Revisiting State Estimation and Weak Detectability of Discrete-Event Systems

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k_{1}$ </tex-math></inline-formula> -length sequence/string, whether or not one can always accurately determine the state of a plant at this moment after at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k_{2}$ </tex-math></inline-formula> 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.