Graph Laplacian Potential and Lyapunov Functions for Multi-Agent Systems

  • Frank L. Lewis
  • Hongwei Zhang
  • Kristian Hengster-Movric
  • Abhijit Das
Chapter
Part of the Communications and Control Engineering book series (CCE)

Abstract

In this chapter we show that for networked multi-agent systems, there is an energy-like function, called the graph Laplacian potential, that depends on the communication graph topology. The Laplacian potential captures the notion of a virtual potential energy stored in the graph. We shall study the Laplacian potential for both undirected graphs and directed graphs. The Laplacian potential is further used here to construct Lyapunov functions that are suitable for the analysis of cooperative control systems on graphs. These Lyapunov functions depend on the graph topology, and based on them a Lyapunov analysis technique is introduced for cooperative multi-agent systems on graphs. Control protocols coming from such Lyapunov functions are distributed in form, depending only on information about the agent and its neighbors.

References

  1. 1.
    Astrom KJ, Wittenmark B (2008) Adaptive Control, 2nd edn. Dover, MineolaGoogle Scholar
  2. 2.
    Bernstein D (2009) Matrix Mathematics: Theory, Facts, and Formulas, 2nd edn. Princeton University Press, PrincetonGoogle Scholar
  3. 3.
    Chopra N, Spong M (2006) Passivity-based control of multi-agent systems. In: Kawamura S, Svinin M (eds) Advances in Robot Control: From Everyday Physics to Human-Like Movements. Springer-Verlag, BerlinGoogle Scholar
  4. 4.
    Godsil C, Royle G (2001) Algebraic Graph Theory. Springer-Verlag, New YorkCrossRefMATHGoogle Scholar
  5. 5.
    Horn R, Johnson C (1990) Matrix Analysis. Cambridge University Press, New YorkMATHGoogle Scholar
  6. 6.
    Igarashi Y, Hatanaka T, Fujita M, Spong M (2008) Passivity-based output synchronization and flocking algorithm in SE (3). In: Proc. Amer. Control Conf., Seattle, WA, pp. 723–728Google Scholar
  7. 7.
    Ioannou P, Fidan B (2006) Adaptive Control Tutorial. SIAM Press, PhiladelphiaGoogle Scholar
  8. 8.
    Jadbabaie A, Lin J, Morse A (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Automat Contr 48(6):988–1001CrossRefMathSciNetGoogle Scholar
  9. 9.
    Khalil H (2002) Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle RiverGoogle Scholar
  10. 10.
    Landau ID, Lozano R, Saad MM (2011) Adaptive Control. Springer-Verlag, BerlinCrossRefMATHGoogle Scholar
  11. 11.
    Olfati-Saber R, Murray R (2003) Consensus protocols for networks of dynamic agents. In: Proc. Amer. Control Conf., Denver, CO, pp. 951–956Google Scholar
  12. 12.
    Olfati-Saber R, Murray R (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Automat Contr 49(9):1520–1533CrossRefMathSciNetGoogle Scholar
  13. 13.
    Qu Z (2009) Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles. Springer-Verlag, LondonGoogle Scholar
  14. 14.
    Ren W, Beard R (2008) Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications. Springer-Verlag, LondonGoogle Scholar
  15. 15.
    Ren W, Beard R, Atkins E (2007) Information consensus in multivehicle cooperative control. IEEE Contr Systs Mag 27(2):71–82CrossRefGoogle Scholar
  16. 16.
    Spong M, Hutchinson S, Vidyasagar M (2006) Robot Modeling and Control. Wiley, New YorkGoogle Scholar
  17. 17.
    Zhang H, Lewis FL, Qu Z (2012) Lyapunov, adaptive, and optimal design techniques for cooperative systems on directed communication graphs. IEEE Trans Ind Elec 59(7):3026–3041CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Frank L. Lewis
    • 1
  • Hongwei Zhang
    • 2
  • Kristian Hengster-Movric
    • 3
  • Abhijit Das
    • 4
  1. 1.UTA Research InstituteUniversity of Texas at ArlingtonFort WorthUSA
  2. 2.School of Electrical EngineeringSouthwest Jiaotong UniversityChengduChina, People’s Republic
  3. 3.UTA Research InstituteUniversity of Texas at ArlingtonFort WorthUSA
  4. 4.Advanced Systems Engineering​Danfoss Power Solutions (US) Company​AmesUSA

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