Abstract
This paper develops an approach for solving the problem of least-cost transition firing sequences estimation in labeled Petri nets with unobservable transitions. Each transition in the net is labeled as either observable or unobservable, and has a nonnegative cost. Additionally, we assume that the net system is bounded, the unobservable subnet is acyclic, and the cost of each unobservable transition is strictly positive. We propose the method mainly on the basis of the notions of basis marking and basis reachability graph (BRG), which is a compact representation of the reachability graph of the net system. By sacrificing extra storage space to keep the BRG and other necessary information in memory, some time-consuming computations are moved off-line. This makes our proposed approach more feasible for on-line applications. Finally, we present a brief survey and comparison of some representative methods that use labeled Petri nets as a formalism in the literature.
Highlights
A discrete event system (DES) is a discrete-state and eventdriven system [45]
With the development of DESs, Petri nets (PNs) have become a conceptual framework and a fundamental model for DESs since they are widely used to address various classes of problems, including the formal representation and development of systems, along with their supervisory control [9], [10], [17], [49], [55], controller synthesis or design [22], [23], [34], [47], state estimation and observability [2], [14], [36], fault detection, diagnosis, and prognosis [11], [16], [19], [25], [37], [48], [56], [57], diagnosability [3], [8], [40], [41], opacity [12], [13], [46], The associate editor coordinating the review of this manuscript and approving it for publication was Zhiwu Li
We reviewed the notions of explanations and minimal firing vectors of explanations, defined the transition sequences consistent with a firing vector at a marking, and developed an algorithm for their computation
Summary
A discrete event system (DES) is a discrete-state and eventdriven system [45]. Automata are intuitive models used to achieve the original contributions and most of the subsequent developments in the literature of supervisory control of DESs [20], [50]. Observed labels represent tasks, each of which can be accomplished via a set of different transitions. Let π : T ∗ → Nn be the function such that, for a transition sequence σ ∈ T ∗, y = π (σ ) is an n-dimensional nonnegative vector and is called the firing vector of σ , where y(i) states the number of ti’s occurrences in σ , i = 1, 2, . Given an observed sequence of labels w ∈ L∗, S(w) = {σ ∈ T ∗|M0[σ , (σ ) = w} is the set of transition firing sequences that are consistent with w, and Z(w) = {M ∈. R(N , M0)|∃σ ∈ S(w) : M0[σ M } is the set of reachable markings consistent with w ∈ L∗
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