Stability analysis of a stochastic port-Hamiltonian car-following model
Authors
Barbara Rüdiger, Antoine Tordeux, Baris E Ugurcan
Abstract
Port-Hamiltonian systems are pertinent representations of many nonlinear physical systems. In this study, we formulate and analyse a general class of stochastic car-following models with a systematic port-Hamiltonian structure. The model class is a generalisation of classical car-following approaches, including the optimal velocity model of Bando et al (1995 Phys. Rev. E 51 1035), the full velocity difference model of Jiang et al (2001 Phys. Rev. E 64 017101), and recent stochastic following models based on the Ornstein–Uhlenbeck process. In contrast to traditional models where the interaction is totally asymmetric (i.e. depending only on the speed and distance to the predecessor), the port-Hamiltonian car-following model also depends on the distance to the follower. We determine the exact stability condition of the finite system with N vehicles and periodic boundaries. The stable system is ergodic with a unique Gaussian invariant measure. Other properties of the model are studied using numerical simulation. It turns out that the Hamiltonian component improves the flow stability and reduces the total energy in the system. Furthermore, it prevents the problematic formation of stop-and-go waves with oscillatory dynamics, even in the presence of stochastic perturbations.
Citation
- Journal: Journal of Physics A: Mathematical and Theoretical
- Year: 2024
- Volume: 57
- Issue: 29
- Pages: 295203
- Publisher: IOP Publishing
- DOI: 10.1088/1751-8121/ad5d2f
BibTeX
@article{R_diger_2024,
title={{Stability analysis of a stochastic port-Hamiltonian car-following model}},
volume={57},
ISSN={1751-8121},
DOI={10.1088/1751-8121/ad5d2f},
number={29},
journal={Journal of Physics A: Mathematical and Theoretical},
publisher={IOP Publishing},
author={Rüdiger, Barbara and Tordeux, Antoine and Ugurcan, Baris E},
year={2024},
pages={295203}
}
References
- Pipes, L. A. An Operational Analysis of Traffic Dynamics. Journal of Applied Physics vol. 24 274–281 (1953) – 10.1063/1.1721265
- Chandler, R. E., Herman, R. & Montroll, E. W. Traffic Dynamics: Studies in Car Following. Operations Research vol. 6 165–184 (1958) – 10.1287/opre.6.2.165
- Herman, R., Montroll, E. W., Potts, R. B. & Rothery, R. W. Traffic Dynamics: Analysis of Stability in Car Following. Operations Research vol. 7 86–106 (1959) – 10.1287/opre.7.1.86
- Gazis, D. C., Herman, R. & Rothery, R. W. Nonlinear Follow-the-Leader Models of Traffic Flow. Operations Research vol. 9 545–567 (1961) – 10.1287/opre.9.4.545
- Bando, M., Hasebe, K., Nakayama, A., Shibata, A. & Sugiyama, Y. Dynamical model of traffic congestion and numerical simulation. Physical Review E vol. 51 1035–1042 (1995) – 10.1103/physreve.51.1035
- Orosz, G., Wilson, R. E., Szalai, R. & Stépán, G. Exciting traffic jams: Nonlinear phenomena behind traffic jam formation on highways. Physical Review E vol. 80 (2009) – 10.1103/physreve.80.046205
- Orosz, G., Wilson, R. E. & Stépán, G. Traffic jams: dynamics and control. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 368 4455–4479 (2010) – 10.1098/rsta.2010.0205
- Nagel, K. & Schreckenberg, M. A cellular automaton model for freeway traffic. Journal de Physique I vol. 2 2221–2229 (1992) – 10.1051/jp1:1992277
- Barlovic, R., Santen, L., Schadschneider, A. & Schreckenberg, M. Metastable states in cellular automata for traffic flow. The European Physical Journal B vol. 5 793–800 (1998) – 10.1007/s100510050504
- Treiber, M. & Helbing, D. Hamilton-like statistics in onedimensional driven dissipative many-particle systems. The European Physical Journal B vol. 68 607–618 (2009) – 10.1140/epjb/e2009-00121-8
- Hamdar, S. H., Mahmassani, H. S. & Treiber, M. From behavioral psychology to acceleration modeling: Calibration, validation, and exploration of drivers’ cognitive and safety parameters in a risk-taking environment. Transportation Research Part B: Methodological vol. 78 32–53 (2015) – 10.1016/j.trb.2015.03.011
- Tordeux, A. & Schadschneider, A. White and relaxed noises in optimal velocity models for pedestrian flow with stop-and-go waves. Journal of Physics A: Mathematical and Theoretical vol. 49 185101 (2016) – 10.1088/1751-8113/49/18/185101
