Discussion on: Decentralized Supervisory Control of Fuzzy Discrete Event Systems

The paper by Liu and Qiu is concerned with the decentralized supervisory control problem for fuzzy discrete event systems (FDESs) with partial and fuzzy observations. On the one hand, this work is a natural continuation of the authors’ paper [8] in which the centralized control under fuzzy observations was discussed; on the other hand, it is a good extension and complement to some related results in [1]. In terms of the decentralized supervisory control, the key point of divergence between the paper and [1] is that the former allows observations to be fuzzy. The fuzziness of observations, originally suggested by Lin and Ying in [4,5], seems an interesting feature of FDESs and desires formal study. In addition to introducing several necessary definitions, the main technical contributions of the paper are a decentralized supervisory control theorem and a computing approach to verifying the fuzzy coobservability condition in this theorem. The theorem shows that there exist non-blocking local fuzzy supervisors if and only if the fuzzy language to be synthesized is fuzzy controllable, fuzzy co-observable, and closed. Determining the existence of local fuzzy supervisors appeals to methods for checking the fuzzy controllability and the fuzzy co-observability. The paper develops a feasible way to test the fuzzy coobservability condition, since another method for checking the fuzzy controllability condition has already been introduced in [7]. Such a way is highly dependent on the max-min fuzzy automaton modeling of FDESs. More explicitly, it is the max-min operator in the model that makes the number of all states reachable from any state in a system finite, which clearly facilitates the use of computing tree.