Skip to main content
SpringerLink
Log in

Consensus in Asynchronous Multiagent Systems. II. Method of Joint Spectral Radius

  • Reviews
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We describe mathematical methods for analyzing the stability, stabilizability and consensus of linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of matrix sets to analyze the rate of convergence of matrix products with factors from the sets of matrices with special properties. This is a continuation of the survey by the same authors named “Consensus in Asynchronous Multiagent Systems”; the first part was published in [1].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kozyakin, V.S., Kuznetsov, N.A., and Chebotarev, P.Yu., Consensus in Asynchronous Multiagent Systems. I. Asynchronous Consensus Models, Autom. Remote Control, 2019, vol. 80, no. 4, pp. 593–623.

    Article  Google Scholar 

  2. Fang, C.-W., On Tracking Systems that Contain Two Impulse Elements with Inequal Repetition Periods, Autom. Remote Control, 1958, vol. 19, no. 10, pp. 917–930.

    Google Scholar 

  3. Tsypkin, Ya. Z., Teoriya lineinykh impul’snykh sistem (Theory of Linear Impulse Systems), Moscow: Fizmatgiz, 1963.

    Google Scholar 

  4. Bertsekas, D.P. and Tsitsiklis, J.N., Parallel and Distributed Computation. Numerical Methods, Engle-wood Cliffs: Prentice Hall, 1989.

    MATH  Google Scholar 

  5. Asarin, E.A., Kozyakin, V.S., Krasnosel’skii, M.A., and Kuznetsov, N.A., Analiz ustoichivosti rassinkhro-nizovannykh diskretnykh sistem (Stability Analysis for Desynchronized Discrete Systems), Moscow: Nauka, 1992. DOI: https://doi.org/10.13140/RG.2.1.3381.6169

    Google Scholar 

  6. Seifullaev, R.E. and Fradkov, A.L., Linear Matrix Inequality-Based Analysis of the Discrete-Continuous Nonlinear Multivariable Systems, Autom. Remote Control, 2015, vol. 76, no. 6, pp. 989–1004.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hetel, L., Fiter, C., Omran, H., et al., Recent Developments on the Stability of Systems with Aperiodic Sampling: An Overview, Automatica J. IFAC, 2017, vol. 76, pp. 309–335. DOI: https://doi.org/10.1016/j.automatica.2016.10.023

    Article  MathSciNet  MATH  Google Scholar 

  8. Kozyakin, V., An Annotated Bibliography on Convergence of Matrix Products and the Theory of Joint/Generalized Spectral Radius, Preprint, Moscow: Institut. Inform. Transmis. Probl., 2013. DOI: https://doi.org/10.13140/RG.2.1.4257.5040/1

    Google Scholar 

  9. Cross, R. and Kozyakin, V.S., Double Exponential Instability of Triangular Arbitrage Systems, Discr. Contin.Dyn.Syst. Ser. B, 2013, vol. 18, no. 2, pp. 349–376. DOI: https://doi.org/10.3934/dcdsb.2013.18.349

    Article  MathSciNet  MATH  Google Scholar 

  10. Bhaya, A., Kaszkurewicz, E., and Kozyakin, V.S., Existence and Stability of a Unique Equilibrium in Continuous-Valued Discrete-Time Asynchronous Hopfield Neural Networks, Proc. 1995 IEEE Int. Sympos. Circuits Syst. ISCAS’95, 1995, vol. 2, pp. 1140–1143. DOI: https://doi.org/10.1109/ISCAS.1995.520345

    Google Scholar 

  11. Blondel, V.D. and Nesterov, Y., Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices, SIAM J. Matrix Anal. Appl., 2009, vol. 31, no. 3, pp. 865–876. DOI: https://doi.org/10.1137/080723764

    Article  MathSciNet  MATH  Google Scholar 

  12. Dybvig, P.H. and Ross, S.A., Arbitrage, in The New Palgrave Dictionary of Economics, Durlauf, S.N. and Blume, L.E., Eds., Basingstoke: Palgrave Macmillan, 2008.

