A necessary and sufficient condition for diagnosability of stochastic discrete event systems

Stochastic discrete event systems (SDES) are systems whose evolution is described by the occurrence of a sequence of events, where each event has a defined probability of occurring from each state. The diagnosability problem for SDES is the problem of determining the conditions under which occurrences of a fault can be detected in finite time with arbitrarily high probability. (IEEE Trans Autom Control 50(4):476–492 2005) proposed a class of SDES and proposed two definitions of stochastic diagnosability for SDES called A- and AA-diagnosability and reported a necessary and sufficient condition for A-diagnosability, but only a sufficient condition for AA-diagnosability. In this paper, we provide a condition that is both necessary and sufficient for determining whether or not an SDES is AA-diagnosable. We also show that verification of AA-diagnosability is equivalent to verification of the termination of the cumulative sum (CUSUM) procedure for hidden Markov models, and that, for a specific class of SDES called fault-immediate systems, the sequential probability ratio test (SPRT) minimizes the expected number of observable events required to distinguish between the normal and faulty modes.