A Game-theoretical Approach for a Finite-time Consensus of Second-order Multi-agent System

  • Lei Xue
  • Changyin SunEmail author
  • Donald C. WunschII
Regular Papers Control Theory and Applications


The second-order consensus problem depends on not only the topology condition but also the coupling strength of the relative positions and velocities between neighboring agents. This paper seeks to solve the finite-time consensus problem of second-order multi-agent systems by games with special structures. Potential game and weakly acyclic game were applied for modeling the second-order consensus problem with different topologies. Furthermore, this paper introduces the event-triggered asynchronous cellular learning automata algorithm for optimizing the decision making process of the agents, which facilitates a convergence with the Nash equilibrium. Finally, numerical examples illustrate the effectiveness of the models.


Event-triggered asynchronous cellular learning automata finite-time second-order consensus graphical games multi-agent system potential game weakly acyclic game 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Subramanian, P. Muthukumar, and Y. Joo, “Leaderfollowing consensus of nonlinear multi-agent systems via reliable control with time-varying communication delay,” International Journal of Control, Automation and Systems, vol. 17, no. 2, pp. 298–306, February 2019.CrossRefGoogle Scholar
  2. [2]
    L. Xue, C. Sun, D. Wunsch, Y. Zhou, and F. Yu, “An adaptive strategy via reinforcement learning for the prisoner’s dilemma game,” IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 1, pp. 301–310, January 2018.MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Mu, Z. Ni, C. Sun, and H. He, “Data-driven tracking control with adaptive dynamic programming for a class of continuous-time nonlinear systems,” IEEE Trans. on Cybernetics, vol. 47, no. 6, pp. 1460–1470, June 2017.CrossRefGoogle Scholar
  4. [4]
    Q. Wang, H. E. Psillakis, and C. Sun, “Cooperative control of multiple agents with unknown highfrequency gain signs under unbalanced and switching topologies,” IEEE Trans. on Automatic Control, 2018. DOI: 10.1109/TAC.2018.2867161Google Scholar
  5. [5]
    X. Zhang and X. Liu, “Consensus tracking of second order multi-agent systems with disturbances under heterogenous position and velocity topologies,” International Journal of Control, Automation and Systems, vol. 16, no. 5, pp. 2334–2342, October 2018.CrossRefGoogle Scholar
  6. [6]
    W. He, Z. Li, and P. Chen, “A survey of human-centered intelligent robots: issues and challenges,” IEEE/CAA Journal of Automatica Sinica, vol. 4, no. 4, pp. 602–609, September 2017.CrossRefGoogle Scholar
  7. [7]
    Q. Wang and C. Sun, “Adaptive consensus of multiagent systems with unknown high-frequency gain signs under directed graphs,” IEEE Trans. on Systems, Man, and Cybernetics: Systems, 2018. DOI: 10.1109/TSMC.2018.2810089Google Scholar
  8. [8]
    W. Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems,” Automatica, vol. 46, no. 6, pp. 1089–1095, March 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    P. Lin and Y. Jia, “Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. on Automatic Control, vol. 55, no. 3, pp. 778–784, February 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Z. Guan, F. Sun, Y. Wang, and T. Li, “Finite-time consensus for leader-following second-order multi-agent networks,” IEEE Trans. on Circuits Systems. I, Reg. Papers, vol. 59, no. 11, pp. 2646–2654, October 2012.MathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Seiffertt, S. Mulder, R. Dua, and D. Wunsch, “Neural networks and Markov models for the iterated prisoner’s dilemma,” International Joint Conference on Neural Networks. IEEE, pp. 1544–1550, 2009.Google Scholar
  12. [12]
    F. Wang, Z. Liu, and Z. Chen, “A novel leader-following consensus of multi-agent systems with smart leader,” International Journal of Control, Automation and Systems, vol. 16, no. 4, pp. 1483–1492, August 2017.CrossRefGoogle Scholar
  13. [13]
