access icon free Consensus of swarm systems with time delays and topology uncertainties

Consensus problems for first-order continuous-time swarm systems in directed networks with time delays and topology uncertainties are dealt with. The state space of a swarm system is decomposed into a null space and a complement null space. By projecting the state of the system onto the two subspaces, a necessary and sufficient condition for consensus is proposed, and an explicit expression of the consensus function is shown based on the impacts of time delays and topology uncertainties. Sufficient conditions in terms of linear matrix inequalities are given for swarm systems with time-varying delays and topology uncertainties to achieve consensus. Numerical examples are shown to demonstrate theoretical results.

Inspec keywords: uncertain systems; directed graphs; multi-robot systems; continuous time systems; delays; linear matrix inequalities

Other keywords: linear matrix inequalities; complement null space; consensus problems; system state space; time-varying delays; first-order continuous-time swarm systems; directed networks; topology uncertainties

Subjects: Robotics; Combinatorial mathematics; Linear algebra (numerical analysis); Distributed parameter control systems

References

    1. 1)
      • 8. Dong, W.J., Guo, Y., Farrell, J.A.: ‘Formation control of nonholonomic mobile robots’. Proc. American Control Conf., 2006, pp. 56025607.
    2. 2)
      • 37. Porfiri, M., Stilwell, D.J., Bollt, E.M.: ‘Synchronization in random weighted directed networks,’ IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., 2008, 55, (10), pp. 31703177.
    3. 3)
      • 2. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: ‘Stable flocking of mobile agents, part I: fixed topology’. Proc. 42nd IEEE Conf. on Decision and Control, 2003, pp. 20102015.
    4. 4)
      • 28. Münz, U., Papachristodoulou, A., Allgöwer, F.: ‘Generalized Nyquist consensus condition for high-order linear multi-agent systems with communication delays’. Proc. IEEE Conf. on Decision and Control, 2009, pp. 47654771.
    5. 5)
      • 22. Xiao, F., Wang, L.: ‘Consensus problems for high-dimensional multi-agent systems’, IET Control Theory Appl., 2007, 1, (3), pp. 830837 (doi: 10.1049/iet-cta:20060014).
    6. 6)
      • 30. Sun, Y.G., Wang, L.: ‘Consensus of multi-agent systems in directed networks with nonuniform time-varying delays’, IEEE Trans. Autom. Control, 2009, 54, (7), pp. 16071613 (doi: 10.1109/TAC.2009.2017963).
    7. 7)
      • 3. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: ‘Stable flocking of mobile agents, part II: dynamic topology’. Proc. 42nd IEEE Conf. on Decision and Control, 2003, pp. 20162021.
    8. 8)
      • 25. Lin, P., Jia, Y.: ‘Average consensus in networks of multi-agents with both switching topology and coupling time-delay’, Physica A, 2008, 387, (1), pp. 303313 (doi: 10.1016/j.physa.2007.08.040).
    9. 9)
      • 15. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: ‘Novel type of phase transition in a system of self-driven particles’, Phys. Rev. Lett., 1995, 75, (6), pp. 12261229 (doi: 10.1103/PhysRevLett.75.1226).
    10. 10)
      • 35. Yang, B., Fang, H.: ‘Forced consensus in networks of double integrator systems with delayed input’, Automatica, 2010, 46, (3), pp. 629632 (doi: 10.1016/j.automatica.2010.01.013).
    11. 11)
      • 1. Olfati-Saber, R.: ‘Flocking for multi-agent dynamic systems: algorithms and theory’, IEEE Trans. Autom. Control, 2006, 51, (3), pp. 401420 (doi: 10.1109/TAC.2005.864190).
    12. 12)
      • 12. Lian, Z., Deshmukh, A.: ‘Performance prediction of an unmanned airborne vehicle multi-agent system’, Euro. J. Oper. Res., 2006, 172, (2), pp. 680695 (doi: 10.1016/j.ejor.2004.10.015).
    13. 13)
      • 23. Xi, J., Cai, N., Zhong, Y.: ‘Consensus problems for high-order linear time-invariant swarm systems’, Physica A, 2010, 389, (24), pp. 56195627 (doi: 10.1016/j.physa.2010.08.038).
    14. 14)
      • 20. Tahbaz-Salehi, A., Jadbabaie, A.: ‘A necessary and sufficient condition for consensus over random networks’, IEEE Trans. Autom. Control, 2008, 53, (3), pp. 791795 (doi: 10.1109/TAC.2008.917743).
    15. 15)
      • 6. Vidal, R., Shakernia, O., Sastry, S.: ‘Formation control of nonholonomic mobile robots omnidirectional visual servoing and motion segmentation’. Proc. IEEE Conf. on Robotics and Automation, 2003, pp. 584589.
    16. 16)
      • 10. Paganini, F., Doyle, J., Low, S.: ‘Scalable laws for stable network congestion control’. Proc. 40th IEEE Conf. on Decision and Control, 2001, pp. 185190.
    17. 17)
      • 19. Ren, W., Beard, R.W.: ‘Consensus seeking in multiagent systems under dynamically changing interaction topologies’, IEEE Trans. Autom. Control, 2005, 50, (5), pp. 655661 (doi: 10.1109/TAC.2005.846556).
