Opacity of discrete-event systems under nondeterministic observation mechanism

For some practical systems modeled by discrete-event systems (DESs), the event observation mechanism may be no longer static/fixed due to sensor faults/failures, packet loss, and/or measurement uncertainties. It is required to redefine the observations of events. Therefore, it is interesting to reconsider the opacity in the context of abnormal conditions as mentioned. In this paper, we extend notions of initial-state opacity (ISO), current-state opacity (CSO), and initial-and-final-state opacity (IFSO) in a standard setting to an advanced framework of nondeterministic observations (i.e., the event-observation mechanism is state-dependent and nondeterministic) by taking the output-observation-equivalence instead of the natural projection-equivalence into account. To efficiently verify them, we construct three new matrix-based information structures in this advanced framework called an initial-state estimator, a current-state estimator, and an initial-and-final-state estimator using the Boolean semi-tensor product (BSTP) theory. Accordingly, the matrix-based criteria for verifying these three types of opacity are given, respectively. Computational complexity analysis as well as illustrative examples are also provided for the proposed results.