通过对偶方法解决随机控制中的概率约束问题，无需近似即可保证安全性。
Strong Duality and Dual Ascent Approach to Continuous-Time Chance-Constrained Stochastic Optimal Control
- 利用退出时间将概率约束转化为期望形式，构建对偶问题。
- 证明了原问题与对偶问题间强对偶性成立，保证解的最优性。
- 结合路径积分法实现数值求解，适用于移动机器人路径规划场景。
本文研究连续时间连续空间下的机会约束随机最优控制问题，明确限制状态约束失效的概率。借助连续时间随机微积分中的退出时间概念，将机会约束转化为指示函数的期望形式，并通过构造对偶问题将其融入目标函数。在系统动力学与代价函数满足特定条件下，证明了原问题与对偶问题之间存在强对偶性。采用路径积分方法，基于开环轨迹样本进行梯度上升求解对偶问题。通过仿真对比了移动机器人空间导航中的路径规划结果，验证了该方法相较于有限差分法的有效性。
原文摘要 · Abstract (English)
The paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem where the probability of failure to satisfy given state constraints is explicitly bounded. We leverage the notion of exit time from continuous-time stochastic calculus to formulate a chance-constrained SOC problem. Without any conservative approximation, the chance constraint is transformed into an expectation of an indicator function which can be incorporated into the cost function by considering a dual formulation. We then express the dual function in terms of the solution to a Hamilton-Jacobi-Bellman partial differential equation parameterized by the dual variable. Under a certain assumption on the system dynamics and cost function, it is shown that a strong duality holds between the primal chance-constrained problem and its dual. The Path integral approach is utilized to numerically solve the dual problem via gradient ascent using open-loop samples of system trajectories. We present simulation studies on chance-constrained motion planning for spatial navigation of mobile robots and the solution of the path integral approach is compared with that of the finite difference method.
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