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doi: 10.3934/naco.2021024
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Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies

1. 

College of Liberal Arts and Sciences, National University of Defense Technology, Hunan, 410000, China

2. 

School of Mathematics and Statistics, Yulin Normal University, Guangxi 537000, China

3. 

College of Mathematics and Statistics, Guangxi Normal University, Guangxi, 537000, China

* Corresponding author: Jianhua Huang

Received  April 2021 Revised  June 2021 Early access June 2021

Fund Project: The first author is supported by the partial supports of the Natural Science Foundation of Guangxi Province (2018JJA110105), the Foundation of Yulin Normal University (G2019ZK12), Middle-aged and Young Teachers’ Basic Ability Promotion Project of Guangxi (2019KY0587), China.
The second author is supported by the partial supports of the Natural Science Foundation of Guangxi Province (2019AC20186), China.

We investigate the consensus stability for linear stochastic multi-agent systems with multiplicative noises and Markovian random graphs and investigate the asymptotic consensus in the mean square sense for the systems. To establish the consensus stability for the systems, we analysis the consensus error systems by developing general stochastic differential equation with jumps, matrix theory and algebraic graph theory, and then show that the error consensus in the mean square sense finally tending to zero as time goes on is determined by the strongly connected property of union of topologies. Finally, we provide an example to demonstrate the effectiveness of our theoretical results.

Citation: 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, doi: 10.3934/naco.2021024
References:
[1]

C. CarliF. FagnaniA. Speranzon and S. Zampieri, Communication constraints in the average consensus problem, Automatica, 44 (2008), 671-684.  doi: 10.1016/j.automatica.2007.07.009.

[2]

D. X. Chen, W. Li, X. L. Liu, W. W. Yu and Y. Z. Sun, Effects of measurement noise on flocking dynamcis of Cucker-Smale systems, IEEE Trans. Circults Syst. II, (2019), 1-5. doi: 10.1109/TCSII.2019.2947788.

[3]

D. V. Dimarogonas and K. H. Johansson, Stability analysis for multi-agent systems using the incidence matrix: Quantized communication and formation control, Automatica, 46 (2010), 695-700.  doi: 10.1016/j.automatica.2010.01.012.

[4]

C. DengM. J. ErG. H. Yang and N. Wang, Event-triggered consensus of linear multiagent systems with time-varying communication delays, IEEE T. Cybernetics, 50 (2020), 2916-2925.  doi: 10.1109/TCYB.2019.2922740.

[5]

J. A Fax and R. M. Nurry, Information flow and cooperative control of vehicle formations, IFAC Proceedings Volumes, 34 (2002), 115-120.  doi: 10.3182/20020721-6-ES-1901.00100.

[6]

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

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L. Gao and D. Cheng, Comment on "coordination of groups of mobile agents using nearest neighbor rules", IEEE Trans. Autom. Control, 52 (2007), 968-969.  doi: 10.1109/TAC.2007.895885.

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F. T. GiancarloA. Buffa and M. Gati, Analysis of coordination multi-agent systems through partial differential equations, IEEE Trans. Autom. Control, 51 (2006), 1058-1063.  doi: 10.1109/TAC.2006.876805.

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Y. HongG. Chen and L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica, 44 (2008), 846-850.  doi: 10.1016/j.automatica.2007.07.004.

[10]

L. L. HuanX. M. Bai and H. M. Li, Consensus of multi-agent systems in random disturbance, Journal of Hefei University of Technology(Natural Science), 42 (2019), 1719-1724.  doi: 10.3969/j.issn.1003-5060.2019.12.024.

[11]

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

[12]

S. Kar and J. M. F. Moura, Distributed consensus algorithms in sensor networks with imperfect communication: Link failures and channel noise, IEEE Trans. Signal Process, 57 (2009), 355-369.  doi: 10.1109/TSP.2008.2007111.

[13]

M. L. Li and F. Q. Deng, Necessary and sufficient conditions for consensus of continuous-time multiagent systems with Markovian switching topologies and communication noise, IEEE T. Cybernetics, 50 (2020), 3264-3270.  doi: 10.1109/TCYB.2019.2919740.

[14]

T. LiF. K. Wu and J. F. Zhang, Multi-agent consensus with relative-state-dependent measurement noises, IEEE Trans. Autom. Control, 59 (2014), 2463-2468.  doi: 10.1109/TAC.2014.2304368.

[15]

T. Li and J. F. Zhang, Mean square average-consensus under measurement noises and fixed topologies: Necessary and sufficient conditions, Automatica, 45 (2009), 1929-1936.  doi: 10.1016/j.automatica.2009.04.017.

[16]

M. Liu and Z. Li, Coherence of noise double integrator networks without velocity measurements, IEEE Trans. Circults Syst. II, 66 (2018), 993-997.  doi: 10.1109/TCSII.2018.2868422.

[17]

N. Lynch, Distributed Algorithm, Morgan Kaufmann, 1996. doi: 10.5555/2821576.

[18]

Q. Ma and G. Y. Miao, Distributed containment control of linear multi-agent systems, Neurocomputing, 133 (2014), 399-403.  doi: 10.1016/j.neucom.2013.12.034.

