Event-Triggered Distributed Cooperative Learning Algorithms over Networks via Wavelet Approximation

  • Hao Dai
  • Jin Xie
  • Weisheng ChenEmail author


This paper investigates the problem of event-triggered distributed cooperative learning (DCL) over networks based on wavelet approximation theory, where each node only has access to local data which are produced by the same and unknown pattern (map or function). All nodes cooperatively learn this unknown pattern by exchanging learned information with their neighboring nodes under event-triggered strategy in order to remove unnecessary communications, so as to avoid the waste of network resources. For the above problem, two novel event-triggered continuous-time and discrete-time DCL algorithms are proposed to approximate the unknown pattern by using wavelet basis function. The proposed event-triggered DCL algorithms are used to train the optimal weight coefficient matrix of wavelet series. Moreover, the convergence of the proposed algorithms are presented by using the Lyapunov method, and the Zeno behavior is excluded as well by the strictly positive sampling interval. The illustrative examples are presented to show the efficiency and convergence of the proposed algorithms.


Event-triggered strategy Distributed cooperative learning (DCL) Wavelet approximation Zeno behavior 



The authors thank the reviewers and the editor for their valuable comments on this paper. This work was supported by the National Natural Science Foundation of China (Grant Numbers: 61503292, 61673308 and 61673014),the Natural Science Foundation of Shaanxi Province (Grant Numbers:2018JM6079) and the Fundamental Research Funds for the Central Universities(Grant No: JB181305).


