On Optimizing the Conditional Value-at-Risk of a Maximum Cost for Risk-Averse Safety Analysis
On Optimizing the Conditional Value-at-Risk of a Maximum Cost for Risk-Averse Safety Analysis
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DOI：
10.1109/tac.2022.3195381
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发表时间：
2021-06
影响因子：
6.8
通讯作者：
Margaret P. Chapman;Michael Fauss;K. Smith
中科院分区：
文献类型：
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作者：
Margaret P. Chapman;Michael Fauss;K. Smith
条件风险价值（CVaR），一个来自金融的风险函数，由于其直观的解释和公理化的基础，在控制系统界越来越受欢迎。我们考虑一个非标准的最优控制问题，其目标是最小化最大随机成本服从Borel空间马尔可夫决策过程的CVaR。该目标表示在最坏情况的给定部分上平均的期望操作区域的最大偏离。这个问题提供了一个随机系统的安全标准，该标准由系统行为的潜在后果的概率和严重性来告知。相比之下，现有的安全分析框架应用阶段性风险约束或评估约束违反的概率，而不量化违反的潜在严重性。据我们所知，利息问题还没有解决。为了解决这个问题，我们提出并研究了增广状态空间上的一类随机动态规划。我们证明了最大随机成本的最优CVaR享有这些动态规划的解决方案在适当的假设下的等价表示。对于每一个动态程序，我们证明了存在的最优策略，取决于一个增广状态的动态假设。在一个数值例子中，我们说明了我们的安全分析框架是如何有用的结合下水道溢流降水不确定性的严重程度进行评估。
The popularity of Conditional Value-at-Risk (CVaR), a risk functional from finance, has been growing in the control systems community due to its intuitive interpretation and axiomatic foundation. We consider a nonstandard optimal control problem in which the goal is to minimize the CVaR of a maximum random cost subject to a Borel-space Markov decision process. The objective represents the maximum departure from a desired operating region averaged over a given fraction of the worst cases. This problem provides a safety criterion for a stochastic system that is informed by both the probability and severity of the potential consequences of the system’s behavior. In contrast, existing safety analysis frameworks apply stagewise risk constraints or assess the probability of constraint violation without quantifying the potential severity of the violation. To the best of our knowledge, the problem of interest has not been solved. To solve the problem, we propose and study a family of stochastic dynamic programs on an augmented state space. We prove that the optimal CVaR of a maximum random cost enjoys an equivalent representation in terms of the solutions to these dynamic programs under appropriate assumptions. For each dynamic program, we show the existence of an optimal policy that depends on the dynamics of an augmented state under the assumptions. In a numerical example, we illustrate how our safety analysis framework is useful for assessing the severity of combined sewer overflows under precipitation uncertainty.