##
**Necessary and sufficient condition for the group consensus of multi-agent systems.**
*(English)*
Zbl 1335.91029

Summary: This paper focuses on the group consensus issue of multi-agent systems, where the agents in a network can reach more than one consistent values asymptotically. A rotation matrix is introduced to an existing consensus algorithm for single-integrator dynamics. Based on algebraic matrix theories, graph theories and the properties of Kronecker product, some necessary and sufficient criteria for the group consensus are derived, where we show that both the eigenvalue distribution of the Laplacian matrix and the Euler angle of the rotation matrix play an important role in achieving group consensus. Simulated results are presented to demonstrate the theoretical results.

### Keywords:

multi-agent system (MAS); group consensus; normal consensus; fixed topology; directed graph
PDF
BibTeX
XML
Cite

\textit{D. Xie} et al., Appl. Math. Comput. 243, 870--878 (2014; Zbl 1335.91029)

Full Text:
DOI

### References:

[1] | R.W. Beard, V. Stepanyan, Synchronization of information in distributed multiple vehicle coordination control, in: IEEE Conf. Decision and Control, Maui, vol. 12, 2003, pp. 2029-2034. |

[2] | T.G. Chu, L. Wang, T.W. Chen, Self-organized motion in a class of anisotropic swarms: convergence vs oscillation, in: Proc. Amer. Control Conf., Portland, vol. 6, 2005, pp. 1345-1351. |

[3] | Vicsek, T.; Czirok, A.; Jacob, E. B.; Cohen, I.; Schochet, O., Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75, 6, 3474-3479, (1995) |

[4] | Zhang, Y.; Tian, Y. P., Consentability and protocol design of multi-agent systems with stochastic switching topology, Automatica, 45, 5, 1195-1201, (2009) · Zbl 1162.94431 |

[5] | F. Xiao, L. Wang, Reaching agreement in finite time via continuous local state feedback, in: Proc. 26th Chinese Control Conf., Zhangjiajie, 2007, pp. 711-715. |

[6] | Cao, M.; Morse, A. S.; Anderson, B. D.O., Agreeing asynchronously, IEEE Trans. Automat. Control, 53, 8, 1826-1838, (2008) · Zbl 1367.93359 |

[7] | Yu, J. Y.; Wang, L., Group consensus in multi-agent systems with switching topologies and communication delays, Syst. Control Lett., 59, 6, 340-348, (2010) · Zbl 1197.93096 |

[8] | Xiao, F.; Wang, L., Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays, IEEE Trans. Autom. Control, 53, 8, 1804-1816, (2008) · Zbl 1367.93255 |

[9] | Hatano, Y.; Mesbahi, M., Agreement over random networks, IEEE Trans. Autom. Control, 50, 11, 1867-1872, (2005) · Zbl 1365.94482 |

[10] | Wu, C. W., Synchronization and convergence of linear dynamics in random directed networks, IEEE Trans. Autom. Control, 51, 7, 1207-1210, (2006) · Zbl 1366.93537 |

[11] | Porfiri, M.; Stiwell, D. J., Consensus seeking over random weighted directed graphs, IEEE Trans. Autom. Control, 52, 9, 1767-1773, (2007) · Zbl 1366.93330 |

[12] | Tahbaz-Salehi, A.; Jadbabaie, A., A necessary and sufficient condition for consensus over random networks, IEEE Trans. Autom. Control, 53, 3, 791-795, (2008) · Zbl 1367.90015 |

[13] | Chen, Y.; Lv, J. H.; Han, F. L.; Yu, X. H., On the cluster consensus of discrete-time multi-agent systems, Syst. Control Lett., 60, 517-523, (2011) · Zbl 1222.93007 |

[14] | J.Y. Yu, L. Wang, Group consensus in multi-agent systems with undirected communication graphs, in: Proc. Seventh Asian Control Conf., Hong Kong, 2009, pp. 105-110. |

[15] | C. Tian, G.P. Liu, G.R. Duan, Group consensus of networked multi-agent systems with directed topology, in: 18th IFAC World Congress, Milano, 2011, pp. 8878-8883. |

[16] | Olfati-Saber, R.; Murray, R. M., Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301 |

[17] | Ren, W.; Cao, Y. C., Distributed coordination of multi-agent networks: emergent problems, models and issues, (2011), Springer · Zbl 1225.93003 |

[18] | W. Ren, Collective motion from consensus with Cartesian coordinate coupling - Part I: Single-integrator kinematics, in: 47th IEEE Conf. Decision and Control, 2008, pp. 1006-1011. |

[19] | Ren, W., Collective motion from consensus with Cartesian coordinate coupling, IEEE Trans. Autom. Control, 54, 6, 1330-1335, (2009) · Zbl 1367.93408 |

[20] | Graham, A., Kronecker products and matrix calculus with applications, (1981), Hasted New York · Zbl 0497.26005 |

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.