Deadlock-Freeness Analysis of Continuous Mono-T-Semiflow Petri Nets

Most verification techniques for highly populated discrete systems suffer from the state explosion problem. The “fluidification” of discrete systems is a classical relaxation technique that aims to avoid the state explosion problem. Continuous Petri nets are the result of fluidifying traditional discrete Petri nets. In continuous Petri nets the firing of a transition is not constrained to the naturals but to the non-negative reals. Unfortunately, some important properties, as liveness, may not be preserved when the discrete net model is fluidified. Therefore, a thorough study of the properties of continuous Petri nets is required. This paper focuses on the study of deadlock-freeness in the framework of mono-T-semiflow continuous Petri nets, i.e., conservative nets with a single repetitive sequence (T-semiflow). The study is developed both on untimed and timed systems. Topological necessary conditions are extracted for this property. Moreover, a bridge relating deadlock-freeness conditions for untimed and timed systems is established.     

离散时间Petri网的模糊模型

离散时间Petri网是一种系统模拟和分析的有效工具，它可以结合图形和分析描述评估离散事件系统的动态执行．基于T—S模糊模型，提出用于描述离散时间Petri网的模糊模型，讨论该模型适应于TtPNs的情况，用以推广文章结论，并且给出用于TtPNs的线性模糊控制规则．同时，文章还给出TtPNs聚合的结论．     

Timed Petri Nets in Hybrid Systems: Stability and Supervisory Control

In this paper, timed Petri nets are used to model and control hybrid systems. Petri nets are used instead of finite automata primarily because of the advantages they offer in dealing with concurrency and complexity issues. A brief overview of existing results on hybrid systems that are based on Petri nets is first presented. A class of timed Petri nets named programmable timed Petri nets (PTPN) is then used to model hybrid systems. Using the PTPN, the stability and supervisory control of hybrid systems are addressed and efficient algorithms are introduced. In particular, we present sufficient conditions for the uniform ultimate boundness of hybrid systems composed of multiple linear time invariant plants which are switched between using a logical rule described by a Petri net. This paper also examines the supervisory control of a hybrid system in which the continuous state is transfered to a region of the state space in a way that respects safety specifications on the plant's discrete and continuous dynamics.     

Estimation of the firing frequencies in discrete and continuous Petri nets models

Production frequencies provide useful information for performance analysis, monitoring and control applications of manufacturing systems. However, such frequencies are not always measurable. In this case, they must be estimated from the observation of other variables. This paper deals with production frequencies estimation for systems modelled by discrete or continuous Petri nets. For this purpose, the content of the buffers is measured on-line with a sampling period, and the evolution equations of the model are reversed. In the case of timed Petri nets, the production frequencies estimation is obtained from the approximation of the firing sequences and in the case of continuous Petri nets it results directly from the variations of the marking vector. The exact and approximated solutions of the estimation problem are described. Accuracy of the estimation is related to measurement error, Petri net structure and sampling period. Moreover, when the estimation provides several solutions, the Petri net model is extended such that a unique solution is obtained.     

Timing and liveness in continuous Petri nets

Fluidification constitutes a relaxation technique for studying discrete event systems through fluidified approximated models, thus avoiding the state explosion problem. Moreover, the class of continuous models thus obtained may be interesting in itself. In Petri nets, fluidification leads to the so-called continuous Petri nets, which are technically hybrid models. Under infinite server semantics, timing a continuous Petri net model preserves the liveness property, but the converse is not necessarily true, and if the autonomous net model is not live, the timing may transform it into a live model. In this paper, we investigate the conditions on the firing rates of timed continuous models that make a given continuous system live.     

Optimal stationary behavior for a class of timed continuous Petri nets

In this paper, we consider a deterministic timed continuous Petri net model where conflicts at places are solved by using stationary routing parameters. We show how to compute the stationary firing rate for all transitions via linear programming, so as to determine the optimal routing parameters that maximize user-defined linear functions of the firing rates. Finally, we discuss the relations with discrete Petri nets.     

