access icon free Some simple criteria for pinning a Lur’e network with directed topology

This study considers the pinning synchronisation in a network of coupled Lur’e dynamical systems under directed topology. By using tools from M-matrix theory, S-procedure and Lyapunov functional method, some simple pinning criteria in terms of linear matrix inequalities, whose dimensions are just determined by the size of a single Lur’e node, are derived for Lur’e networks with fixed and designed inner coupling matrices, respectively. A selective pinning scheme is proposed for a directed Lur’e network such that the network can be globally asymptotically pinned to a homogeneous state. Simulation results are provided to illustrate the effectiveness of the theoretical analysis.

Inspec keywords: network topology; matrix algebra; network theory (graphs); complex networks; synchronisation; Lyapunov methods

Other keywords: Lur’e network pinning; inner coupling matrices; directed Lur’e network; homogeneous state; S-procedure; M-matrix theory; linear matrix inequalities; Lyapunov functional method; directed topology; coupled Lur’e dynamical systems

Subjects: Combinatorial mathematics; Algebra; Combinatorial mathematics; Algebra; Combinatorial mathematics; Algebra, set theory, and graph theory; Algebra

References

    1. 1)
      • 21. Song, Q., Cao, J., Yu, W.: ‘Second-order leader-following consensus of nonlinear multi-agent systems via pinning control’, Syst. Control Lett., 2010, 59, pp. 553562 (doi: 10.1016/j.sysconle.2010.06.016).
    2. 2)
      • 16. Song, Q., Cao, J.: ‘On pinning synchronization of directed and undirected complex dynamical networks’, IEEE Trans. Circuits Syst. I, 2010, 57, (3), pp. 672680 (doi: 10.1109/TCSI.2009.2024971).
    3. 3)
      • 33. Yalçin, M.E., Suykens, J.A.K., Vandewalle, J.: ‘Master-slave synchronization of Lur’e systems with time-delay’, Int. J. Bifurcation Chaos, 2001, 11, (6), pp. 17071722 (doi: 10.1142/S021812740100295X).
    4. 4)
      • 23. Song, Q., Liu, F., Cao, J., Yu, W.: ‘M-matrix strategies for pinning-controlled leader-following consensus in multi-agent systems with nonlinear dynamics’, IEEE Trans. Cybern., 2013, doi:10.1109/TSMCB.2012.2227723, in press.
    5. 5)
      • 9. Wang, X.F., Chen, G.: ‘Pinning control of scale-free dynamical networks’, Physica A, 2002, 310, pp. 521531 (doi: 10.1016/S0378-4371(02)00772-0).
    6. 6)
      • 35. Han, Q.L., Yue, D.: ‘Absolute stability of Lur’e systems with time-varying delay’, IET Control Theory Appl., 2007, 1, (3), pp. 854859 (doi: 10.1049/iet-cta:20060213).
    7. 7)
      • 37. Zhang, Q., Li, Z.: ‘Pinning control of complex Lur’e networks’, Chinese Phys. B, 2009, 18, (6), pp. 21762183 (doi: 10.1088/1674-1056/18/6/011).
    8. 8)
      • 2. Barabási, A.L., Albert, R.: ‘Emergence of scaling in random networks’, Science, 1999, 286, pp. 509512 (doi: 10.1126/science.286.5439.509).
    9. 9)
      • 4. Pecora, L.M., Carroll, T.L.: ‘Master stability functions for synchronized coupled systems’, Phys. Rev. Lett., 1998, 80, (10), pp. 21092112 (doi: 10.1103/PhysRevLett.80.2109).
    10. 10)
      • 19. Horn, R.A., Johnson, C.R.: ‘Topics in matrix analysis’ (Cambridge University Press, Cambridge, UK, 1991).
    11. 11)
      • 31. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: ‘Linear matrix inequalities in system and control theory’ (SIAM, Philadelphia, PA, 1994).
    12. 12)
      • 8. Wang, P., Lü, J., Ogorzalek, M. J.: ‘Global relative parameter sensitivities of the feed-forward loops in genetic networks’, Neurocomputing, 2012, 78, pp. 155165 (doi: 10.1016/j.neucom.2011.05.034).
    13. 13)
      • 36. Gonzaga, C.A.C., Jungers, M., Daafouz, J.: ‘Stability analysis of discrete-time Lur’e systems’, Automatica, 2012, 48, pp. 22772283 (doi: 10.1016/j.automatica.2012.06.034).
    14. 14)
      • 28. Gazi, V., Passino, K.M.: ‘Stability analysis of swarms’, IEEE Trans. Autom. Control, 2003, 48, (4), pp. 692697 (doi: 10.1109/TAC.2003.809765).
    15. 15)
      • 22. Wieland, P., Kim, J.S., Allgöwer, F.: ‘On topology and dynamics of consensus among linear high-order agents’, Int. J. Syst. Sci., 2011, 42, (10), pp. 18311842 (doi: 10.1080/00207721003658202).
    16. 16)
      • 6. Lü, J., Chen, G.: ‘A time-varying complex dynamical network model and its controlled synchronization criteria’, IEEE Trans. Autom. Control, 2005, 50, (6), pp. 841846 (doi: 10.1109/TAC.2005.849233).
    17. 17)
      • 24. Olfati-Saber, R., Murray, R.M.: ‘Consensus problems in networks of agents with switching topology and time-delays’, IEEE Trans. Autom. Control, 2004, 49, (9), pp. 15201533 (doi: 10.1109/TAC.2004.834113).
    18. 18)
