Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.

翻译：区分性及由此延伸的可观性是动力系统的关键属性。当缺乏解析模型且需要直接从测量数据推断这些属性时，建立这些性质颇具挑战性。噪声的存在进一步复杂化了此类分析，因为标准区分性概念是针对确定性系统设计的。我们基于分布区分性展开研究，该方法通过比较随机系统输出的分布来拓展确定性概念。首先证明对于包含线性系统的一类系统，两种概念等价。随后提出一种从输出数据评估并量化分布区分性的方法：具体而言，我们的量化指标衡量区分两个初始状态所需的数据量，由此引出连续谱系的区分性。我们提出一种统计检验方法，用于确定两个状态能否以高置信度被区分的阈值。通过仿真在状态空间上计算区分性图谱来展示这些工具的有效性，并利用该检验在硬件上比较传感器配置方案。