Abstract
We solve the problem of synchronizing a network of linear agents with unknown parameters and unknown network topology given that the Laplacian that defines it has no complex eigenvalues. To solve this problem, we use a modified high order adaptation algorithm. We obtain conditions for reaching consensus with the proposed algorithm. We show modeling results that demonstrate the efficiency of the proposed approach.
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Amelina, N.O., Anan’evskii, M.S., Andrievskii, B.R., et al., Problemy setevogo upravleniya (Network Control Problems), Fradkov, A.L., Ed., Moscow–Izhevsk: Izhevsk. Inst. Comp. Issled., 2015.
Cheng, Z., Zhang, H.-T., and Fan, M.-C., Consensus and Rendezvous Predictive Control for Multi-agent Systems with Input Constraints, 33rd Chinese Control Conf. (CCC), 2014, pp. 1438–1443.
Wu, Z., Guan, Z., Wu, X., and Li, T., Consensus Based Formation Control and Trajectory Tracing of Multi-Agent Robot Systems, J. Intelligent Robot. Syst., 2007, vol. 48, no. 3, pp. 397–410.
Amelina, N., Fradkov, A., Jiang, Y., and Vergados, D.J., Approximate Consesus in Stochastic Networks with Application to Load Balancing, IEEE Trans. Inform. Theory, 2015, vol. 61, no. 4, pp. 1739–1752.
Leonard, N.E. and Fiorelli, E., Virtual Leaders, Artificial Potentials, and Coordinated Control of Groups, Proc. 40 IEEE Conf. Decision Control, 2001, pp. 2968–2973.
Amelina, N.O., Multiagent Technologies, Adaptation, Self-Organization, and Reaching Consensus, Stokhast. Optim. Informat., 2011, vol. 7, no. 1, pp. 149–185.
Olfati-Saber, R. and Murray, R., Consensus Problems in Networks of Agents with Switching Topology and Time-Delays, IEEE Trans. Autom. Control, 2004, vol. 49, no. 9, pp. 1520–1533.
Ren, W. and Beard, R.W., Distributed Consensus in Multi-vehicle Cooperative Control, London: Springer-Verlag, 2008.
Olfati-Saber, R., Fax, A., and Murray, R., Consensus and Cooperation in Networked Multi-agent Systems, Proc. IEEE, 2007, vol. 95, no. 1, pp. 215–233.
Agaev, R.P. and Chebotarev, P.Yu., Coordination in Multiagent Systems and Laplacian Spectra of Digraphs, Autom. Remote Control, 2009, vol. 70, no. 3, pp. 469–483.
Fradkov, A.L., Kiberneticheskaya fizika (Cybernetic Physics), St. Petersburg: Nauka, 2003.
Gazis, D.C., Herman, R., and Rothery, R.W., Nonlinear Follow-the-Leader Models of Traffic Flow Operations Research, INFORMS, 1961, no. 9, pp. 545–567.
Chandler, R.E., Herman, R., and Montroll, E.W., Traffic Dynamics: Studies in Car Following, Oper. Res. Informs., 1958, no. 6, pp. 165–184.
Newell, G.F., Nonlinear Effects in the Dynamics of Car Following, Oper. Res., 1961, vol. 9, no. 2, pp. 209–229.
Agaev, R.P. and Chebotarev, P.Yu., The Matrix of Maximal Outgoing Forests of a Digraph and Its Applications, Autom. Remote Control, 2000, vol. 61, no. 9, pp. 1424–1450.
Agaev, R.P. and Chebotarev, P.Yu., Spanning Forests of a Digraph and Their Applications, Autom. Remote Control, 2001, vol. 62, no. 3, pp. 443–466.
Fax, J.A. and Murray, R.M., Information Flow and Cooperative Control of Vehicle Formations, IEEE Trans. Autom. Control, 2004, no. 8, pp. 1465–1476.
Proskurnikov, A.V., The Popov Criterion for Consensus Between Delayed Agents, Proc. 9 IFAC Nonlinear Control Syst. Sympos. NOLCOS-2013, Toulouse, France, pp. 693–698.
Li, Z., Duan, Z., and Chen, G., Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint, IEEE Transact. Circuits Systems I, Regular Papers, 2010, no. 1, pp. 213–224.
Polyak, B.T. and Tsypkin, Ya.Z., Stability and Robust Stability of Uniform System, Autom. Remote Control, 1996, vol. 57, no. 11, pp. 1606–1617.
Hara, S., Shimizu, H., and Kim, T.-H., Consensus in Hierarchical Multi-agent Dynamical Systems with Low-rank Interconnections: Analysis of Stability and Convergence Rates, Am. Control Conf., 2009, pp. 5192–5197.
Proskurnikov, A.V., Frequency-Domain Criteria for Consensus in Multiagent Systems with Nonlinear Sector-shaped Couplings, Autom. Remote Control, 2014, vol. 75, no. 11, pp. 1982–1995.
Godsil, C. and Royle, G., Algebraic Graph Theory, New York: Springer-Verlag, 2001.
Tomashevich, S. and Andrievsky, B., Stability and Performance of Networked Control of Quadrocopters Formation Flight, Proc. 6 Int. Congr. Ultra Modern Telecommun. Control Syst. Workshops (ICUMT 2014), St. Petersburg, Russia, 2014, pp. 331–336.
Furtat, I.B., Adaptive Control for a Dynamic Network with Linear Subsystems, Vestn. Astrakhan. Gos. Tekh. Univ., Ser. Upravlen., Vychisl. Tekh. Informatika, 2012, no. 1, pp. 69–78.
