Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

Authors

Julia Ackermann, Matthias Ehrhardt, Thomas Kruse, Antoine Tordeux

Abstract

This study revisits a recently proposed symmetric port-Hamiltonian single-file model in one dimension. The uniform streaming solutions are stable in the deterministic model. However, the introduction of white noise into the dynamics causes the model to exhibit divergence. In response, we add a relaxation term that draws the agents’ speed to a desired constant speed and plays the role of the input in the port-Hamiltonian framework. Our results show that this relaxation term effectively stabilises the dynamics even in the presence of stochastic noise.

Keywords

Port-Hamiltonian Systems; Stability; Stochastic Modeling and Stochastic Systems Theory

Citation

• Journal: IFAC-PapersOnLine
• Year: 2024
• Volume: 58
• Issue: 17
• Pages: 145–150
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2024.10.128
• Note: 26th International Symposium on Mathematical Theory of Networks and Systems MTNS 2024- Cambridge, United Kingdom, August 19-23, 2024

BibTeX

@article{Ackermann_2024,
  title={{Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems}},
  volume={58},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2024.10.128},
  number={17},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Ackermann, Julia and Ehrhardt, Matthias and Kruse, Thomas and Tordeux, Antoine},
  year={2024},
  pages={145--150}
}

Download the bib file

References

• 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
• 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
• Dai, Safety analysis of integrated adaptive cruise and lane keeping control using multi-modal port-Hamiltonian systems. Nonlinear Analysis: Hybrid Systems (2020)
• Ehrhardt, M., Kruse, T. & Tordeux, A. The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension. ESAIM: Mathematical Modelling and Numerical Analysis vol. 58 515–544 (2024) – 10.1051/m2an/2024004
• 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
• Gardiner, (1985)
• Gasser, I., Sirito, G. & Werner, B. Bifurcation analysis of a class of ‘car following’ traffic models. Physica D: Nonlinear Phenomena vol. 197 222–241 (2004) – 10.1016/j.physd.2004.07.008
• 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
• Jacob, Port-Hamiltonian structure of interacting particle systems and its mean-field limit. arXiv preprint (2023)
• 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
• 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
• Matei, Inferring particle interaction physical models and their dynamical properties. (2019)
• Orosz, Traffic jams: dynamics and control. Proceedings of the Royal Society A (2010)
• 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
• Pavliotis, (2014)
• Pipes, L. A. An Operational Analysis of Traffic Dynamics. Journal of Applied Physics vol. 24 274–281 (1953) – 10.1063/1.1721265
• Rashad, Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. Journal of Geometry and Physics (2021)
• Rashad, Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible fow. Journal of Geometry and Physics (2021)
• 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
• Reuschel, Fahrzeugbewegungen in der Kolonne. Österreichisches Ingenieur Archiv (1950)
• Rüdiger, Stability analysis of a stochastic port-Hamiltonian car-following model. arXiv preprint (2022)
• 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
• Tordeux, A., Costeseque, G., Herty, M. & Seyfried, A. From Traffic and Pedestrian Follow-the-Leader Models with Reaction Time to First Order Convection-Diffusion Flow Models. SIAM Journal on Applied Mathematics vol. 78 63–79 (2018) – 10.1137/16m110695x
• 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. & 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
• Tordeux, A. & Totzeck, C. Multi-scale description of pedestrian collective dynamics with port-Hamiltonian systems. Networks and Heterogeneous Media vol. 18 906–929 (2023) – 10.3934/nhm.2023039
• 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
• 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
• Wilson, R. E. & Ward, J. A. Car-following models: fifty years of linear stability analysis – a mathematical perspective. Transportation Planning and Technology vol. 34 3–18 (2011) – 10.1080/03081060.2011.530826