» Download Article

Authors
Rifat Sipahi; Silviu-Iulian Niculescu

Digital Object Identifier (DOI)
10.3182/20060710-3-IT-4901.00031

Page Numbers:
187-192

Index Terms
car following model,delay,crossing curves,stability,sensitivity

Abstract
Analytical stability study of some deterministic car following models under time-delay influences is presented and various case studies are demonstrated. Interestingly, for some control law deployed by human drivers, more than one stability interval in the domain of time delay is revealed. Physical interpretations along with comparisons conclude the study.

References

[1] R. W. Allen, T. D. Marcotte, T. J. Rosenthal, B. L.
    Aponso, "Driver Assesment with Measure of
    Continuous Control", Proc. 3rd Int. Driving
    Symp. on Human Factors in Driver Assess.,
    Training and Vehicle Design, Rockport, Maine,
    USA, 2005.
[2] M. Bando, K. Hasebe, K. Nakanishi, A. Nakayama,
    "Delay of Vehicle Motion in Traffic Dynamics",
    Aichi University, 1996, Internal Report.
[3] R. E. Chandler, R. Herman, and E. W. Montroll, "Traffic
     Dynamics: Analysis of Stability in Car Following.
     Operational Research", Vol. 7, No. 1,
    pp. 165-184, 1958.
[4] M. Green, "How Long Does It Take to Stop?"
    Methodological Analysis of Driver Perception-Brake
     Times," Transportation Human Factors,
    vol. 2, pp. 195-216, 2000.
[5] K. Gu, S.-I. Niculescu, and J. Chen, "On Stability
    Crossing Curves for General Systems with Two
    Delays," J. Math. Anal. Appl., vol. 311, pp. 231-
    253, 2005.
[6] J. K. Hale and S. M. Verduyn Lunel, An Introduction
    to Functional Differential Equations. New York:
    Springer-Verlag, 1993.
[7] J. K. Hale and W. Huang, "Global Geometry of the
    Stable Regions for Two Delay Differential Equations,"
     J. Math. Anal. Appl., vol. 178, pp. 344-
    362, 1993.
[8] D. Helbing, "Traffic and Related Self-Driven Many-Particle
     Systems," Reviews of Modern Physics,
    vol. 73, pp. 1067-1141, 2001.
[9] Hoogendoorn, S. P. and Ossen, S., "Parameter Estimation
     and Analysis of Car-Following Models". In
    HS Mahmassani (ed.) Flow, Dynamics and Human
     Interaction (Transportation and Traffic Theory,
     16), (pp. 245-266). Elsevier, The Netherlands,
     2005.
[10] B. S. Kerner, "Experimental Features of Self-Organization
      in Traffic Flow," Physical Review
     Letters, vol. 81, pp. 3797-3800, 1998.
[11] W. Michiels, Stability and stabilization of time-delay
     systems, Ph.D. Thesis, KU Leuven, Belgium,
     May 2002.
[12] S.-I. Niculescu, Delay Effects on Stability, Springer-Verlag,
      2001.
[13] T. Nagatani, "Stabilization and Enhancement of Traffic
      Flow by the Next-Nearest-Neighbor Interaction,"
      Physical Review E, vol. 60, 1999.
[14] A. Nakayama, Y. Sugiyama, and K. Hasebe, "Effect
     of Looking at the Car That Follows in an Optimal
     Velocity Model of Traffic Flow," Physical Review
     E, vol. 65, 2001.
[15] N. Olgac and R. Sipahi, "An Exact Method for the
     Stability Analysis of Time Delayed Lti Systems,"
     IEEE Trans. Auto. Control, vol. 47, pp. 793-797,
     2002.
[16] G. Orosz and G. Stepan, "Hopf bifurcation calculations
      in delayed systems with translational symmetry",
      J. Nonlin. Sci., 14 (2004) 505-528.
[17] G. Orosz, R. E. Wilson and B. Krauskopf, "Bifurcations
      in a Car-Following Model with Delay",
     IFAC TDS, Leuven, Belgium, 2004.
[18] R. W. Rothery, "Transportation Research Board (Trb)
     Special Report 165," in "Traffic Flow Theory",
     2nd Edition, N. H. Gartner, C. J. Messner, and A.
     J. Rathi, Eds., 1998.
[19] R. Sipahi, Cluster Treatment of Characteristic Roots,
     CTCR, A Unique Methodology for the Complete
     Stability Robustness Analysis of Linear Time Invariant
      Multiple Time Delay Systems Against
     Delay Uncertainties, Ph.D. Thesis, University
     of Connecticut, Mechanical Engineering Department,
      August 2005.
[20] R. Sipahi and N. Olgac, "Complete Stability Robustness
      of Third-Order Lti Multiple Time-Delay
     Systems," Automatica, vol. 41, pp. 1413-1422,
     2005.
[21] R. Sipahi and S.-I. Niculescu, "Some Remarks on the
     Characterization of Delay Interactions in Deterministic
      Car Following Models", MTNS Proc.,
     July 2006 in Kyoto (Japan).
[22] G. Stepan, Retarded Dynamical Systems: Stability and
     Characteristic Function. Longman, New York,
     US, 1989.
[23] M. Treiber, A. Hennecke, and D. Helbing, "Congested
     traffic states in empirical observations and microscopic
      simulations", Physical review E, vol. 62,
     pp. 1805-1824, 2000.
[24] M. Treiber, A. Kesting and D. Helbing, "Delays, Inaccuracies
      and Anticipation in Microscopic Traffic
     Models", Physica A, Volume 360, 71-88 (2006).