A characterization of the stochastic process underlying astochastic Petri net

Stochastic Petri nets (SPN's) with generally distributed firing times can model a large class of systems, but simulation is the only feasible approach for their solution. We explore a hierarchy of SPN classes where modeling power is reduced in exchange for an increasingly efficient solution. Generalized stochastic Petri nets (GSPN's), deterministic and stochastic Petri nets (DSPN's), semi-Markovian stochastic Petri nets (SM-SPN's), timed Petri nets (TPN's), and generalized timed Petri nets (GTPN's) are particular entries in our hierarchy. Additional classes of SPN's for which we show how to compute an analytical solution are obtained by the method of the embedded Markov chain (DSPN's are just one example in this class) and state discretization, which we apply not only to the continuous-time case (PH-type distributions), but also to the discrete case     

Piecewise constant timed continuous PNs for the steady state estimation of stochastic PNs

Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. The computation of numerical approximations is also time consuming due to the slow convergence of stochastic simulations. For these reasons, fluidification can be investigated to estimate the asymptotic behaviour of stochastic processes. The contributions of this paper are to point out that timed continuous Petri nets may lead to biased estimators of the stochastic steady state and to introduce fluid Petri nets with piecewise-constant maximal firing speeds and sufficient conditions in order to obtain unbiased estimators.     

Fluid approximation of Petri net models with relatively small populations

Fluidization is an appealing relaxation technique based on the removal of integrality constraints in order to ease the analysis of discrete Petri nets. The result of fluidifying discrete Petri nets are the so called Fluid or Continuous Petri nets. As with any relaxation technique, discrepancies among the behaviours of the discrete and the relaxed model may appear. Moreover, such discrepancies may have a comparatively bigger effect when the population of the system, the marking in Petri net terms, is “relatively” small. This paper proposes two complementary approaches to obtain a better fluid approximation of discrete Petri nets. The first one focuses on untimed systems and is based on the addition of places that are implicit in the untimed discrete system but not in the continuous. The idea is to cut undesired spurious solutions whose existence worsens the fluidization. The second one focuses on a particular situation that can severely affect the quality of fluidization in timed systems. Namely, such a situation arises when the enabling degree of a transition is equal to 1. This last approach aims to alleviate such a state of affairs, which is termed the bound reaching problem, on systems under infinite servers semantics.     

动态模糊Petri网模型及其应用研究

以模糊Petri网的基本定义，动态模糊集和动态模糊逻辑为基础，讨论了动态模糊Petri网的基本模型，建立了动态模糊Petri网与模糊Petri网之间的映射关系和转移算法，解决了与动态模糊知识的动态模糊Petri网表示相关的几个问题，最后给出了动态模糊Petri网中动态模糊性的计算方法和相应的推理方法。     

混合Petri网交叉口信号灯建模的模糊优化

城市车流量急剧增加和道路通行能力之间的矛盾日益激化,交通拥堵已成为亟待解决的社会问题。道路交叉口信号灯优化控制是解决该问题的有效方法。基于城市道路交通系统随机性强、离散性连续性混杂、难以用数学模型精确建模等特点,提出了一种用连续Petri网建立道路交通流模型,用离散Petri网对道路交叉口信号灯进行控制的方案。根据车流量的动态变化,采用模糊控制对交叉口绿灯时间进行自适应优化。仿真结果表明,该方案能提高交叉口通行能力,减少车辆延误,优于传统的控制方法。     

Correctness verification and performance analysis of real-time systems using stochastic preemptive time Petri nets

Time Petri nets describe the state of a timed system through a marking and a set of clocks. If clocks take values in a dense domain, state space analysis must rely on equivalence classes. These support verification of logical sequencing and quantitative timing of events, but they are hard to be enriched with a stochastic characterization of nondeterminism necessary for performance and dependability evaluation. Casting clocks into a discrete domain overcomes the limitation, but raises a number of problems deriving from the intertwined effects of concurrency and timing. We present a discrete-time variant of time Petri nets, called stochastic preemptive time Petri nets, which provides a unified solution for the above problems through the adoption of a maximal step semantics in which the logical location evolves through the concurrent firing of transition sets. We propose an analysis technique, which integrates the enumeration of a succession relation among sets of timed states with the calculus of their probability distribution. This enables a joint approach to the evaluation of performance and dependability indexes as well as to the verification of sequencing and timeliness correctness. Expressive and analysis capabilities of the model are demonstrated with reference to a real-time digital control system.     

