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Allocating Minimum Number of Leaders for Seeking Consensus over Directed Networks with Time-varying Nonlinear Multi-agents

  • Leitao Gao
  • Guangshe ZhaoEmail author
  • Guoqi LiEmail author
  • Yuming Liu
  • Jiangshuai Huang
  • Changyun Wen
Regular Papers Control Theory and Applications
  • 19 Downloads

Abstract

In this paper, we consider how to determine the minimum number of leaders with allocation and how to achieve consensus over directed networks consisting of time-varying nonlinear multi-agents. Firstly, the problem of finding minimum number of leaders is formulated as a minimum spanning forest problem, i.e., finding the minimum population of trees in the network. By introducing a toll station connecting with each agent, this problem is converted to a minimum spanning tree problem. In this way, the minimum number of leaders is determined and these leaders are found locating at the roots of each tree in the obtained spanning forest. Secondly, we describe a virtual leader connected with the allocated leaders, which indicates that the number of edges connected the follower agents with the virtual leader is the least in an arbitrary directed network. This method is different from the existing consensus problem of redundant leaders or edges that connect the follower with one leader in special networks. A distributed consensus protocol is revisited for achieving final global consensus of all agents. It is theoretically shown that such a protocol indeed ensures consensus. Simulation examples in real-life networks are also provided to show the effectiveness of the proposed methodology. Our works enable studying and extending application of consensus problems in various complex networks.

