Detectability in Stochastic Discrete Event Systems

In this paper we define and analyze notions of detectability in stochastic discrete event systems (SDES). More specifically, we introduce the notions of A-detectability and AA-detectability which focus on characterizing our ability to estimate the true current state of a given SDES with increasing certainty as we observe more output symbols. We discuss observer-based techniques that can be used to verify A-detectability, and provide associated necessary and sufficient conditions for this property to hold. We also present and discuss the notion of AA-detectability, which encompasses a broader class of SDES than A-detectability. Our approach is differentiated from the existing notion of stochastic detectability because it takes into account the probability of problematic observation sequences (that do not allow us to perfectly know the system state), whereas stochastic detectability considers each observation sequence that can be generated by the underlying system even if this observation sequence can only be generated with very small probability. The notions of A- and AA-detectability that are introduced are inspired by combining existing notions for detectability in deterministic discrete event systems and diagnosability in SDES.