    C. Mu, Y. Tang, and H. He, “Improved sliding mode design for load frequency control of power system integrated an adaptive learning strategy,” IEEE Trans. on Industrial Electronics, vol. 64, no. 8, pp. 6742–6751, August 2017.CrossRefGoogle Scholar
  14. [14]
    C. Mu, D. Wang, and H. He, “Data-driven finite-horizon approximate optimal control for discrete-time nonlinear systems using iterative HDP approach” IEEE Trans. on Cybernetics, vol. 48, no. 10, pp. 2948–2961, October 2017.CrossRefGoogle Scholar
  15. [15]
    Z. Tang, “Event-triggered consensus of linear discretetime multi-agent systems with time-varying topology,” The IEEE Computational Intelligence Magazine, vol. 16, no. 3, pp. 1179–1185, June 2018.Google Scholar
  16. [16]
    J. Li, P. Hingston, and G. Kendall, “Engineering design of strategies for winning iterated prisoner’s dilemma competitions,” IEEE Trans. on Computational Intelligence and AI in Games, vol. 3, no. 4, pp. 348–360, December 2011.CrossRefGoogle Scholar
  17. [17]
    D. Wunsch II, and S. Mulder, “Evolutionary algorithms, Markov decision processes, adaptive critic designs, and clustering: commonalities, hybridization, and performance,” International Conference on Intelligent Sensing and Information Processing, pp. 477–482, 2004.Google Scholar
  18. [18]
    E. Semsar-Kazerooni and K. Khorasani, “Multi-agent team cooperation: a game theory approach,” Automatica, vol. 45, no. 10, pp. 2205–2213, June 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    K. G. Vamvoudakis, F. L. Lewis, and G. R. Hudas, “Multiagent differential graphical games: Online adaptive learning solution for synchronization with optimality,” Automatica, vol. 48, no. 8, pp. 1598–1611, May 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. I. Abouheaf, F. L. Lewis, K. G. Vamvoudakis, S. Haesaert, and R. Babuska, “Multi-agent discrete-time graphical games and reinforcement learning solutions,” Automatica, vol. 50, no. 12, pp. 3038–3053, October 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, Philadelphia, 1999.zbMATHGoogle Scholar
  22. [22]
    R. Murray, “Recent research in cooperative control of multi-vehicle systems,” Journal of Dynamic Systems Measurement and Control, vol. 129, no. 5, pp. 571–598, September 2007.CrossRefGoogle Scholar
  23. [23]
    J. Shamma, Cooperative Control of Distributed Multiagent Systems, Wiley-Interscience, Hoboken, NJ, 2008.Google Scholar
  24. [24]
    F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks, Ser. Applied Mathematics Series, Princeton Univ, Princeton, NJ, 2008.zbMATHGoogle Scholar
  25. [25]
    J. R. Marden, G. Arslan, and J. S. Shamma, “Cooperative control and potential games,” IEEE Trans. on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 39, no. 6, pp. 1393–1407, December 2009.CrossRefGoogle Scholar
  26. [26]
    W. Ren and E. Atkins, “Second-order consensus protocols in multiple vehicle systems with local interactions,” Proc. of AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 1998–2006, 2005.Google Scholar
  27. [27]
    Y. Y. Liu, J. J. Slotine, and A. L. Barabasi, “Controllability of complex networks,” Nature, vol. 473, no. 7346, pp. 167–173, May 2011.CrossRefGoogle Scholar
  28. [28]
    H. Beigy and M. R. Meybodi, “Asynchronous cellular learning automata,” Automatica, vol. 44, no. 5, pp. 1350–1357, September 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    H. Beigy and M. R. Meybodi, “A mathematical framework for cellular learning automata,” Journal of Advances in Complex Systems, vol. 7, pp. 295–320, July 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    C. Nowzari and J. Cortés, “Self-triggered optimal servicing in dynamic environments with acyclic structure,” IEEE Trans. on Automatic Control, vol. 58, no. 5, pp. 1236–1249, May 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D. Monderer and L. Shapley, “Potential games,” Games and Economics Behavior, vol. 14, no. 1, pp. 124–143, May 1996.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, School of AutomationSoutheast UniversityNanjingChina
  2. 2.Department of Electrical and Computer EngineeringMissouri University of Science and TechnologyRollaUSA

Personalised recommendations