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Discrete-time Deployment of Agents on a Line Segment: Delays and Switches Do Not Matter

  • Robust, Adaptive, and Network Control
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Abstract

This paper considers the discrete-time problem of deploying agents on a line segment under delay in communication channels and switching and also under incomplete information about the value of the delay and the law of switching. It is shown that neither delay nor switching affect the convergence of the agents' states to the uniform allocation on the line segment. The theoretical results are illustrated by numerical simulation. The evidence is based on the well-known and new approaches to the stability analysis of positive systems.

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References

  1. Ren, W. and Cao, W., Distributed Coordination of Multi-Agent Networks, London: Springer-Verlag, 2011.

    Book  Google Scholar 

  2. Mesbahi, M. and Egerstedt, M., Graph Theoretic Methods in Multiagent Networks, Princeton and Oxford: Princeton Univ. Press, 2010.

    Book  Google Scholar 

  3. Martinez, S. and Bullo, F., Optimal Sensor Placement and Motion Coordination for Target Tracking, Automatica, 2006, vol. 42, pp. 661–668.

    Article  MathSciNet  Google Scholar 

  4. Proskurnikov, A.V. and Fradkov, A.L., Problems and Methods of Network Control, Autom. Remote Control, 2016, vol. 77, no. 10, pp. 1711–1740.

    Article  MathSciNet  Google Scholar 

  5. Wagner, I.A. and Bruckstein, A.M., Row Straightening via Local Interactions, Circuits Syst. Signal Process., 1997, vol. 16, no. 2, pp. 287–305.

    Article  MathSciNet  Google Scholar 

  6. Shcherbakov, P.S., Formation Control. The Van Loan Scheme and Other Algorithms, Autom. Remote Control, 2011, vol. 72, no. 10, pp. 681–696.

    Article  Google Scholar 

  7. Kvinto, Y.I. and Parsegov, S.E., Equidistant Arrangement of Agents on Line. Analysis of the Algorithm and Its Generalization, Autom. Remote Control, 2012, vol. 73, no. 11, pp. 1784–1793.

    Article  MathSciNet  Google Scholar 

  8. Parsegov, S.E., Polyakov, A.E., and Shcherbakov, P.S., Nonlinear Fixed-Time Control Protocol for Uniform Allocation of Agents on a Segment, Proc. 51st IEEE Conf. Decision Control, 2012, pp. 7732–7737.

    Google Scholar 

  9. Parsegov, S.E., Polyakov, A.E., and Shcherbakov, P.S., Nonlinear Fixed-Time Control Protocol for Uniform Allocation of Agents on a Segment, Dokl. Math., 2013, vol. 87, pp. 133–136.

    Article  MathSciNet  Google Scholar 

  10. Darboux, J.G., Sur un probleme de geometrie elementaire, Bull. Sci. Math., 1878, vol. 2, no. 1, pp. 298–304.

    MATH  Google Scholar 

  11. Proskurnikov, A.V. and Parsegov, S.E., Problem of Uniform Deployment on a Line Segment for Second-Order Agents, Autom. Remote Control, 2016, vol. 77, no. 7, pp. 1248–1258.

    Article  MathSciNet  Google Scholar 

  12. Cortes, J., Martinez, S., Karatas, T., and Bullo, F., Coverage Control for Mobile Sensing Networks, IEEE Trans. Robot. Autom., 2004, vol. 20, no. 2, pp. 243–255.

    Article  Google Scholar 

  13. Farina, L. and Rinaldi, S., Positive Linear Systems: Theory and Applications, New York: Wiley, 2000.

    Book  Google Scholar 

  14. Hofbauer, J. and So, J.W., Diagonal Dominance and Harmless Off-Diagonal Delays, Proc. Am. Math. Soc., 2000, vol. 128, pp. 2675–2682.

    Article  MathSciNet  Google Scholar 

  15. Wu, L., Lam, J., Shu, Z., and Du, B., On Stability and Stabilizability of Positive Delay Systems, Asian J. Control, 2009, vol. 11, no. 2, pp. 226–234.

    Article  MathSciNet  Google Scholar 

  16. Aleksandrov, A. and Mason, O., Diagonal Stability of a Class of Discrete-Time Positive Switched Systems with Delay, IET Control Theory Appl., 2018, vol. 12, no. 6, pp. 812–818.