- Treiber, M. & Kesting, A. The Intelligent Driver Model with Stochasticity -New Insights Into Traffic Flow Oscillations. Transportation Research Procedia vol. 23 174–187 (2017) – 10.1016/j.trpro.2017.05.011
- Wang, Y., Li, X., Tian, J. & Jiang, R. Stability Analysis of Stochastic Linear Car-Following Models. Transportation Science vol. 54 274–297 (2020) – 10.1287/trsc.2019.0932
- Friesen, M., Gottschalk, H., Rüdiger, B. & Tordeux, A. Spontaneous Wave Formation in Stochastic Self-Driven Particle Systems. SIAM Journal on Applied Mathematics vol. 81 853–870 (2021) – 10.1137/20m1315567
- Ngoduy, D., Lee, S., Treiber, M., Keyvan-Ekbatani, M. & Vu, H. L. Langevin method for a continuous stochastic car-following model and its stability conditions. Transportation Research Part C: Emerging Technologies vol. 105 599–610 (2019) – 10.1016/j.trc.2019.06.005
- Xu, T. & Laval, J. A. Analysis of a Two-Regime Stochastic Car-Following Model: Explaining Capacity Drop and Oscillation Instabilities. Transportation Research Record: Journal of the Transportation Research Board vol. 2673 610–619 (2019) – 10.1177/0361198119850464
- Stern, R. E. et al. Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments. Transportation Research Part C: Emerging Technologies vol. 89 205–221 (2018) – 10.1016/j.trc.2018.02.005
- Gunter, G. et al. Are Commercially Implemented Adaptive Cruise Control Systems String Stable? IEEE Transactions on Intelligent Transportation Systems vol. 22 6992–7003 (2021) – 10.1109/tits.2020.3000682
- Makridis, M., Mattas, K., Anesiadou, A. & Ciuffo, B. OpenACC. An open database of car-following experiments to study the properties of commercial ACC systems. Transportation Research Part C: Emerging Technologies vol. 125 103047 (2021) – 10.1016/j.trc.2021.103047
- Ciuffo, B. et al. Requiem on the positive effects of commercial adaptive cruise control on motorway traffic and recommendations for future automated driving systems. Transportation Research Part C: Emerging Technologies vol. 130 103305 (2021) – 10.1016/j.trc.2021.103305
- Treiber, M., Kesting, A. & Helbing, D. Delays, inaccuracies and anticipation in microscopic traffic models. Physica A: Statistical Mechanics and its Applications vol. 360 71–88 (2006) – 10.1016/j.physa.2005.05.001
- Wang, T., Li, G., Zhang, J., Li, S. & Sun, T. The effect of Headway Variation Tendency on traffic flow: Modeling and stabilization. Physica A: Statistical Mechanics and its Applications vol. 525 566–575 (2019) – 10.1016/j.physa.2019.03.116
- Khound, P., Will, P., Tordeux, A. & Gronwald, F. Extending the adaptive time gap car-following model to enhance local and string stability for adaptive cruise control systems. Journal of Intelligent Transportation Systems vol. 27 36–56 (2021) – 10.1080/15472450.2021.1983810
- van der Schaft, Port-Hamiltonian systems: an introductory survey. (2006)
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- van der Schaft, A. Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noether’s theorem. Systems & Control Letters vol. 1 108–115 (1981) – 10.1016/s0167-6911(81)80046-1
- Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
- Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
- Knorn, S., Chen, Z. & Middleton, R. H. Overview: Collective Control of Multiagent Systems. IEEE Transactions on Control of Network Systems vol. 3 334–347 (2016) – 10.1109/tcns.2015.2468991
- Wang, Output synchronization of multi-agent port-Hamiltonian systems with link dynamics. Kybernetika (2016)
- Cristofaro, A., Giunta, G. & Giordano, P. R. Fault-Tolerant Formation Control of Passive Multi-Agent Systems Using Energy Tanks. IEEE Control Systems Letters vol. 6 2551–2556 (2022) – 10.1109/lcsys.2022.3169308
- van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Dynamics on Graphs: Consensus and Coordination Control Algorithms. IFAC Proceedings Volumes vol. 43 175–178 (2010) – 10.3182/20100913-2-fr-4014.00012
- LI, C.-S. & WANG, Y.-Z. Protocol Design for Output Consensus of Port-controlled Hamiltonian Multi-agent Systems. Acta Automatica Sinica vol. 40 415–422 (2014) – 10.1016/s1874-1029(14)60004-5
- Jafarian, M., Vos, E., De Persis, C., van der Schaft, A. J. & Scherpen, J. M. A. Formation control of a multi-agent system subject to Coulomb friction. Automatica vol. 61 253–262 (2015) – 10.1016/j.automatica.2015.08.021
