On Stability Analysis of Car-Following Models with Various Discrete Operators
Source abstract
Abstract
This paper investigates a car-following model that incorporates both classical and fractional discrete operators. While classical models have been extensively studied, the influence of discrete fractional operators on the stability of such systems has not yet been systematically analyzed. Stability conditions are derived and rigorously proven for systems employing three widely used fractional h-difference operators—Grünwald–Letnikov, Riemann–Liouville, and Caputo—as well as the classical h-difference operator. The analysis reveals that the established conditions are independent of the specific operator used. Furthermore, a comprehensive numerical study validates the theoretical findings and demonstrates that the fractional models can significantly extend the stability bound for the step size from h<6.67 (classical case) to h<22.3 (fractional case).
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Research context
OpenAlex topics: Traffic control and management · Transportation Planning and Optimization · Autonomous Vehicle Technology and Safety
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Affiliations
Faculty of Computer Science, Bialystok University of Technology, 15-351 Bialystok, Poland
Funding
Bialystok University of Technology
References (38)
Show source-provided references
1. An analysis of traffic flow10.1287/opre.7.1.79 ↗
2. Shock waves on the highway10.1287/opre.4.1.42 ↗
3. Traffic jams induced by fluctuation of a leading car10.1103/PhysRevE.61.3534 ↗
4. Greenshields, B.D., Thompson, J., Dickinson, H., and Swinton, R. (1933). The Photographic Method of Studying Traffic Behavior. [Ph.D. Thesis, University of Michigan].
5. Greenshields, B.D., Bibbins, J.R., Channing, W., and Miller, H.H. (1935, January 7–10). A study of traffic capacity. Proceedings of the Highway Research Board Proceedings, Washington, DC, USA.
6. Kühne, R. (2008, January 8–10). Foundation of traffic flow theory I: Greenshields legacy highway traffic. Proceedings of the Proceedings Symposium on the Fundamental Diagram: 75 Years, Woods Hole, MA, USA.
7. On the optimal error regulation of a string of moving vehicles10.1109/TAC.1966.1098376 ↗
8. Smart cars on smart roads: Problems of control10.1109/9.250509 ↗
9. Charlottin, T., Varotto, S., and Buisson, C. (2024, January 18–20). Detecting car-following and platoons within traffic using the dynamic time warping algorithm. Proceedings of the Heart 2024, 12th Symposium of the European Association for Research in Transportation, Espoo, Finland.
10. On the stability analysis of mixed traffic with vehicles under car-following and bilateral control10.1109/TAC.2019.2945888 ↗
11. Horn, B.K. (2013, January 6–9). Suppressing traffic flow instabilities. Proceedings of the 16th International IEEE Conference on Intelligent Transportation Systems (ITSC 2013), The Hague, The Netherlands.10.1109/ITSC.2013.6728204 ↗
12. Wave equation of suppressed traffic flow instabilities10.1109/TITS.2017.2767595 ↗
13. Eigenvalue and eigenvector analysis of stability for a line of traffic10.1111/sapm.12144 ↗
14. Cao, X., Wang, J., and Chen, C. (2020). A modified car-following model considering traffic density and acceleration of leading vehicle. Appl. Sci., 10.10.3390/app10041268 ↗
15. Chen, K., Gu, Y., Lin, H., Zhang, Z., Zhou, X., and Wang, X. (2024). Guaranteed performance event-triggered adaptive consensus control for multiagent systems under time-varying actuator faults. Mathematics, 12.10.3390/math12101528 ↗
16. Song, C.J., and Jia, H.F. (2022). Car-following model optimization and simulation based on cooperative adaptive cruise control. Sustainability, 14.10.3390/su142114067 ↗
17. Data driven scalability and profitability analysis in free floating electric car sharing systems10.1016/j.ins.2022.11.116 ↗
18. Optimal pricing and pricing policy selection for a B2C car-sharing platform during peak and off-peak hours10.1016/j.ins.2022.05.009 ↗
19. A new definition of fractional derivative without singular kernel
20. Kaczorek, T. (2011). Selected Problems of Fractional Systems Theory, Springer.10.1007/978-3-642-20502-6 ↗
21. Kilbas, A. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Elsevier.
22. Ostalczyk, P. (2015). Discrete Fractional Calculus: Applications in Control and Image Processing, World Scientific.10.1142/9833 ↗
23. Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier.
24. Girejko, E., Mozyrska, D., and Wyrwas, M. Fractional Cucker-Smale Type Models with the Caputo Variable-Order Operator. Proceedings of the Conference on Non-Integer Order Calculus and Its Applications.
25. Behaviour of fractional discrete-time consensus models with delays for summator dynamics
26. Uber “begrente” Derivationen und deren Anwedung
27. Theory of differentiation of fractional order
28. Rosas-Jaimes, O., Quezada Téllez, L.A., and Fernandez-Anaya, G. (2017, January 4–6). A Fractional Approach to Carfollowing Pipe’s Model. Proceedings of the Congreso Nacional de Control Automático, Monterrey, Mexico.
29. Zhang, Y., and Zhao, J. (2018). Fractional order car-following model and its simulation. Mod. Phys. Lett. B, 32.10.1142/S0217984918502135 ↗
30. Dubey, V.P., Kumar, D., Alshehri, H.M., Dubey, S., and Singh, J. (2022). Computational analysis of local fractional LWR model occurring in a fractal vehicular traffic flow. Fractal Fract., 6.10.3390/fractalfract6080426 ↗
31. A fractional-order car-following model with random delay optimized by an enhanced whale optimization algorithm10.1007/s11071-025-11271-4 ↗
32. Fractional-order-based ACC/CACC algorithm for improving string stability10.1016/j.trc.2018.07.026 ↗
33. Iso-damping fractional-order control for robust automated car-following10.1016/j.jare.2020.05.013 ↗
34. Elgezouli, D.E., Eltayeb, H., and Abdoon, M.A. (2024). Novel GPID: Grünwald–Letnikov fractional PID for enhanced adaptive cruise control. Fractal Fract., 8.10.3390/fractalfract8120751 ↗
35. Stability of discrete fractional linear systems with positive orders10.1016/j.ifacol.2017.08.1250 ↗
36. Mozyrska, D., and Wyrwas, M. (2018, January 4–6). Stability of linear systems with Caputo fractional-, variable-order difference operator of convolution type. Proceedings of the 2018 41st International Conference on Telecommunications and Signal Processing (TSP), Athens, Greece.10.1109/TSP.2018.8441360 ↗
37. Numerical analysis of behaviour of the Cucker–Smale type models with fractional operators10.1016/j.cam.2017.12.013 ↗
38. Brown, R. (2025, April 26). ECE504: Lecture 8. Available online: https://spinlab.wpi.edu/courses/ece504_2008/8.pdf.