Consensus Theory in Networked Systems

  • Angel StanoevEmail author
  • Daniel Smilkov
Part of the Understanding Complex Systems book series (UCS)


This chapter provides a theoretical analysis for consensus reaching in networked systems. Convergence analysis is carried out for distributed algorithms on directed weighted networks for continuous- and discrete-time cases. Also, systems where consensus can not be reached from every initial state have been studied. We describe the connections between spectral and structural properties of complex networks and the convergence rate of consensus algorithms. Theoretical results regarding consensus-seeking under dynamically changing communication topologies and communication time delays are summarized. Consensus algorithms for double-integrator dynamics are described in the context of cooperative control of multivehicle systems.



The work is supported by the European Commission (ERC Grant #266722) and ONR (Grant No. 62909-10-1-7074). The authors thank the editor for his invaluable support and insightful discussions.


  1. 1.
    Agaev, R.P., Chebotarev, P.: On the spectra of nonsymmetric laplacian matrices. Linear Algebra Appl. 399, 157–168 (2005)Google Scholar
  2. 2.
    Benediktsson, J.A., Swain, P.H.: Consensus theoretic classification methods. IEEE Trans. Syst. Man Cybernet. 22, 688–704 (1992)Google Scholar
  3. 3.
    Borkar, V., Varaiya, P.: Asymptotic agreement in distributed estimation. IEEE Trans. Automat. Contr. AC-27, 650–655 (1982)Google Scholar
  4. 4.
    Boyd, S.: Convex optimization of graph laplacian eigenvalues. Proc. Int. Congr. Math.: Madrid 3, 1311–1320 (2006)Google Scholar
  5. 5.
    Chebotarev, P., Agaev, R.P.: Forest matrices around the Laplacian matrix. Linear Algebra Appl. 356, 253–274 (2002)Google Scholar
  6. 6.
    Chebotarev, P., Agaev, R.P.: Coordination in multiagent systems and laplacian spectra of digraphs. Autom. Rem. Contr. 70, 469–483 (2009)Google Scholar
  7. 7.
    DeGroot, M.H.: Reaching a consensus. J. Am. Statist. Assoc. 69, 118–121 (1974)Google Scholar
  8. 8.
    Dimakis, A., Kar, S., Moura, J.M.F., Rabbat, M.G., Scaglione, A.: Gossip algorithms for distributed signal processing. Proc. IEEE 98, 1847–1864 (2010)Google Scholar
  9. 9.
    Dimarogonas, D.V., Kyriakopoulos, K.J.: On the rendezvous problem for multiple nonholonomic agents. IEEE Trans. Autom. Contr. 52, 916–922 (2007)Google Scholar
  10. 10.
    Fang, L., Antsaklis, P.J.: On communication requirements for multiagent consensus seeking. In: Antsaklis, P.J., Tabuada, P. (eds.) Networked Embedded Sensing and Control. Lecture Notes in Control and Information Sciences, vol. 331, pp. 53–67. Springer, Berlin (2006)Google Scholar
  11. 11.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23, 298–305 (1973)Google Scholar
  12. 12.
    Gazi, V., Passino, K.M.: Stability analysis of swarms. IEEE Trans. Autom. Contr. 48, 692–697 (2003)Google Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1987)Google Scholar
  14. 14.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Automat. Contr. 48, 988–1001 (2003)Google Scholar
  15. 15.
    Kim, Y., Mesbahi, M.: On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian. IEEE Trans. Autom. Contr. 51, 116–120 (2006)Google Scholar
  16. 16.
    Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionelly coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819 (1996)Google Scholar
  17. 17.
    Lin, Z., Francis, B.A., Maggiore, M.: State agreement for continuous-time coupled nonlinear systems. SIAM J. Contr. Optim. 46, 288–307 (2007)Google Scholar
  18. 18.
    Lin, J., Morse, A.S., Anderson, B.D.O.: The multi-agent rendezvous problem. Part 1: The synchronous case. SIAM J. Contr. Optim. 46, 2096–2119 (2007a)Google Scholar
  19. 19.
    Lin, J., Morse, A.S., Anderson, B.D.O.: The multi-agent rendezvous problem. Part 2: The asynchronous case. SIAM J. Contr. Optim. 46, 2120–2147 (2007b)Google Scholar
  20. 20.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, San Francisco, CA (1996)Google Scholar
