Differential evolution-based parameter estimation and synchronization of heterogeneous uncertain nonlinear delayed fractional-order multi-agent systems with unknown leader

  • Wei Hu
  • Guoguang WenEmail author
  • Ahmed Rahmani
  • Yongguang Yu
Original Paper


In this paper, under a fixed directed graph, the distributed cooperative synchronization of heterogeneous uncertain nonlinear chaotic delayed fractional-order multi-agent systems (FOMASs) with a leader of bounded unknown input is investigated, where the fractional orders and system parameters are uncertain and the controller gains are heterogeneous due to imperfect implementation. It should be noted that the study is more general by considering the FOMASs with time delays, unknown leader, heterogeneity, and unknown nonlinear dynamics. Firstly, a differential evolution-based parameter estimation method is proposed to identify the uncertain parameters. Then based on the identified parameters, by using the matrix theory, graph theory, fractional derivative inequality, and comparison principle of linear fractional equation with delay, a heterogeneous discontinuous controller is designed to achieve the distributed cooperative synchronization asymptotically. Thirdly, a heterogeneous continuous controller is further constructed to suppress the undesirable chattering behavior, where uniformly ultimately bounded synchronization tracking errors can be achieved and tuned as small as desired. Finally, numerical simulations are provided to validate the effectiveness of the proposed parameter estimation scheme and the designed control algorithms.


Distributed cooperative synchronization Fractional-order multi-agent systems (FOMASs) Time delays Unknown leader Differential evolution (DE) 



