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

TL;DR

The paper develops a Sum-of-Squares framework to certify safe invariant subsets of the Region of Attraction (ROA) for a seven-state vehicle-with-driver system, explicitly accounting for driver action and state constraints. By formulating ROA estimation as a Lyapunov-based SOS optimization with S-procedure constraints, it computes safe regions that remain within prescribed safety bounds and are amenable to real-time safety checks. The approach is demonstrated on a two-state benchmark and a seven-state vehicle-with-driver model with polynomial approximations, showing close alignment with simulation-based safety envelopes while providing formal guarantees. Limitations include the need for polynomial approximations and the associated domain restrictions, but the method shows promise for integrating certified safety regions as supervisory layers in active vehicle control systems.

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.

Figures (7)

• Figure 1: Phase portraits of time-reversed Van der Pol Oscillator with ROA estimates.
• Figure 2: Classical ROA estimate with no constraints (pink region, red boundary). State constraints $x^2\leq 1$ are introduced (dashed black lines). The boundary of the true safe ROA is illustrated in solid black lines. The safe ROA estimate computed enforcing state constraints is given by the blue region (blue boundary). The safe ROA estimate given by the green region (green boundary) is obtained introducing also anchor points as described in the hybrid approach.
• Figure 3: Vehicle-with-driver model. The delayed steering logic mimics a human driver by correcting the lateral error $e$ with respect to the reference path (grey dashed line) of the lookahead point $P$ (red).
• Figure 4: Polynomial fitting of axle characteristics in the understeering setup. Dark blue/red thin curves show the Magic Formula (MF) for the front/rear axles ($\alpha_i$ in rad, $F_{y,i}$ in N), while the light counterparts depict the cubic fits (poly), solid within the $[-\bar{\alpha}_i,\bar{\alpha}_i]$ range used for fitting and dashed outside it. Vertical blue/red lines mark the axle slip angles limits corresponding to $95\%$ of the respective Magic Formula peak forces.
• Figure 5: Oversteering scenario: safe subset of the ROA estimated via SOS, SOS-S-ROA (blue region), state constraints on axle slip angles (dashed black lines), simulation-based ROA, SB-ROA, boundaries (dashed red lines) and safe subset of the simulation-based ROA, SB-S-ROA, boundaries (red for stability limit, yellow for $\alpha_f$ limit)