[19]

X. R. Mao, Stochastic Differential Quations and Applications, Elsevier, 2007.

[20]

X. MaoY. Shen and A. Gray, Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math., 235 (2011), 1213-1226.  doi: 10.1016/j.cam.2010.08.006.

[21]

X. Mu and X. Wu, Tracking consensus for stochastic hybrid multi-agent systems with partly unknown transition rates via sliding mode control, IET Control Theory & Applicstions, 14 (2020), 1091-1103.  doi: 10.1049/iet-cta.2019.0782.

[22]

I. MateiJ. S. Baras and C. Somarakis, Convergence results for the linear consensus problem under Markovian random graphs, SIAM J. Control Optim., 52 (2013), 1574-1591.  doi: 10.1137/100816870.

[23]

B. NingQ. L. Han and Z. Y. Zuo, Distributed optimization of multiagent systems with preserved network connectivity, IEEE T. Cybernetics, 49 (2021), 3980-3990.  doi: 10.1109/TCYB.2018.2856508.

[24]

R. Olfati-Saber, Distributed Kalman filter with embedded consensus filters, In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Conferece, Seville, Spain, (2005), 8179-8184. doi: 10.1109/CDC.2005.1583486.

[25]

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.

[26]

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

[27]

C. W. Reynolds, Flocks, herds, and schools: a distributed behavioral model, Computer Graphics, SIGGRAPH '87 Conference Proceedings, 21 (1987), 25-34. 

[28]

G. D. Shi and K. H. Johansson, Robust consensus for continuous-time multiagent dynamics, SIAM J. Control. Optim., 51 (2013), 3673-3691.  doi: 10.1137/110841308.

[29]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995) 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[30]

J. Wang and N. Elia, Distributed averaging under constraints on information exchange: Emergece of Levy flights, IEEE Trans. Autom. Control, 57 (2012), 2435-2449.  doi: 10.1109/TAC.2012.2186093.

[31]

K. Y. YouZ. K. Li and L. H. Xie, Consensus condition for linear-agent systems over randomly switching topologies, Automatica, 19 (2013), 3125-3132.  doi: 10.1016/j.automatica.2013.07.024.

[32]

X. F. ZongT. Li and J. F. Zhang, Consensus control of discrete-time multi-agent systems with time delays and multiplicative measurement noises (in Chinese), Sci. Sin. Math., 46 (2016), 1617-1636.  doi: 10.1360/N012015-00398.

[33]

X. F. ZongT. Li and J. F. Zhang, Consensus conditions of continuous-time multi-agent systems with additice and multiplicative measurement noise, SIAM J. Control. Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.

[34]

Q. Zhang and J. F. Zhang, Distributed consensus of continuous-time multi-agent systems with Markovian switching topologies and stochastic communication noises, J. Sys. Sci. & Math. Scis., 21 (2011), 1097-1110. 

[35]

J. H. Ren and X. F. Zong, Containment control of multi-agent systems with multiplicative noises, J. Syst. Sci. & Complex, 34 (2021), 1-21. 

show all references

References:
[1]

C. CarliF. FagnaniA. Speranzon and S. Zampieri, Communication constraints in the average consensus problem, Automatica, 44 (2008), 671-684.  doi: 10.1016/j.automatica.2007.07.009.

[2]

D. X. Chen, W. Li, X. L. Liu, W. W. Yu and Y. Z. Sun, Effects of measurement noise on flocking dynamcis of Cucker-Smale systems, IEEE Trans. Circults Syst. II, (2019), 1-5. doi: 10.1109/TCSII.2019.2947788.

[3]

D. V. Dimarogonas and K. H. Johansson, Stability analysis for multi-agent systems using the incidence matrix: Quantized communication and formation control, Automatica, 46 (2010), 695-700.  doi: 10.1016/j.automatica.2010.01.012.

[4]

C. DengM. J. ErG. H. Yang and N. Wang, Event-triggered consensus of linear multiagent systems with time-varying communication delays, IEEE T. Cybernetics, 50 (2020), 2916-2925.  doi: 10.1109/TCYB.2019.2922740.

[5]

J. A Fax and R. M. Nurry, Information flow and cooperative control of vehicle formations, IFAC Proceedings Volumes, 34 (2002), 115-120.  doi: 10.3182/20020721-6-ES-1901.00100.

[6]

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

[7]

L. Gao and D. Cheng, Comment on "coordination of groups of mobile agents using nearest neighbor rules", IEEE Trans. Autom. Control, 52 (2007), 968-969.  doi: 10.1109/TAC.2007.895885.

[8]

F. T. GiancarloA. Buffa and M. Gati, Analysis of coordination multi-agent systems through partial differential equations, IEEE Trans. Autom. Control, 51 (2006), 1058-1063.  doi: 10.1109/TAC.2006.876805.

[9]

Y. HongG. Chen and L. Bushnell, Distributed observers design for leader-following control of multi-agent networks, Automatica, 44 (2008), 846-850.  doi: 10.1016/j.automatica.2007.07.004.

[10]

L. L. HuanX. M. Bai and H. M. Li, Consensus of multi-agent systems in random disturbance, Journal of Hefei University of Technology(Natural Science), 42 (2019), 1719-1724.  doi: 10.3969/j.issn.1003-5060.2019.12.024.

[11]

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

[12]

S. Kar and J. M. F. Moura, Distributed consensus algorithms in sensor networks with imperfect communication: Link failures and channel noise, IEEE Trans. Signal Process, 57 (2009), 355-369.  doi: 10.1109/TSP.2008.2007111.

[13]

M. L. Li and F. Q. Deng, Necessary and sufficient conditions for consensus of continuous-time multiagent systems with Markovian switching topologies and communication noise, IEEE T. Cybernetics, 50 (2020), 3264-3270.  doi: 10.1109/TCYB.2019.2919740.

[14]

T. LiF. K. Wu and J. F. Zhang, Multi-agent consensus with relative-state-dependent measurement noises, IEEE Trans. Autom. Control, 59 (2014), 2463-2468.  doi: 10.1109/TAC.2014.2304368.

[15]

T. Li and J. F. Zhang, Mean square average-consensus under measurement noises and fixed topologies: Necessary and sufficient conditions, Automatica, 45 (2009), 1929-1936.  doi: 10.1016/j.automatica.2009.04.017.

[16]

M. Liu and Z. Li, Coherence of noise double integrator networks without velocity measurements, IEEE Trans. Circults Syst. II, 66 (2018), 993-997.  doi: 10.1109/TCSII.2018.2868422.

[17]

N. Lynch, Distributed Algorithm, Morgan Kaufmann, 1996. doi: 10.5555/2821576.

[18]

Q. Ma and G. Y. Miao, Distributed containment control of linear multi-agent systems, Neurocomputing, 133 (2014), 399-403.  doi: 10.1016/j.neucom.2013.12.034.

[19]

X. R. Mao, Stochastic Differential Quations and Applications, Elsevier, 2007.

[20]

X. MaoY. Shen and A. Gray, Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math., 235 (2011), 1213-1226.  doi: 10.1016/j.cam.2010.08.006.

[21]

X. Mu and X. Wu, Tracking consensus for stochastic hybrid multi-agent systems with partly unknown transition rates via sliding mode control, IET Control Theory & Applicstions, 14 (2020), 1091-1103.  doi: 10.1049/iet-cta.2019.0782.

[22]

I. MateiJ. S. Baras and C. Somarakis, Convergence results for the linear consensus problem under Markovian random graphs, SIAM J. Control Optim., 52 (2013), 1574-1591.  doi: 10.1137/100816870.

[23]

B. NingQ. L. Han and Z. Y. Zuo, Distributed optimization of multiagent systems with preserved network connectivity, IEEE T. Cybernetics, 49 (2021), 3980-3990.  doi: 10.1109/TCYB.2018.2856508.

[24]

R. Olfati-Saber, Distributed Kalman filter with embedded consensus filters, In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Conferece, Seville, Spain, (2005), 8179-8184. doi: 10.1109/CDC.2005.1583486.

[25]

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.

[26]

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

[27]

C. W. Reynolds, Flocks, herds, and schools: a distributed behavioral model, Computer Graphics, SIGGRAPH '87 Conference Proceedings, 21 (1987), 25-34. 

[28]

G. D. Shi and K. H. Johansson, Robust consensus for continuous-time multiagent dynamics, SIAM J. Control. Optim., 51 (2013), 3673-3691.  doi: 10.1137/110841308.

[29]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995) 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[30]

J. Wang and N. Elia, Distributed averaging under constraints on information exchange: Emergece of Levy flights, IEEE Trans. Autom. Control, 57 (2012), 2435-2449.  doi: 10.1109/TAC.2012.2186093.

[31]

K. Y. YouZ. K. Li and L. H. Xie, Consensus condition for linear-agent systems over randomly switching topologies, Automatica, 19 (2013), 3125-3132.  doi: 10.1016/j.automatica.2013.07.024.

[32]

X. F. ZongT. Li and J. F. Zhang, Consensus control of discrete-time multi-agent systems with time delays and multiplicative measurement noises (in Chinese), Sci. Sin. Math., 46 (2016), 1617-1636.  doi: 10.1360/N012015-00398.

[33]

X. F. ZongT. Li and J. F. Zhang, Consensus conditions of continuous-time multi-agent systems with additice and multiplicative measurement noise, SIAM J. Control. Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.

[34]

Q. Zhang and J. F. Zhang, Distributed consensus of continuous-time multi-agent systems with Markovian switching topologies and stochastic communication noises, J. Sys. Sci. & Math. Scis., 21 (2011), 1097-1110. 

[35]

J. H. Ren and X. F. Zong, Containment control of multi-agent systems with multiplicative noises, J. Syst. Sci. & Complex, 34 (2021), 1-21. 

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