    Article  MathSciNet  Google Scholar 

  17. Aleksandrov, A., Fradkov, A., and Semenov, A., Delayed and Switched Control of Formations on a Line Segment: Delays and Switches Do Not Matter, IEEE Trans. Autom. Control., 2020, vol. 65, no. 2, pp. 794–800. https://doi.org/10.1109/TAC.2019.2918995

    Article  Google Scholar 

  18. Proskurnikov, A.V. and Matveev, A.S., Tsypkin and Jury-Lee Criteria for Synchronization and Stability of Discrete-time Multiagent Systems, Autom. Remote Control, 2018, vol. 79, no. 6, pp. 1057–1073.

    Article  MathSciNet  Google Scholar 

  19. Lin, P. and Ren, W., Constrained Consensus in Unbalanced Networks with Communication Delays, IEEE Trans. Autom. Control, 2014, vol. 59, no. 3, pp. 775–781.

    Article  MathSciNet  Google Scholar 

  20. Xiong, Q., Lin, P., Ren, W., Yang, Ch., and Gui, W., Containment Control for Discrete-time Multiagent Systems with Communication Delays and Switching Topologies, IEEE Trans. Cybernet., 2019, vol. 49, no. 10, pp. 3827–3830.

    Article  Google Scholar 

  21. Niculescu, S., Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Science, New York: Springer, 2001.

    MATH  Google Scholar 

  22. Shorten, R., Wirth, F., Mason, O., Wulf, K., and King, C., Stability Criteria for Switched and Hybrid Systems, SIAM Rev., 2007, vol. 49, no. 4, pp. 545–592.

    Article  MathSciNet  Google Scholar 

  23. Berman, A. and Plemmons, R.J., Nonnegative Matrices in the Mathematical Sciences, Philadelphia: SIAM, 1994.

    Book  Google Scholar 

  24. Blanchini, F., Colaneri, P., and Valcher, M.E., Switched Positive Linear Systems, Foundat. Trends Syst. Control, 2015, vol. 2, no. 2, pp. 101–273.

    Article  Google Scholar 

  25. Kazkurewicz, E. and Bhaya, A., Matrix Diagonal Stability in Systems and Computation, Boston: Birkhauser, 1999.

    Google Scholar 

  26. Aleksandrov, A.Yu., Aleksandrova, E.B., and Platonov, A.V., Ultimate Boundedness Conditions for a Hybrid Model of Population Dynamics, Proc. 21st Mediterranean Conf. Control Autom., 2013, pp. 622–627.

    Google Scholar 

  27. Valcher, M.E. and Zorzan, I., On the Consensus of Homogeneous Multiagent Systems with Positivity Constraints, IEEE Trans. Autom. Control, 2017, vol. 62, no. 10, pp. 5096–5110.

    Article  MathSciNet  Google Scholar 

  28. Shorten, R.N., Wirth, F., and Leith, D., A Positive Systems Model of TCP-like Congestion Control, IEEE Trans. Networking, 2006, vol. 14, no. 3, pp. 616–629.

    Article  Google Scholar 

  29. Aleksandrov, A.Yu. and Platonov, A.V., On Stability of Solutions for a Class of Nonlinear Difference Systems with Switching, Autom. Remote Control, 2016, vol. 77, no. 5, pp. 779–788.

    Article  MathSciNet  Google Scholar 

  30. Aleksandrov, A. and Mason, O., Diagonal Lyapunov-Krasovskii Functionals for Discrete-time Positive Systems with Delay, Syst. Control Lett., 2014, vol. 63, pp. 63–67.

    Article  MathSciNet  Google Scholar 

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Funding

This work was implemented under the State support of the leading universities of the Russian Federation, subsidy no. 08-08. This work was supported in part by the Russian Foundation for Basic Research, project no. 19-01-00146-a.

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    A. Yu. Aleksandrov

  2. ITMO University, St. Petersburg, Russia

    A. D. Semenov

  3. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    A. L. Fradkov

Authors
  1. A. Yu. Aleksandrov
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  2. A. D. Semenov
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  3. A. L. Fradkov
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Corresponding authors

Correspondence to A. Yu. Aleksandrov, A. D. Semenov or A. L. Fradkov.

Additional information

This paper was recommended for publication by P.V. Pakshin, a member of the Editorial Board

Russian Text © The Author(s), 2020, published in Avtomatika i Telemekhanika, 2020, No. 4, pp. 79-93.

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Aleksandrov, A.Y., Semenov, A.D. & Fradkov, A.L. Discrete-time Deployment of Agents on a Line Segment: Delays and Switches Do Not Matter. Autom Remote Control 81, 637–648 (2020). https://doi.org/10.1134/S0005117920040062

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  • DOI: https://doi.org/10.1134/S0005117920040062

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