Modelling car-following dynamics with stochastic input-state-output port-Hamiltonian systems

TL;DR

This paper introduces a stochastic port-Hamiltonian framework for car-following models, analyzing stability, long-term behavior, and the impact of open-loop and closed-loop speed controls on traffic dynamics.

Contribution

It presents a novel port-Hamiltonian modeling approach for car-following dynamics, deriving stability conditions and demonstrating the effects of feedback control on system behavior.

Findings

Uncontrolled dynamics are unstable and exhibit random collective behavior.

Open-loop speed control stabilizes the system and leads to Gaussian distributions.

Closed-loop control can reproduce realistic stop-and-go traffic patterns.

Abstract

In this contribution, we introduce a general class of car-following models with an input-state-output port-Hamiltonian structure. We derive stability conditions and long-term behavior of the finite system with periodic boundaries and quadratic interaction potential by spectral analysis and using asymptotic properties of multivariate Ornstein-Uhlenbeck processes. The uncontrolled dynamics exhibit instability and random collective behavior under stochastic perturbations. By implementing an open-loop speed control, the system stabilizes and weakly converges to Gaussian limit distributions. The convergence is unconditional for constant speed control. However, a stability condition arises for the closed-loop system where the speed control acts as a dynamic feedback depending on the distance ahead. The results are illustrated by numerical simulations. Interestingly, only the closed-loop system is able to reproduce, at least transiently, realistic stop-and-go behavior that can be resolved using the Hamiltonian component of the model.