      • 5. Lü, J., Yu, X., Chen, G., Cheng, D.: ‘Characterizing the synchronizability of small-world dynamical networks’, IEEE Trans. Circuits Syst. I, 2004, 51, (4), pp. 787796 (doi: 10.1109/TCSI.2004.823672).
    19. 19)
      • 12. Chen, T., Liu, X., Lu, W.: ‘Pinning complex networks by a single controller’, IEEE Trans. Circuits Syst. I, 2007, 54, (6), pp. 13171326 (doi: 10.1109/TCSI.2007.895383).
    20. 20)
      • 29. Olfati-Saber, R.: ‘Flocking for multi-agent dynamic systems: algorithms and theory’, IEEE Trans. Autom. Control, 2006, 51, (3), pp. 401420 (doi: 10.1109/TAC.2005.864190).
    21. 21)
      • 32. Park, P.: ‘A revisited Popov criterion for nonlinear Lur’e systems with sector-restrictions’, Int. J. Control, 1997, 68, (3), pp. 461469 (doi: 10.1080/002071797223479).
    22. 22)
      • 1. Watts, D.J., Strogatz, S.H.: ‘Collective dynamics of ‘small-world’ networks’, Nature, 1998, 393, pp. 440442 (doi: 10.1038/30918).
    23. 23)
      • 20. Berman, A., Plemmons, R.J.: ‘Nonnegative matrices in the mathematical sciences’ (SIAM, Philadelphia, PA, 1994).
    24. 24)
      • 17. Lu, W., Li, X., Rong, Z.: ‘Global stabilization of complex networks with diagraph topologies via a local pinning algorithm’, Automatica, 2010, 46, pp. 116121 (doi: 10.1016/j.automatica.2009.10.006).
    25. 25)
      • 15. Lu, J., Ho, D.W.C., Wang, Z.: ‘Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers’, IEEE Trans. Neural Netw., 2009, 20, (10), pp. 16171629 (doi: 10.1109/TNN.2009.2027810).
    26. 26)
      • 40. DeLellis, P., Di Bernardo, M.: ‘Adaptive pinning control of complex networks of Lur’e systems’. Proc. 51st IEEE Conf. on Decision and Control, Maui, Hawaii, USA, 2012, pp. 60606064.
    27. 27)
      • 25. Ren, W., Beard, R.W.: ‘Consensus seeking in multiagent systems under dynamically changing interaction topologies’, IEEE Trans. Autom. Control, 2005, 50, (5), pp. 655661 (doi: 10.1109/TAC.2005.846556).
    28. 28)
      • 18. Song, Q., Liu, F., Cao, J., Yu, W.: ‘Pinning-controllability analysis of complex networks: An M-Matrix approach’, IEEE Trans. Circuits Syst. I, 2012, 59, (11), pp. 26922701 (doi: 10.1109/TCSI.2012.2190573).
    29. 29)
      • 3. Wu, C.W., Chua, L.O.: ‘Synchronization in an array of linearly coupled dynamical systems’, IEEE Trans. Circuits Syst. I, 1995, 42, (8), pp. 430447 (doi: 10.1109/81.404047).
    30. 30)
      • 27. Wen, G., Duan, Z., Yu, W., Chen, G.: ‘Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach’, Int. J. Robust Nonlinear Control, 2013, 23, (6), pp. 602619 (doi: 10.1002/rnc.2779).
    31. 31)
      • 26. Chen, Y., Lü, J., Lin, Z.: ‘Consensus of discrete-time multi-agent systems with transmission nonlinearity’, Automatica, 2013, 49, pp. 17681775 (doi: 10.1016/j.automatica.2013.02.021).
    32. 32)
      • 11. Yu, W., Chen, G., Lü, J.: ‘On pinning synchronization of complex dynamical networks’, Automatica, 2009, 45, pp. 429435 (doi: 10.1016/j.automatica.2008.07.016).
    33. 33)
      • 30. Zhu, J., Lü, J., Yu, X.: ‘Flocking of multi-agent non-holonomic systems with proximity graphs’, IEEE Trans. Circuits Syst. I, 2013, 60, (1), pp. 199210 (doi: 10.1109/TCSI.2012.2215715).
    34. 34)
      • 39. Guo, L., Nian, X., Zhao, Y., Duan, Z.: ‘Cluster synchronisation of Lur’e dynamical networks’, IET Control Theory Appl., 2012, 6, (16), pp. 24992508 (doi: 10.1049/iet-cta.2012.0467).
    35. 35)
      • 7. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: ‘Synchronization in complex networks’, Phys. Rep., 2008, 469, pp. 93153 (doi: 10.1016/j.physrep.2008.09.002).
    36. 36)
      • 10. Li, X., Wang, X.F., Chen, G.: ‘Pinning a complex dynamical network to its equilibrium’, IEEE Trans. Circuits Syst. I, 2004, 51, (10), pp. 20742087 (doi: 10.1109/TCSI.2004.835655).
    37. 37)
      • 14. Zhou, J., Lu, J., Lü, J.: ‘Pinning adaptive synchronization of a general complex dynamical network’, Automatica, 2008, 44, pp. 9961003 (doi: 10.1016/j.automatica.2007.08.016).
    38. 38)
      • 34. Khalil, H.K.: ‘Nonlinear systems’ (Prentice-Hall, Englewood Cliffs, NJ, 2002, 3rd edn.).
    39. 39)
      • 13. Lu, Y.Y., Wang, X.F.: ‘Pinning control of directed dynamical networks based on ControlRank’, Int. J. Comput. Math., 2008, 85, (8), pp. 12791286 (doi: 10.1080/00207160701665948).
    40. 40)
      • 38. Li, Z., Duan, Z., Chen, G.: ‘Global synchronised regions of linearly coupled Lur’e systems’, Int. J. Control, 2011, 84, (2), pp. 216227 (doi: 10.1080/00207179.2010.546882).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2013.0422
Loading

Related content

content/journals/10.1049/iet-cta.2013.0422
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading