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Dealing with Uncertainty in Consensus Protocols

  • Dario Bauso
  • Laura Giarré
  • Raffaele Pesenti
Part of the Understanding Complex Systems book series (UCS)

Abstract

Recent results on consensus protocols for networks are presented. The basic tools and the main contribution available in the literature are considered, together with some of the related challenging aspects: estimation in networks and how to deal with disturbances is considered. Motivated by applications to sensor, peer-to-peer, and ad hoc networks, many papers have considered the problem of estimation in a consensus fashion. Here, the Unknown But Bounded (UBB) noise affecting the network is addressed in details. Because of the presence of UBB disturbances convergence to equilibria with all equal components is, in general, not possible. The solution of the ε-consensus problem, where the states converge in a tube of ray ε asymptotically or in finite time, is described. In solving the ε-consensus problem a focus on linear protocols and a rule for estimating the average from a compact set of candidate points, the lazy rule, is shown.

Keywords

Laplacian Matrix Consensus Problem Consensus Algorithm Cooperative Control Consensus Protocol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Angeli, D., Bliman, P.-A.: Stability of leaderless discrete-time multi-agent systems. Mathematics of Control, Signals & Systems 18(4), 293–322 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arslan, G., Marden, J.R., Shamma, J.S.: Autonomous vehicle-target assignment: A game theoretical formulation. ASME Journal of Dynamic Systems, Measurement, and Control, special issue on Analysis and Control of Multi-Agent Dynamic Systems, 584–596 (2007)Google Scholar
  3. 3.
    Bauso, D., Giarré, L., Pesenti, R.: Nonlinear Protocols for the Optimal Distributed Consensus in Networks of Dynamic Agents. Systems and Control Letters 55(11), 918–928 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bauso, D., Giarrè, L., Pesenti, R.: Consensus for switched networks with unknown but bounded disturbances. SIAM Journal on Control and Optimization (2006) arXiv: math.OC/0612834v1Google Scholar
  5. 5.
    Bauso, D., Giarrè, L., Pesenti, R.: Lazy consensus for networks with unknown but bounded disturbances. In: Proceedings of IEEE CDC, New Orleans, pp. 2283–2288 (2007)Google Scholar
  6. 6.
    Bauso, D., Giarré, L., Pesenti, R.: Distributed consensus in noncooperative inventory games. European Journal of Operational Research (in press) (corrected proof, 17 October 2007)Google Scholar
  7. 7.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized Gossip Algorithms. IEEE Trans. on Information Theory 52(6), 2508–2530 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Borkar, V., Varaiya, P.: Asymptotic agreement in distributed estimation. IEEE Trans. Automat. Control 27, 650–655 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bertsekas, D.P., Rhodes, I.: Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. on Automatic Control 16(2), 117–128 (1971)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and distributed computation. Prentice-Hall International, Englewood Cliffs (1989); republished by Athena Scientific (1997), https://dspace.mit.edu/handle/1721.1/3719 zbMATHGoogle Scholar
  11. 11.
    Broy, M.: Software engineering — from auxiliary to key technologies. In: Broy, M., Dener, E. (eds.) Software Pioneers, pp. 10–13. Springer, Heidelberg (2002)Google Scholar
  12. 12.
    Geromel, J., Colaneri, P.: RMS gain with dwell time for discrete-time switched linear systems. In: Proc. of IEEE Mediterrean Conference, Ajaccio (July 2008)Google Scholar
  13. 13.
    Huang, M., Manton, J.H.: Stochasting approximation for consensus seeking: mean square and almost sure convergence. In: Proceedings of IEEE CDC, New Orleans, pp. 306–311 (2007)Google Scholar
  14. 14.
    Fax, A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. on Automatic Control 49(9), 1565–1576 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Jadbabaie, A., Lin, J., Morse, A.: Coordination of Groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. on Automatic Control 48(6), 988–1001 (2003)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kashyap, A., Basar, T., Srikant, R.: Quantized Consensus. Automatica 43, 1192–1203 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Liberzon, D.: Switching in Systems and Control. Volume in series Systems and Control: Foundations and Applications. Birkhauser, Boston (2003)zbMATHGoogle Scholar
  18. 18.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann Publishers, Inc., San Francisco (1996)zbMATHGoogle Scholar
  19. 19.
    Moreau, L.: Leaderless coordination via bidirectional and unidirectional time-dependent communication. In: Proc. of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, pp. 3070–3075 (2003)Google Scholar
  20. 20.
    Nedić, A., Olshevsky, A., Ozdaglar, A., Tsitsiklis, J.N.: On Distributed Averaging Algorithms and Quantization Effects?, LIDS Report 2778 (November 2007)Google Scholar
  21. 21.
    Olfati-Saber, R., Murray, R.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. on Automatic Control 49(9), 1520–1533 (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and Cooperation in Networked Multi-Agent Systems. Proceedings of the IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  23. 23.
    Olfati-Saber, R.: Distributed Kalman Filtering for Sensor Networks. In: Proc. of IEEE CDC, New Orleans, pp. 5492–5498 (2007)Google Scholar
  24. 24.
    Ren, W., Beard, R., Atkins, E.M.: A survey of consensus problems in multi-agent coordination. In: Proc. of the American Control Conference, Portland, OR, USA, pp. 1859–1864 (2005)Google Scholar
  25. 25.
    Ren, W., Beard, R.W., Atkins, E.: Information Consensus in Multivehicle Cooperative Control: Collective Group Behavior through Local Interaction. IEEE Control Systems Magazine 27(2), 71–82 (2005)CrossRefGoogle Scholar
  26. 26.
    Ren, W., Beard, R.: Consensus seeking in multi-agent systems under dynamically changing interaction topologies. IEEE Trans. on Automatic Control 50(5), 655–661 (2005)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Schizas, I.D., Ribeiro, A., Giannakis, G.B.: Consensus in Ad Hoc WSNs With Noisy Links- Part I: Distributed Estimation of Deterministic Signals. IEEE Trans. on Signal Processing 56(1) (January 2008)Google Scholar
  28. 28.
    Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability Criteria for Switched and Hybrid Systems? SIAM Review 49(4), 545–592 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Tsitsiklis, J.N., Athans, M.: Convergence and asymptotic agreement in distributed decision problems. IEEE Trans. Automat. Control 29(1), 42–50 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tsitsiklis, J.N., Bertsekas, D.P., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Automat. Contr. 31(9), 803–812 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Stable flocking of mobile agents, part ii: Dynamic topology. In: Proc. of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, pp. 2016–2021 (2003)Google Scholar
  32. 32.
    Xiao, L., Boyd, S.: Fast linear iterations for distributed averaging. Systems and Control Letters 53(1), 65–78 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Xiao, L., Boyd, S., Kim, S.-J.: Distributed average consensus with least-mean-square deviation. Journal of Parallel and Distributed Computing 67, 33–46 (2007)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Dario Bauso
    • 1
  • Laura Giarré
    • 1
  • Raffaele Pesenti
    • 2
  1. 1.Universitá di PalermoPalermoItaly
  2. 2.Università di Venezia Cà FoscariVeneziaItaly

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