Delayed Detectability of Discrete Event Systems

In this paper, we extend detectability to delayed detectability in order to answer the following question. After observing k₁+k₂ observable events, can we determine the state of a system at the time when the k₁th event was observed? If the answer is yes, we say that the system is delayed detectable. It involves two delays: the prior delay k₁ before the estimation is attempted and the post delay k₂ after which the estimation is completed. We investigate various properties of delayed detectability. We also provide polynomial algorithms to check delayed detectability. The usefulness of delayed detectability is not only evident by the state estimation problems it solves, which include current state estimation problem and initial state estimation problem, but also evident by the fact that two important properties in discrete event systems, namely observability and diagnosability, can both be shown to be special cases of delayed detectability.