##
**Distributed containment control of networked fractional-order systems with delay-dependent communications.**
*(English)*
Zbl 1251.93096

Summary: We are concerned with a containment problem of networked fractional-order system with multiple leaders under a fixed directed interaction graph. Based on the neighbor rule, a distributed protocol is proposed in delayed communication channels. By employing the algebraic graph theory, matrix theory, Nyquist stability theorem, and frequency domain method, it is analytically proved that the whole follower agents will flock to the convex hull which is formed by the leaders. Furthermore, a tight upper bound on the communication time-delay that can be tolerated in the dynamic network is obtained. As a special case, the interconnection topology under the undirected case is also discussed. Finally, some numerical examples with simulations are presented to demonstrate the effectiveness and correctness of the theoretical results.

### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

34A08 | Fractional ordinary differential equations |

PDF
BibTeX
XML
Cite

\textit{X. Liu} et al., J. Appl. Math. 2012, Article ID 840873, 13 p. (2012; Zbl 1251.93096)

Full Text:
DOI

### References:

[1] | F. Xiao, L. Wang, J. Chen, and Y. Gao, “Finite-time formation control for multi-agent systems,” Automatica, vol. 45, no. 11, pp. 2605-2611, 2009. · Zbl 1180.93006 |

[2] | R. Vidal, O. Shakernia, and S. Sastry, “Formation control of nonholonomic mobile robots with omnidirectional visual servoing and motion segmentation,” in Proceedings of the IEEE Conference on Robotics and Automation (ICRA ’03), pp. 584-589, Taipei, Taiwan, September 2003. |

[3] | J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465-1476, 2004. · Zbl 1365.90056 |

[4] | F. Cucker and S. Smale, “Emergent behavior in flocks,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 852-862, 2007. · Zbl 1366.91116 |

[5] | R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory,” IEEE Transactions on Automatic Control, vol. 51, no. 3, pp. 401-420, 2006. · Zbl 1366.93391 |

[6] | L. Xiao, S. Boyd, and S. Lall, “A scheme for robust distributed sensor fusion based on average consensus,” in Proceedings of the 4th International Conference on Information Process in Sensor Networks, pp. 63-70, Los Angeles, Calif, USA, April 2005. |

[7] | R. Saber and J. Shamma, “Consensus filters for sensor networks and distributed sensor fusion,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC ’05), pp. 6698-6703, Seville, Spain, 2005. |

[8] | T. Li and J. Zhang, “Mean square average-consensus under measurement noise and fixed topologies: necessary and sufficient conditions,” Automatica, vol. 45, no. 8, pp. 1929-1936, 2009. · Zbl 1185.93006 |

[9] | J. Hu and Y. Hong, “Leader-following coordination of multi-agent systems with coupling time delays,” Physica A, vol. 374, no. 2, pp. 853-863, 2007. |

[10] | W. Ren, R. W. Beard, and D. Kingston, “Multi-agent kalman consensus with relative uncertainty,” in Proceedings of the American Control Conference, pp. 1865-1870, Portland, Ore, USA, June 2005. |

[11] | 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. · Zbl 1365.93301 |

[12] | W. Yu, G. Chen, and J. Cao, “Adaptive synchronization of uncertain coupled stochastic complex networks,” Asian Journal of Control, vol. 13, no. 3, pp. 418-429, 2011. · Zbl 1221.93268 |

[13] | M. Barahona and L. Pecora, “Synchronization in small-world systems,” Physical Review Letters, vol. 89, no. 5, Article ID 054101-4, 2002. |

[14] | N. A. Lynch, Distributed Algorithms, The Morgan Kaufmann Series in Data Management Systems, Morgan Kaufmann, San Francisco, Calif, USA, 1996. · Zbl 0877.68061 |

[15] | W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control, Springer, London, UK, 2008. · Zbl 1144.93002 |

[16] | M. Cao, A. S. Morse, and B. D. O. Anderson, “Agreeing asynchronously,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1826-1838, 2008. · Zbl 1367.93359 |

[17] | I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008 |

[18] | K. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011 |

[19] | Y. Cao, Y. Li, W. Ren, and Y. Chen, “Distributed coordination algorithms for multiple fractional-order systems,” in Proceedings of the IEEE Conference on Decision and Control (CDC ’08), pp. 2920-2925, Cancun, Mexico, December 2008. |

[20] | Y. Cao, Y. Li, W. Ren, and Y. Chen, “Distributed coordination of networked fractional-order systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 2, pp. 362-370, 2010. |

[21] | W. Sun, Y. Li, C. Li, and Y. Chen, “Convergence speed of a fractional order consensus algorithm over undirected scale-free networks,” Asian Journal of Control, vol. 13, no. 6, pp. 936-946, 2011. · Zbl 1263.93013 |

[22] | J. Shen, J. D. Cao, and J. Lu, “Consenus of fractional-order systems with non-uniform input and communication delays,” Journal of Systems and Control Engineering, vol. 226, no. 2, pp. 271-283, 2012. |

[23] | J. Shen and J. Cao, “Necessary and sufficient conditions for consenus of delayed fractional-order systems over directed graph,” 2011, http://www.paper.edu.cn/. |

[24] | Y. Cao and W. Ren, “Distributed formation control for fractional-order systems: dynamic interaction and absolute/relative damping,” Systems & Control Letters, vol. 59, no. 3-4, pp. 233-240, 2010. · Zbl 1222.93006 |

[25] | D. Matignon, “Stability result on fractional differential equations with applications to control processing,” in Proceedings of the IMACSSMC, pp. 963-968, Lille, France, 1996. |

[26] | Y. Cao and W. Ren, Distributed Coordination of Multi-agent Networks Emergent Problems, Models, and Issues, Springer, London, UK, 2011. · Zbl 1225.93003 |

[27] | W. Deng, C. Li, and J. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409-416, 2007. · Zbl 1185.34115 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.