An extremum timed extended reachability graph for temporal analysis of time Petri nets

Abstract

In this paper, a type of graph, called an extremum timed extended reachability graph, is designed to abstract the temporal specifications and represent the feasible trajectories of time Petri nets. This graph improves the timed extended reachability graph recently proposed for time Petri nets (Lefebvre. Discrete Event Dynamic Systems 29(1):31–56. (2019); Zhou et al. IEEE Trans Autom Control 67(7):3693–3698. (2022)) by replacing the earliest-firing policy with a more general policy. In detail, when a transition is preselected for the next firing, the firing can be delayed for a certain period after its minimal residual time has elapsed, rather than immediately firing once its minimal residual time has elapsed. Then, a sampled timed extended reachability graph is designed, wherein, for a transition preselected to fire next, a finite number of time points within a time interval, starting at minimal residual time and ending at maximal residual time, are selected as the firing time instants for the preselected transition. Furthermore, a special case of the sampled timed extended reachability graph, called an extremum timed extended reachability graph that details only the minimal and maximal residual times of the transitions, is also proposed. For a feasible sequence, the corresponding feasible trajectories with minimal and maximal durations are easy to compute with this graph. Thus, an end-to-end delay of a feasible sequence can be obtained by directly searching the graph. Finally, the scheduling of a typical flexible manufacturing system illustrates the advantages and applications of the proposed approach.