:274–284MathSciNetMATHGoogle Scholar
  6. 6.
    Duarte-Mermoud MA, Aguila-Camacho N, Gallegos JA, Castro-Linares R (2015) Using general quadratic lyapunov functions to prove lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simul 22(1):650–659MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Wu H, Wang L, Wang Y, Niu P, Fang B (2016) Global mittag-leffler projective synchronization for fractional-order neural networks: an lmi-based approach. Adv Differ Equ 2016(1):1–18MathSciNetCrossRefGoogle Scholar
  8. 8.
    Wu H, Zhang X, Xue S, Wang L, Wang Y (2016) Lmi conditions to global mittag-leffler stability of fractional-order neural networks with impulses. Neurocomputing 193:148–154CrossRefGoogle Scholar
  9. 9.
    Liu H, Li S, Cao J, Li G, Alsaedi A, Alsaadi FE (2017) Adaptive fuzzy prescribed performance controller design for a class of uncertain fractional-order nonlinear systems with external disturbances. Neurocomputing 219:422–430CrossRefGoogle Scholar
  10. 10.
    Liu H, Li SG, Sun YG, Wang HX (2015) Adaptive fuzzy synchronization for uncertain fractional-order chaotic systems with unknown non-symmetrical control gain. Acta Physica Sinaca 64(7):070503Google Scholar
  11. 11.
    Liu H, Li S-G, Sun Y-G, Wang H-X (2015) Prescribed performance synchronization for fractional-order chaotic systems. Chin Phys B 24(9):090505CrossRefGoogle Scholar
  12. 12.
    Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. SpringerGoogle Scholar
  13. 13.
    Liu H, Pan Y, Li S, Chen Y. Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems. doi: 10.1109/TSMC.2016.2640950
  14. 14.
    Liu H, Li S, Wang H, Huo Y, Luo J (2015) Adaptive synchronization for a class of uncertain fractional-order neural networks. Entropy 17(10):7185–7200MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yu J, Hu C, Jiang H (2012) \(\alpha\)-stability and \(\alpha\)-synchronization for fractional-order neural networks. Neural Netw 35:82–87CrossRefMATHGoogle Scholar
  16. 16.
    Wu H, Zhang X, Li R, Yao R (2015) Adaptive exponential synchronization of delayed cohen-grossberg neural networks with discontinuous activations. Int J Mach Learn Cybern 6(2):253–263CrossRefGoogle Scholar
  17. 17.
    Chen J, Zeng Z, Jiang P (2014) Global mittag-leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51:1–8CrossRefMATHGoogle Scholar
  18. 18.
    Bao H-B, Cao J-D (2015) Projective synchronization of fractional-order memristor-based neural networks. Neural Netw 63:1–9CrossRefMATHGoogle Scholar
  19. 19.
    Wang F, Yang Y, Hu M (2015) Asymptotic stability of delayed fractional-order neural networks with impulsive effects. Neurocomputing 154:239–244CrossRefGoogle Scholar
  20. 20.
    Chen L, Wu R, Cao J, Liu J-B (2015) Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw 71:37–44CrossRefGoogle Scholar
  21. 21.
    Wu H, Li R, Yao R, Zhang X (2015) Weak, modified and function projective synchronization of chaotic memristive neural networks with time delays. Neurocomputing 149:667–676CrossRefGoogle Scholar
  22. 22.
    Ding Z, Shen Y, Wang L (2016) Global mittag-leffler synchronization of fractional-order neural networks with discontinuous activations. Neural Netw 73:77–85CrossRefGoogle Scholar
  23. 23.
    Wu H, Li R, Zhang X, Yao R (2015) Adaptive finite-time complete periodic synchronization of memristive neural networks with time delays. Neural Process Lett 42(3):563–583CrossRefGoogle Scholar
  24. 24.
    Velmurugan G, Rakkiyappan R, Cao J (2016) Finite-time synchronization of fractional-order memristor-based neural networks with time delays. Neural Netw 73:36–46CrossRefMATHGoogle Scholar
  25. 25.
