# American Institute of Mathematical Sciences

December  2020, 25(12): 4535-4551. doi: 10.3934/dcdsb.2020111

## Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain

 1 School of Management, Tianjin University of Technology, Tianjin 300384, China 2 Department of Basic Education, Tianjin City Vocational College, Tianjin 300250, China

* Corresponding author: Xi Zhu

Received  April 2019 Revised  October 2019 Published  December 2020 Early access  March 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 61440058, 11501412 and 11401073

In this paper, we study consensus problem in a discrete-time multi-agent system with uncertain topologies and random delays governed by a Markov chain. The communication topology is assumed to be directed but interrupted by system uncertainties. Furthermore, the system delays are modeled by a Markov chain. We first use a reduced-order system featuring the error dynamics to transform the consensus problem of the original one into the stabilization of the error dynamic system. By using the linear matrix inequality method and the stability theory in stochastic systems with time-delay, several sufficient conditions are established for the mean square stability of the error dynamics which guarantees consensus. By redesigning its adjacency matrices, we develop a switching control scheme which is delay-dependent. Finally, simulation results are worked out to illustrate the theoretical results.

Citation: Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111
##### References:
 [1] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777. [2] W. He, G. Chen, Q.-L. Han and F. Qian, Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control, Information Sciences, 380 (2017), 145-158.  doi: 10.1016/j.ins.2015.06.005. [3] Y. M. Chen and W.-Y. Wu, Cooperative electronic attack for groups of unmanned air vehicles based on multi-agent simulation and evaluation, International Journal of Computer Science Issues, 9 (2012), 6 pp. [4] O. L. V. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov jump linear systems, Springer-Verlag London, Ltd., London, 2005. doi: 10.1007/b138575. [5] P. Lin, W. Lu and Y. Song, Distributed nested rotating consensus problem of multi-agent systems, The 26th Chinese Control and Decision Conference, (2014 CCDC), (2014), 1785–1789. [6] S. Liu, L. Xie and F. L. Lewis, Synchronization of multi-agent systems with delayed control input information from neighbors, Automatica J. IFAC, 47 (2011), 2152-2164.  doi: 10.1016/j.automatica.2011.03.015. [7] J. Monteil, R. Billot, J. Sau, F. Armetta, S. Hassas and N.-E. E. Faouzi, Cooperative highway traffic: Multi-agent modeling and robustness assessment of local perturbations, Transportation Research Record, 2391 (2013), 1-10. [8] R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113. [9] W. Ren, R. W. Beard and E. M. Atkins, A survey of consensus problems in multi-agent coordination, Proceedings of the 2005, American Control Conference, 2005, 3 (2005), 1859-1864. [10] W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Control, 50 (2005), 655-661.  doi: 10.1109/TAC.2005.846556. [11] H. J. Savino, C. R. P. dos Santos, F. O. Souza, L. C. A. Pimenta, M. de Oliveira and R. M. Palhares, Conditions for consensus of multi-agent systems with time-delays and uncertain switching topology, IEEE Transactions on Industrial Electronics, 63 (2016), 1258-1267. [12] Y. Shang, Average consensus in multi-agent systems with uncertain topologies and multiple time-varying delays, Linear Algebra Appl., 459 (2014), 411-429.  doi: 10.1016/j.laa.2014.07.019. [13] A. Soriano, E. J. Bernabeu, A. Valera and M. Vallés, Multi-agent systems platform for mobile robots collision avoidance, in International Conference on Practical Applications of Agents and Multi-Agent Systems (eds. Y. Demazeau, T. Ishida, J. M. Corchado and J. Bajo), Springer, Berlin, Heidelberg, 7879 (2013), 320–323. [14] Y. Sun, Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays, J. Franklin Inst., 349 (2012), 1061-1073.  doi: 10.1016/j.jfranklin.2011.12.007. [15] Y. G. Sun and L. Wang, Consensus problems in networks of agents with double-integrator dynamics and time-varying delays, Internat. J. Control, 82 (2009), 1937-1945.  doi: 10.1080/00207170902838269. [16] T. Y. Teck and M. Chitre, Hierarchical multi-agent command and control system for autonomous underwater vehicles, 2012 IEEE/OES Autonomous Underwater Vehicles (AUV), (2012), 1–6. [17] G. Wen, Z. Duan, W. Yu and G. Chen, Consensus of second-order multi-agent systems with delayed nonlinear dynamics and intermittent communications, Internat. J. Control, 86 (2013), 322-331.  doi: 10.1080/00207179.2012.727473. [18] J. Wu and Y. Shi, Consensus in multi-agent systems with random delays governed by a Markov chain, Systems Control Lett., 60 (2011), 863-870.  doi: 10.1016/j.sysconle.2011.07.004. [19] G. Yang, Q. Yang, V. Kapila, D. Palmer and R. Vaidyanathan, Fuel optimal manoeuvres for multiple spacecraft formation reconfiguration using multi-agent optimization, International Journal of Robust and Nonlinear Control, 12 (2002), 243-283.

