Positive Consensus Problem: The Case of Complete Communication

  • Maria Elena ValcherEmail author
  • Irene Zorzan
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)


In this chapter the positive consensus problem for homogeneous multi-agent systems is investigated, by assuming that agents are described by positive single-input and continuous-time systems, and that each agent communicates with all the other agents. Under certain conditions on the Laplacian of the communication graph, that arise only when the graph is complete, some of the main necessary conditions for the problem solvability derived in [17, 18, 19] do not hold, and this makes the problem solution more complex. In this chapter we investigate this specific problem, by providing either necessary or sufficient conditions for its solvability and by analysing some special cases.


Multi agent system Continuous time positive system Consensus Complete communication graph 


  1. 1.
    Chaudhary, D.D., Nayse, S.P., Waghmare, L.M.: Application of wireless sensor networks for greenhouse parameter control in precision agriculture. Int. J. Wirel. Mobile Netw. (IJWMN) 3(1) (2011)Google Scholar
  2. 2.
    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465–1476 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23, 298–305 (1973)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fu, E., Markham, T.: On the eigenvalues and diagonal entries of a hermitian matrix. Linear Algebra Appl. 179, 7–10 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goldberg, F.: Bounding the gap between extremal Laplacian eigenvalues of graphs. Linear Algebra Appl. 416, 68–74 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hinrichsen, D., Plischke, E.: Robust stability and transient behaviour of positive linear systems. Vietnam J. Math. 35, 429–462 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Knorn, F., Corless, M.J., Shorten, R.N.: A result on implicit consensus with application to emissions control. In: Proceedings of the 2011 CDC-ECC, pp. 1299–1304, Orlando, FL (2011)Google Scholar
  9. 9.
    Lin, J., Morse, A.S., Anderson, B.D.O.: The multi-agent rendezvous problem. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 1508–1513, Maui, Hawaii (2003)Google Scholar
  10. 10.
    Mohar, B.: The Laplacian spectrum of graphs. Graph Theory Comb. Appl. 2, 871–898 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  12. 12.
    Pejic, S.: Algebraic graph theory in the analysis of frequency assignment problems. PhD thesis, London School of Economics and Political Science (2008)Google Scholar
  13. 13.
    Ren, W., Beard, R.W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41. AMS, Mathematical Surveys and Monographs, Providence, RI (1995)Google Scholar
  15. 15.
    Son, N.K., Hinrichsen, D.: Robust stability of positive continuous time systems. Numer. Funct. Anal. Optimiz. 17(5–6), 649–659 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tsitsiklis, J.N.: Problems in Decentralized Decision Making and Computation. PhD thesis, Department of EECS, MIT (1984)Google Scholar
  17. 17.
    Valcher, M.E., Zorzan, I.: New results on the solution of the positive consensus problem. In: Proceedings of the 55th IEEE Conf. on Decision and Control, pp. 5251–5256, Las Vegas, Nevada, December 12–14 (2016)Google Scholar
  18. 18.
    Valcher, M.E., Zorzan, I.: On the consensus of homogeneous multi-agent systems with positivity constraints (2017, under review)Google Scholar
  19. 19.
    Valcher, M.E., Zorzan, I.: On the consensus problem with positivity constraints. In: Proceedings of the 2016 American Control Conference, pp. 2846–2851, Boston, MA (2016)Google Scholar
  20. 20.
    Wieland, P., Kim, J.-S., Allgover, F.: On topology and dynamics of consensus among linear high-order agents. Int. J. Syst. Sci. 42(10), 1831–1842 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dip. di Ingegneria dell’InformazioneUniv. di PadovaPadovaItaly

Personalised recommendations