Leader-following Consensus of Nonlinear Delayed Multi-agent Systems with Randomly Occurring Uncertainties and Stochastic Disturbances under Impulsive Control Input

  • Yunjian Xu
  • Shiguo Peng
  • Aiyin Guo
Regular Paper Control Theory and Applications


This paper investigates the leader-following consensus problem for a class of nonlinear delayed multiagent systems with randomly occurring uncertainties and stochastic disturbances under impulsive control inputs. For this class of multi-agent system, we present a novel impulsive control protocol which can effectively reduce the control cost and is easy to implement. Two consensus criteria are derived for ensuring global exponential consensus of nonlinear delayed multi-agent systems under non-uniformly distributed impulsive control signals based on comparison principle and average impulsive interval. Compared with the consensus criteria which are derived by the upper bound or lower bound of the impulse intervals in existing results, the obtained criteria are proved to be easier to be satisfied. Simulation results illustrate the effectiveness of the theoretical results.


Delayed multi-agent systems leader-following consensus stochastic disturbances impulsive control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Bertsekas and J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, Englewood Cliffs, NJ, 1989.zbMATHGoogle Scholar
  2. [2]
    R. Olfati-Saber, J. Fax, and R. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, January 2007. [click]CrossRefzbMATHGoogle Scholar
  3. [3]
    W. Ren and R. Beard, Communications and Control Engineering Series, Distributed Consensus in Multi-vehicle Cooperative Control, Springer Verlag, London, 2008.zbMATHGoogle Scholar
  4. [4]
    Z. Qu, Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles, Springer Verlag, London, 2009.zbMATHGoogle Scholar
  5. [5]
    Z. Meng, Z. Zhao, and Z. Lin, “On global leader-following consensus of identical linear dynamic systems subject to actuator saturation,” Syst. Control Lett., vol. 62, no. 2, pp. 132–142, February 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Q. Song, J. Cao, and W. Yu, “Second-order leaderfollowing consensus of nonlinear multi-agent systems via pinning control,” Syst. Control Lett., vol. 59, no. 9, pp. 553–562, September 2010.CrossRefzbMATHGoogle Scholar
  7. [7]
    W. Ni, D.-Y. Zhao, Y.-H. Ni, and X.-L. Wang, “Stochastic averaging approach to leader-following consensus of linear multi-agent systems,” Journal of the Franklin Institute, vol. 352, no. 12, pp. 2650–2669, August 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    X. Xu, S. Chen, W. Huang, and L. Gao, “Leaderfollowing consensus of discrete-time multi-agent systems with observer- based protocols,” Neurocomputing, vol. 118, pp. 334–341, October 2013. [click]CrossRefGoogle Scholar
  9. [9]
    C.-C. Hua, X. You, and X.-P. Guan, “Leader-following consensus for a class of high-order nonlinear multi-agent systems,” Automatica, vol. 73, pp. 138–144, November 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    K. Peng and Y. Yang, “Leader-following consensus problem with a varying-velocity leader and time-varying delays,” Physica A, vol. 388, no. 2-3, pp. 193–208, January 2009. [click]CrossRefGoogle Scholar
  11. [11]
    Y. Feng, S. Xu, and B. Zhang, “Group consensus control for double-integrator dynamic multi-agent systems with fixed communication topology,” International Journal of Robust and Nonlinear Control, vol. 24, no. 3, pp. 532–547, February 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.-M. Xie and L. Teng, “Second-order group consensus for multi-agent systems with time delays,” Neurocomputing, vol. 153, pp. 133–139, April 2015. [click]CrossRefGoogle Scholar
  13. [13]
    H. Hu, L. Yu, W. Zhang, and H. Song, “Group consensus in multi-agent systems with hybrid protocol,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 575–597, April 2013. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    L.-H Ji, X.-F Liao, and Q. Liu, “Group consensus analysis of multi-agent systems with delays,” Acta Phys. Sin., vol. 61, no. 22, 220202(1)–220202(7), 2012.Google Scholar