    Google Scholar 

  13. Marshall, B.R., Treepongkaruna, S., and Young, M., Exploitable Arbitrage Opportunities Exist in the Foreign Exchange Market, Discussion Paper, 10 September, Massey Univ., Palmerston North, New Zealand, 2007.

    Google Scholar 

  14. Akram, Q.F., Rime, D., and Sarno, L., Arbitrage in the Foreign Exchange Market: Turning on the Microscope, J. Int. Econom., 2008, December, vol. 76, no. 2, pp. 237–253. DOI: https://doi.org/10.1016/j.jinteco.2008.07.004

    Article  Google Scholar 

  15. Eun, C.S. and Resnick, B.G., International Financial Management, Irwin Series in Finance, Insurance, and Real Estate, Columbus: McGraw-Hill/Irwin, 2011, 6th ed.

    Google Scholar 

  16. Cassel, G., The Present Situation of the Foreign Exchanges. I, Econom. J., 1916, vol. 26, no. 1, pp. 62–65.

    Google Scholar 

  17. Ross, S.A., A Simple Approach to the Valuation of Risky Streams, J. Business, 1978, vol. 51, pp. 453–475.

    Article  Google Scholar 

  18. Cross, R., Kozyakin, V., O’Callaghan, B., et al., Periodic Sequences of Arbitrage: A Tale of Four Currencies, Metroeconomica, 2012, vol. 63, no. 2, pp. 250–294. DOI: https://doi.org/10.1111/j.1467-999X.2011.04140.x

    Article  MATH  Google Scholar 

  19. Kozyakin, V.S., Absolute Stability of Discrete Desynchronized Systems, Dokl. Akad. Nauk USSR, 1990, vol. 312, no. 5, pp. 1066–1070.

    MathSciNet  MATH  Google Scholar 

  20. Baudet, G.M., Asynchronous Iterative Methods for Multiprocessors, J. Assoc. Comput. Mach., 1978, vol. 25, no. 2, pp. 226–244.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleptsyn, A.F., Kozyakin, V.S., Krasnoselskii, M.A., and Kuznetsov, N.A., Effect of Small Synchronization Errors on Stability of Complex Systems. I, Autom. Remote Control, 1983, vol. 44, no. 7, pp. 861–867.

    MATH  Google Scholar 

  22. Kleptsyn, A.F., Kozyakin, V.S., Krasnoselskii, M.A., and Kuznetsov, N.A., Effect of Small Synchronization Errors on Stability of Complex Systems. II, Autom. Remote Control, 1984, vol. 45, no. 3, pp. 309–314.

    MATH  Google Scholar 

  23. Kleptsyn, A.F., Kozyakin, V.S., Krasnoselskii, M.A., and Kuznetsov, N.A., Effect of Small Synchronization Errors on Stability of Complex Systems. III, Autom. Remote Control, 1984, vol. 45, no. 8, pp. 1014–1018.

    MATH  Google Scholar 

  24. Kleptsyn, A.F., Kozyakin, V.S., Krasnosel’skii, M.A., and Kuznetsov, N.A., Stability of Desynchronized Systems, Dokl. Akad. Nauk USSR, 1984, vol. 274, no. 5, pp. 1053–1056.

    MathSciNet  MATH  Google Scholar 

  25. Aizerman, M.A. and Gantmakher, F.R., Absolyutnaya ustoichivost’ reguliruemykh sistem (Absolute Stability of Controllable Systems), Moscow: Akad. Nauk SSSR, 1963.

    MATH  Google Scholar 

  26. Asarin, Y.A., Kozyakin, V.S., Krasnosel’skii, M.A., Kuznetsov, N.A., and Pokrovskii, A.V., On Some New Types of Mathematical Models of Complex Systems, in Modelling and Adaptive Control, Lecture Notes in Control and Information Sciences, vol. 105, (Sopron, 1986), Berlin: Springer, 1988, pp. 10–26. DOI: https://doi.org/10.1007/BFb0043174

    Google Scholar 

  27. Asarin, E.A., Kozyakin, V.S., Krasnosel’skii, M.A., and Kuznetsov, N.A., Mathematical Models of Stability for Desynchronized Systems, in Intellektual’nye Protsessy I Ikh Modelirovanie. Informatsionnye Seti, Moscow: Ross. Akad. Nauk, 1994, pp. 3–17.

