In practical systems, due to reasons such as sensor limitations, sensor faults, and packet losses in networks, the observation of events becomes nondeterministic. In this article, we extend strong detectability and weak detectability to the case of nondeterministic observations and denote them as A-( $k_{1}$ , $k_{2}$ )-detectability and O-( $k_{1}$ , $k_{2}$ )-detectability, respectively. A-( $k_{1}$ , $k_{2}$ )-detectability says that, for any string, we can distinguish state pairs in the specification for all possible observations of the string. O-( $k_{1}$ , $k_{2}$ )-detectability says that, for at least one string, we can distinguish state pairs in the specification for all possible observations of the string. For A-( $k_{1}$ , $k_{2}$ )-detectability, we construct a transformed automaton and then translate the A-( $k_{1}$ , $k_{2}$ )-detectability problem into the traditional detectability problem that has been solved. We show that A-( $k_{1}$ , $k_{2}$ )-detectability can be used to solve the deterministic supervisory control problem. For O-( $k_{1}$ , $k_{2}$ )-detectability, we construct an augmented automaton that includes all the information of the given automaton and its state estimates. Based on the augmented automaton, we propose a depth-first search (DFS)-based algorithm to check O-( $k_{1}$ , $k_{2}$ )-detectability. Note to Practitioners —Nowadays, practical engineering systems become more and more complex. In these systems, the observation of events often becomes nondeterministic due to reasons such as sensor limitations, sensor faults, and packet losses in networks. Consider a mobile robot as an example. The availability of a sensor output may depend on the current location of the mobile robot. If the mobile robot is in an area where the wireless network is unreliable, the sensor output may not be received by the supervisor. In this article, we investigate the state estimation problem for practical engineering systems under nondeterministic observations within a discrete-event system framework. The results in this article provide not only insights for engineers in the automatic control field to understand nondeterministic observations in practical systems but also the methodology to estimate the current discrete state that is always an important issue. Therefore, we believe that the engineers in the automatic control field should be interested in this article and can benefit from it.

中文翻译：

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非确定性观测下离散事件系统的可检测性

在实际系统中，由于网络中传感器限制、传感器故障和数据包丢失等原因，对事件的观察变得不确定。在本文中，我们将强可检测性和弱可检测性扩展到非确定性观察的情况，并将它们表示为 A-( $k_{1}$ , $k_{2}$ )-可检测性和 O-( $k_{1}$ , $k_{2}$ )-可检测性，分别。一种-（ $k_{1}$ , $k_{2}$ )-detectability 表示，对于任何字符串，我们可以区分规范中所有可能的字符串观察的状态对。哦-( $k_{1}$ , $k_{2}$ )-detectability 表示，对于至少一个字符串，我们可以区分规范中所有可能的字符串观察的状态对。为一个-（ $k_{1}$ , $k_{2}$ )-可检测性，我们构建了一个变换的自动机，然后将 A-( $k_{1}$ , $k_{2}$ )-可检测性问题转化为已解决的传统可检测性问题。我们证明 A-( $k_{1}$ , $k_{2}$ )-可检测性可用于解决确定性监督控制问题。对于 O-( $k_{1}$ , $k_{2}$ )-可检测性，我们构建了一个增强自动机，其中包括给定自动机的所有信息及其状态估计。基于增强自动机，我们提出了一种基于深度优先搜索 (DFS) 的算法来检查 O-( $k_{1}$ , $k_{2}$ )-可检测性。 从业者须知 ——如今，实际工程系统变得越来越复杂。在这些系统中，由于传感器限制、传感器故障和网络中的数据包丢失等原因，对事件的观察通常变得不确定。以移动机器人为例。传感器输出的可用性可能取决于移动机器人的当前位置。如果移动机器人处于无线网络不可靠的区域，则监控器可能无法接收到传感器输出。在本文中，我们研究了离散事件系统框架内在非确定性观察下实际工程系统的状态估计问题。本文的结果不仅为自动控制领域的工程师提供了理解实际系统中非确定性观察的见解，而且为估计始终是一个重要问题的当前离散状态提供了方法。因此，我们认为自动控制领域的工程师应该对本文感兴趣，并能从中受益。