基于有限划分观测的递归滤波与随机控制
Recursive Filtering and Stochastic Control under Finite Partition-Based Observations
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中文总结 AI 辅助
针对隐状态空间有限划分观测的部分可观测随机系统，构建滤波与最优控制框架，将其转化为完全可观测马尔可夫决策过程，提出类相关有限维近似方法并推导误差界，通过参考模型验证构建过程。
中文摘要 AI 辅助
我们针对部分可观测随机系统开发了滤波与最优控制框架，其中每个观测值对应隐状态空间的有限可测划分的一个类。该结构涵盖与阈值、量化、删失、事件触发及间歇观测相关的区域信息机制，且允许观测类包含原子、连续或混合分量。该公式首先在测度层面构建：对每个观测类，我们定义类受限的未归一化条件测度，后验分布通过其预测概率归一化得到。基于此递归关系，我们引入由观测类及其上支撑的条件测度构成的信息状态，从而将原问题转化为完全可观测的马尔可夫决策过程。我们构建了折扣代价准则，推导了贝尔曼方程，并建立了平稳最优策略的存在条件。为解决信息空间的无限维性质，我们提出了类相关的有限维近似方法，该方法能够同时保留连续分量与原子质量。我们还推导了一个抽象界，将近似滤波器的误差与值函数的误差关联起来。一个参考模型通过基于直方图的近似说明了该构建过程。
英文摘要
We develop a filtering and optimal-control framework for partially observable stochastic systems in which each observation identifies a class of a finite measurable partition of the hidden state space. This structure covers regional-information mechanisms associated with threshold, quantized, censored, event-triggered, and intermittent observations, and allows observable classes with atomic, continuous, or mixed components. The formulation is constructed first at the level of measures: for each observable class, we define a class-restricted unnormalized conditional measure, and the posterior distribution is obtained by normalizing with its predictive probability. Based on this recursion, we introduce an information state consisting of the observed class and the conditional measure supported on it, thereby transforming the original problem into a fully observable Markov decision process. We formulate the discounted-cost criterion, derive the Bellman equation, and establish conditions for the existence of stationary optimal policies. To address the infinite-dimensional nature of the information space, we propose class-dependent finite-dimensional approximations capable of preserving both continuous components and atomic masses. We also derive an abstract bound linking the error of the approximate filter to the error of the value function. A reference model illustrates the construction through histogram-based approximations
