The Over-Damped String Stability Condition for a Platooning System
Keywords:over-damped string stability, classical string stability, platooning control, adaptive cruise control
The over-damped string stability criterion is a very strong stability condition that not only addresses the stability in a stricter sense but also adequately captures the safety performance of a platoon. However, the mathematical representation of this criterion is incomplete in the literature. Here, this representation is completely described. Moreover, this article presents the mathematical test method to evaluate this stability condition for linear or linearized systems from the transfer function. The classical sting stability condition does not address the transient undesired convergent dynamics of a platoon, such as over-shooting, under-shooting or damped oscillating dynamics. This paper demonstrates that the over-damped string stability characteristic significantly attenuates these undesired convergent dynamics in the upstream direction. Thus, the advantage of this condition over the classical criterion for linear system is clarified theoretically and by simulation. Later, the numerical method to analyze the over-damped string stability criterion for nonlinear systems is discussed. Additionally, numerical simulations of an over-damped string stable adaptive cruise control (ACC) vehicle model are compared with that of some experimental test results on platoons of commercially implemented ACC equipped vehicles.
R. Rajamani, Vehicle dynamics and control, 2nd ed. New York, NY, USA: Springer Science & Business Media, 2012, pp. 141-200.
P. Khound, P. Will, and F. Gronwald, “Local and string stability conditions of a generalized adaptive cruise control system,” in AmE 2020 - Automotive meets Electronics; 11th GMM-Symp., Dortmund, Germany, Mar. 2020, pp. 29–36.
J. Zhou and H. Peng, “String stability conditions of adaptive cruise control algorithms,” in IFAC Symp. Advances in Automotive Control, Salerno, Italy, Apr. 2004, pp. 649–654.
J. Zhou and H. Peng, “Range policy of adaptive cruise control vehicles for improved flow stability and string stability,” IEEE Trans. Intell. Transp. Syst., vol. 6, no. 2, pp. 229-237, Jun. 2005.
C. Wu, Z. Xu, Y. Liu, C. Fu, K. Li, and M. Hu, “Spacing policies for adaptive cruise control: a survey,” IEEE Access, vol. 8, pp. 50149-50162, 2020.
H. Xing, J. Ploeg, and H. Nijmeijer, “Compensation of communication delays in a cooperative ACC system,” IEEE Trans. Veh. Tech., vol. 69, no. 2, pp. 1177-1189, Feb. 2020.
P. Khound, P. Will, and F. Gronwald, “Design methodology to derive over-damped string stable adaptive cruise control systems,” IEEE Trans. Intell. Veh., vol. 7, no. 1, pp. 32-44, Mar. 2022.
J. Lunze, “Adaptive cruise control with guaranteed collision avoidance,” IEEE Trans. Intell. Transp. Syst., vol. 20, no. 5, pp. 1897-1907, May. 2019.
J. Lunze, “Design of the communication structure of cooperative adaptive cruise controllers,” IEEE Trans. Intell. Transp. Syst., vol. 21, no. 10, pp. 4378-4387, Oct. 2020.
A. Schwab and J. Lunze, “Design of platooning controllers that achieve collision avoidance by external positivity,” IEEE Trans. Intell. Transp. Syst., vol. 23, no. 9, pp. 14883-14892, Sep. 2022.
P. Khound, P. Will, A. Tordeux, and F. Gronwald, “Extending the adaptive time gap car-following model to enhance local and string stability for adaptive cruise control systems,” J. Intell. Transp. Syst., vol. 27, no. 1, pp. 36-56, Jan. 2023.
M. Montanino, J. Monteil, and V. Punzo, “From homogeneous to heterogeneous traffic flows: Lp String stability under uncertain model parameters,” Transp. Res. Part B Meth., vol. 146, pp. 136-254, Apr. 2021.
P. Khound, P. Will, A. Tordeux, and F. Gronwald, “Unified framework for over-damped string stable adaptive cruise control systems,” Transp. Res. Part C Emerg. Technol., vol. 148, pp. 1-24, Mar. 2023.
A. Tordeux, S. Lassarre, and M. Roussignol, “An adaptive time gap carfollowing model,” Transp. Res. Part B Meth., vol. 44, pp. 1115-1131, 2010.
P. Khound, P. Will, A. Tordeux, and F. Gronwald, “The importance of the over-damped string stability criterion for a platooning control system,” in 26th Internat. Conf. Syst. Theory, Contrl. and Comp. (ICSTCC), IEEE, Sinaia, Romania, Oct. 2022, pp. 1-6.
G. Gunter, D. Gloudemans, R. E. Stern, S. McQuade, R. Bhadani, M. Bunting, M. L. D. Monache, R. Lysecky, B. Seibold, J. Sprinkle, B. Piccoli, and D. B. Work, “Are commercially implemented adaptive cruise control systems string stable?,” IEEE Trans. Intell. Transp. Syst., vol. 22, no. 11, pp. 6992-7003, Nov. 2021.
A. Anesiadou, M. Makridis, B. Ciuffo, and K. Mattas, Open ACC Database. European Commission, Joint Research Centre (JRC), 2020, [Dataset] PID: http://data.europa.eu/89h/9702c950-c80f-4d2f-982f-44d06ea0009f
S. Darbha, S. Konduri, and P. R. Pagilla, “Vehicle platooning with constant spacing strategies and multiple vehicle look ahead information”, Intell. Transport Syst., vol. 14, no. 6, pp. 589-600, Jun. 2020.
H. Lutz andW.Wendt, Taschenbuch der Regelungstechnik, vol. 2, Verlag Harri Deutsch, 1995.
N. F. Macia and G. J. Thaler, Modeling and control of dynamic systems, Cengage Learning, 2005, pp. 215-220.
S. Janson, “Roots of polynomials of degrees 3 and 4”, arXiv:1009.2373v1, Sep. 2010.
V. Milan´es and S. Shaldover, “Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data,” Transp. Res. Part C Emerg. Technol., vol. 48, pp. 285-300, Nov. 2014.
M. Makridis, K. Mattas, A. Anesiadou, and B. Ciuffo, “OpenACC. An open database of car-following experiments to study the properties of commercial ACC systems,” Transp. Res. Part C Emerg. Technol., vol. 125, no. 103047, pp. 1-19, Apr. 2021.
B. Ciuffo, K. Mattas, M. Makridis, G. Albano, A. Anesiadou, Y. He, S. Josvai, D. Komnos, M. Pataki, S. Vass, Z. Szalay, “Requiem on the positive effects of commercial adaptive cruise control on motorway traffic and recommendations for future automated driving systems,” Transp. Res. Part C Emerg. Technol., vol. 130, pp. 1-31, Sep. 2021.
G. Gunter, et al. (2019), Experimental Data: Platoon data [Online]. Available: https://vanderbilt.app.box.com/v/accData (accessed Nov. 20th, 2022).
A. Tordeux (2022), Online simulation with NetLogo: ACC equipped vehicles: Classical stability VS Overdamped stability [Online]. Available: https://www.vzu.uni-wuppertal.de/fileadmin/site/vzu/Damped_stability_VS_Overdamped_stability.html?speed=0.5 (accessed Nov. 30th, 2022).
A. Tordeux (2022), Online simulation with NetLogo: ACC equipped vehicles: European OpenACC experiment [Online]. Available: https://www.vzu.uni-wuppertal.de/fileadmin/site/vzu/OpenACC_experiment_2019.html?speed=0.5 (accessed Dec. 05th, 2022).