Quantifying and Managing Uncertainty in Piecewise-Deterministic Markov Processes
Quantifying and Managing Uncertainty in Piecewise-Deterministic Markov Processes
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DOI：
10.1137/20m1357275
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发表时间：
2020-08
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通讯作者：
E. Cartee;Antonio Farah;April Nellis;Jacob Van Hook;A. Vladimirsky
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作者：
E. Cartee;Antonio Farah;April Nellis;Jacob Van Hook;A. Vladimirsky
在分段确定性马尔可夫过程 (PDMP) 中，有限维系统的状态动态演化，但演化方程可能因离散切换而随机变化。沿着相应的分段确定性轨迹对运行成本进行积分直至终止，以产生过程的累积成本。我们解决了与 PDMP 模型累积成本不确定性相关的三个自然问题：（1）当切换率完全已知时如何计算累积分布函数（CDF）； (2)当切换率不确定时如何准确绑定CDF； (3) 假设 PDMP 受到控制，如何选择控制来优化 CDF。在所有三种情况下，我们的方法都需要提出一个合适的双曲偏微分方程组（弱耦合），然后在增广状态空间上进行数值求解。我们使用不确定性下轨迹规划的简单例子来说明我们的方法。
In piecewise-deterministic Markov processes (PDMP) the state of a finite-dimensional system evolves dynamically, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a (weakly-coupled) system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty.