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.

翻译：本研究针对七状态车辆-驾驶员系统，估计其吸引域（ROA）的安全不变子集，同时捕捉渐近稳定性及系统轨迹上状态安全边界的影响。通过原创的迭代平方和（SOS）程序优化李雅普诺夫函数，计算得到安全集合。该方法首先在双状态基准测试中得到验证，精确恢复了多项式李雅普诺夫函数1-水平集所规定的安全区域。随后，我们阐述了所研究的车辆-驾驶员系统的显著特征：控制动力学通过延迟预览跟踪模型模拟人类驾驶员行为，该模型在适当参数选择下亦可模拟数字控制器。为实现SOS优化，推导了非线性车辆模型的多项式近似及其运行包络约束。该框架随后应用于不足转向和过度转向场景，并将估计的安全集合与通过穷举仿真获得的参考边界进行比较。结果表明，SOS技术能够高效地提供李雅普诺夫定义的安全区域，为其在实时安全评估（例如作为主动车辆控制的监督层）中的潜在应用提供了支持。