Keywords

Distributed consensus leader allocation spanning forest 

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References

  1. [1]
    P. Ogren, E. Fiorelli, and N. E. Leonard, “Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment,” IEEE Transactions on Automatic Control, vol. 49, no. 8, pp. 1292–1302, August 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465–1476, September 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 988–1001, June 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. V. Proskurnikov, A. S. Matveev, and M. Cao, “Opinion dynamics in social networks with hostile camps: Consensus vs. polarization,” IEEE Transactions on Automatic Control, vol. 61, no. 6, pp. 1524–1536, June 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Basu, U. Maulik, and O. Chatterjee, “Stability of consensus node orderings under imperfect network data,” IEEE Transactions on Computational Social Systems, vol. 3, no. 3, pp. 120–131, September 2016.CrossRefGoogle Scholar
  6. [6]
    X. Xu, L. Liu, and G. Feng, “Consensus of discrete–time linear multi–agent systems with communication, input and output delays,” IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 492–497, February 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Y. Sun, Y. Tian, and X. J. Xie, “Stabilization of positive switched linear systems and its application in consensus of multiagent systems,” IEEE Transactions on Automatic Control, vol. 62, no. 12, pp. 6608–6613, December 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    W. Ni and D. Cheng, “Leader–following consensus of multiagent systems under fixed and switching topologies,” Systems and Control Letters, vol. 59, no. 3, pp. 209–217, March 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    R. Olfati–Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and timedelays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, September 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Li, G. Feng, X. Luo, and X. Guan, “Output consensus of heterogeneous linear discrete–time multiagent systems with structural uncertainties,” IEEE Transactions on Cybernetics, vol. 45, no. 12, pp. 2868–2879, December 2015.CrossRefGoogle Scholar
  11. [11]
    Z. Li, X. Liu, W. Ren, and L. Xie, “Distributed tracking control for linear multiagent systems with a leader of bounded unknown input,” IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 518–523, February 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Z. Li, G. Wen, Z. Duan, and W. Ren, “Designing fully distributed consensus protocols for linear multi–agent systems with directed graphs,” IEEE Transactions on Automatic Control, vol. 60, no. 4, pp. 1152–1157, April 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    W. Xu, D.W. C. Ho, L. Li, and J. Cao, “Event–triggered schemes on leader–following consensus of general linear multiagent systems under different topologies,” IEEE Transactions on Cybernetics, vol. 47, no. 1, pp. 212–223, January 2017.CrossRefGoogle Scholar
  14. [14]
    X. Zhang, L. Liu, and G. Feng, “Leader–follower consensus of time–varying nonlinear multi–agent systems,” Automatica, vol. 52, no. C, pp. 8–14, February 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C. Hua, X. You, and X. Guan, “Adaptive leader–following consensus for second–order time–varying nonlinear multiagent systems,” IEEE Transactions on Cybernetics, vol. 47, no. 6, pp. 1532–1539, June 2017.CrossRefGoogle Scholar
  16. [16]
    C. Wang, X. Wang, and H. Ji, “Leader–following consensus for a class of second–order nonlinear multi–agent systems,” Systems and Control Letters, vol. 89, pp. 61–65, March 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Lu, F. Chen, and G. Chen, “Nonsmooth leader–following formation control of nonidentical multi–agent systems with directed communication topologies,” Automatica, vol. 64, pp. 112–120, February 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Huang, Y. Song, W. Wang, C. Wen, and G. Li, “Smooth control design for adaptive leader–following consensus control of a class of high–order nonlinear systems with timevarying reference,” Automatica A Journal of Ifac the Internation, vol. 83, no. C, pp. 361–367, 2017.CrossRefzbMATHGoogle Scholar
  19. [19]
    Z. Wang, H. Zhang, X. Song, and H. Zhang, “Consensus problems for discrete–time agents with communication delay,” International Journal of Control Automation and Systems, vol. 15, no. 7, pp. 1–9, June 2017.Google Scholar
  20. [20]
    H. Krishnan and N. Mcclamroch, “Tracking in nonlinear differential–algebraic control systems with applications to constrained robot systems,” Automatica, vol. 30, no. 12, pp. 1885–1897, December 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    R. Akmeliawati and I. M. Y. Mareels, “Nonlinear energybased control method for aircraft automatic landing systems,” IEEE Transactions on Control Systems Technology, vol. 18, no. 4, pp. 871–884, July 2010.CrossRefGoogle Scholar
  22. [22]
    F. Berrezzek, W. Bourbia, and B. Bensaker, “Flatness based nonlinear sensorless control of induction motor systems,” International Journal of Power Electronics and Drive System, vol. 7, no. 1, pp. 265–278, March 2016.Google Scholar
  23. [23]