Relevant Timed Schedules/Clock Vectors for Constructing Time Petri Net Reachability Graphs

We consider here the time Petri nets (the TPN model) and its state space abstractions. We show that only some timed schedules/clock vectors (one per enabled transition) of the clock/firing domains are relevant to construct reachability graphs for the TPN. Moreover, we prove formally that the resulting graphs are smaller than the TPN reachability graphs proposed in the literature. Furthermore, these results establish a relation between dense time and discrete time analysis of time Petri nets and allow also improving discrete time analysis by considering only some elements of the clock/firing domains.     

Asymptotic continuous Petri nets

A Petri net is basically a discrete model. However, continuous Petri nets, such that the markings are real numbers have been defined. Two continuous Petri net models involving time have been drawn up. They differ by the calculation of the instantaneous firing speeds of the transitions. Both can be used to approximate a timed Petri net. The former considers constant firing speeds (CCPN) and is very easy to simulate (few events have to be considered, even when it approximates a timed Petri net with many reachable markings). The latter considers firing speeds depending on the marking (VCPN). Although it provides a better approximation, its simulation is longer because the markings and speeds are given by differential equations. This paper introduces a third model (ACPN) which presents the advantages of the two preceding ones. In most cases, this model represents the asymptotic behavior of the VCPN. Then their precisions are similar. Since the firing speeds of the ACPN are constant, it is as easy to simulate as the CCPN.     

基于模糊Petri网的瓦斯突出空间模型

瓦斯突出相关影响因素具有明显的时空特性,煤层突出危险程度的判别是一个多因素决定的模糊事件,模糊Petri网在离散事件动态建模具有突出的优势。在分析瓦斯突出空间数据分布及模糊性应用基础上,对比分析基本Petri网和模糊Petri网的知识表示,结合瓦斯突出各因素空间分布的特点,扩展了模糊Petrie网的相关组件成分,建立基于模糊Petri网的瓦斯突出空间模型,使瓦斯突出影响因素在不同的空间位置通过空间模型表示出来,进一步描述了瓦斯突出空间位置动态转换,在实际预测中具有现实的意义和可操作性。     

Feasibility of piecewise-constant control sequences for timed continuous Petri nets

In this study, the determination of control actions for timed continuous Petri nets is investigated by the characterisation of attractive regions in marking space. In particular, attraction in finite time, which is important for practical issues, is considered. Based on the characterisation of attractive regions, the domain of admissible piecewise constant control actions is computed, and sufficient conditions to verify the feasibility of the control objectives are proposed. As a consequence, an iterative procedure is presented to compute piecewise constant control actions that correspond to local minimum time control for timed continuous Petri nets.     

Fuzzy Petri net-based programmable logic controller

Programmable logic controllers (PLCs) are able to directly implement control sequences specified by means of standard languages such as Grafcet or formal models such as Petri nets. In the case of simple regulation problems between two steps it could be of great interest to introduce a notion of “fuzzy events” in order to denote a continuous evolution from one state to another. This could result from a linear interpolation between the commands attached to two control steps represented by two Petri net (PN) places. This paper is an attempt to develop fuzzy PN-based PLCs in a similar way as fuzzy controllers (regulators). Our approach is based on a combination of Petri nets with possibility theory (Petri nets with fuzzy markings)     

New NP-Complete Problems in Performance Evaluation of Concurrent Systems Using Petri Nets

Timed Petri nets are useful in performance evaluation of concurrent systems. The maximum computation rate is achieved for minimal cycle time of timed Petri net. It is known that minimal cycle time problem for P-invariant Petri nets is NP-complete. In this paper we prove that the minimal cycle time problem, for non-P-invariant Petri nets and for a small subclass of P-invariant Petri nets called free-choice nets having live and safe marking, is NP-complete.     

Optimal Model Predictive Control of Timed Continuous Petri Nets

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.