Stability analysis of a stochastic port-Hamiltonian car-following model

TL;DR

This paper introduces a stochastic port-Hamiltonian car-following model that generalizes classical models, analyzes its stability, and demonstrates how the Hamiltonian structure enhances traffic flow stability and prevents stop-and-go waves.

Contribution

It formulates a novel stochastic port-Hamiltonian framework for car-following models, providing exact stability conditions and showing improved flow stability over traditional models.

Findings

The model's stability condition is explicitly derived.

Hamiltonian component reduces total energy and prevents stop-and-go waves.

The system exhibits ergodicity with a Gaussian invariant measure.

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), the full velocity difference model of Jiang et al. (2001), 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.