American Institute of Mathematical Sciences

Advanced
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

Xi Zhu 1,, , Meixia Li 2, and  Chunfa Li 1,

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.

Keywords: Multi-agent system, consensus, random delay, Markov chain, uncertain topology.
Mathematics Subject Classification: Primary: 37B25, 39A30; Secondary: 60J10.
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. 

show all references

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. 

Figure 1.  Communication topology with a directed spanning tree
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Time delay ($ d_k $) over time
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  State of all nodes in the original system
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  State of all nodes under the control scheme in [18] when $ \alpha_0 = 0.26 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Consensus with switching adjacency matrices when $ \alpha_0 = 0.26 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Mei Luo, Jinrong Wang, Yumei Liao. Bounded consensus of double-integrator stochastic multi-agent systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022088
[2]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks and Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012
[3]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[4]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489
[5]

Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022
[6]

Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396
[7]

Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021024
[8]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105
[9]

Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2022, 14 (1) : 1-28. doi: 10.3934/jgm.2021005
[10]

Hong Man, Yibin Yu, Yuebang He, Hui Huang. Design of one type of linear network prediction controller for multi-agent system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 727-734. doi: 10.3934/dcdss.2019047
[11]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001
[12]

Nadia Loy, Andrea Tosin. Boltzmann-type equations for multi-agent systems with label switching. Kinetic and Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027
[13]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623
[14]

Zhiyong Sun, Toshiharu Sugie. Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 297-318. doi: 10.3934/naco.2019020
[15]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations and Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
[16]

Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1395-1414. doi: 10.3934/dcdss.2020379
[17]

Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061
[18]

Yan Zhou, Chi Kin Chan, Kar Hung Wong. The impacts of retailers' regret aversion on a random multi-period supply chain network. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021086
[19]

Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
[20]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (481)
  • HTML views (408)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy