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Continuous-time stochastic consensus: stochastic approximation and Kalman-Bucy filtering based protocols. (English) Zbl 1327.93027
Summary: This paper investigates the continuous-time multi-agent consensus with stochastic communication noises. Each agent can only use its own and neighbors’ information corrupted by random noises to design its control input. To attenuate the communication noises, we consider the stochastic approximation type and the Kalman-Bucy filtering based protocols. By using the tools of stochastic analysis and algebraic theory, the asymptotic properties of these two kinds of protocols are analyzed. Firstly, for the stochastic approximation type protocol, we clarify the relationship between the convergence rate of the consensus error and a representative class of consensus gains in both mean square and probability one. Secondly, we propose Kalman-Bucy filtering based consensus protocols. Each agent uses Kalman-Bucy filters to get asymptotically unbiased estimates of neighbors’ states and the control input is designed based on the protocol with precise communication and the certainty equivalence principle. The iterated logarithm law of estimation errors is developed. It is shown that if the communication graph has a spanning tree, then the consensus error is bounded above by \(O(t^{- 1})\) in mean square and by \(O(t^{- 1 / 2}(\log \log t)^{1 / 2})\) almost surely. Finally, the superiority of the Kalman-Bucy filtering based protocol over the stochastic approximate type protocol is studied both theoretically and numerically.

93A14 Decentralized systems
93E03 Stochastic systems in control theory (general)
68T42 Agent technology and artificial intelligence
93E11 Filtering in stochastic control theory
93E10 Estimation and detection in stochastic control theory
Full Text: DOI
[1] Chen, H.; Guo, L., Identification and stochastic adaptive control, (1991), Birkhäuser Boston, MA
[2] Cheng, L.; Hou, Z. G.; Tan, M., A mean square consensus protocol for linear multi-agent systems with communication noises and fixed topologies, IEEE Transactions on Automatic Control, 59, 1, 261-267, (2014) · Zbl 1360.93020
[3] Cheng, L.; Hou, Z. G.; Tan, M.; Wang, X., Necessary and sufficient conditions for consensus of double-integrator multi-agent systems with measurement noises, IEEE Transactions on Automatic Control, 56, 8, 1958-1963, (2011) · Zbl 1368.93659
[4] Fax, J. A.; Murray, R. M., Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, 49, 9, 1465-1476, (2004) · Zbl 1365.90056
[5] Friedman, A., Stochastic differential equations and applications. vol. 1, (1975), Academic Press New York · Zbl 0323.60056
[6] Hu, J.; Feng, G., Distributed tracking control of leader-follower multi-agent systems under noisy measurement, Automatica, 46, 8, 1382-1387, (2010) · Zbl 1204.93011
[7] Huang, M.; Manton, J. H., Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior, SIAM Journal on Control and Optimization, 48, 1, 134-161, (2009) · Zbl 1182.93108
[8] Huang, M.; Manton, J. H., Stochastic consensus seeking with noisy and directed inter-agent communication: fixed and randomly varying topologies, IEEE Transactions on Automatic Control, 55, 1, 235-241, (2010) · Zbl 1368.94002
[9] Kallianpur, G., Stochastic filtering theory, (1980), Springer New York · Zbl 0458.60001
[10] Kar, S.; Moura, J. M., Distributed consensus algorithms in sensor networks: quantized data and random link failures, IEEE Transactions on Signal Processing, 58, 3, 1383-1400, (2010) · Zbl 1392.94269
[11] Li, T.; Zhang, J. F., Mean square average-consensus under measurement noises and fixed topologies: necessary and sufficient conditions, Automatica, 45, 8, 1929-1936, (2009) · Zbl 1185.93006
[12] Li, T.; Zhang, J. F., Consensus conditions of multi-agent systems with time-varying topologies and stochastic communication noises, IEEE Transactions on Automatic Control, 55, 9, 2043-2057, (2010) · Zbl 1368.93548
[13] Liu, J.; Liu, X.; Xie, W.-C.; Zhang, H., Stochastic consensus seeking with communication delays, Automatica, 47, 12, 2689-2696, (2011) · Zbl 1235.93013
[14] Ma, C. Q; Li, T.; Zhang, J. F., Consensus control for leader-following multi-agent systems with measurement noises, Journal of Systems Science and Complexity, 23, 1, 35-49, (2010) · Zbl 1298.93028
[15] Nedic, A.; Ozdaglar, A., Distributed subgradient methods for multi-agent optimization, IEEE Transactions on Automatic Control, 54, 1, 48-61, (2009) · Zbl 1367.90086
[16] Nourian, M.; Caines, P. E.; Malhame, R. P., A mean field game synthesis of initial mean consensus problems: A continuum approach for non-Gaussian behaviour, IEEE Transactions on Automatic Control, 59, 2, 449-455, (2014) · Zbl 1360.93049
[17] Øksendal, B., Stochastic differential equations: an introduction with applications, (2010), Springer-Verlag Berlin, Heidelberg, New York
[18] Olfati-Saber, R., Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Transactions on Automatic Control, 51, 3, 401-420, (2006) · Zbl 1366.93391
[19] Olfati-Saber, R.; Murray, R. M., Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301
[20] Pasqualetti, F.; Borra, D.; Bullo, F., Consensus networks over finite fields, Automatica, 50, 2, 349-358, (2014) · Zbl 1364.93029
[21] Ren, W., Beard, R. W., & Atkins, E. M. (2005). A survey of consensus problems in multi-agent coordination. In Proceedings of the 2005 American control conference. Portland, OR, USA (pp. 1859-1864).
[22] Su, Y.; Huang, J., Two consensus problems for discrete-time multi-agent systems with switching network topology, Automatica, 48, 9, 1988-1997, (2012) · Zbl 1258.93015
[23] Vicsek, T.; Czirok, A.; Ben-Jacob, E.; Cohen, I.; Sochet, O., Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75, 6, 1226-1229, (1995)
[24] Wang, Y.; Cheng, L.; Hou, Z. G.; Tan, M.; Wang, M., Containment control of multi-agent systems in a noisy communication environment, Automatica, 50, 7, 1922-1928, (2014) · Zbl 1296.93012
[25] Wang, B. C; Zhang, J. F., Consensus conditions of multi-agent systems with unbalanced topology and stochastic disturbances, Journal of Systems Science and Mathematical Sciences, 29, 10, 1353-1365, (2009), (in Chinese) · Zbl 1212.90072
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