提出新方法求解复杂系统的可达与稳定避障问题
Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability
- 设计连续的可达值函数，精确刻画可达区域
- 证明该函数是唯一黏性解，确保算法收敛
- 结合稳定性理论，实现目标到达与长期稳定
本文研究一般非线性连续时间系统在无限时域下的可达-避障（RA）与稳定-避障（SA）零和博弈问题，目标是找到在最坏扰动下仍能控制至目标集且不违反约束的状态集合。基于哈密顿-雅可比可达性方法，我们设计了一个新的 Lipschitz 连续的 RA 值函数，其零次水平集恰好表征了 RA 集。我们证明相关贝尔曼更新算子为压缩算子，且该值函数是哈密顿-雅可比变分不等式的唯一黏性解。最后，通过将我们的 RA 策略与近期提出的鲁棒控制李亚普诺夫-值函数相结合，构建了 SA 问题的两步框架，从而同时保证目标可达性和长期稳定性。我们在三维杜宾斯汽车系统上数值验证了所提框架的有效性。
原文摘要 · Abstract (English)
In this article, we consider the infinite-horizon reach-avoid (RA) and stabilize-avoid (SA) zero-sum game problems for general nonlinear continuous-time systems, where the goal is to find the set of states that can be controlled to reach or stabilize to a target set, without violating constraints even under the worst-case disturbance. Based on the Hamilton-Jacobi reachability method, we address the RA problem by designing a new Lipschitz continuous RA value function, whose zero sublevel set exactly characterizes the RA set. We establish that the associated Bellman backup operator is contractive and that the RA value function is the unique viscosity solution of a Hamilton-Jacobi variational inequality. Finally, we develop a two-step framework for the SA problem by integrating our RA strategies with a recently proposed Robust Control Lyapunov-Value Function, thereby ensuring both target reachability and long-term stability. We numerically verify our RA and SA frameworks on a 3D Dubins car system to demonstrate the efficacy of the proposed approach.
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