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##### References:
 [1] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777. [2] W. He, G. Chen, Q.-L. Han and F. Qian, Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control, Information Sciences, 380 (2017), 145-158.  doi: 10.1016/j.ins.2015.06.005. [3] Y. M. Chen and W.-Y. Wu, Cooperative electronic attack for groups of unmanned air vehicles based on multi-agent simulation and evaluation, International Journal of Computer Science Issues, 9 (2012), 6 pp. [4] O. L. V. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markov jump linear systems, Springer-Verlag London, Ltd., London, 2005. doi: 10.1007/b138575. [5] P. Lin, W. Lu and Y. Song, Distributed nested rotating consensus problem of multi-agent systems, The 26th Chinese Control and Decision Conference, (2014 CCDC), (2014), 1785–1789. [6] S. Liu, L. Xie and F. L. Lewis, Synchronization of multi-agent systems with delayed control input information from neighbors, Automatica J. IFAC, 47 (2011), 2152-2164.  doi: 10.1016/j.automatica.2011.03.015. [7] J. Monteil, R. Billot, J. Sau, F. Armetta, S. Hassas and N.-E. E. Faouzi, Cooperative highway traffic: Multi-agent modeling and robustness assessment of local perturbations, Transportation Research Record, 2391 (2013), 1-10. [8] R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113. [9] W. Ren, R. W. Beard and E. M. Atkins, A survey of consensus problems in multi-agent coordination, Proceedings of the 2005, American Control Conference, 2005, 3 (2005), 1859-1864. [10] W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Control, 50 (2005), 655-661.  doi: 10.1109/TAC.2005.846556. [11] H. J. Savino, C. R. P. dos Santos, F. O. Souza, L. C. A. Pimenta, M. de Oliveira and R. M. Palhares, Conditions for consensus of multi-agent systems with time-delays and uncertain switching topology, IEEE Transactions on Industrial Electronics, 63 (2016), 1258-1267. [12] Y. Shang, Average consensus in multi-agent systems with uncertain topologies and multiple time-varying delays, Linear Algebra Appl., 459 (2014), 411-429.  doi: 10.1016/j.laa.2014.07.019. [13] A. Soriano, E. J. Bernabeu, A. Valera and M. Vallés, Multi-agent systems platform for mobile robots collision avoidance, in International Conference on Practical Applications of Agents and Multi-Agent Systems (eds. Y. Demazeau, T. Ishida, J. M. Corchado and J. Bajo), Springer, Berlin, Heidelberg, 7879 (2013), 320–323. [14] Y. Sun, Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays, J. Franklin Inst., 349 (2012), 1061-1073.  doi: 10.1016/j.jfranklin.2011.12.007. [15] Y. G. Sun and L. Wang, Consensus problems in networks of agents with double-integrator dynamics and time-varying delays, Internat. J. Control, 82 (2009), 1937-1945.  doi: 10.1080/00207170902838269. [16] T. Y. Teck and M. Chitre, Hierarchical multi-agent command and control system for autonomous underwater vehicles, 2012 IEEE/OES Autonomous Underwater Vehicles (AUV), (2012), 1–6. [17] G. Wen, Z. Duan, W. Yu and G. Chen, Consensus of second-order multi-agent systems with delayed nonlinear dynamics and intermittent communications, Internat. J. Control, 86 (2013), 322-331.  doi: 10.1080/00207179.2012.727473. [18] J. Wu and Y. Shi, Consensus in multi-agent systems with random delays governed by a Markov chain, Systems Control Lett., 60 (2011), 863-870.  doi: 10.1016/j.sysconle.2011.07.004. [19] G. Yang, Q. Yang, V. Kapila, D. Palmer and R. Vaidyanathan, Fuel optimal manoeuvres for multiple spacecraft formation reconfiguration using multi-agent optimization, International Journal of Robust and Nonlinear Control, 12 (2002), 243-283.
Communication topology with a directed spanning tree
Time delay ($d_k$) over time
State of all nodes in the original system
State of all nodes under the control scheme in [18] when $\alpha_0 = 0.26$
Consensus with switching adjacency matrices when $\alpha_0 = 0.26$
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