    Google Scholar 

  28. Walters, P., An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982.

    Book  MATH  Google Scholar 

  29. Furstenberg, H. and Kesten, H., Products of Random Matrices, Ann. Math. Statist., 1960, vol. 31, pp. 457–469. DOI: https://doi.org/10.1214/aoms/1177705909

    Article  MathSciNet  MATH  Google Scholar 

  30. Lancaster, P., Theory of Matrices, New York: Academic, 1969.

    MATH  Google Scholar 

  31. Kozyakin, V., Hourglass Alternative and the Finiteness Conjecture for the Spectral Characteristics of Sets of Non-Negative Matrices, Linear Algebra Appl., 2016, vol. 489, pp. 167–185. DOI: https://doi.org/10.1016/j.laa.2015.10.017

    Article  MathSciNet  MATH  Google Scholar 

  32. Rota, G.-C. and Strang, G., A Note on the Joint Spectral Radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 1960, vol. 22, pp. 379–381.

    Article  MathSciNet  MATH  Google Scholar 

  33. Daubechies, I. and Lagarias, J.C., Sets of Matrices All Infinite Products of Which Converge, Linear Algebra Appl., 1992, vol. 161, pp. 227–263. DOI: https://doi.org/10.1016/0024-3795(92)90012-Y

    Article  MathSciNet  MATH  Google Scholar 

  34. Daubechies, I. and Lagarias, J.C., Corrigendum/Addendum to: “Sets of Matrices All Infinite Products of Which Converge” [Linear Algebra Appl., vol. 161 (1992), pp. 227–263; MR1142737 (93f:15006)], Linear Algebra Appl., 2001, vol. 327, no. 1–3, pp. 69–83. DOI: https://doi.org/10.1016/S0024-3795(00)00314-1

    Article  MathSciNet  MATH  Google Scholar 

  35. Berger, M.A. and Wang, Y., Bounded Semigroups of Matrices, Linear Algebra Appl., 1992, vol. 166, pp. 21–27. DOI: https://doi.org/10.1016/0024-3795(92)90267-E

    Article  MathSciNet  MATH  Google Scholar 

  36. Gurvits, L., Stability of Discrete Linear Inclusion, Linear Algebra Appl., 1995, vol. 231, pp. 47–85. DOI: https://doi.org/10.1016/0024-3795(95)90006-3

    Article  MathSciNet  MATH  Google Scholar 

  37. Theys, J., Joint Spectral Radius: Theory and Approximations, PhD Dissertation, Faculté des sciences appliquees, Departement d’ingénierie mathématique, Center for Systems Engineering and Applied Mechanics., Université Catholique de Louvain, 2005.

    Google Scholar 

  38. Czornik, A., On the Generalized Spectral Subradius, Linear Algebra Appl., 2005, vol. 407, pp. 242–248. DOI: https://doi.org/10.1016/j.laa.2005.05.006

    Article  MathSciNet  MATH  Google Scholar 

  39. Jungers, R., The Joint Spectral Radius, Berlin: Springer-Verlag, 2009, vol. 385 of Lecture Notes in Control and Information Sciences. DOI: https://doi.org/10.1007/978-3-540-95980-9

    Book  Google Scholar 

  40. Lagarias, J.C. and Wang, Y., The Finiteness Conjecture for the Generalized Spectral Radius of a Set of Matrices, Linear Algebra Appl., 1995, vol. 214, pp. 17–42. DOI: https://doi.org/10.1016/0024-3795(93)00052-2

    Article  MathSciNet  MATH  Google Scholar 

  41. Bousch, T. and Mairesse, J., Asymptotic Height Optimization for Topical IFS, Tetris Heaps, and the Finiteness Conjecture, J. Am. Math. Soc., 2002, vol. 15, no. 1, pp. 77–111. DOI: https://doi.org/10.1090/S0894-0347-01-00378-2