    18. 18)
      • 34. Cepeda-Gomez, R., Olgac, N.: ‘An exact method for the stability analysis of linear consensus protocols with time delay’, IEEE Trans. Autom. Control, 2011, 56, (7), pp. 17341740 (doi: 10.1109/TAC.2011.2152510).
    19. 19)
      • 40. Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: ‘LMI control toolbox user's guide’, (The Math Works, Natick, MA, 1995).
    20. 20)
      • 33. Sipahi, R., Qiao, W.: ‘Responsible eigenvalue concept for the stability of a class of single-delay consensus dynamics with fixed topology’, IET Control Theory Appl., 2011, 5, (4), pp. 622629 (doi: 10.1049/iet-cta.2010.0202).
    21. 21)
      • 36. Godsil, C., Royal, G.: ‘Algebraic graph theory’, (Springer-Verlag, New York, 2001).
    22. 22)
      • 5. Lafferriere, G., Williams, A., Caughman, J., Veerman, J.J.P.: ‘Decentralized control of vehicle formations’, Syst. Control Lett., 2005, 54, (9), pp. 889910 (doi: 10.1016/j.sysconle.2005.02.004).
    23. 23)
      • 14. Reynolds, C.W.: ‘Flocks, herds, and schools: a distributed behavioral model’, Comput. Graph., 1987, 21, (4), pp. 2534 (doi: 10.1145/37402.37406).
    24. 24)
      • 31. Lin, P., Jia, Y., Li, L.: ‘Distributed robust H consensus control in directed networks of agents with time-delay’, Syst. Control Lett., 2008, 57, (8), pp. 643653 (doi: 10.1016/j.sysconle.2008.01.002).
    25. 25)
      • 9. Lawton, J.R., Beard, R.W.: ‘Synchronized multiple spacecraft rotations’, Automatica, 2002, 38, (8), pp. 13591364 (doi: 10.1016/S0005-1098(02)00025-0).
    26. 26)
      • 11. Tian, Y.P., Yang, H.Y.: ‘Stability of the internet congestion control with diverse delays’, Automatica, 2004, 40, (9), pp. 15331541 (doi: 10.1016/j.automatica.2004.03.015).
    27. 27)
      • 16. Jadbabaie, A., Lin, J., Morse, A.S.: ‘Coordination of groups of mobile autonomous agents using nearest neighbor rules’, IEEE Trans. Autom. Control, 2003, 48, (6), pp. 9881001 (doi: 10.1109/TAC.2003.812781).
    28. 28)
      • 29. Sun, Y.G., Wang, L., Xie, G.: ‘Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays’, Syst. Control Lett., 2008, 57, (2), pp. 175183 (doi: 10.1016/j.sysconle.2007.08.009).
    29. 29)
      • 7. Olfati-Saber, R., Murray, R.M.: ‘Distributed cooperative control of multiple vehicle formations using structural potential functions’, Proc. 15th IFAC World Congress, 2002, http://citeseerx.ist.psu.edu/messages/downloadsexceeded.html.
    30. 30)
      • 38. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: ‘Linear matrix inequalities in system and control theory’, (SIAM, Philadelphia, PA, 1994).
    31. 31)
      • 24. Cai, N., Zhong, Y.: ‘Formation controllability of high-order linear time-invariant swarm systems’, IET Control Theory Appl., 2010, 4, (4), pp. 646654 (doi: 10.1049/iet-cta.2008.0202).
    32. 32)
      • 32. Mo, L., Jia, Y.: ‘H consensus control of a class of high-order multi- agent systems’, IET Control Theory Appl., 2011, 5, (1), pp. 247253 (doi: 10.1049/iet-cta.2009.0365).
    33. 33)
      • 17. Moreau, L.: ‘Stability of multiagent systems with time-dependent communication links’, IEEE Trans. Autom. Control, 2005, 50, (2), pp. 169182 (doi: 10.1109/TAC.2004.841888).
    34. 34)
      • 4. Fax, J.A., Murray, R.M.: ‘Information flow and cooperative control of vehicle formations’, IEEE Trans. Autom. Control, 2004, 49, (9), pp. 14651476 (doi: 10.1109/TAC.2004.834433).
    35. 35)
      • 21. Liu, B., Chen, T.: ‘Consensus in networks of multiagents with cooperation and competition via stochastically switching topologies’, IEEE Trans. Neural Netw., 2008, 19, (11), pp. 19671973 (doi: 10.1109/TNN.2008.2004404).
    36. 36)
      • 27. Bliman, P.A., Ferrari-Trecate, G.: ‘Average consensus problems in networks of agents with delayed communications’, Automatica, 2008, 44, (8), pp. 19851995 (doi: 10.1016/j.automatica.2007.12.010).
    37. 37)
      • 39. Xie, L.: ‘Output feedback H control of systems with parameter uncertainty’, Int. J. Control, 1996, 63, (4), pp. 741750 (doi: 10.1080/00207179608921866).
    38. 38)
      • 26. Hu, J., Lin, Y.S.: ‘Consensus control for multi-agent systems with double-integrator dynamics and time delays’, IET Control Theory Appl., 2010, 4, (1), pp. 109118 (doi: 10.1049/iet-cta.2008.0479).
    39. 39)
      • 13. Lynch, N.A.: ‘Distributed algorithms’ (Morgan Kaufmann, San Mateo, CA, 1997).
    40. 40)
      • 18. Olfati-Saber, R., Murray, R.M.: ‘Consensus problems in networks of agents with switching topology and time-delays’, IEEE Trans. Autom. Control, 2004, 49, (9), pp. 15201533 (doi: 10.1109/TAC.2004.834113).
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