In this paper, we revisit state estimation and weak detectability verification for discrete event systems (DES) from a span-new perspective. Specifically, using the semi-tensor product (STP) technique, we construct two new matrix-based information structures called a current-state estimator (C-estimator) and an initial-state estimator (I-estimator) for computing three fundamental types of state estimates, namely, current-state estimate (CSE), initial-state estimate (ISE), and delayed-state estimate (DSE). The complexity of building C-estimator and I-estimator is polynomial time with respect to the size of a plant. A notion of weak delayed detectability is introduced, which captures that, after observing a $k_{1}$ -length sequence/string, whether or not one can always accurately determine the state of a plant at this moment after at most $k_{2}$ steps of delays for some trajectories. Further, using the proposed C-estimator and I-estimator, we discuss the different types of detectability verification problems, including, but not restricted to, weak current-state detectability (C-detectability), weak initial-state detectability (I-detectability), and weak delayed detectability. Accordingly, several necessary and sufficient criteria are derived for verifying the aforementioned different types of detectability. Our approaches are numerically tractable and only involve some basic matrix manipulations. Finally, some examples are given to illustrate the obtained results. Note to Practitioners—State estimation is one of the most fundamental problems in many practical engineering systems. For instance, one needs to infer the state of a manufacturing system before a failure occurs. For a communication system, can we guarantee that whether important information remains secret to outsiders for security requirements? Finding an alternative and efficient approach to capture the state of a plant based on imperfect observations is still crucial for engineers. To solve these problems, in this paper we develop a novel methodology to tackle simultaneously three fundamental categories of state estimation for practical engineering systems that are inherently abstracted as partially-observed discrete-event systems. Our approaches are technically quite different from the existing ones. The novel results obtained in this paper are all of matrix-based characterization, which can be implemented algorithmically by means of the user-friendly STP software package. We believe that the alternative methodology provides an innovative insight for engineers in the field of automatic control.

中文翻译：

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重温离散事件系统的状态估计和弱可检测性

在本文中，我们从一个全新的角度重新审视了离散事件系统 (DES) 的状态估计和弱可检测性验证。具体来说，使用半张量积 (STP) 技术，我们构建了两个新的基于矩阵的信息结构，称为当前状态估计器（C-估计器）和初始状态估计器（I-估计器），用于计算三种基本类型状态估计，即当前状态估计（CSE）、初始状态估计（ISE）和延迟状态估计（DSE）。构建 C 估计器和 I 估计器的复杂性是关于工厂规模的多项式时间。引入了弱延迟可检测性的概念，在观察到 $k_{1}$ -length序列/字符串，是否总能准确判断一株植物此时的状态至多 $k_{2}$某些轨迹的延迟步骤。此外，使用提出的 C 估计器和 I 估计器，我们讨论了不同类型的可检测性验证问题，包括但不限于弱当前状态可检测性（C-detectability）、弱初始状态可检测性（I-detectability） ), 弱的延迟检测能力。据此，推导出几个必要和充分的准则来验证上述不同类型的可检测性。我们的方法在数值上易于处理，并且只涉及一些基本的矩阵操作。最后，给出了一些例子来说明所获得的结果。从业者须知——状态估计是许多实际工程系统中最基本的问题之一。例如，需要在故障发生之前推断制造系统的状态。对于一个通信系统，我们能否保证重要的信息是否对外保密？寻找一种替代的、有效的方法来根据不完美的观察来捕捉工厂的状态对工程师来说仍然是至关重要的。为了解决这些问题，在本文中，我们开发了一种新的方法来同时处理实际工程系统的状态估计的三个基本类别，这些系统本质上被抽象为部分观察到的离散事件系统。我们的方法在技术上与现有方法完全不同。本文获得的新结果都是基于矩阵的表征，可以通过用户友好的 STP 软件包通过算法实现。