Self-tuning consensus on directed graph in the case of time-varying nonhomogeneous input gains

Document Type : English


Department of Mathematics, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran


In this paper, the problem of self-tuning of coupling parameters in multi-agent systems is considered. Agent dynamics are described by a discrete-time double integrator with time-varying nonhomogeneous input gain. The coupling parameters defining the strength of agents interactions are locally self-tuning by each node based on the velocities of its neighbors. The cost function is equal to the square of the local error between the agent velocity and the weighted average of the velocities of interacting neighbors. So, the proposed algorithm is the normalized gradient algorithm which is minimized the square of the local error between the agent velocity and the one step delayed average of the velocities of its neighbors. Provided that the underlying graph is strongly connected, it is shown that the sequence of the inter-agent coupling parameters generated by the proposed algorithm is convergent. Also, assuming the suitable initial condition on coupling parameters, it is proved that the network achieves average consensus. In other words, the agent velocities converge toward the average of the initial velocities values. Furthermore, the distance among agents converges to a finite limit. Simulation results illustrate effectiveness of the proposed method.


[1] Lewis, M. A., & Tan, K. H. (1997). High precision formation control of mobile robots using virtual structures, Autonomous Robots, 4(4), 387-403.
[2] Zhu, J., Lu, J., & Yu, X. (2012). Flocking of multi-agent non-holonomic systems with proximity graphs. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(1),199-210.
[3] Das, A. K., Fierro, R., Kumar, V., Ostrowski, J. P., Spletzer, J., & Taylor, C. J. (2002). A vision-based formation control framework. IEEE Transactions on Robotics and Automation, 18(5), 813-825.
[4] Yamchi, M. H., & Esfanjani, R. M. (2017). Formation control of networked mobile robots with guaranteed obstacle and collision avoidance. Robotica, 35(6), 1365-1377.
[5] Xiwang, D. (2016). Formation Control of Swarm Systems. Formation and Containment Control for High-order Linear Swarm Systems. Springer, Berlin, Heidelberg, 53-103.
[6] Qiao, W., & Sipahi, R. (2016). Consensus control under communication delay in a three robot system: design and experiments. IEEE transactions on control systems technology, 24, 687-694.
[7] Zong, X., Zhang, J. F., & Yin, G. (2021). Stochastic Consensus Control of Multi-agent Systems under General Noises and Delays. In Developments in Advanced Control and Intelligent Automation for Complex Systems, Springer, Cham, 225-254.
[8] Hu, G. (2012). Robust consensus tracking of a class of second-order multi-agent dynamic systems. Systems & Control Letters, 61(1), 134-142.
[9]   Vicsek, T., Czirok, A., Jacob, E. B., Cohen, I., & Schochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75, 1226-1229.
[10]   Jadbabaie, A., Lin, J., & Morse, S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, l48, 988-1001.
[11]   Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520-1533.
[12] Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE transactions on automatic control, 50(5), 655-661.
[13] Li, Y., & Tan, C. (2019). A survey of the consensus for multi-agent systems. Systems Science & Control Engineering, 7(1), 468-482.
[14]   Li, Z., Ren, W., Liu, X., & Fu, M. (2013). Consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics using distributed adaptive protocols. IEEE Transactions on Automatic Control, 58(7), 1786-1791.
[15]   Li, Z., Ren, W., Liu, X., & Xie, L. (2013). Distributed consensus of linear multi-agent systems with adaptive dynamic protocols. Automatica, 49, 1986-1995.
[16]   Mei, J., Ren, W., & Chen, J. (2014). Consensus of second order heterogeneous multi-agent system under a directed graph. Proceeding of the 2014 American Control Conference, Portland, Oregon, 802-807.
[17]   Chen, W., Li, X., Ren, W., & Wen, C. (2014). Adaptive consensus of multi-agent systems with unknown identical control directions based on a novel Nussbaum-type function. IEEE Transactions on Automatic Control, 59 (7), 1887-1892.
[18]   Liu, Y., Min, H., Liu, Z. & Liao, S. (2014). Distributed adaptive consensus for multiple mechanical systems with switching topologies and time-varying delay. Systems & Control Letters, 64, 119-126.
[19]  Radenkovic, M. S., & Tadi, M. (2015). Self-tuning average consensus in complex networks. Journal of the Franklin Institute, 352, 1152-1168.
[20]  Radenkovic, M. S., & Golkowski, M. (2015). Distributed self-tuning consensus and synchronization. System & Control Letters, 76, 66-73.
[21] Radenkovic M. S., & Krstic, M. (2018). Distributed adaptive consensus and synchronization in complex networks of dynamical systems. Automatica, 91, 233-243.
[22] Shi, Y., Yin, Y., Liu, C., Liu, C., Liu, F. (2018).  Consensus for heterogeneous multi-agent systems with second-order linear and nonlinear dynamics. 2018 Chinese Control And Decision Conference (CCDC). IEEE, 6068-6071.
[23] Zhao, J., Dai, F., & Song, Y. (2022). Consensus of Heterogeneous Mixed-Order Multi-agent Systems Including UGV and UAV. Proceedings of 2021 Chinese Intelligent Systems Conference, Springer, Singapore, 202-210.
[24] Cardei, M., & Wu, J. (2004). Coverage in wireless sensor networks. Handbook of Sensor Networks, M. Ilyas, Ed. CRC Press, West Palm Beach, FL.
[25] Horn, R., & Johnson, C. (1985). Matrix analysis. Cambridge University Press, Cambridge, UK.
[26] Goodwin, G., & Sin, K. S. (1985). Adaptive filtering, prediction and control. Prentice Hall, New Jersey.
[27]   Mareels I., & Polderman, J. W. (1996). Adaptive systems: An introduction. Birkh user.