This note extends previous results on the supervision of Petri nets (PNs) to the decentralized setting. While focusing on the extension of supervision based on place invariants (SBPI), the proposed approach is more general and could be applied to other types of supervision as well. We begin by introducing d-admissibility as an extension to the decentralized setting of the centralized admissibility concept. We define also structural d-admissibility, as the counterpart of the simple sufficient conditions for centralized admissibility in the context of the SBPI. Note that (structural) d-admissibility is only sufficient for a specification to be enforcible with the same permissiveness as in the centralized setting with full controllability and observability. However, structural d-admissibility can be checked with low polynomial complexity. Based on the d-admissibility concept, we propose two suboptimal methods to design decentralized supervisors. The first method is to find a centralized solution, and then distribute the centralized supervisory policy by means of communication. The amount of communication can be minimized by means of an integer linear program (ILP). The second method is to transform the specification to a (more restrictive) d-ad-missible specification by means of an ILP. In the case of decentralized supervision with communication, the ILP can be used to minimize the amount of communication required by the solution.
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