A deadlock prevention approach for flexible manufacturing systems with uncontrollable transitions in their Petri net models

Deadlocks are a highly undesired situation in a fully automated flexible manufacturing system, whose occurrences are tied to the existence of shared resources that are competed by different production processes. In the last two decades, a fair amount of research has been done on deadlock analysis and control for flexible manufacturing systems, leading to a variety of strategies in the literature. Petri nets are a promising mathematical tool to handle deadlock problems in flexible manufacturing systems. However, most deadlock control policies based on a Petri net formalism assume that all the transitions in a plant model are controllable. However, uncontrollability of events are a natural feature in a real‐world production system. This paper proposes a deadlock prevention policy for a class of Petri nets by considering the existence of uncontrollable transitions. Deadlocks are prevented by adding monitors to a plant Petri net model, whose addition does not inhibit the firings of uncontrollable transitions. Linear programming techniques are employed to find transitions to which a monitor points in order that a more permissive liveness‐enforcing Petri net supervisor can be found. A number of manufacturing examples are used to demonstrate the proposed methods. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Suboptimal liveness-enforcing supervisor design for a class of generalised Petri nets using partial siphon enumeration and mathematical programming

This article develops a deadlock prevention policy for a class of generalised Petri nets, which can well model a large class of flexible manufacturing systems. The analysis of such a system leads us to characterise the deadlock situations in terms of the insufficiently marked siphons in its generalised Petri-net model. The proposed policy is carried out in an iterative way. At each step a minimal siphon is derived from a maximal deadly marked siphon that is found by solving a mixed integer programming (MIP) problem. An algorithm is formalised that can efficiently compute such a minimal siphon from a maximal one. A monitor is added for a derived minimal siphon such that it is max-controlled if it is elementary with respect to the siphons that have been derived. The liveness of the controlled system is decided by the fact that no siphon can be derived due to the MIP solution. After a liveness-enforcing net supervisor computed without complete siphon enumeration, the output-arcs of the additional monitors are rearranged such that the monitors act while restricting the system less. Examples are presented to demonstrate the proposed method.     

一种改进型的S4PR网活性条件

研究了顺序资源共享分配系统的建模模型S4PR (Systems of sequential systems with shared resources)网的活性问题. 已有的研究成果表明, 一个S4PR网在所有信标都满足max, max'或max"-controlled 时能保持活性, 但现有的活性条件对信标的限制严格且不适用于某些网系统, 本文提出了一类名为max*-controlled的改进型条件, 并证明了当一个S4PR网的所有信标都满足max*-controlled条件时, 网系统能保持活性. 与现有的其他条件相比, 新的活性条件更加宽松, 为设计更高允许度的死锁预防或者活性保持监控器提供了理论支撑.     

Necessary and sufficient liveness condition of GS3PR Petri nets

Structural analysis is one of the most important and efficient methods to investigate the behaviour of Petri nets. Liveness is a significant behavioural property of Petri nets. Siphons, as structural objects of a Petri net, are closely related to its liveness. Many deadlock control policies for flexible manufacturing systems (FMS) modelled by Petri nets are implemented via siphon control. Most of the existing methods design liveness-enforcing supervisors by adding control places for siphons based on their controllability conditions. To compute a liveness-enforcing supervisor with as much as permissive behaviour, it is both theoretically and practically significant to find an exact controllability condition for siphons. However, the existing conditions, max, max′, and max″-controllability of siphons are all overly restrictive and generally sufficient only. This paper develops a new condition called max*-controllability of the siphons in generalised systems of simple sequential processes with resources (GS³PR), which are a net subclass that can model many real-world automated manufacturing systems. We show that a GS³PR is live if all its strict minimal siphons (SMS) are max*-controlled. Compared with the existing conditions, i.e., max-, max′-, and max″-controllability of siphons, max*-controllability of the SMS is not only sufficient but also necessary. An example is used to illustrate the proposed method.     

A deadlock prevention approach for flexible manufacturing systems without complete siphon enumeration of their Petri net models

Siphons are very important in the analysis and control of deadlocks in a Petri net. However, it is quite time-consuming or even impossible to get the complete siphon enumeration of a Petri net. This paper focuses on the deadlock prevention problems in flexible manufacturing systems that are modeled with S⁴PR, a general class of Petri nets. The analysis of S⁴PR leads us to characterize deadlock situations in terms of insufficiently marked siphons. The method proposed in this paper is an iterative approach. At each iteration, a non-max-marked siphon is computed by solving a mixed integer linear programming problem. Then the siphon is max-marked through a P-invariant by adding a monitor place. This process is carried out until no non-max-marked siphon can be found in the net. As a result all the siphons in the net are max-controlled. Then the net becomes live. Without computing all the siphons, a monitor-based liveness-enforcing Petri net supervisor can be found with more permissive behavior. A number of flexible manufacturing examples are used to demonstrate the proposed methods.     