  21. 21.
    Merris, R.: Laplacian matrices of graphs: A survey. Linear Algebra Appl. 197, 143–176 (1994)Google Scholar
  22. 22.
    Mohar, B.: The Laplacian spectrum of graphs. In: Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J. (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 871–898. Wiley, New York (1991)Google Scholar
  23. 23.
    Moreau, L.: Stability of continuous-time distributed consensus algorithms. In: Proceedings of the IEEE Conference on Decision Control, pp. 3998–4003, Paradise Island, Bahamas (2004)Google Scholar
  24. 24.
    Newman, M.E.J.: Networks – An Introduction. Oxford University Press, Oxford (2010)Google Scholar
  25. 25.
    Nijmeijer, H., Mareels, I.: An observer looks at synchronization. IEEE Trans. Circ. Syst. I 44, 882–890 (1997)Google Scholar
  26. 26.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95, 215–233 (2007)Google Scholar
  27. 27.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Contr. 49, 1520–1533 (2004)Google Scholar
  28. 28.
    Pecora, L., Carroll, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–823 (1990)Google Scholar
  29. 29.
    Ren, W.: Distributed attitude alignment in spacecraft formation flying. Int. J. Adapt. Contr. Signal Process. 21, 95–113 (2007)Google Scholar
  30. 30.
    Ren, W.: Consensus strategies for cooperative control of vehicle formations. IET Contr. Theory Appl. 1, 505–512 (2007)Google Scholar
  31. 31.
    Ren, W., Atkins, E.M.: Distributed multi-vehicle coordinated control via local information exchange. Int. J. Robust Nonlin. Contr. 17, 1002–1033 (2007)Google Scholar
  32. 32.
    Ren, W., Beard, R.W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Automat. Contr. 50, 655–661 (2005)Google Scholar
  33. 33.
    Ren, W., Beard, R.W., Kingston, D.B.: Multi-agent Kalman consensus with relative uncertainty. In: Proceedings of the American Control Conference, pp. 1865–1870, Portland, OR (2005)Google Scholar
  34. 34.
    Ren, W., Beard, R.W., Atkins, E.M.: Information consensus in multivehicle cooperative control: Collective group behavior through local interaction. IEEE Contr. Syst. Mag. 27, 71–82 (2007)Google Scholar
  35. 35.
    Ren, W., Cao, Y.: Distributed coordination of multi-agent networks. Communications and Control Engineering Series. Springer, Berlin (2011)Google Scholar
  36. 36.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996)Google Scholar
  37. 37.
    Saber, R.O.: Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Trans. Autom. Contr. 51, 401–420 (2006)Google Scholar
  38. 38.
    Tanner, H.G., Christodoulakis, D.K.: State synchronization in local interaction networks is robust with respect to time delays. In: Proceedings of the IEEE Conference on Decision Control, European Control Conference, pp. 4945–4950, Seville, Spain (2005)Google Scholar
  39. 39.
    Tsitsiklis, J.N., Bertsekas, D.P., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Autom. Contr. 31, 803–812 (1986)Google Scholar
  40. 40.
    Weller, S.C., Mann, N.C.: Assessing rater performance without a ‘gold standard’ using consensus theory. Med. Decis. Making 17, 71–79 (1997)Google Scholar
  41. 41.
    Wolfowitz, J.: Products of indecomposable, aperiodic, stochastic matrices. Proc. Am. Math. Soc. 15, 733–736 (1963)Google Scholar
  42. 42.
    Xiao, L., Boyd, S.: Fast linear iterations for distributed averaging. Syst. Contr. Lett. 53, 65–78 (2004)Google Scholar
  43. 43.
    Xiao, F., Wang, L.: State consensus for multi-agent systems with switching topologies and time-varying delays. Int. J. Contr. 79, 1277–1284 (2006)Google Scholar
  44. 44.
    Xie, G., Wang, L.: Consensus control for a class of networks of dynamic agents. Int. J. Robust Nonlin. Contr. 17, 941–959 (2007)Google Scholar
  45. 45.
    Yu, W., Chen, G., Cao, M., Kurths, J.: Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man Cybernet. B: Cybernet. 40, 881–891 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Macedonian Academy of Sciences and ArtsSkopjeMacedonia

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