American Institute of Mathematical Sciences

Advanced
September  2014, 9(3): 553-573. doi: 10.3934/nhm.2014.9.553

Group pinning consensus under fixed and randomly switching topologies with acyclic partition

Yilun Shang 1,

1. 

Department of Mathematics, Tongji University, Shanghai 200092, China

Received  December 2013 Revised  June 2014 Published  October 2014

This paper addresses group consensus problems in generic linear multi-agent systems with directed information flow over (i) fixed topology and (ii) randomly switching topology governed by a continuous-time homogeneous Markov process. We propose two types of pinning control protocols to ensure group consensus regardless of the magnitude of the coupling strengths among the agents. In the case of randomly switching topology, we show that the group consensus behavior is unrelated to the magnitude of the couplings among agents if the union of the topologies corresponding to the positive recurrent states of the Markov process possesses an acyclic partition. Sufficient conditions for achieving group consensus are presented in terms of simple graphic conditions, which are easy to be checked compared to conventional algebraic criteria. Simulation examples are also presented to validate the effectiveness of the theoretical results.
Keywords: linear time-invariant system, Network of multi-agents, directed acyclic graph., pinning control, Markovian switching.
Mathematics Subject Classification: Primary: 93C05, 37H10, 05C20; Secondary: 05C50, 60J2.
Citation: Yilun Shang. Group pinning consensus under fixed and randomly switching topologies with acyclic partition. Networks and Heterogeneous Media, 2014, 9 (3) : 553-573. doi: 10.3934/nhm.2014.9.553
References:
[1]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153. doi: 10.1016/j.physrep.2008.09.002.

[2]

J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithm and Applications, 2nd Ed., Springer-Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.

[3]

V. N. Belykh, I. V. Belykh and M. Hasler, Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems, Phys. Rev. E, 62 (2000), 6332-6345. doi: 10.1103/PhysRevE.62.6332.

[4]

V. Borkar and P. P. Varaiya, Asymptotic agreement in distributed estimation, IEEE Trans. Automat. Control, 27 (1982), 650-655. doi: 10.1109/TAC.1982.1102982.

[5]

T. Chen, X. Liu and W. Lu, Pinning complex networks by a single controller, IEEE Trans. Circuit Syst. I, 54 (2007), 1317-1326. doi: 10.1109/TCSI.2007.895383.

[6]

O. Costa and M. Fragoso, A unified approach for stochastic and mean square stability of continuous-time linear systems with Markovian jumping parameters and additive disturbances, SIAM J. Contr. Optim., 44 (2005), 1165-1191. doi: 10.1137/S0363012903434753.

[7]

T. Dahms, J. Lehnert and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), 016202. doi: 10.1103/PhysRevE.86.016202.

[8]

X. Feng and K. A. Loparo, Stability of linear Markovian jump systems, Proc. of the 29th IEEE Conf. Decision and Control, Honolulu, HI, (1990), 1408-1413. doi: 10.1109/CDC.1990.203842.

[9]

Y. Z. Feng, J. Lu, S. Xu and Y. Zou, Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamcis, J. Franklin Institute, 350 (2013), 3277-3292. doi: 10.1016/j.jfranklin.2013.07.004.

[10]

Y. Han, W. Lu and T. Chen, Cluster consensus in discrete-time networks of multiagents with inter-cluster nonidentical inputs, IEEE Trans. Neural Networks and Learning Syst., 24 (2013), 566-578.

[11]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[12]

Z. Li, Z. Duan and G. Chen, Dynamic consensus of linear multi-agent systems, IET Control Theory Appl., 5 (2011), 19-28. doi: 10.1049/iet-cta.2009.0466.

[13]

W. Lu, F. M. Atay and J. Jost, Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Netw. Heterog. Media, 6 (2011), 329-349. doi: 10.3934/nhm.2011.6.329.

[14]

I. Matei and J. S. Baras, Convergence results for the linear consensus problem under Markovian random graphs, SIAM J. Control Optim., 51 (2013), 1574-1591. doi: 10.1137/100816870.

[15]

G. Miao, S. Xu and Y. Zou, Necessary and sufficient conditions for mean square consensus under Markov switching topologies, Int. J. Syst. Sci., 44 (2013), 178-186. doi: 10.1080/00207721.2011.598961.

[16]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.

[17]

R. Olfati-Saber and R. M. Murray, Consensus problem in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[18]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.

[19]

J. Qin and C. Yu, Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition, Automatica, 49 (2013), 2898-2905. doi: 10.1016/j.automatica.2013.06.017.

[20]

W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interation topologies, IEEE Trans. Autom. Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.

[21]

A. H. Roger and R. J. Charles, Matrix Analysis, Cambridge University Press, Cambridge, 1985. doi: 10.1017/CBO9780511810817.

