In this paper, we investigate how to verify delay co-observability for a given discrete event system and its specification language. Delay co-observability is introduced to solve the decentralized nonblocking networked control problem for discrete event systems under nondeterministic communication delays in our previous paper. We first define a set of indistinguishable string collections and transform the verification problem of delay co-observability to the problem of checking if all of the indistinguishable string collections are valid or not. We then construct a set of verifiers to track all of these string collections. Checking if all of the indistinguishable string collections are valid or not can be done by checking if the states in these verifiers are “good” or “bad”. The specification language is delay co-observable if and only if there are no “bad” states in all verifiers. The computational complexity of the proposed algorithm is polynomial with respect to the number of states and events of the system and exponential with respect to the number of local supervisors and the upper bounds on the delays.

中文翻译：

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离散事件系统的延迟协同可观性验证

在本文中，我们研究了如何验证给定离散事件系统及其规范语言的延迟可观察性。在我们先前的论文中，引入了延迟协同可观察性以解决离散事件系统在不确定通信延迟下的分散式无阻塞网络控制问题。我们首先定义了一组不可区分的字符串集合，并将延迟可观察性的验证问题转换为检查所有不可区分的字符串集合是否有效的问题。然后，我们构造一组验证器来跟踪所有这些字符串集合。可以通过检查这些验证器中的状态是“好”还是“坏”来检查所有无法区分的字符串集合是否有效。当且仅当在所有验证者中都没有“不良”状态时，规范语言才是延迟可共同观察的。所提出算法的计算复杂度相对于系统状态和事件的数量是多项式的，而相对于本地监管者的数量和时延的上限则是指数级的。