Chen, Y. and Yuping Tian, Y., Cooperative Control of Multi-agent Moving Along a Set of Given Curves, J. Syst. Sci. Complex., 2011, vol. 24, pp. 631–646.
Selivanov, A., Fradkov, A.L., and Junessov, I., Robust and Adaptive Passification Based Consensus Control of Dynamical Networks, IFAC Int. Workshop Adapt. Learn. Control Signal Proc., 2013, pp. 707–711.
Parsegov, S.E., Joining Coordinates and Hierarchical Algorithms in the Problem of Equidistant Location of Agents on a Segment, Upravlen. Bol’shimi Sist., 2012, no. 39, pp. 264–287.
Dzhunusov, I.A. and Fradkov, A.L., Synchronization in Networks of Linear Agents with Output Feedbacks, Autom. Remote Control, 2011, vol. 72, no. 8, pp. 1615–1626.
Furtat, I.B., Consensus Output Control for a Linear Dynamical Network with Disturbance Compensation, Mekhatronika, Avtomatiz., Upravlen., 2011, no. 4, pp. 12–18.
Li, Z., Ren, W., Liu, X., and Fu, M., Consensus of Multi-Agent Systems With General Linear and Lipschitz Nonlinear Dynamics Using Distributed Adaptive Protocols, IEEE Transact. Automat. Control, 2012, vol. 58, no. 7, pp. 1786–1791.
Li, Z., Liu, X., Ren, W., and Xie, L., Distributed Consensus of Linear Multi-Agent Systems with Adaptive Dynamic Protocols, Automatica, 2013, vol. 49, no. 7, pp. 1986–1995.
Selivanov, A.A., Control over Synchronization of Networks with Nonlinearities and Delayed Connections, Vestn. Nizhegorod. Univ. im. N.I. Lobachevskogo, 2014, no. 1 (3), pp. 265–271.
Hara, S. and Tsubakino, D., Eigenvector-Based Intergroup Connection of Low Rank for Hierarchical Multi-agent Dynamical Systems, Syst. Control Lett., 2012, vol. 61, no. 2, pp. 354–361.
Morse, A.S., High-order Parameter Tuners for the Adaptive Control of Nonlinear Systems, in Systems, Models and Feedback: Theory Appl., Isidori, A. and Tarn, T.J., Eds., Basel: Birkhauser, 1992, pp. 339–364.
Tsykunov, A.M., A Modified High-order Adaptive Output Feedback Control Algorithm for Linear Plants, Autom. Remote Control, 2006, vol. 67, no. 8, pp. 1311–1321.
Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nelineinoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami (Nonlinear and Adaptive Control over Complex Dynamical Systems), St. Petersburg: Nauka, 2000.
Nikiforov, V.O. and Fradkov, A.L., Adaptive Control Schemes with Extended Error, Autom. Remote Control, 1994, vol. 55, no. 9, pp. 1239–1255.
Chung, F.R.K., Spectral Graph Theory, Ser. Regional Conference Series in Mathematics, Providence: Am. Math. Soc., 1997, vol. 92.
Tomashevich, S.I., Stability of Multiagent Linear Scalar Systems and Its Dependence on the Graph of Connections, Nauch.-tekhn. Vestn. Inform. Tekhnol., Mekh. Optiki, 2014, vol. 14, no. 2, pp. 72–78.
Furtat, I.B. and Tsykunov, A.M., A Modified Algorithm of High Order Adaptation for Systems with State Delay, Vestn. Astrakhan. Gos. Tekh. Univ., 2006, no. 1, pp. 24–33.
Atassi, A.N. and Khalil, H.K., A Separation Principle for the Stabilization of Class of Nonlinear Systems, IEEE Trans. Autom. Control, 1999, vol. 44, no. 9, pp. 1672–1687.
Fax, J.A., Optimal and Cooperative Control of Vehicle Formations, PhD Dissertation, California Inst. Technol., Pasadena, CA, 2002, pp. 55–58.
Amelin, K., Tomashevich, S., and Andrievsky, B., Recursive Identification of Motion Model Parameters for Ultralight UAV, IFAC-PapersOnLine 1st IFAC Conf. Modell., Identificat. Control Nonlin. Syst. (MICNON 2015), Russia, St. Petersburg, 2015, vol. 48, no. 11, pp. 233–237.
Fradkov, A.L. and Furtat, I.B., Robust Control for a Network of Electric Power Generators, Autom. Remote Control, 2013, vol. 74, no. 11, pp. 18851–1862.
Furtat, I.B. and Fradkov, A.L., Robust Control of Multi-Machine Power Systems with Compensation of Disturbances, Electr. Power Energy Syst., 2015, no. 73, pp. 584–590.
Furtat, I.B., Robust Control for a Certain Class of Nonminimally–Phase Dynamical Networks, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2014, no. 1, pp. 35–48.
Brusin, V.A., On a Class of Singular Perturbed Adaptive Systems, Autom. Remote Control, 1995, vol. 56, no. 4, pp. 552–559.
Furtat, I.B. and Tsykunov, A.M., Adaptive Control of Plants of Unknown Relative Degree, Autom. Remote Control, 2010, vol. 71, no. 6, pp. 1076–1084.
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Original Russian Text © S.I. Tomashevich, 2017, published in Avtomatika i Telemekhanika, 2017, No. 2, pp. 99–114.
This paper was recommended for publication by A.A. Fradkov, a member of the Editorial Board
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Tomashevich, S.I. Control for a system of linear agents based on a high order adaptation algorithm. Autom Remote Control 78, 276–288 (2017). https://doi.org/10.1134/S0005117917020072
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DOI: https://doi.org/10.1134/S0005117917020072