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Mathematical Problems in Engineering
Volume 2015, Article ID 453072, 13 pages
http://dx.doi.org/10.1155/2015/453072
Research Article

Consensus of Noisy Multiagent Systems with Markovian Switching Topologies and Time-Varying Delays

Department of Mathematics, Tongji University, Shanghai 200092, China

Received 25 March 2015; Revised 26 May 2015; Accepted 21 July 2015

Academic Editor: Son Nguyen

Copyright © 2015 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. N. Tsitsiklis, Problems in decentralized decision making and computation [Ph.D. thesis], Massachusetts Institute of Technology, Cambridge, Mass, USA, 1984.
  2. S. Chatterjee and E. Seneta, “Towards consensus: some convergence theorems on repeated averaging,” Journal of Applied Probability, vol. 14, no. 1, pp. 89–97, 1977. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. 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, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 655–661, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. Y. Hong, G. Chen, and L. Bushnell, “Distributed observers design for leader-following control of multi-agent networks,” Automatica, vol. 44, no. 3, pp. 846–850, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. W. Ren, “On consensus algorithms for double-integrator dynamics,” IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1503–1509, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. Xie and L. Wang, “Consensus control for a class of networks of dynamic agents,” International Journal of Robust and Nonlinear Control, vol. 17, no. 10-11, pp. 941–959, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. K. You, Z. Li, and L. Xie, “Consensus condition for linear multi-agent systems over randomly switching topologies,” Automatica, vol. 49, no. 10, pp. 3125–3132, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. C.-Q. Ma and J.-F. Zhang, “Necessary and sufficient conditions for consensusability of linear multi-agent systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1263–1268, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. K. You and L. Xie, “Network topology and communication data rate for consensusability of discrete-time multi-agent systems,” IEEE Transactions on Automatic Control, vol. 56, no. 10, pp. 2262–2275, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z. Li, Z. Duan, and G. Chen, “Dynamic consensus of linear multi-agent systems,” IET Control Theory & Applications, vol. 5, no. 1, pp. 19–28, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Y. Shang, “Group pinning consensus under fixed and randomly switching topologies with acyclic partition,” Networks and Heterogeneous Media, vol. 9, no. 3, pp. 553–573, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. Y. Hatano and M. Mesbahi, “Agreement over random networks,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1867–1872, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. M. Porfiri and D. J. Stilwell, “Consensus seeking over random weighted directed graphs,” IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1767–1773, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Tahbaz-Salehi and A. Jadbabaie, “Consensus over ergodic stationary graph processes,” IEEE Transactions on Automatic Control, vol. 55, no. 1, pp. 225–230, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. T. Li and J.-F. Zhang, “Consensus conditions of multi-agent systems with time-varying topologies and stochastic communication noises,” IEEE Transactions on Automatic Control, vol. 55, no. 9, pp. 2043–2057, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. B. Touri and A. Nedić, “On ergodicity, infinite flow, and consensus in random models,” IEEE Transactions on Automatic Control, vol. 56, no. 7, pp. 1593–1605, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. G. Miao, S. Xu, and Y. Zou, “Necessary and sufficient conditions for mean square consensus under Markov switching topologies,” International Journal of Systems Science, vol. 44, no. 1, pp. 178–186, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. I. Matei, J. S. Baras, and C. Somarakis, “Convergence results for the linear consensus problem under Markovian random graphs,” SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 1574–1591, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  23. U. Münz, A. Papachristodoulou, and F. Allgöwer, “Delay robustness in consensus problems,” Automatica, vol. 46, no. 8, pp. 1252–1265, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. U. Münz, A. Papachristodoulou, and F. Allgöwer, “Delay robustness in non-identical multi-agent systems,” IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1597–1603, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. X. Wang, A. Saberi, A. A. Stoorvogel, H. F. Grip, and T. Yang, “Consensus in the network with uniform constant communication delay,” Automatica, vol. 49, no. 8, pp. 2461–2467, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. Liu, X. Liu, W.-C. Xie, and H. Zhang, “Stochastic consensus seeking with communication delays,” Automatica, vol. 47, no. 12, pp. 2689–2696, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. Y.-Z. Sun and J. Ruan, “Leader-follower consensus problems of multi-agent systems with noise perturbation and time delays,” Chinese Physics Letters, vol. 25, no. 9, pp. 3493–3495, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. Y. Shang, “Group consensus of multi-agent systems in directed networks with noises and time delays,” International Journal of Systems Science, vol. 46, no. 14, pp. 2481–2492, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  29. J. Lai, S. Chen, and X. Lu, “Tracking consensus of nonlinear MASs with asymmetric communication delays in noisy environments,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 7, pp. 2334–2344, 2014. View at Publisher · View at Google Scholar · View at Scopus
