Positive Systems pp 339-347

Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 389) | Cite as

A Problem in Positive Systems Stability Arising in Topology Control

  • Florian Knorn
  • Rade Stanojevic
  • Martin Corless
  • Robert Shorten

Abstract

We present a problem in the stability of switched positive systems that arises in network topology control. Preliminary results are given that guarantee stability of a network topology control problem under certain assumptions. Roughly speaking, these assumptions reduce the underlying stability problem to a nonlinear consensus problem with a driving term, that eventually becomes a Lur’e problem. Simulation results are given to illustrate our algorithm. While these results indicate that our assumptions can be removed, a proof of the general stability problem remains open.

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References

  1. 1.
    Chung, F.R.K. (ed.): Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society, Province (1997)MATHGoogle Scholar
  2. 2.
    Estrin, D., Girod, L.D., Pottie, G.J., Srivastava, M.: Instrumenting the world with wireless sensor networks. In: Proc. of the Int. Conf. on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, vol. 4, pp. 2033–2036 (2001)Google Scholar
  3. 3.
    Fiedler, M.: Algebraic connectivity of graphs. Czech. Math. J. 23(98), 298–305 (1973)MathSciNetGoogle Scholar
  4. 4.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Khalil, H.K.: Nonlinear Systems. Macmillan Publishing Co., New York (1992)MATHGoogle Scholar
  6. 6.
    Knorn, F., Stanojevic, R., Corless, M., Shorten, R.: A framework for decentralised feedback connectivity control with application to sensor networks. Int. J. Control (to appear, 2009)Google Scholar
  7. 7.
    Moreau, L.: Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Reynolds, C.W.: Flocks, herds, and schools: A distributed behavioral model. In: Proc. of the 14th Annual Conf. on Computer Graphics and Interactive Techniques, Anaheim, CA, USA, pp. 25–34 (1987)Google Scholar
  10. 10.
    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995)CrossRefGoogle Scholar
  11. 11.
    Zavlanos, M.M., Pappas, G.J.: Controlling connectivity of dynamic graphs. In: Proc. of the Joint 44th IEEE Conf. on Decision and Control, and the European Control Conf., Seville, Spain, pp. 6388–6393 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Florian Knorn
    • 1
  • Rade Stanojevic
    • 1
  • Martin Corless
    • 2
  • Robert Shorten
    • 1
  1. 1.Hamilton InstituteNational University of Ireland MaynoothMaynoothIreland
  2. 2.School of Aeronautics & AstronauticsPurdue UniversityWest LafayetteUSA

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