  15. [15]
    H.-P. Zhang, D. Yue, and X.-X. Yin, “Finite-Time distributed event-triggered consensus control for multi-agent systems,” Information Sciences, vol. 339, pp. 132–142, April 2016. [click]CrossRefGoogle Scholar
  16. [16]
    F. Sun, J. Chen, Z.-H. Guan, L. Ding, and T. Li, “Leaderfollowing finite-time consensus for multi-agent systems with jointly-reachable leader,” Nonlinear Anal. Real World Appl., vol. 13, no. 5, pp. 2271–2284, October 2012. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T. Gao, J. Huang, and Y. Zhou, “Finite-time consensus control of second-order nonlinear systems with input saturation,” Transactions of the Institute of Measurement and Control, vol. 38, no. 11, pp. 1381–1391, November 2016.CrossRefGoogle Scholar
  18. [18]
    Z.-H. Guan, Z.-W. Liu, and G. Feng, “Impulsive consensus algorithms for the second-order multi-agent networks with sampled information,” Automatica, vol. 48, no. 7, pp. 1397–1404, July 2012. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T.-D. Ma, L.-Y. Zhang, and Z.-Y. Gu, “Further studies on impulsive consensus of multi-agent nonlinear systems with control gain error,” Neurocomputing, vol. 190, pp. 140–146, March 2016.CrossRefGoogle Scholar
  20. [20]
    Z. Ye, Y. Chen, and H. Zhang, “Leader-following consensus of multiagent systems with time-varying delays via impulsive control,” Mathematical Problems in Engineering, vol. 2014, no. 240503, pp. 1–10, March 2014.MathSciNetGoogle Scholar
  21. [21]
    S.-S. Yang, X.-F. Liao, and Y.-B. Liu, “Second-order consensus in directed networks of identical nonlinear dynamics via impulsive control,” Neurocomputing, vol. 179, pp. 290–297, February 2016.CrossRefGoogle Scholar
  22. [22]
    J. Hu and Z.-D. Wang, “Robust sliding mode control for discrete stochastic systems with mixed tine delays, randomly occurring uncertainties, and randomly occurring nonlinearities,” IEEE Trans. Indus. Elect., vol. 59, no. 7, pp. 3008–3015, July 2012. [click]CrossRefGoogle Scholar
  23. [23]
    Y. Li, S. Tong, and T. Li, “Observer-Based Adaptive Fuzzy Tracking Control of MIMO Stochastic Nonlinear Systems With Unknown Control Directions and Unknown Dead Zones,” IEEE Transactions on Fuzzy Systems, vol. 23, no. 4, pp. 1228–1241, August 2015. [click]CrossRefGoogle Scholar
  24. [24]
    Y. Li, S. Sui, and S. Tong, “Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics,” IEEE Transactions on Cybernetics, vol. 47, no. 2, pp. 403–414, February 2013. [click]Google Scholar
  25. [25]
    Y. Li and S. Tong, “Adaptive fuzzy output constrained control design for multi-input multi-output stochastic nonstrict-feedback nonlinear systems,” IEEE Transactions on Cybernetics, vol. 47, no. 12, pp. 4086–4095, December 2017.CrossRefGoogle Scholar
  26. [26]
    Y. Li, Z. Ma, and S. Tong, “Adaptive fuzzy outputconstrained fault-tolerant control of nonlinear stochastic large-scale systems with actuator faults,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2362–2376, September 2017.CrossRefGoogle Scholar
  27. [27]
    Y. Tang, H.-J. Gao, and W.-B. Zhang, “Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses,” Automatica, vol. 53, pp. 346–354, March 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. P. Hespanha, D. Liberzon, and A. R. Teel, “Lyapunov conditions for input-tostate stability of impulsive systems,” Automatica, vol. 44, no. 11, pp. 2735–2744, November 2008. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Liu and D. J. Hill, “Impulsive consensus for complex dynamical networks with non-identical nodes and coupling time-delays,” SIAM Journal on Control and Optimization, vol. 49, no. 2, pp. 315–338, 2011. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. R. Teel, A. Subbaramana, and A. Sferlazza, “Stability analysis for stochastic hybrid systems: a survey,” Automatica, vol. 50, no. 10, pp. 2435–2456, October 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Z.