    Article  MathSciNet  MATH  Google Scholar 

  42. Blondel, V.D., Theys, J., and Vladimirov, A.A., An Elementary Counterexample to the Finiteness Conjecture, SIAM J. Matrix Anal. Appl., 2003, vol. 24, no. 4, pp. 963–970. DOI: https://doi.org/10.1137/S0895479801397846

    Article  MathSciNet  MATH  Google Scholar 

  43. Kozyakin, V., A Dynamical Systems Construction of a Counterexample to the Finiteness Conjecture, Proc. 44 IEEE Conf. Decision Control 2005 and 2005 Eur. Control Conf. CDC-ECC05, 2005, pp. 2338–2343. DOI: https://doi.org/10.1109/CDC.2005.1582511

    Google Scholar 

  44. Kozyakin, V.S., Structure of Extremal Trajectories of Discrete Linear Systems and the Finiteness Conjecture, Autom. Remote Control, 2007, vol. 68, no. 1, pp. 174–209. DOI: https://doi.org/10.1134/S0005117906040171

    Article  MATH  Google Scholar 

  45. Hare, K.G., Morris, I.D., Sidorov, N., and Theys, J., An Explicit Counterexample to the Lagarias-Wang Finiteness Conjecture, Adv. Math., 2011, vol. 226, no. 6, pp. 4667–4701. DOI: https://doi.org/10.1016/j.aim.2010.12.012

    Article  MathSciNet  MATH  Google Scholar 

  46. Morris, I. and Sidorov, N., On a Devil’s Staircase Associated to the Joint Spectral Radii of a Family of Pairs of Matrices, J. Eur. Math. Soc. (JEMS), 2013, vol. 15, no. 5, pp. 1747–1782. DOI: https://doi.org/10.4171/JEMS/402

    Article  MathSciNet  MATH  Google Scholar 

  47. Jenkinson, O. and Pollicott, M., Joint Spectral Radius, Sturmian Measures, and the Finiteness Conjecture, Ergodic Theory Dynam. Syst., 2017, pp. 1–39. DOI: https://doi.org/10.1017/etds.2017.18

  48. Jungers, R.M., On Asymptotic Properties of Matrix Semigroups with an Invariant Cone, Linear Algebra Appl., 2012, vol. 437, no. 5, pp. 1205–1214. DOI: https://doi.org/10.1016/j.laa.2012.04.006

    Article  MathSciNet  MATH  Google Scholar 

  49. Bochi, J. and Morris, I.D., Continuity Properties of the Lower Spectral Radius, Proc. Lond. Math. Soc. (3), 2015, vol. 110, no. 2, pp. 477–509. DOI: https://doi.org/10.1112/plms/pdu058

    Article  MathSciNet  MATH  Google Scholar 

  50. Czornik, A. and Jurgaś, P., Falseness of the Finiteness Property of the Spectral Subradius, Int. J. Appl. Math. Comput. Sci., 2007., vol. 17, no. 2, pp. 173–178. DOI: https://doi.org/10.2478/v10006-007-0016-1

    Article  MathSciNet  MATH  Google Scholar 

  51. Plischke, E. and Wirth, F., Duality Results for the Joint Spectral Radius and Transient Behavior, Linear Algebra Appl., 2008, vol. 428, no. 10, pp. 2368–2384. DOI: https://doi.org/10.1016/j.laa.2007.12.009

    Article  MathSciNet  MATH  Google Scholar 

  52. Jungers, R.M. and Blondel, V.D., On the Finiteness Property for Rational Matrices, Linear Algebra Appl., 2008, vol. 428, no. 10, pp. 2283–2295. DOI: https://doi.org/10.1016/j.laa.2007.07.007

    Article  MathSciNet  MATH  Google Scholar 

  53. Cicone, A., Guglielmi, N., Serra-Capizzano, S., and Zennaro, M., Finiteness Property of Pairs of 2×2 Sign-Matrices via Real Extremal Polytope Norms, Linear Algebra Appl., 2010, vol. 432, no. 2–3, pp. 796–816. DOI: https://doi.org/10.1016/j.laa.2009.09.022