On Controllability of Dependent Siphons for Deadlock Prevention in Generalized Petri Nets

A fair amount of research has shown the importance of siphons in the analysis and control of deadlocks in a variety of resource allocation systems by using a Petri net formalism. In this paper, siphons in a generalized Petri net are classified into elementary and dependent ones, as done for ordinary nets in our previous work. Conditions are derived under which a dependent siphon is controlled by properly supervising its elementary siphons, which indicates that the controllability of dependent siphons in an ordinary Petri net is a special case of that in a generalized one. The application of the controllability of dependent siphons is shown by considering the deadlock prevention problem for a class of resource allocation systems, namely, G-system that allows multiple resource acquisitions and flexible routings in a flexible manufacturing system with machining, assembly, and disassembly operations. We develop a monitor-based deadlock prevention policy that first adds monitors for elementary siphons only to a G-system plant model such that the resultant net system satisfies the maximal controlled-siphon property (maximal cs-property). Then, by linear programming, initial tokens in the additional monitors are decided such that liveness is enforced to the supervised system. Also, a simplified live marking relationship for a G-system between the initial tokens of the source places and those of the resource places is derived. Finally, the proposed deadlock prevention methods are illustrated by using an example.     

A deadlock prevention approach for a class of timed Petri nets using elementary siphons

To solve the problem of deadlock prevention for timed Petri nets, an effective deadlock prevention policy based on elementary siphons is proposed in this paper. Without enumerating reachable markings, deadlock prevention is achieved by adding monitors for elementary siphons, increasing control depth variables when necessary, and removing implicit, liveness‐restricted and redundant control places. The final supervisor is live. First, a timed Petri net is stretched into a stretched Petri net (SPN). Unchanging the system performance, each transition in the SPN has a unit delay time. Then the siphon‐control‐based approach is applied. Monitors computed according to the marking constraints are added to the SPN model to ensure all strict minimal siphons in the net invariant‐controlled. A liveness‐enforcing supervisor with simple structure can be obtained by reverting the SPN into a TdPN. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Optimal deadlock avoidance Petri net supervisors for automated manufacturing systems

Deadlock avoidance problems are investigated for automated manufacturing systems with flexible routings. Based on the Petri net models of the systems, this paper proposes, for the first time, the concept of perfect maximal resourcetransition circuits and their saturated states. The concept facilities the development of system liveness characterization and deadlock avoidance Petri net supervisors. Deadlock is characterized as some perfect maximal resource-transition circuits reach their saturated states. For a large class of manufacturing systems, which do not contain center resources, the optimal deadlock avoidance Petri net supervisors are presented. For an general manufacturing system, a method is proposed for reducing the system Petri net model so that the reduced model does not contain center resources and, hence, has optimal deadlock avoidance Petri net supervisor. The controlled reduced Petri net model can then be used as the liveness supervisor of the system.     

Optimal deadlock avoidance Petri net supervisors for automated manufacturing systems

Deadlock avoidance problems are investigated for automated manufacturing systems with flexible routings. Based on the Petri net models of the systems, this paper proposes, for the first time, the concept of perfect maximal resource-transition circuits and their saturated states. The concept facilitates the development of system liveness characterization and deadlock avoidance Petri net supervisors. Deadlock is characterized as some perfect maximal resource-transition circuits reaching their saturated states. For a large class of manufacturing systems, which do not contain center resources, the optimal deadlock avoidance Petri net supervisors are presented. For a general manufacturing system, a method is proposed for reducing the system Petri net model so that the reduced model does not contain center resources and, hence, has optimal deadlock avoidance Petri net supervisor. The controlled reduced Petri net model can then be used as the liveness supervisor of the system.     

Design of liveness‐enforcing supervisors via transforming plant petri net models of FMS

This paper focuses on the deadlock prevention problems in a class of Petri nets, systems of simple sequential process with resources, S³PR for short. By structure analysis, we propose an approach that can transform a plant net model into a weighted S³PR (WS³PR) that is behaviorally equivalent to the plant model. The WS³PR is made to be live by properly reconfiguring its weight distribution such that its all strict minimal siphons are self‐max'‐controlled. The resulting WS³PR can serve as a liveness‐enforcing Petri net supervisor for the plant model after removing some idle and operation places. A live controlled system can be accordingly obtained by synchronizing a plant model and the places whose weights are regulated. This research shows that a small number of monitors is obtained, leading to more permissive behavior of the controlled system. Examples are used to demonstrate the proposed concepts and methods. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Design of a maximally permissive liveness-enforcing supervisor with a compressed supervisory structure for flexible manufacturing systems

