As a preliminary overview, this work provides first a broad tutorial on the fluidization of discrete event dynamic models, an efficient technique for dealing with the classical state explosion problem. Even if named as continuous or fluid, the relaxed models obtained are frequently hybrid in a technical sense. Thus, there is plenty of room for using discrete, hybrid and continuous model techniques for logical verification, performance evaluation and control studies. Moreover, the possibilities for transferring concepts and techniques from one modeling paradigm to others are very significant, so there is much space for synergy. As a central modeling paradigm for parallel and synchronized discrete event systems, Petri nets (PNs) are then considered in much more detail. In this sense, this paper is somewhat complementary to David and Alla (2010). Our presentation of fluid views or approximations of PNs has sometimes a flavor of a survey, but also introduces some new ideas or techniques. Among the aspects that distinguish the adopted approach are: the focus on the relationships between discrete and continuous PN models, both for untimed, i.e., fully non-deterministic abstractions, and timed versions; the use of structure theory of (discrete) PNs, algebraic and graph based concepts and results; and the bridge to Automatic Control Theory. After discussing observability and controllability issues, the most technical part in this work, the paper concludes with some remarks and possible directions for future research.
This paper addresses the optimal control problem of timed continuous Petri nets under infinite servers semantics. In particular, our goal is to find a control input optimizing a certain cost function that permits the evolution from an initial marking (state) to a desired steady-state. The solution we propose is based on a particular discrete-time representation of the controlled continuous Petri net system, as a certain linear constrained system. An upper bound on the sample period is given in order to preserve important information of the timed continuous net, in particular the positiveness of the markings. The reachability space of the sampled system in relation to autonomous continuous Petri nets is also studied.Based on the resulting linear constrained model, the optimal control problem is studied through Model Predictive Control (MPC). Implicit and explicit procedures are presented together with a comparison between the two schemes. Stability of the closed-loop system is also studied.
In this paper we propose an automated method for planning a team of mobile robots such that a Booleanbased mission is accomplished. The task consists of logical requirements over some regions of interest for the agents' trajectories and for their final states. In other words, we allow combinatorial specifications defining desired final states whose attainment includes visits to, avoidance of, and ending in certain regions. The path planning approach should select such final states that optimize a certain global cost function. In particular, we consider minimum expected traveling distance of the team and reduce congestions. A Petri net (PN) with outputs models the movement capabilities of the team and the regions of interest. The imposed specification is translated to a set of linear restrictions for some binary variables, the robot movement capabilities are formulated as linear constraints on PN markings, and the evaluations of the binary variables are linked with PN markings via linear inequalities. This allows us to solve an Integer Linear Programming problem whose solution yields robotic trajectories satisfying the task.
When discrete event systems are used to model systems with a large number of possible (reachable) states, many problems such as simulation, optimization, and control, may become computationally prohibitive because they require some enumeration of such states. A common way to effectively address this issue is fluidization. The goal of this paper is that of studying the effect of fluidization on fault diagnosis. In particular, we focus on the purely logic Petri net model that results in the untimed continuous Petri net model after fluidization. In accordance to most of the literature on discrete event systems, we define three diagnosis states, namely N , U and F , corresponding respectively to no fault, uncertain and fault state. We prove that, given an observation, the resulting diagnosis state can be computed solving linear programming problems rather than integer programming problems as in the discrete case. The main advantage of fluidization is that it enables to deal with much more general Petri net structures. In particular, the unobservable subnet needs not be acyclic as in the discrete case. Moreover, the compact representation of the set of consistent markings using convex polytopes can be seen in some cases as an improvement in terms of computational complexity.
Continuous Petri nets were introduced as an approximation to deal with the state explosion problem which can appear in discrete event models. When time is introduced, the flow through a fluidified transition can be defined in many ways. The most used in literature are infinite and finite servers semantics. Both can be seen as derived from stochastic Petri nets. The practical problems addressed in this contribution are:(1) a sufficient condition for the performance monotonicity, and (2) a study of the transition semantics, always related to continuous Petri nets. We prove that under some conditions, the subclass of mono-T-semiflow is monotone and also for the same class of nets we prove a property for which infinite servers semantics offers a better approximation than finite servers semantics for the discrete model.
Continuous Petri nets were introduced as an approximation to deal with the state explosion problem which can appear in discrete event models. When time is introduced, the flow through a fluidified transition can be defined in many ways. The most used in literature are constant and variable speed [8], which can be seen as some kind of finite and infinite server interpretations of the transitions behavior, and derived from stochastic (discrete) Petri nets [18]. In this paper we will compare the results obtained with both relaxations for the broad class of mono-T-semiflow reducible nets, and prove that, under some frequently true conditions, infinite server semantics offers a throughput which is closer to the throughput of the discrete system in steady state. In the second part, it will be proved that the throughput of mono-T-semiflow reducible net systems is monotone with respect to the speed of the transitions and the initial marking of the net.
This paper proposes an online approach for fault diagnosis of timed discrete event systems modeled by Time Petri Net (TPN). The set of transitions is partitioned into two subsets containing observable and unobservable transitions, respectively. Faults correspond to a subset of unobservable transitions. In accordance with most of the literature on discrete event systems, we define three diagnosis states, namely normal, faulty and uncertain states, respectively. The proposed approach uses a fault diagnosis graph, which is incrementally computed using the state class graph of the unobservable TPN. After each observation, if the part of FDG necessary to compute the diagnosis states is not available, the state class graph of the unobservable TPN is computed starting from the consistent states. This graph is then optimized and added to the partial FDG keeping only the necessary information for computation of the diagnosis states. We provide algorithms to compute the FDG and the diagnosis states. The method is implemented as a software package and simulation results are included.Index Terms-Discrete event system (DES), fault diagnosis, Petri net, timed systems.
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