    Article  MathSciNet  MATH  Google Scholar 

  54. Dai, X., Huang, Y., Liu, J., and Xiao, M., The Finite-Step Realizability of the Joint Spectral Radius of a Pair of d × d Matrices One of Which Being Rank-One, Linear Algebra Appl., 2012, vol. 437, no. 7, pp. 1548–1561. DOI: https://doi.org/10.1016/j.laa.2012.04.053

    Article  MathSciNet  MATH  Google Scholar 

  55. Liu, J. and Xiao, M., Computation of Joint Spectral Radius for Network Model Associated with Rank-One Matrix Set, Neural Inform. Proc. Proc. 19 Int. Conf., ICONIP 2012, Doha, Qatar, November 12–15, 2012, part III., Springer Berlin Heidelberg, 2012, vol. 7665 of Lecture Notes Comput. Sci, pp. 356–363. DOI: https://doi.org/10.1007/978-3-642-34487-9_44

    Google Scholar 

  56. Liu, J. and Xiao, M., Rank-One Characterization of Joint Spectral Radius of Finite Matrix Family, Linear Algebra Appl., 2013, vol. 438, no. 8, pp. 3258–3277. DOI: https://doi.org/10.1016/j.laa.2012.12.032

    Article  MathSciNet  MATH  Google Scholar 

  57. Morris, I.D., Rank One Matrices Do Not Cntribute to the Failure of the Finiteness Property, ArXiv.org e-Print archive, 2011, arXiv: 1109.4648.

    Google Scholar 

  58. Wang, S. and Wen, J., The Finiteness Conjecture for the Joint Spectral Radius of a Pair of Matrices, Proc. 9 Int. Conf. Comput. Intelligence Security (CIS), 2013, Emeishan, China, December 14–15, 2013, pp. 798–802.

    Google Scholar 

  59. Nesterov, Y. and Protasov, V.Y., Optimizing the Spectral Radius, SIAM J. Matrix Anal. Appl., 2013, vol. 34, no. 3, pp. 999–1013. DOI: https://doi.org/10.1137/110850967

    Article  MathSciNet  MATH  Google Scholar 

  60. Asarin, E., Cervelle, J., Degorre, A., et al., Entropy Games and Matrix Multiplication Games, 33 Symp.Theor.Aspects Comp.Sci. (STACS 2016), Ollinger, N. and Vollmer, H., Eds., vol. 47 of LIPIcs, Leibniz Int. Proc. Inform., Dagstuhl, Germany: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016, pp. 11:1–11:14. DOI: https://doi.org/10.4230/LIPIcs.STACS.2016.11

    Google Scholar 

  61. Golan, J.S., Semirings and Their Applications, Dordrecht: Kluwer, 1999. DOI: https://doi.org/10.1007/978-94-015-93335

    Book  MATH  Google Scholar 

  62. Borisovich, Y.G., Gel’man, B.D., Myshkis, A.D., and Obukhovskii, V.V., Multivalued Mappings, J. Soviet Math., 1984, vol. 24, no. 6, pp. 719–791. DOI: https://doi.org/10.1007/BF01305758

    Article  MATH  Google Scholar 

  63. Protasov, V.Yu., Spectral Simplex Method, Math. Program., 2016, vol. 156, no. 1–2, Ser. A, pp. 485–511. DOI: https://doi.org/10.1007/s10107-015-0905-2

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    V. S. Kozyakin

  2. Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, Russia

    V. S. Kozyakin, N. A. Kuznetsov & P. Yu. Chebotarev

  3. Moscow Institute of Physics and Technology, Moscow, Russia

    N. A. Kuznetsov & P. Yu. Chebotarev

  4. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    P. Yu. Chebotarev

Authors
  1. V. S. Kozyakin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Yu. Chebotarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. S. Kozyakin, N. A. Kuznetsov or P. Yu. Chebotarev.

Additional information

Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 5, pp. 3–31.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozyakin, V.S., Kuznetsov, N.A. & Chebotarev, P.Y. Consensus in Asynchronous Multiagent Systems. II. Method of Joint Spectral Radius. Autom Remote Control 80, 791–812 (2019). https://doi.org/10.1134/S0005117919050011

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919050011

Keywords

Search

Navigation