用数学方法计算车辆与驾驶员系统的安全行驶范围，确保行车稳定。
Global stability of vehicle-with-driver dynamics via Sum-of-Squares programming
- 通过迭代平方和优化法构造李雅普诺夫函数，确定系统安全区域。
- 在转向不足和过度场景下，估算的安全集与仿真结果高度吻合。
- 适合自动驾驶安全验证、实时控制系统设计人员参考。
本文针对七维车辆-驾驶员系统，估计其吸引域中安全不变子集，兼顾渐近稳定性与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，计算安全集。首先在两维基准问题上验证：准确恢复了指定安全区域作为多项式李雅普诺夫函数的1-水平集。所研究系统采用带延迟预览跟踪的控制模型，可模拟人类驾驶行为，也可对应数字控制器。为支持SOS优化，推导出非线性车辆模型的多项式近似及其工作包络约束。该框架应用于转向不足与过度场景，估算的安全集与穷举仿真所得参考边界进行对比，结果显示SOS方法能高效生成李雅普诺夫定义的安全区域，具备用于实时安全评估的潜力，例如作为主动车辆控制的监督层。
原文摘要 · Abstract (English)
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.
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