    Wu R, Lu Y, Chen L (2015) Finite-time stability of fractional delayed neural networks. Neurocomputing 149:700–707CrossRefGoogle Scholar
  26. 26.
    Lundstrom BN, Higgs MH, Spain WJ, Fairhall AL (2008) Fractional differentiation by neocortical pyramidal neurons. Nature Neurosci 11(11):1335–1342CrossRefGoogle Scholar
  27. 27.
    Stamova I (2014) Global mittag-leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn 77(4):1251–1260MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    He Q, Liu D, Wu H, Ding S (2014) Robust exponential stability analysis for interval cohen-grossberg type bam neural networks with mixed time delays. Int J Mach Learn Cybern 5(1):23–38CrossRefGoogle Scholar
  29. 29.
    Aghababa MP (2012) Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique. Nonlinear Dyn 69(1–2):247–261MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rakkiyappan R, Cao J, Velmurugan G (2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 26(1):84–97MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wang H, Yu Y, Wen G, Zhang S, Yu J (2015) Global stability analysis of fractional-order hopfield neural networks with time delay. Neurocomputing 154:15–23CrossRefGoogle Scholar
  32. 32.
    Xiao M, Zheng WX, Jiang G, Cao J (2015) Undamped oscillations generated by hopf bifurcations in fractional-order recurrent neural networks with caputo derivative. IEEE Trans Neural Netw Learn Syst 26(12):3201–3214MathSciNetCrossRefGoogle Scholar
  33. 33.
    Bao H, Park JH, Cao J (2015) Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn 82(3):1343–1354MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wu A, Zeng Z. Global mittag-leffler stabilization of fractional-order memristive neural networks. IEEE Transactions on Neural Networks and Learning Systems. doi: 10.1109/TNNLS.2015.2506738Google Scholar
  35. 35.
    Ding Z, Shen Y. Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller. Neural Networks. doi: 10.1016/j.neunet.2016.01.006Google Scholar
  36. 36.
    Wu H, Zhang X, Xue S, Niu P. Quasi-uniform stability of caputo-type fractional-order neural networks with mixed delay. International Journal of Machine Learning and Cybernetics. doi: 10.1007/s13042-016-0523-1Google Scholar
  37. 37.
    Levant A (1993) Sliding order and sliding accuracy in sliding mode control. Int J Control 58(6):1247–1263MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Utkin V, Guldner J, Shi J (2009) Sliding mode control in electro-mechanical systems, vol. 34. CRC pressGoogle Scholar
  39. 39.
    Goyal V, Deolia VK, Sharma TN (2015) Robust sliding mode control for nonlinear discrete-time delayed systems based on neural network. Intell Control Autom 6(1):75CrossRefGoogle Scholar
  40. 40.
    Pan Y, Yu H (2016) Composite learning from adaptive dynamic surface control 61(9):2603–2609Google Scholar
  41. 41.
    Karimi HR (2012) A sliding mode approach to \(h_\infty\) synchronization of master-slave time-delay systems with markovian jumping parameters and nonlinear uncertainties. J Franklin Insti 349(4):1480–1496MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Pan Y, Yu H (2015) Dynamic surface control via singular perturbation analysis. Automatica 57:29–33MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Pan Y, Sun T, Yu H (2016) Composite adaptive dynamic surface control using online recorded data. Int J Robust Nonlinear Control 26(18):3921–3936MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Yang Y-S, Chang J-F, Liao T-L, Yan J-J (2009) Robust synchronization of fractional chaotic systems via adaptive sliding mode control. Int J Nonlinear Sci Numerical Simul 10(9):1237–1244CrossRefGoogle Scholar
  45. 45.
    Tavazoei MS, Haeri M (2008) Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A Stat Mech Appl 387(1):57–70CrossRefGoogle Scholar
  46. 46.
    Pisano A, Rapaić M, Jeličić Z, Usai E (2010) Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. Int J Robust Nonlinear Control 20(18):2045–2056MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Gao Z, Liao X (2013) Integral sliding mode control for fractional-order systems with mismatched uncertainties. Nonlinear Dyn 72(1–2):27–35MathSciNetCrossRefGoogle Scholar
  48. 48.