-H. Guan, D. J. Hill, and J. Yao, “A hybrid impulsive and switching control strategy for synchronization of nonlinear systems and application to Chua’s chaotic circuit,” International Journal of Bifurcation and Chaos, vol. 16, no. 1, pp. 229–238, January 2006. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    B. Liu, X.-Z. Liu, and G.-R. Chen, “Robust impulsive synchronization of uncertain dynamical networks,” IEEE Transactions on Circuits and Systems I-Regular Papers, vol. 52, no. 7, pp. 1431–1441, July 2005. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J. Zhou, L. Xiang, and Z. Liuc, “Synchronization in complex delayed dynamical networks with impulsive effects,” Physica A-Statistical Mechanics and Its Applications, vol. 384, no. 2, pp. 684–692, October 2007.CrossRefGoogle Scholar
  34. [34]
    X.-P. Han, J.-A. Lu, and X.-Q. Wu, “Synchronization of impulsively coupled systems,” International of Journal of Bifurcation and Chaos, vol. 18, no. 5, pp. 1539–1549, March 2008. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J.-A, Lu, H. Liu, and J. Chen, Synchronization in Complex Dynamical Networks, Higher education press, Beijing, 2016.Google Scholar
  36. [36]
    J. Lu, D. W. C. Ho, and J.-D. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatic, vol. 46, no. 7, pp. 1215–1221, July 2010. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Z.-C. Yang and D.-Y. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Transactions on Automatic Control, vol. 52, no. 8, pp. 1448–1454, August 2007. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    C.-R. Xie, Y.-H. Xu, and D.-B. Tong, “Synchronization of time varying delayed complex networks via impulsive control,” Optik, vol. 125, no. 15, pp. 3781–3787, 2014.CrossRefGoogle Scholar
  39. [39]
    D.-D. Li, J. Ma, and H.-M. Zhu, “The consensus of multiagent systems with uncertainties and randomly occurring nonlinearities via impulsive control,” International Journal of Control Automation and Systems, vol. 14, no. 4, pp. 1005–1011, August 2016. [click]CrossRefGoogle Scholar
  40. [40]
    Z.-Y. Ye, Y.-G. Chen, and H. Zhang, “Leader-following consensus of multiagent systems with time-varying delays via impulsive control,” Mathematical Problems in Engineering, 240503, 2014.Google Scholar
  41. [41]
    S. Djaidja and Q. Wu, “Leader-following consensus of single-integrator multi-agent systems under noisy and delayed communication,” International Journal of Control, Automation and Systems, vol. 14, no. 2, pp. 357–366, April 2016. [click]CrossRefGoogle Scholar
  42. [42]
    B. Cui, C. Zhao, T. Ma, and C. Feng, “Leaderless and leader-following consensus of multi-agent chaotic systems with unknown time delays and switching topologies,” Nonlinear Analysis: Hybrid Systems, vol. 24, pp. 115–131, Janurary 2017. [click]MathSciNetzbMATHGoogle Scholar
  43. [43]
    Z. Wang, S. Lauria, J. Fang, and X. Liu, “Exponential stability of uncertain stochastic neural networks with mixed time-delays,” Chaos Solitons Fractals, vol. 32, no. 1, pp. 62–72, April 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    X.-T. Wu, Y. Tang, and W.-B. Zhang, “Input-to-state stability of impulsive stochastic delayed systems under linear assumptions,” Automatica, vol. 66 pp. 195–204, April 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    R. Khasminskill, Stochastic Stability of Differnetial Equations, Springer-Verlag, Heidelberg, Berlin, 2012.CrossRefGoogle Scholar
  46. [46]
    L. P. Shilnikov, “Chua’s circuit: rigorous result and future problems,” Int. J. Bifurc. Chaos, vol. 4, no. 4, pp. 784–786, 2011.Google Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AutomationGuangdong University of TechnologyGuangzhouChina
  2. 2.School of Information Science and EngineeringHunan International Economics UniversityChangshaChina

Personalised recommendations