##
**A new approach to average consensus problems with multiple time-delays and jointly-connected topologies.**
*(English)*
Zbl 1254.93009

Summary: This paper investigates average consensus problem in networks of continuous-time agents with delayed information and jointly-connected topologies. A lemma is derived by extending Barbalat’s Lemma to piecewise continuous functions, which provides a new analysis approach for switched systems. Then based on this lemma, a sufficient condition in terms of linear matrix inequalities is given for average consensus of the system by employing a Lyapunov approach, where the communication structures vary over time and the corresponding graphs may not be connected. Finally, simulation results are provided to demonstrate the effectiveness of our theoretical results.

### MSC:

93A14 | Decentralized systems |

93C15 | Control/observation systems governed by ordinary differential equations |

94C15 | Applications of graph theory to circuits and networks |

68T42 | Agent technology and artificial intelligence |

### Keywords:

approach for switched systems; networks of continuous-time agents; problems with multiple time-delays; Barbalat’s Lemma; linear matrix inequalities; piecewise continuous functions; average consensus problem; jointly-connected topologies; communication structures; simulation results; Lyapunov approach; corresponding graphs
PDF
BibTeX
XML
Cite

\textit{P. Lin} et al., J. Franklin Inst. 349, No. 1, 293--304 (2012; Zbl 1254.93009)

Full Text:
DOI

### References:

[1] | Benediktsson, J.A.; Swain, P.H., Consensus theoretic classification methods, IEEE transactions on systems, man, and cybernetics, 22, 4, 688-704, (1992) · Zbl 0775.62149 |

[2] | Weller, S.C.; Mann, N.C., Assessing rater performance without a gold standard using consensus theory, Medical decision making, 17, 1, 71-79, (1997) |

[3] | Borkar, V.; Varaiya, P., Asymptotic agreement in distributed estimation, IEEE transactions on automatic control, AC-27, 3, 650-655, (1982) · Zbl 0497.93037 |

[4] | J.N. Tsitsiklis, Problems in Decentralized Decision Making and Computation, Ph.D. Dissertation, Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 1984. |

[5] | Tsitsiklis, J.N.; Bertsekas, D.P.; Athans, M., Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE transactions on automatic control, 31, 9, 803-812, (1986) · Zbl 0602.90120 |

[6] | Olfati-Saber, R.; Murray, R.M., Consensus problems in networks of agents with switching topology and time-delays, IEEE transactions on automatic control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301 |

[7] | Ren, W.; Beard, R.W., Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE transactions on automatic control, 50, 5, 655-661, (2005) · Zbl 1365.93302 |

[8] | Bliman, P.; Ferrari-Trecate, G., Average consensus problems in networks of agents with delayed communications, Automatica, 44, 8, 1985-1995, (2008) · Zbl 1283.93013 |

[9] | Hong, Y.; Gao, L.; Cheng, D.; Hu, J., Lyapunov-based approach to multiagent systems with switching jointly connected interconnection, IEEE transactions on automatic control, 52, 5, 943-948, (2007) · Zbl 1366.93437 |

[10] | Cheng, D.; Wang, J.; Hu, X., An extension of Lasalle’s invariance principle and its application to multi-agent consensus, IEEE transactions on automatic control, 53, 7, 1765-1770, (2008) · Zbl 1367.93427 |

[11] | Hu, J.; Hong, Y., Leader-following coordination of multi-agent systems with coupling time delays, Physica A, 374, 2, 853-863, (2007) |

[12] | Xie, G.; Wang, L., Consensus control for a class of networks of dynamic agents, International journal of robust and nonlinear control, 17, 10-11, 941-959, (2007) · Zbl 1266.93013 |

[13] | Shi, H.; Wang, L.; Chu, T., Virtual leader approach to coordinated control of multiple mobile agents with asymmetric interactions, Physica D, 213, 1, 51-65, (2006) · Zbl 1131.93354 |

[14] | Hu, J.; Yuan, H., Collective coordination of multi-agent systems guided by multiple leaders, Chinese physics, 18, 9, 3777-3782, (2009) |

[15] | Guan, X.; Li, Y., Nonlinear consensus protocols for multi-agent systems based on centre manifold reduction, Chinese physics, 18, 8, 3355-3366, (2009) |

[16] | Pei, W.; Chen, Z.; Yuan, Z., A dynamic epidemic control model on uncorrelated complex networks, Chinese physics, 17, 2, 373-379, (2008) |

[17] | Lin, P.; Jia, Y., Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica, 45, 9, 2154-2158, (2009) · Zbl 1175.93078 |

[18] | Lin, P.; Jia, Y., Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies, IEEE transactions on automatic control, 55, 3, 778-784, (2010) · Zbl 1368.93275 |

[19] | Lin, P.; Jia, Y.; Du, J.; Yu, F., Average consensus for networks of continuous-time agents with delayed information and jointly-connected topologies, (), 3884-3889 |

[20] | Chen, C.; Lee, C., Delay-independent stabilization of linear systems with time-varying delayed state and uncertainties, Journal of the franklin institute, 346, 4, 378-390, (2009) · Zbl 1166.93369 |

[21] | Liu, P., Robust exponential stability for uncertain time-varying delay systems with delay dependence, Journal of the franklin institute, 346, 10, 958-968, (2009) · Zbl 1192.34086 |

[22] | Chen, W.; Guan, Z.; Lu, X., Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays, Journal of the franklin institute, 341, 5, 419-430, (2004) · Zbl 1055.93054 |

[23] | Hien, L.V.; Vu, N., Exponential stability and stabilization of a class of uncertain linear time-delay systems, Journal of the franklin institute, 346, 6, 611-625, (2009) · Zbl 1169.93396 |

[24] | Song, Q.; Cao, J., Global exponential robust stability of cohen – grossberg neural network with time-varying delays and reaction-diffusion terms, Journal of the franklin institute, 343, 7, 705-719, (2006) · Zbl 1135.93026 |

[25] | Godsil, C.; Royle, G., Algebraic graph theory, (2001), Springer-Verlag New York · Zbl 0968.05002 |

[26] | Hale, J., Theory of functional differential equations, (1977), Springer-Verlag New York |

[27] | Liberzon, D., Theory of functional differential equations, (2003), Birkhäuser Boston |

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.