    H. Sarand and B. Karimi, “Adaptive consensus tracking of nonsquare mimo nonlinear systems with input saturation and input gain matrix under directed graph,” Neural Computing and Applications, pp. 1–12, August 2017.Google Scholar
  24. [24]
    Z. Yan, K. H. Reza, L. Bing, and H. Gao, “Consensus of nonlinear multi–agent systems based on t–s fuzzy models,” Proceeding of the 31st Chinese Control Conference, pp. 3499.3504, January 2012.Google Scholar
  25. [25]
    W. Xiong, W. Yu, J. Lu, and X. Yu, “Fuzzy modelling and consensus of nonlinear multiagent systems with variable structure,” IEEE Transactions on Circuits and Systems, vol. 61, no. 4, pp. 1183–1191, April 2014.CrossRefGoogle Scholar
  26. [26]
    G. S. Seyboth, G. S. Schmidt, and F. Allower, “Cooperative control of linear parameter–varying systems,” Proc. of American Control Conference, June 2012.Google Scholar
  27. [27]
    J. Chen, W. Zhang, Y. Y. Cao, and H. Chu, “Observerbased consensus control against actuator faults for linear parameter–varying multiagent systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 7, pp. 1336–1347, July 2017.CrossRefGoogle Scholar
  28. [28]
    Q. Wang, J. Fu, and J. Wang, “Fully distributed containment control of high–order multi–agent systems with nonlinear dynamics,” Systems and Control Letters, vol. 99, pp. 33–39, January 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Li, Z. Q. Chen, C. Y. Zhang, Z. X. Liu, and Q. Zhang, “Containment control for directed networks multi–agent system with nonlinear dynamics and communication timedelays,” International Journal of Control Automation and Systems, vol. 15, no. 3, pp. 1181–1188, July 2017.CrossRefGoogle Scholar
  30. [30]
    X. H. Wang and H. B. Ji, “Leader–follower consensus for a class of nonlinear multi–agent systems,” International Journal of Control Automation and Systems, vol. 10, no. 1, pp. 27–35, February 2012.CrossRefGoogle Scholar
  31. [31]
    J. Edmonds, “Optimum branching,” Journal of Research of the National Bureau of Standards, vol. 71B, no. 4, pp. 233–240, 1967.Google Scholar
  32. [32]
    D. Baird and R. E. Ulanowicz, “The seasonal dynamics of the chesapeake bay ecosystem,” Ecological Monographs, vol. 59, no. 4, pp. 329–364, December 1989.CrossRefGoogle Scholar
  33. [33]
    Y. J. Chu and T. H. Liu, “On the shortest arborescence of a directed graph,” Science Sinica, vol. 14, pp. 1396.1400, January 1965.Google Scholar
  34. [34]
    H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,” Combinatorica, vol. 6, no. 2, pp. 109–122, June 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    P. Ming, J. Liu, S. Tan, S. Li, L. Shang, and X. Yu, “Consensus stabilization in stochastic multi–agent systems with markovian switching topology, noises and delay,” Neurocomputing, vol. 200, no. C, pp. 1–10, August 2016.CrossRefGoogle Scholar
  36. [36]
    J. Dai and G. Guo, “Event–based consensus for secondorder multi–agent systems with actuator saturation under fixed and markovian switching topologies,” Journal of the Franklin Institute, vol. 354, no. 4, pp. 6098–6118, September 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen–Orr, I. Ayzenshtat, M. Sheffer, and U. Alon, “Superfamilies of evolved and designed networks,” Science, vol. 303, no. 5663, pp. 1538–1542, March 2004.CrossRefGoogle Scholar
  38. [38]
    R. R. Christian and J. J. Luczkovich, “Organizing and understanding a winters seagrass foodweb network through effective trophic levels,” Ecological Modelling, vol. 117, no. 1, pp. 99–124, April 1999.CrossRefGoogle Scholar
  39. [39]
    J. Almunia, G. Basterretxea, J. Aristegui, and R. E. Ulanowicz, “Benthic–pelagic switching in a coastal subtropical lagoon, Estuarine,” Estuarine Coastal and Shelf Science, vol. 49, no. 3, pp. 363–384, September 1999.CrossRefGoogle Scholar
  40. [40]
    D. Baird, J. Luczkovich, and R. R. Christian, “Assessment of spatial and temporal variability in ecosystem attributes of the st marks national wildlife refuge, apalachee bay, florida, Estuarine,” Estuarine Coastal and Shelf Science, vol. 47, no. 3, pp. 329–349, September 1998.CrossRefGoogle Scholar
  41. [41]
    J. Leskovec, J. Kleinberg, and C. Faloutsos, “Graph evolution,” ACM Transactions on Knowledge Discovery from Data, vol. 1, no. 1, March 2007.Google Scholar
  42. [42]
    R. S. Bur, “Social contagion and innovation: Cohesion versus structural equivalence,” American Journal of Sociology, vol. 92, no. 6, pp. 1287–1335, May 1987.CrossRefGoogle Scholar
  43. [43]
    M. P. Young, “The organization of neural systems in the primate cerebral cortex,” Proceedings Biological Sciences, vol. 252, no. 1333, pp. 13–18, May 1993.CrossRefGoogle Scholar
  44. [44]
    T. Opsahl, F. Agneessens, and J. Skvoretzc, “Node centrality in weighted networks: Generalizing degree and shortest paths,” Social Networks, vol. 32, no. 3, pp. 245–251, July 2010.CrossRefGoogle 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 2019

Authors and Affiliations

  1. 1.School of Electronic and Information EngineeringXi’an Jiaotong UniversityXi’anChina
  2. 2.Qingdao R&D InstituteXi’an Jiaotong UniversityQingdaoChina
  3. 3.Department of Precision InstrumentTsinghua UniversityBeijingChina
  4. 4.School of AutomationChongqing UniversityChongqingChina
  5. 5.School of EEENanyang Technological UniversitySingaporeSingapore

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