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Closed-form solution of controller synthesis for infinitely large systems of resource sharing systems of a subclass of Petri nets

Little literature has dealt with infinite systems where an arbitrarily large number of resources is shared between larger systems with many stages due to the associated state explosion problem. Current most advanced controlled techniques for flexible manufacturing systems (FMS) cannot handle very large systems. We proposed earlier a method without reachability analysis to find the closed-form solutions of all reachable, forbidden, and live markings for the so-called kth-order system. This is the first reported closed-form solution of arbitrarily large systems. This paper further extends the closed-form solutions to the controller synthesis of infinite kth-order systems, which is a subclass of FMS. The place-invariant-based deadlock prevention controls are adopted to reduce the computational burden by considering only the minimal set of all first-met bad markings (FBMs). No live states are lost by considering also only the minimal set of all live markings. The above requires solving integer linear programming problems (ILPPs), which is NP-hard and quite time-consuming. By merging several monitors into a single one while not losing live states, this paper is able to achieve the same best results in the literature while avoiding the time-consuming reachability analysis and complete siphon computation which does not scale well with the size of the nets. Furthermore, closed-form solution of arbitrarily large systems can be derived, which has never been attained so far. Application to large Gadara resource allocation systems is also mentioned. Further work on handling a very large system with resources shared between more than two processes is reported. Moreover, the initial markings of the two terminal resource places are allowed to vary above one.