## Abstract and Applied Analysis

### ${H}_{\infty }$ Consensus for Multiagent Systems with Heterogeneous Time-Varying Delays

#### Abstract

We apply the linear matrix inequality method to consensus and ${H}_{\infty }$ consensus problems of the single integrator multiagent system with heterogeneous delays in directed networks. To overcome the difficulty caused by heterogeneous time-varying delays, we rewrite the multiagent system into a partially reduced-order system and an integral system. As a result, a particular Lyapunov function is constructed to derive sufficient conditions for consensus of multiagent systems with fixed (switched) topologies. We also apply this method to the ${H}_{\infty }$ consensus of multiagent systems with disturbances and heterogeneous delays. Numerical examples are given to illustrate the theoretical results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 576290, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449968

Digital Object Identifier
doi:10.1155/2013/576290

Mathematical Reviews number (MathSciNet)
MR3129365

Zentralblatt MATH identifier
1291.93006

#### Citation

Wang, Beibei; Sun, Yuangong. ${H}_{\infty }$ Consensus for Multiagent Systems with Heterogeneous Time-Varying Delays. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 576290, 8 pages. doi:10.1155/2013/576290. https://projecteuclid.org/euclid.aaa/1393449968

#### References

• W. Ren and R. W. Beard, Distributed Consensus in Multi-Vehicle Cooperative Control, Springer, London, UK, 2008.
• 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.
• W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Systems Magazine, vol. 27, no. 2, pp. 71–82, 2007.
• V. Borkar and P. P. Varaiya, “Asymptotic agreement in distributed estimation,” IEEE Transactions on Automatic Control, vol. 27, no. 3, pp. 650–655, 1982.
• J. N. Tsitsiklis and M. Athans, “Convergence and asymptotic agreement in distributed decision problems,” IEEE Transactions on Automatic Control, vol. 29, no. 1, pp. 42–50, 1984.
• T. Vicsek, A. Czirk, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles,” Physical Review Letters, vol. 75, no. 6, pp. 1226–1229, 1995.
• 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.
• 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.
• 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.
• Q. Hui and W. M. Haddad, “Distributed nonlinear control algorithms for network consensus,” Automatica, vol. 44, no. 9, pp. 2375–2381, 2008.
• X. Liu, T. Chen, and W. Lu, “Consensus problem in directed networks of multi-agents via nonlinear protocols,” Physics Letters A, vol. 373, no. 35, pp. 3122–3127, 2009.
• D. Bauso, L. Giarré, and R. Pesenti, “Non-linear protocols for optimal distributed consensus in networks of dynamic agents,” Systems & Control Letters, vol. 55, no. 11, pp. 918–928, 2006.
• C. Tan and G.-P. Liu, “Consensus of networked multi-agent systems via the networked predictive control and relative outputs,” Journal of the Franklin Institute, vol. 349, no. 7, pp. 2343–2356, 2012.
• T. Li and J.-F. Zhang, “Mean square average-consensus under measurement noises and fixed topologies: necessary and sufficient conditions,” Automatica, vol. 45, no. 8, pp. 1929–1936, 2009.
• Y. Zheng and L. Wang, “Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements,” Systems & Control Letters, vol. 61, no. 8, pp. 871–878, 2012.
• P. Lin and W. Ren, “Constrained consensus in unbalanced networks with communication delays,” IEEE Transactions on Automatic Control, 2013.
• P. Lin and W. Ren, “Distributed constrained consensus in the presence of unbalanced switching graphs and communication delays,” IEEE Conference on Decision and Control, vol. 51, pp. 2238–2243, 2012.
• K. Liu, G. Xie, W. Ren, and L. Wang, “Consensus for multi-agent systems with inherent nonlinear dynamics under directed topologies,” Systems & Control Letters, vol. 62, no. 2, pp. 152–162, 2013.
• P. Lin and W. Ren, “Distributed shortest distance consensus problem in multi-agent systems,” IEEE Conference on Decision and Control, vol. 51, pp. 4696–4701, 2012.
• Z. Li, W. Ren, X. Liu, and L. Xie, “Distributed consensus of linear multi-agent systems with adaptive dynamic protocols,” Automatica, vol. 49, no. 7, pp. 1986–1995, 2013.
• 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.
• Y. G. Sun and L. Wang, “Consensus of multi-agent systems in directed networks with uniform time-varying delays,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1607–1613, 2009.
• Y. Sun, “Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays,” Journal of the Franklin Institute, vol. 349, no. 3, pp. 1061–1073, 2012.
• 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.
• P. Lin and Y. Jia, “Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies,” IEEE Transactions on Automatic Control, vol. 55, no. 3, pp. 778–784, 2010.
• P. Lin and Y. Jia, “Multi-agent consensus with diverse time-delays and jointly-connected topologies,” Automatica, vol. 47, no. 4, pp. 848–856, 2011.
• P. Lin, K. Qin, H. Zhao, and M. Sun, “A new approach to average consensus problems with multiple time-delays and jointly-connected topologies,” Journal of the Franklin Institute, vol. 349, no. 1, pp. 293–304, 2012.
• Y. Gao, J. Ma, M. Zuo, T. Jiang, and J. Du, “Consensus of discrete-time second-order agents with time-varying topology and time-varying delays,” Journal of the Franklin Institute, vol. 349, no. 8, pp. 2598–2608, 2012.
• L. Moreau, “Stability of continuous-time distributed consensus algorithms,” in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC '04), pp. 3998–4003, December 2004.
• F. Xiao and L. Wang, “Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1804–1816, 2008.
• A. Papachristodoulou, A. Jadbabaie, and U. Münz, “Effects of delay in multi-agent consensus and oscillator synchronization,” IEEE Transactions on Automatic Control, vol. 55, no. 6, pp. 1471–1477, 2010.
• U. Münz, A. Papachristodoulou, and F. Allgöwer, “Delay robustness in consensus problems,” Automatica, vol. 46, no. 8, pp. 1252–1265, 2010.
• 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.
• P. Lin, Y. Jia, and L. Li, “Distributed robust ${H}_{\infty }$ consensus control in directed networks of agents with time-delay,” Systems & Control Letters, vol. 57, no. 8, pp. 643–653, 2008.
• V. Ugrinovskii, “Distributed robust filtering with ${H}_{\infty }$ consensus of estimates,” Automatica, vol. 47, no. 1, pp. 1–13, 2011.
• Y. Sun and L. Wang, “${H}_{\infty }$ consensus of second-order multi-agent systems with asymmetric delays,” Systems & Control Letters, vol. 61, no. 8, pp. 857–862, 2012.