    Mohadeszadeh M, Delavari H (2015)Synchronization of fractional-order hyper-chaotic systems based on a new adaptive sliding mode control. International Journal of Dynamics and Control, pp 1–11Google Scholar
  49. 49.
    Djeghali N, Djennoune S, Bettayeb M, Ghanes M, Barbot J-P. Observation and sliding mode observer for nonlinear fractional-order system with unknown input.ISA Transactions. doi: 10.1016/j.isatra.2016.02.015
  50. 50.
    Mobayen S (2015) Fast terminal sliding mode controller design for nonlinear second-order systems with time-varying uncertainties. Complexity 21(2):239–244MathSciNetCrossRefGoogle Scholar
  51. 51.
    Chen L, Wu R, He Y, Chai Y (2015) Adaptive sliding-mode control for fractional-order uncertain linear systems with nonlinear disturbances. Nonlinear Dyn 80(1–2):51–58MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Balasubramaniam P, Muthukumar P, Ratnavelu K (2015) Theoretical and practical applications of fuzzy fractional integral sliding mode control for fractional-order dynamical system. Nonlinear Dyn 80(1–2):249–267MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Jakovljević B, Pisano A, Rapaić M, Usai E (2015) On the sliding-mode control of fractional-order nonlinear uncertain dynamics. Int J Robust Nonlinear Control 26(4):782–798MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Ke Z, Zhi-Hui W, Li-Ke G, Yue S, Tie-Dong M (2015) Robust sliding mode control for fractional-order chaotic economical system with parameter uncertainty and external disturbance. Chin Phys B 24(3):030504CrossRefGoogle Scholar
  55. 55.
    Corradini ML, Giambò R, Pettinari S (2015) On the adoption of a fractional-order sliding surface for the robust control of integer-order lti plants. Automatica 51:364–371MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Trigeassou J-C, Maamri N, Sabatier J, Oustaloup A (2011) A lyapunov approach to the stability of fractional differential equations. Signal Process 91(3):437–445CrossRefMATHGoogle Scholar
  57. 57.
    Li Y, Chen Y, Podlubny I (2009) Mittag-leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8):1965–1969MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Yu J, Hu C, Jiang H, Fan X (2014) Projective synchronization for fractional neural networks. Neural Netw 49:87–95CrossRefMATHGoogle Scholar
  59. 59.
    Zhou S, Li H, Zhu Z (2008) Chaos control and synchronization in a fractional neuron network system. Chaos, Solitons Fractals 36(4):973–984MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Roohi M, Aghababa MP, Haghighi AR (2015) Switching adaptive controllers to control fractional-order complex systems with unknown structure and input nonlinearities. Complexity 21(2):211–223MathSciNetCrossRefGoogle Scholar
  61. 61.
    Sutha S, Lakshmi P, Sankaranarayanan S (2015) Fractional-order sliding mode controller design for a modified quadruple tank process via multi-level switching. Comput Electr Eng 45:10–21CrossRefGoogle Scholar
  62. 62.
    Aghababa MP (2015) A fractional sliding mode for finite-time control scheme with application to stabilization of electrostatic and electromechanical transducers. Appl Math Model 39(20):6103–6113MathSciNetCrossRefGoogle Scholar
  63. 63.
    Binazadeh T, Shafiei M (2013) Output tracking of uncertain fractional-order nonlinear systems via a novel fractional-order sliding mode approach. Mechatronics 23(7):888–892CrossRefGoogle Scholar
  64. 64.
    Pashaei S, Badamchizadeh M. A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances, ISA transactions, vol. 63, pp 39–48Google Scholar
  65. 65.
    Zhang B-L, Ma L, Han Q-L (2013) Sliding mode h\(_\infty\) control for offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbance. Nonlinear Anal Real World Appl 14(1):163–178MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  2. 2.Department of Applied MathematicsHuainan Normal UniversityHuainanChina
  3. 3.Department of Biomedical EngineeringNational University of SingaporeSingaporeSingapore

Personalised recommendations