  30. Y. Shang, “Synchronization in networks of coupled harmonic oscillators with stochastic perturbation and time delays,” Annals of the Academy of Romanian Scientists—Series on Mathematics and its Applications, vol. 4, no. 1, pp. 44–58, 2012. View at Google Scholar · View at MathSciNet
  31. Y. Shang and R. Bouffanais, “Influence of the number of topologically interacting neighbors on swarm dynamics,” Scientific Reports, vol. 4, article 4184, 2014. View at Publisher · View at Google Scholar · View at Scopus
  32. M. Cao, A. S. Morse, and B. D. O. Anderson, “Reaching a consensus in a dynamically changing environment: convergence rates, measurement delays, and asynchronous events,” SIAM Journal on Control and Optimization, vol. 47, no. 2, pp. 601–623, 2008. View at Publisher · View at Google Scholar · View at Scopus
  33. N. Chopra, D. M. Stipanović, and M. W. Spong, “On synchronization and collision avoidance for mechanical systems,” in Proceedings of the American Control Conference (ACC '08), pp. 3713–3718, Seattle, Wash, USA, June 2008. View at Publisher · View at Google Scholar · View at Scopus
  34. A. Papachristodoulou and A. Jadbabaie, “Synchonization in oscillator networks with heterogeneous delays, switching topologies and nonlinear dynamics,” in Proceedings of the 45th IEEE Conference on Decision and Control, pp. 4307–4312, December 2006. View at Scopus
  35. L. Moreau, “Stability of multiagent systems with time-dependent communication links,” IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 169–182, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. Y. Shang, “Average consensus in multi-agent systems with uncertain topologies and multiple time-varying delays,” Linear Algebra and its Applications, vol. 459, pp. 411–429, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. Y. G. Sun, L. Wang, and G. Xie, “Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays,” Systems & Control Letters, vol. 57, no. 2, pp. 175–183, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, USA, 1980. View at MathSciNet
  39. O. L. V. Costa, M. D. Fragoso, and R. P. Marques, Discrete-Time Markov Jump Linear Systems, Springer, London, UK, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  40. X. Feng and K. A. Loparo, “Stability of linear Markovian jump systems,” in Proceedings of the 29th IEEE Conference on Decision and Control, pp. 1408–1413, Honolulu, Hawaii, USA, December 1990. View at Scopus
  41. E. Seneta, Non-Negative Matrices and Markov Chains, Springer, New York, NY, USA, 2006. View at MathSciNet
  42. X. Mao, “Robustness of stability of stochastic differential delay equations with Markovian switching,” Stability and Control: Theory and Applications, vol. 3, no. 1, pp. 48–61, 2000. View at Google Scholar · View at MathSciNet
  43. Z. Zeng, J. Wang, and X. Liao, “Global asymptotic stability and global exponential stability of neural networks with unbounded time-varying delays,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 3, pp. 168–173, 2005. View at Publisher · View at Google Scholar · View at Scopus
  44. Q. Zhu, J. Cao, and R. Rakkiyappan, “Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays,” Nonlinear Dynamics, vol. 79, no. 2, pp. 1085–1098, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. Q. Zhu and J. Cao, “Mean-square exponential input-to-state stability of stochastic delayed neural networks,” Neurocomputing, vol. 131, pp. 157–163, 2014. View at Publisher · View at Google Scholar · View at Scopus
  46. Q. Zhu and J. Cao, “Stability analysis of markovian jump stochastic BAM neural networks with impulse control and mixed time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 3, pp. 467–479, 2012. View at Publisher · View at Google Scholar · View at Scopus
  47. X. Mao, A. Matasov, and A. B. Piunovskiy, “Stochastic differential delay equations with Markovian switching,” Bernoulli, vol. 5, pp. 1–18, 1999. View at Google Scholar
  48. J. S. Caughman and J. J. P. Veerman, “Kernels of directed graph Laplacians,” Electronic Journal of Combinatorics, vol. 13, article R39, 2006. View at Google Scholar · View at MathSciNet
  49. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1994.
  50. H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),” Mathematische Annalen, vol. 71, no. 4, pp. 441–479, 1912. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  51. J. G. VanAntwerp and R. D. Braatz, “Tutorial on linear and bilinear matrix inequalities,” Journal of Process Control, vol. 10, no. 4, pp. 363–385, 2000. View at Publisher · View at Google Scholar · View at Scopus
  52. B. Bollobás, Random Graphs, Cambridge University Press, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  53. Y. Zheng and L. Wang, “Consensus of heterogeneous multi-agent systems without velocity measurements,” International Journal of Control, vol. 85, no. 7, pp. 906–914, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus