基于单侧Vysochanskij-Petunin不等式的扰动LTI系统的机会约束随机最优控制
Chance Constrained Stochastic Optimal Control for Arbitrarily Disturbed LTI Systems Via the One-Sided Vysochanskij-Petunin Inequality
摘要：
虽然已经开发了许多技术的机会约束随机最优控制与高斯扰动过程，远少于已知的计算有效的方法来处理非高斯过程。本文研究了具有有限矩和单峰机会约束的一般加性扰动线性定常系统的机会约束随机最优控制问题。我们提出了一个开环控制方案的多车辆规划，既有目标集和防撞约束。我们的方法依赖于单侧Vysochanskij-Petunin不等式，这是一种用于约束单峰随机变量尾概率的统计工具。使用单侧Vysochanskij-Petunin不等式，我们重新制定每个机会约束的期望和标准差。虽然重新制定的界限是保守的，相对于原来的界限，他们有一个简单的和封闭的形式，并服从凸优化技术的差异。我们证明了我们的方法上的多卫星交会问题。
英文摘要：
While many techniques have been developed for chance constrained stochastic optimal control with Gaussian disturbance processes, far less is known about computationally efficient methods to handle non-Gaussian processes. In this paper, we develop a method for solving chance constrained stochastic optimal control problems for linear time-invariant systems with general additive disturbances with finite moments and unimodal chance constraints. We propose an open-loop control scheme for multi-vehicle planning, with both target sets and collision avoidance constraints. Our method relies on the one-sided Vysochanskij-Petunin inequality, a tool from statistics used to bound tail probabilities of unimodal random variables. Using the one-sided Vysochanskij-Petunin inequality, we reformulate each chance constraint in terms of the expectation and standard deviation. While the reformulated bounds are conservative with respect to the original bounds, they have a simple and closed form, and are amenable to difference of convex optimization techniques. We demonstrate our approach on a multi-satellite rendezvous problem.
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