  1. 1.
    Predd JB, Kulkarni SR, Poor HV (2006) Distributed learning in wireless sensor networks. IEEE Signal Process Mag 23(4):56–69CrossRefGoogle Scholar
  2. 2.
    Georgopoulos L, Hasler M (2014) Distributed machine learning in networks by consensus. Neurocomputing 124(2):2–12CrossRefGoogle Scholar
  3. 3.
    Chen JS, Sayed AH (2012) Diffusion adaptation strategies for distributed optimization and learning over networks. IEEE Trans Signal Process 60(8):4289–4305MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen WS, Hua SY, Ge SS (2014) Consensus-based distributed cooperative learning control for a group of discrete-time nonlinear multi-agent systems using neural networks. Automatica 50(1):2254–2268MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen WS, Hua SY, Zhang HG (2015) Consensus-based distributed cooperative learning from closed-loop neural control systems. IEEE Trans Neural Netw Learn Syst 26(2):331–345MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ren PF, Chen WS, Dai H, Zhang HG (2017) Distributed cooperative learning over networks via fuzzy logic systems: performance analysis and comparison. IEEE Trans Fuzzy Syst 26:2075–2088CrossRefGoogle Scholar
  7. 7.
    Xie J, Chen WS, Dai H (2017) Distributed cooperative learning algorithms using wavelet neural network. Neural Comput Appl. Google Scholar
  8. 8.
    Lim C, Lee S, Choi JH, Chang JH (2014) Efficient implementation of statistical model-based voice activity detection using Taylor series approximation. IEICE Trans Fundam Electron Commun Comput Sci E97.A(3):865–868CrossRefGoogle Scholar
  9. 9.
    Sharapudinov II (2014) Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier–Haar series. Rus Acad Sci Sb Math 205(205):145–160MathSciNetzbMATHGoogle Scholar
  10. 10.
    Yang C, Yi Z, Zuo L (2008) Function approximation based on twin support vector machines. In Cybernetics and intelligent systems IEEE conference on, pp 259–264Google Scholar
  11. 11.
    Huang GB, Saratchandran P, Sundararajan N (2005) A generalized growing and pruning RBF (GGAP-RBF) neural network for function approximation. IEEE Trans Neural Netw 16(1):57–67CrossRefGoogle Scholar
  12. 12.
    Yang C, Jiang K, Li Z, He W, Su CY (2017) Neural control of bimanual robots with guaranteed global stability and motion precision. IEEE Trans Ind Inf 13(3):1162–1171CrossRefGoogle Scholar
  13. 13.
    Cui R, Yang C, Li Y, Sharma S (2017) Adaptive neural network control of AUVs with control input nonlinearities using reinforcement learning. IEEE Trans Syst Man Cybern Syst 47(6):1019–1029CrossRefGoogle Scholar
  14. 14.
    Wu S, Er MJ (2000) Dynamic fuzzy neural networks-a novel approach to function approximation. IEEE Trans Syst Man Cybern Part B Cybern A Publ IEEE Syst Man Cybern Soc 30(2):358–364CrossRefGoogle Scholar
  15. 15.
    Ferrari S, Stengel RF (2005) Smooth function approximation using neural networks. IEEE Trans Neural Netw 16(1):24–38CrossRefGoogle Scholar
  16. 16.
    Pavez E, Silva JF (2012) Analysis and design of Wavelet-Packet Cepstral coefficients for automatic speech recognition. Speech Commun 54(6):814–835CrossRefGoogle Scholar
  17. 17.
    Yan R, Gao RX, Chen X (2014) Wavelets for fault diagnosis of rotary machines: a review with applications. Signal Process 96(5):1–15CrossRefGoogle Scholar
  18. 18.
    Siddiqi MH, Lee SW, Khan AM (2014) Weed image classification using wavelet transform, stepwise linear discriminant analysis, and support vector machines for an automatic spray control system. J Inf Sci Eng 30(4):1227–1244Google Scholar
  19. 19.
    Zainuddin Z, Ong P (2016) Optimization of wavelet neural networks with the firefly algorithm for approximation problems. Neural Comput Appl 28:1–14CrossRefGoogle Scholar
  20. 20.
    Hou MZ, Han XL, Gan YX (2009) Constructive approximation to real function by wavelet neural networks. Neural Comput Appl 18(8):883–889CrossRefGoogle Scholar
  21. 21.
    Cao J, Lin Z, Huang GB (2011) Composite function wavelet neural networks with differential evolution and extreme learning machine. Neural Process Lett 33(3):251–265CrossRefGoogle Scholar
  22. 22.
    Cordova J, Yu W (2012) Two types of haar wavelet neural networks for nonlinear system identification. Neural Process Lett 35(3):283–300CrossRefGoogle Scholar
  23. 23.
    Alexandridis AK, Zapranis AD (2013) Wavelet neural networks: a practical guide. Neural Netw 42:1–27CrossRefzbMATHGoogle Scholar
  24. 24.
    Courroux S, Chevobbe S, Darouich M, Paindavoine M (2013) Use of wavelet for image processing in smart cameras with low hardware resources. J Syst Archit 59(10):826–832CrossRefGoogle Scholar
  25. 25.
    Chen S, Zhao HC, Zhang SN, Yang YX (2014) Study of ultra-wideband fuze signal processing method based on wavelet transform. IET Radar Sonar Navig 8(3):167–172CrossRefGoogle Scholar
  26. 26.
    Ganjefar S, Tofighi M (2015) Single-hidden-layer fuzzy recurrent wavelet neural network: applications to function approximation and system identification. Inf Sci 294:269–285MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nejad HC, Farshad M, Khayat O, Rahatabad FN (2016) Performance verification of a fuzzy wavelet neural network in the first order partial derivative approximation of nonlinear functions. Neural Process Lett 43(1):219–230CrossRefGoogle Scholar
  28. 28.
    Sibel S, Ali MS, Vadivel R, Arik S (2017) Decentralized event-triggered synchronization of uncertain Markovian jumping neutral-type neural networks with mixed delays. Neural Netw 86:32–41CrossRefGoogle Scholar
  29. 29.
    Wang AJ, Dong T, Liao XF (2016) Event-triggered synchronization strategy for complex dynamical networks with the Markovian switching topologies. Neural Netw 74:52–57CrossRefzbMATHGoogle Scholar
  30. 30.
    Han YJ, Lu WL, Chen TP (2015) Consensus analysis of networks with time-varying topology and event-triggered diffusions. Neural Netw 71:196–203CrossRefzbMATHGoogle Scholar
  31. 31.
    Li HQ, Liao XF, Chen G, Hill DJ, Dong ZY, Huang TW (2015) Event-triggered asynchronous intermittent communication strategy for synchronization in complex dynamical networks. Neural Netw 66:1–10CrossRefzbMATHGoogle Scholar
  32. 32.
    Mazo M, Tabuada P (2011) Decentralized event-triggered control over wireless sensor/actuator networks. IEEE Trans Autom Control 56(10):2456–2461MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hu SL, Yue D (2012) Event-triggered control design of linear networked systems with quantizations. ISA Trans 51:153–162CrossRefGoogle Scholar
  34. 34.
    Fan Y, Feng G, Wang Y, Song C (2013) Distributed event-triggered control of multi-agent systems with combinational measurements. Automatica 49(2):671–675MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Seyboth GS, Dimarogonas DV, Johansson KH (2013) Event-based broadcasting for multi-agent average consensus. Automatica 49(1):245–252MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Aranda-Escolastico E, Guinaldo M, Gordillo F, Dormido S (2016) A novel approach to periodic event-triggered control: design and application to the inverted pendulum. ISA Trans 65:327–338CrossRefGoogle Scholar
  37. 37.
    Mahmoud MS, Sabih M, Elshafei M (2016) Event-triggered output feedback control for distributed networked systems. ISA Trans 60:294–302CrossRefGoogle Scholar
  38. 38.
    Zainuddin Z, Pauline O (2011) Modified wavelet neural network in function approximation and its application in prediction of time-series pollution data. Appl Soft Comput 11(8):4866–4874CrossRefGoogle Scholar
  39. 39.
    Cattani C (2012) Fractional Calculus and Shannon wavelet. Mathe Probl Eng. Article ID 502812, p 26Google Scholar
  40. 40.
    Bazaraa MS, Goode JJ (1973) On symmetric duality in nonlinear programming. Op Res 21(1):1–9MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lu J, Tang CY (2012) Zero-gradient-sum algorithms for distributed convex optimization: the continuous-time case. IEEE Trans Autom Control 57(9):2348–2354MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.School of Aerospace Science and TechnologyXidian UniversityXi’anPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China

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