Supervisory control of quantitative Petri nets for fixed-initial-credit energy problems using a game structure

This work investigates quantitative supervisory control of discrete event systems modeled with Petri nets under the fixed-initial-credit energy objective. A weight function referred to as an energy function is defined on a Petri net to characterize the energy level of a transition. The proposed fixed-initial-credit energy problem aims to design a supervisor such that the energy level of a transition sequence in a supervised system is higher than 0 under a given initial energy level. The problem is eventually transformed into a two-player game between a system and a supervisor; supervisor synthesis is reduced to finding a winning strategy in the two-player game. Instead of enumerating the complete state space of the underlying Petri net, two information structures are utilized, namely the conventional basis reachability graph and the newly proposed essential marking graph, to construct two-player games based on each of them. It is shown that a winning strategy for a supervisor decoded from the game based on the basis reachability graph of the Petri net is a solution to the problem but is in general restrictive. Further, it is shown that the set of strategies for a supervisor in the game based on the essential marking graph is consistent with that from the game based on the reachability graph of a Petri net. The two developed approaches do not require an exhaustive exploration of the state space of a plant, thus achieving higher efficiency.