[22]

E. Seneta, Non-negative Matrices and Markov Chains, Springer, New York, 2006.

[23]

Y. Shang, Multi-agent coordination in directed moving neighborhood random networks, Chin. Phys. B, 19 (2010), 070201.

[24]

Y. Shang, Finite-time consensus for multi-agent systems with fixed topologies, Int. J. Syst. Sci., 43 (2012), 499-506. doi: 10.1080/00207721.2010.517857.

[25]

Y. Shang, $L^1$ group consensus of multi-agent systems with stochastic inputs under directed interaction topology, Int. J. Control, 86 (2013), 1-8. doi: 10.1080/00207179.2012.715753.

[26]

Y. Shang, Continuous-time average consensus under dynamically changing topologies and multiple time-varying delays, Appl. Math. Comput., 244 (2014), 457-466. doi: 10.1016/j.amc.2014.07.019.

[27]

Y. Shang, Group consensus of multi-agent systems in directed networks with noises and time delays, Int. J. Syst. Sci. doi: 10.1080/00207721.2013.862582.

[28]

Y. Shang, Group consensus in generic linear multi-agent sytesms with inter-group non-identical inputs, Cogent Engineering, 1 (2014), 947761. doi: 10.1080/23311916.2014.947761.

[29]

F. Sorrentino and E. Ott, Network synchronization of groups, Phys. Rev. E, 76 (2007), 056114. doi: 10.1103/PhysRevE.76.056114.

[30]

R. Stanley, Acyclic orientations of graphs, Discrete Math., 5 (1973), 171-178. doi: 10.1016/0012-365X(73)90108-8.

[31]

W. Sun, Y. Q. Bai, R. Jia, R. Xiong and J. Chen, Multi-group consensus via pinning control with non-linear heterogeneous agents, Proc. 8th Asian Control Conference, Taiwan, (2011), 323-328.

[32]

C. Tan, G.-P. Liu and G.-R. Duan, Couple-group consensus of multi-agent systems with directed and fixed topology, Proc. 30th Chinese Contr. Conf., Yantai, (2011), 6515-6520.

[33]

B. Touri and A. Nedić, On ergodicity, infinite flow, and consensus in random models, IEEE Trans. Autom. Control, 56 (2011), 1593-1605. doi: 10.1109/TAC.2010.2091174.

[34]

J. N. Tsitsiklis, Problems in Decentralized Decision Making and Computation, Ph.D. Dissertation, Massachusetts Inst. Technol., Cambridge, MA, 1984.

[35]

F. Xiao and L. Wang, Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays, IEEE Trans. Autom. Control, 53 (2008), 1804-1816. doi: 10.1109/TAC.2008.929381.

[36]

K. You, Z. Li and L. Xie, Consensus condition for linear multi-agent systems over randomly switching topologies, Automatica, 49 (2013), 3125-3132. doi: 10.1016/j.automatica.2013.07.024.

[37]

W. Yu, G. Chen and J. Lü, On pinning control synchronization of complex dynamical networks, Automatica, 45 (2009), 429-435. doi: 10.1016/j.automatica.2008.07.016.

[38]

J. Yu and L. Wang, Group consensus of multi-agent sytems with undirected communication graphs, Proc. 7th Asian Control Conf., (2009), 105-110.

[39]

J. Yu and L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Syst. Control Lett., 59 (2010), 340-348. doi: 10.1016/j.sysconle.2010.03.009.

show all references

References:
[1]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93-153. doi: 10.1016/j.physrep.2008.09.002.

[2]

J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithm and Applications, 2nd Ed., Springer-Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.

[3]

V. N. Belykh, I. V. Belykh and M. Hasler, Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems, Phys. Rev. E, 62 (2000), 6332-6345. doi: 10.1103/PhysRevE.62.6332.

[4]

V. Borkar and P. P. Varaiya, Asymptotic agreement in distributed estimation, IEEE Trans. Automat. Control, 27 (1982), 650-655. doi: 10.1109/TAC.1982.1102982.

[5]

T. Chen, X. Liu and W. Lu, Pinning complex networks by a single controller, IEEE Trans. Circuit Syst. I, 54 (2007), 1317-1326. doi: 10.1109/TCSI.2007.895383.

[6]

O. Costa and M. Fragoso, A unified approach for stochastic and mean square stability of continuous-time linear systems with Markovian jumping parameters and additive disturbances, SIAM J. Contr. Optim., 44 (2005), 1165-1191. doi: 10.1137/S0363012903434753.

[7]

T. Dahms, J. Lehnert and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), 016202. doi: 10.1103/PhysRevE.86.016202.

[8]

X. Feng and K. A. Loparo, Stability of linear Markovian jump systems, Proc. of the 29th IEEE Conf. Decision and Control, Honolulu, HI, (1990), 1408-1413. doi: 10.1109/CDC.1990.203842.

[9]

Y. Z. Feng, J. Lu, S. Xu and Y. Zou, Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamcis, J. Franklin Institute, 350 (2013), 3277-3292. doi: 10.1016/j.jfranklin.2013.07.004.