This work was supported by China Scholarship Council and the National Natural Science Foundation of China under Grants 61403019 and 61772063 and the Fundamental Research Funds for the Central Universities under Grant 2017JBM067.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Li, P., Xu, S., Chen, W., Wei, Y., Zhang, Z.: A connectivity preserving rendezvous for unicycle agents with heterogeneous input disturbances. J. Frank. Inst. 355(10), 4248–4267 (2018)zbMATHGoogle Scholar
  2. 2.
    Peng, Z., Wen, G., Yang, S., Rahmani, A.: Distributed consensus-based formation control for nonholonomic wheeled mobile robots using adaptive neural network. Nonlinear Dyn. 86(1), 605–622 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Yazdani, S., Haeri, M.: Robust adaptive fault-tolerant control for leader-follower flocking of uncertain multi-agent systems with actuator failure. ISA Trans. 71, 227–234 (2017)Google Scholar
  4. 4.
    Jiang, W., Wen, G., Peng, Z., Huang, T., Rahmani, R.A.: Fully distributed formation-containment control of heterogeneous linear multi-agent systems. IEEE Trans. Autom. Control (2018). Google Scholar
  5. 5.
    Jain, A., Ghose, D.: Synchronization of multi-agent systems with heterogeneous controllers. Nonlinear Dyn. 89(2), 1433–1451 (2017)zbMATHGoogle Scholar
  6. 6.
    Jain, A., Ghose, D.: Stabilization of collective formations with speed and controller gain heterogeneity and saturation. J. Frank. Inst. 354(14), 5964–5995 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Guo, S., Mo, L., Yu, Y.: Mean-square consensus of heterogeneous multi-agent systems with communication noises. J. Frank. Inst. 355(8), 3717–3736 (2018)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Wang, F., Wen, G., Peng, Z., Huang, T., Yu, Y.: Event-triggered consensus of general linear multiagent systems with data sampling and random packet losses. IEEE Trans. Syst. Man Cybern. Syst. (2019). Google Scholar
  9. 9.
    Li, Z., Wen, G., Duan, Z., Ren, W.: Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs. IEEE Trans. Autom. Control 60(4), 1152–1157 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cao, Y., Yu, W., Ren, W., Chen, G.: An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans. Industr. Inform. 9(1), 427–438 (2013)Google Scholar
  11. 11.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)zbMATHGoogle Scholar
  12. 12.
    Cao, Y., Ren, W.: Distributed formation control for fractional-order systems: dynamic interaction and absolute/relative damping. Syst. Control Lett. 59(3–4), 233–240 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cao, Y., Li, Y., Ren, W., Chen, Y.: Distributed coordination of networked fractional-order systems. IEEE Trans. Syst. Man Cybern. B Cybern. 40(2), 362–370 (2010)Google Scholar
  14. 14.
    Yu, Z., Jiang, H., Hu, C., Yu, J.: Necessary and sufficient conditions for consensus of fractional-order multiagent systems via sampled-data control. IEEE Trans. Cybern. 47(8), 1892–1901 (2017)Google Scholar
  15. 15.
    Chen, Y., Wen, G., Peng, Z., Rahmani, A.: Consensus of fractional-order multiagent system via sampled-data event-triggered control. J. Frank. Inst. (2018). Google Scholar
  16. 16.
    Gong, Y., Wen, G., Peng, Z., Huang, T., Chen, Y.: Observer-based time-varying formation control of fractional-order multi-agent systems with general linear dynamics. IEEE Trans. Circ. Syst. II Exp. Briefs (2019). Google Scholar
  17. 17.
    Wang, H., Yu, Y., Wen, G., Zhang, S., Yu, J.: Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154, 15–23 (2015)Google Scholar
  18. 18.
    Rakkiyappan, R., Cao, J., Velmurugan, G.: Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 26(1), 84–97 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw. 32, 245–256 (2012)zbMATHGoogle Scholar
  20. 20.
    Huang, X., Zhao, Z., Wang, Z., Li, Y.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)Google Scholar
  21. 21.
    Fan, Y., Huang, X., Wang, Z., Li, Y.: Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function. Nonlinear Dyn. 93(2), 611–627 (2018)zbMATHGoogle Scholar
  22. 22.
    Cao, J., Li, H.X., Ho, D.W.: Synchronization criteria of Lur’e systems with time-delay feedback control. Chaos Solitons Fractals 23(4), 1285–1298 (2005)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bao, H., Park, J.H., Cao, J.: Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn. 82(3), 1343–1354 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Chen, L., Wu, R., Cao, J., Liu, J.B.: Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw. 71, 37–44 (2015)zbMATHGoogle Scholar
  25. 25.
    Zhang, L., Yang, Y.: Lag synchronization for fractional-order memristive neural networks with time delay via switching jumps mismatch. J. Frank. Inst. 355(3), 1217–1240 (2018)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Chen, J., Chen, B., Zeng, Z.: Global asymptotic stability and adaptive ultimate Mittag-Leffler synchronization for a fractional-order complex-valued memristive neural networks with delays. IEEE Trans. Syst. Man Cybern. Syst. 99, 1–17 (2018)Google Scholar
  27. 27.
    Liu, P., Zeng, Z., Wang, J.: Global synchronization of coupled fractional-order recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. (2018). Google Scholar
  28. 28.
    Lakshmanan, S., Prakash, M., Lim, C.P., Rakkiyappan, R., Balasubramaniam, P., Nahavandi, S.: Synchronization of an inertial neural network with time-varying delays and its application to secure communication. IEEE Trans. Neural Netw. Learn. Syst. 29(1), 195–207 (2018)MathSciNetGoogle Scholar
  29. 29.
    Zhou, L., Tan, F.: A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks. Nonlinear Dyn. (2019). Google Scholar
  30. 30.
    Li, Z., Duan, Z., Chen, G., Huang, L.: Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans. Circ. Syst. I: Regul. Pap. 57(1), 213–224 (2010)MathSciNetGoogle Scholar
  31. 31.
    Wen, G., Yu, W., Zhao, Y., Cao, J.: Pinning synchronisation in fixed and switching directed networks of Lorenz-type nodes. IET Control Theory Appl. 7(10), 1387–1397 (2013)MathSciNetGoogle Scholar
  32. 32.
