Global stability of vehicle-with-driver dynamics via Sum-of-Squares programming

TL;DR

This paper develops an SOS-based method to estimate safe invariant regions for vehicle-with-driver systems, accounting for stability and safety bounds, with potential applications in real-time vehicle safety supervision.

Contribution

It introduces an iterative SOS procedure to compute Lyapunov functions for complex vehicle-driver models, including nonlinearities and control constraints.

Findings

Accurately recovers safe regions in benchmark tests

Effectively models human driver behavior and vehicle dynamics

Demonstrates potential for real-time safety assessment

Abstract

This work estimates safe invariant subsets of the Region of Attraction (ROA) for a seven-state vehicle-with-driver system, capturing both asymptotic stability and the influence of state-safety bounds along the system trajectory. Safe sets are computed by optimizing Lyapunov functions through an original iterative Sum-of-Squares (SOS) procedure. The method is first demonstrated on a two-state benchmark, where it accurately recovers a prescribed safe region as the 1-level set of a polynomial Lyapunov function. We then describe the distinguishing characteristics of the studied vehicle-with-driver system: the control dynamics mimic human driver behavior through a delayed preview-tracking model that, with suitable parameter choices, can also emulate digital controllers. To enable SOS optimization, a polynomial approximation of the nonlinear vehicle model is derived, together with its operating-envelope constraints. The framework is then applied to understeering and oversteering scenarios, and the estimated safe sets are compared with reference boundaries obtained from exhaustive simulations. The results show that SOS techniques can efficiently deliver Lyapunov-defined safe regions, supporting their potential use for real-time safety assessment, for example as a supervisory layer for active vehicle control.