[10]

Y. Han, W. Lu and T. Chen, Cluster consensus in discrete-time networks of multiagents with inter-cluster nonidentical inputs, IEEE Trans. Neural Networks and Learning Syst., 24 (2013), 566-578.

[11]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[12]

Z. Li, Z. Duan and G. Chen, Dynamic consensus of linear multi-agent systems, IET Control Theory Appl., 5 (2011), 19-28. doi: 10.1049/iet-cta.2009.0466.

[13]

W. Lu, F. M. Atay and J. Jost, Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Netw. Heterog. Media, 6 (2011), 329-349. doi: 10.3934/nhm.2011.6.329.

[14]

I. Matei and J. S. Baras, Convergence results for the linear consensus problem under Markovian random graphs, SIAM J. Control Optim., 51 (2013), 1574-1591. doi: 10.1137/100816870.

[15]

G. Miao, S. Xu and Y. Zou, Necessary and sufficient conditions for mean square consensus under Markov switching topologies, Int. J. Syst. Sci., 44 (2013), 178-186. doi: 10.1080/00207721.2011.598961.

[16]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.

[17]

R. Olfati-Saber and R. M. Murray, Consensus problem in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[18]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.

[19]

J. Qin and C. Yu, Cluster consensus control of generic linear multi-agent systems under directed topology with acyclic partition, Automatica, 49 (2013), 2898-2905. doi: 10.1016/j.automatica.2013.06.017.

[20]

W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interation topologies, IEEE Trans. Autom. Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.

[21]

A. H. Roger and R. J. Charles, Matrix Analysis, Cambridge University Press, Cambridge, 1985. doi: 10.1017/CBO9780511810817.

[22]

E. Seneta, Non-negative Matrices and Markov Chains, Springer, New York, 2006.

[23]

Y. Shang, Multi-agent coordination in directed moving neighborhood random networks, Chin. Phys. B, 19 (2010), 070201.

[24]

Y. Shang, Finite-time consensus for multi-agent systems with fixed topologies, Int. J. Syst. Sci., 43 (2012), 499-506. doi: 10.1080/00207721.2010.517857.

[25]

Y. Shang, $L^1$ group consensus of multi-agent systems with stochastic inputs under directed interaction topology, Int. J. Control, 86 (2013), 1-8. doi: 10.1080/00207179.2012.715753.

[26]

Y. Shang, Continuous-time average consensus under dynamically changing topologies and multiple time-varying delays, Appl. Math. Comput., 244 (2014), 457-466. doi: 10.1016/j.amc.2014.07.019.

[27]

Y. Shang, Group consensus of multi-agent systems in directed networks with noises and time delays, Int. J. Syst. Sci. doi: 10.1080/00207721.2013.862582.

[28]

Y. Shang, Group consensus in generic linear multi-agent sytesms with inter-group non-identical inputs, Cogent Engineering, 1 (2014), 947761. doi: 10.1080/23311916.2014.947761.

[29]

F. Sorrentino and E. Ott, Network synchronization of groups, Phys. Rev. E, 76 (2007), 056114. doi: 10.1103/PhysRevE.76.056114.

[30]

R. Stanley, Acyclic orientations of graphs, Discrete Math., 5 (1973), 171-178. doi: 10.1016/0012-365X(73)90108-8.

[31]

W. Sun, Y. Q. Bai, R. Jia, R. Xiong and J. Chen, Multi-group consensus via pinning control with non-linear heterogeneous agents, Proc. 8th Asian Control Conference, Taiwan, (2011), 323-328.

[32]

C. Tan, G.-P. Liu and G.-R. Duan, Couple-group consensus of multi-agent systems with directed and fixed topology, Proc. 30th Chinese Contr. Conf., Yantai, (2011), 6515-6520.

[33]

B. Touri and A. Nedić, On ergodicity, infinite flow, and consensus in random models, IEEE Trans. Autom. Control, 56 (2011), 1593-1605. doi: 10.1109/TAC.2010.2091174.

[34]

J. N. Tsitsiklis, Problems in Decentralized Decision Making and Computation, Ph.D. Dissertation, Massachusetts Inst. Technol., Cambridge, MA, 1984.

[35]

F. Xiao and L. Wang, Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays, IEEE Trans. Autom. Control, 53 (2008), 1804-1816. doi: 10.1109/TAC.2008.929381.

[36]

K. You, Z. Li and L. Xie, Consensus condition for linear multi-agent systems over randomly switching topologies, Automatica, 49 (2013), 3125-3132. doi: 10.1016/j.automatica.2013.07.024.

[37]

W. Yu, G. Chen and J. Lü, On pinning control synchronization of complex dynamical networks, Automatica, 45 (2009), 429-435. doi: 10.1016/j.automatica.2008.07.016.

[38]

J. Yu and L. Wang, Group consensus of multi-agent sytems with undirected communication graphs, Proc. 7th Asian Control Conf., (2009), 105-110.

[39]

J. Yu and L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Syst. Control Lett., 59 (2010), 340-348. doi: 10.1016/j.sysconle.2010.03.009.

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