    Ma, T., Lewis, F.L., Song, Y.: Exponential synchronization of nonlinear multi-agent systems with time delays and impulsive disturbances. Int. J. Robust Nonlinear Control 26(8), 1615–1631 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Cui, B., Zhao, C., Ma, T., Feng, C.: Leaderless and leader-following consensus of multi-agent chaotic systems with unknown time delays and switching topologies. Nonlinear Anal. Hybrid Syst. 24, 115–131 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shen, J., Cao, J.: Necessary and sufficient conditions for consensus of delayed fractional-order systems. Asian J. Control 14(6), 1690–1697 (2012)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Yang, H.Y., Zhu, X.L., Cao, K.C.: Distributed coordination of fractional order multi-agent systems with communication delays. Fract. Calc. Appl. Anal. 17(1), 23–37 (2014)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhu, W., Chen, B., Yang, J.: Consensus of fractional-order multi-agent systems with input time delay. Fract. Calc. Appl. Anal. 20(1), 52–70 (2017)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Hu, W., Wen, G., Rahmani, A., Yu, Y.: Distributed consensus tracking of unknown nonlinear chaotic delayed fractional-order multi-agent systems with external disturbances based on ABC algorithm. Commun. Nonlinear Sci. Numer. Simul. 71, 101–117 (2019)MathSciNetGoogle Scholar
  38. 38.
    Yuan, C., He, H.: Cooperative output regulation of heterogeneous multi-agent systems with a leader of bounded inputs. IET Control Theory Appl. 12(2), 233–242 (2017)MathSciNetGoogle Scholar
  39. 39.
    Yu, J., Dong, X., Li, Q., Ren, Z.: Time-varying formation tracking for high-order multi-agent systems with switching topologies and a leader of bounded unknown input. J. Frank. Inst. 355(5), 2808–2825 (2018)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Gong, P.: Distributed tracking of heterogeneous nonlinear fractional-order multi-agent systems with an unknown leader. J. Frank. Inst. 354(5), 2226–2244 (2017)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Devasia, S.: Iterative control for networked heterogeneous multi-agent systems with uncertainties. IEEE Trans. Autom. Control 62(1), 431–437 (2017)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Meng, X., Xie, L., Soh, Y.C.: Event-triggered output regulation of heterogeneous multiagent networks. IEEE Trans. Autom. Control. 63(12), 4429–4434 (2018)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Khalili, M., Zhang, X., Polycarpou, M.M., Parisini, T., Cao, Y.: Distributed adaptive fault-tolerant control of uncertain multi-agent systems. Automatica 87, 142–151 (2018)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Chen, C., Wen, C., Liu, Z., Xie, K., Zhang, Y., Chen, C.P.: Adaptive consensus of nonlinear multi-agent systems with non-identical partially unknown control directions and bounded modelling errors. IEEE Trans. Autom. Control 62(9), 4654–4659 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Gong, P., Lan, W.: Adaptive robust tracking control for uncertain nonlinear fractional-order multi-agent systems with directed topologies. Automatica 92, 92–99 (2018)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Gong, P., Lan, W.: Adaptive robust tracking control for multiple unknown fractional-order nonlinear systems. IEEE Trans. Cybern. 99, 1–12 (2018)Google Scholar
  47. 47.
    Parlitz, U.: Estimating model parameters from time series by autosynchronization. Phys. Rev. Lett. 76(8), 1232 (1996)Google Scholar
  48. 48.
    Konnur, R.: Synchronization-based approach for estimating all model parameters of chaotic systems. Phys. Rev. E 67(2), 027204 (2003)Google Scholar
  49. 49.
    Gu, Y., Yu, Y., Wang, H.: Synchronization-based parameter estimation of fractional-order neural networks. Physica A 483, 351–361 (2017)MathSciNetGoogle Scholar
  50. 50.
    Hu, W., Yu, Y., Zhang, S.: A hybrid artificial bee colony algorithm for parameter identification of uncertain fractional-order chaotic systems. Nonlinear Dyn. 82(3), 1441–1456 (2015)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Panahi, S., Jafari, S., Pham, V.T., Kingni, S.T., Zahedi, A., Sedighy, S.H.: Parameter identification of a chaotic circuit with a hidden attractor using Krill herd optimization. Int. J. Bifurc. Chaos 26(13), 1650221 (2016)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Ahandani, M.A., Ghiasi, A.R., Kharrati, H.: Parameter identification of chaotic systems using a shuffled backtracking search optimization algorithm. Soft Comput. 22(24), 8317–8339 (2018)Google Scholar
  53. 53.
    Storn, R., Price, K.: Differential evolution-simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Das, S., Suganthan, P.N.: Differential evolution: a survey of the state-of-the-art. IEEE Trans. Evol. Comput. 15(1), 4–31 (2011)Google Scholar
  55. 55.
    Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 650–659 (2015)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Bhalekar, S.A.C.H.I.N., Daftardar-Gejji, V.A.R.S.H.A.: A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fract. Calc. Appl. 1(5), 1–9 (2011)zbMATHGoogle Scholar
  57. 57.
    Young, K.D., Utkin, V.I., Ozguner, U.: A control engineer’s guide to sliding mode control. IEEE Trans. Control Syst. Technol. 7(3), 328–342 (1999)Google Scholar
  58. 58.
    Shahamatkhah, E., Tabatabaei, M.: Leader-following consensus of discrete-time fractional-order multi-agent systems. Chin. Phys. B 27(1), 010701 (2018)Google Scholar
  59. 59.
    Wyrwas, M., Mozyrska, D., Girejko, E.: Fractional discrete-time consensus models for single-and double-summator dynamics. Int. J. Syst. Sci. 49(6), 1212–1225 (2018)MathSciNetGoogle Scholar
  60. 60.
    Stanisławski, R., Latawiec, K.J.: Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: new stability criterion for FD-based systems. Bull. Pol. Acad. Technol. 61(2), 363–370 (2013)Google Scholar
  61. 61.
    Yuan, L., Yang, Q.: Parameter identification of fractional-order chaotic systems without or with noise: reply to comments. Commun. Nonlinear Sci. Numer. Simul. 67, 506–516 (2019)MathSciNetGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CRIStAL, UMR CNRS 9189, Centrale LilleVilleneuve-d’AscqFrance
  2. 2.School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

Personalised recommendations