Fakultät für Maschinenbau und Sicherheitstechnik

Stability analysis

Stability properties of car-following models are expected features of adaptive cruise control systems in traffic engineering. Indeed, several recent experiments have shown that current ACC systems describe unstable behaviors.

Local stability

In the local stability analysis (also platoon stability), some vehicles follow a leader with assigned speed. The local stability is called over-damped when the convergence is oscillation-free. Such a feature allows ensuring (at least partially) the absence of collision in the dynamics.

Global stability

The global stability analysis (also string stability) tackles vehicles on an infinite line or with periodic boundary conditions. String stability conditions contain inductive and convective perturbations that may locally vanish. They are, in general, more restrictive than local conditions.

Local stability analysis generally relies on Laplace transform, while circulant properties or exponential ansatz allow tackling the string stability for finite periodic and infinite systems, respectively. The stability conditions are determined by solving the characteristic equation in the complex plane. Stable solutions only exist in the left half-plane with overdamped solutions living on the real semi-axis. The characteristic equations are polynomials for ODE systems while they are exponential-polynomials for delayed systems. Bifurcation analysis can allow determining the stability conditions in this last case. However, only few delayed models in the literature have explicit stability conditions.


G. Gunter et al. Are Commercially Implemented Adaptive Cruise Control Systems String Stable?. IEEE Trans. Intell. Transp. Syst. 22, 6992 (2021).

R.E. Wilson, J.A. Ward. Car-Following Models: Fifty Years of Linear Stability Analysis – A Mathematical Perspective. Transport. Plan. Techn. 34, 3 (2011).

G. Orosz, R.E. Wilson, G. Stépán (2010). Traffic jams: dynamics and control. Phil. Trans. Roy. Soc. A, 368(1928), 4455-4479.

A. Tordeux, S. Lassarre, M. Roussignol. Linear Stability Analysis of First-Order Delayed Car-Following Models on a Ring. Phys. Rev. E 86, 036207 (2012).

A. Tordeux, M. Chraibi, A. Schadschneider, A. Seyfried. Influence of the number of predecessors in interaction with acceleration-based flow models. J. Phys. A 54, 345102 (2017).

NetLogo online simulation module of car-following models


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