- Wei, J., Everts, A. R. F., Camlibel, M. K. & van der Schaft, A. J. Consensus dynamics with arbitrary sign-preserving nonlinearities. Automatica vol. 83 226–233 (2017) – 10.1016/j.automatica.2017.06.001
- Xue, D., Hirche, S. & Cao, M. Opinion Behavior Analysis in Social Networks Under the Influence of Coopetitive Media. IEEE Transactions on Network Science and Engineering vol. 7 961–974 (2020) – 10.1109/tnse.2019.2894565
- Sharf, M. & Zelazo, D. Analysis and Synthesis of MIMO Multi-Agent Systems Using Network Optimization. IEEE Transactions on Automatic Control vol. 64 4512–4524 (2019) – 10.1109/tac.2019.2908258
- Matei, Inferring particle interaction physical models and their dynamical properties. (2019)
- Mavridis, Detection of dynamically changing leaders in complex swarms from observed dynamic data. (2020)
- Ma, Y., Chen, J., Wang, J., Xu, Y. & Wang, Y. Path-Tracking Considering Yaw Stability With Passivity-Based Control for Autonomous Vehicles. IEEE Transactions on Intelligent Transportation Systems vol. 23 8736–8746 (2022) – 10.1109/tits.2021.3085713
- Knorn, S., Donaire, A., Agüero, J. C. & Middleton, R. H. Passivity-based control for multi-vehicle systems subject to string constraints. Automatica vol. 50 3224–3230 (2014) – 10.1016/j.automatica.2014.10.038
- Knorn, S., Donaire, A., Agüero, J. C. & Middleton, R. H. Scalability of Bidirectional Vehicle Strings with Measurement Errors. IFAC Proceedings Volumes vol. 47 9171–9176 (2014) – 10.3182/20140824-6-za-1003.00741
- Dai, Safety analysis of integrated adaptive cruise control and lane keeping control using discrete-time models of port-Hamiltonian systems. (2017)
- Dai, S. & Koutsoukos, X. Safety analysis of integrated adaptive cruise and lane keeping control using multi-modal port-Hamiltonian systems. Nonlinear Analysis: Hybrid Systems vol. 35 100816 (2020) – 10.1016/j.nahs.2019.100816
- Bansal, H. et al. Port-Hamiltonian formulation of two-phase flow models. Systems & Control Letters vol. 149 104881 (2021) – 10.1016/j.sysconle.2021.104881
- Clemente-Gallardo, Geometric discretization of fluid dynamics. (2002)
- Rashad, R., Califano, F., Schuller, F. P. & Stramigioli, S. Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. Journal of Geometry and Physics vol. 164 104201 (2021) – 10.1016/j.geomphys.2021.104201
- Rashad, R., Califano, F., Schuller, F. P. & Stramigioli, S. Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow. Journal of Geometry and Physics vol. 164 104199 (2021) – 10.1016/j.geomphys.2021.104199
- Sugiyama, Y. et al. Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam. New Journal of Physics vol. 10 033001 (2008) – 10.1088/1367-2630/10/3/033001
- Tordeux, A. & Seyfried, A. Collision-free nonuniform dynamics within continuous optimal velocity models. Physical Review E vol. 90 (2014) – 10.1103/physreve.90.042812
- Helly, Simulation of bottlenecks in single lane traffic flow. (1959)
- Jiang, R., Wu, Q. & Zhu, Z. Full velocity difference model for a car-following theory. Physical Review E vol. 64 (2001) – 10.1103/physreve.64.017101
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- Teschl, (2012)
- Da Prato, (1992)
- Abramovich, (2002)
- Da Prato, (1996)
- Gibbs, A. L. & Su, F. E. On Choosing and Bounding Probability Metrics. International Statistical Review vol. 70 419–435 (2002) – 10.1111/j.1751-5823.2002.tb00178.x
- Klenke, (2014)
- Karatzas, (1991)
- Frank, E. On the zeros of polynomials with complex coefficients. Bulletin of the American Mathematical Society vol. 52 144–157 (1946) – 10.1090/s0002-9904-1946-08526-2
- Tordeux, A., Roussignol, M. & Lassarre, S. Linear stability analysis of first-order delayed car-following models on a ring. Physical Review E vol. 86 (2012) – 10.1103/physreve.86.036207
- Tordeux, A., Chraibi, M., Schadschneider, A. & Seyfried, A. Influence of the number of predecessors in interaction within acceleration-based flow models. Journal of Physics A: Mathematical and Theoretical vol. 50 345102 (2017) – 10.1088/1751-8121/aa7fca
- Cordes, J., Chraibi, M., Tordeux, A. & Schadschneider, A. Single-File Pedestrian Dynamics: A Review of Agent-Following Models. Modeling and Simulation in Science, Engineering and Technology 143–178 (2023) doi:10.1007/978-3-031-46359-4_6 – 10.1007/978-3-031-46359-4_6
- Kloeden, (2011)