Matrix approach to detectability of discrete event systems under partial observation

In this paper, we investigate the problem of detectability of nondeterministic discrete event systems (DESs) with partial event observation and partial state observation (partially-observed DESs for short). Concretely, it include several aspects below. First, we assume that we do not know initially which state the system is in. To discuss detectability property of partially-observed DESs (i.e., how to determine the current and subsequent states of a partially-observed DES after a finite number of observations), we introduce two key notions, namely, unobservable reach and detector for a partially-observed DES. Second, the dynamics of a detector, under the frameworks of the Boolean semi-tensor product (STP) of matrices, are converted into an algebraic representation. Using it, necessary and sufficient conditions are presented to verify whether a partially-observed DES is detectable or not. Compared with the existing approaches, the proposed approach is easier and more direct since it involve only straightforward matrix manipulations. Finally, we illustrate the application of the proposed approach to the verification of detectability property of partially-observed DESs by means of two examples.

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