In this paper, a deadlock prevention policy for flexible manufacturing systems (FMS) is proposed, which can obtain a maximally permissive liveness-enforcing Petri net supervisor while the number of control places is compressed. By using a vector covering approach, the sets of legal markings and first-met bad markings (FBM) are reduced to two small ones, i.e., the minimal covering set of legal markings and the minimal covered set of FBM. A maximally permissive control purpose can be achieved by designing control places such that all markings in the minimal covered set of FBM are forbidden and no marking in the minimal covering set of legal markings is forbidden. An integer linear programming problem is designed to minimize the number of control places under an assumption that a control place is associated with a P-semiflow. The resulting net has the minimal number of control places on the premise that the assumption holds, and possesses all permissive states of a plant. The only problem of the proposed method is its computational complexity that makes it inapplicable to large-scale Petri net models. An FMS example from the literature is presented to illustrate the proposed method.     

An iterative method for synthesizing non-blocking supervisors for a class of generalized Petri nets using mathematical programming

This paper presents a novel and computational deadlock prevention policy for a class of generalized Petri nets, namely G-systems, which allows multiple resource acquisitions and flexible routings with machining, assembly and disassembly operations. In this research, a mixed integer programming (MIP)-based deadlock detection technique is used to find an insufficiently marked minimal siphon from a maximal deadly marked siphon for generalized Petri nets. In addition, two-stage control method is employed for deadlock prevention in Petri net model. Such proposed method is an iterative approach consisting of two stages. The first one is called siphons control, which adds a control place to the original net for each insufficiently marked minimal siphon. The objective is to prevent minimal siphons from being insufficiently marked. The second one, called control-induced siphons control, is to add a control place to the augmented net with its output arcs connecting to source transitions, which assures that there is no new insufficiently marked siphon generated due to the addition of the monitors. Compared with the existing approaches, the proposed deadlock prevention policy can usually lead to a non-blocking supervisor with more permissive behavior and high computational efficiency for a sizeable plant model due to avoiding complete siphon enumeration. Finally, a practical flexible manufacturing system (FMS) example is utilized to illustrate the proposed method.     

一类Petri网—S4R的死锁预防策略

S4R网作为一种特殊的Petri网子类,与S' PR网相比可以建模更为复杂的、拥有多个并行加工进程的资源分配系统。针对S4R网提出了一种综合的死锁预防策略。利用MIP检验由S4R网建模的柔性制造系统的活性,在新的信标控制概念的基础上对需要控制的系统进行控制。再利用MIP检验受控网系统的活性,进一步控制不活的网系统。避免了对一些网不必要的控制以及一些网过于保守的控制,得到许可行为较多的控制器。     

Design of liveness-enforcing supervisors with simpler structures for deadlock-free operations in flexible manufacturing systems using necessary siphons

Siphons can be used to characterize deadlock states and solve deadlock problems in Petri nets that model flexible manufacturing systems. This paper presents an iterative siphon-based control (ISC) deadlock prevention policy for Petri nets via the combination of mixed integer programming (MIP) and the concept of necessary siphons (NSs). At each iteration in this ISC policy, a maximal deadly marked siphon that is closely related to deadlocks in a Petri net can be conveniently found by an MIP-based deadlock detection method. Then the places in it are classified and an NS is derived from the classified places. For each NS found, depending on its complementary set, the proposed policy adds a proper control place (CP) to make it marked (max-controlled). Moreover, during the ISC procedure, a test for redundant NSs is carried out under a certain condition in order to avoid the addition of the corresponding CPs. The siphon control process proceeds iteratively until the controlled system is live. Compared with the existing approaches, the proposed policy usually leads to a structurally simple liveness-enforcing supervisor by adding as few CPs as possible and achieves better control results. Some examples are introduced to illustrate the proposed approach.     

一类FMS的最佳活Petri网模型的综合

利用Petri网为一类柔性制造系统建模,并讨论避免系统死锁问题.通过Petri网模 型的结构分析,证明了系统产生死锁的一个充分必要条件.给出了避免死锁的最佳控制器,它 可以通过给系统的Petri网模型增加一些新的位置与相应的弧来实现.从而导出了这类制造 系统的最佳活Petri网模型.     

Extended Elementary Siphons and Their Application to Liveness‐Enforcement of Generalized Petri Nets

As a significant structural object, siphons are extensively employed to implement a large number of deadlock prevention and liveness‐enforcing methods for flexible manufacturing systems modeled by Petri nets. By linear combinations, a set of elementary siphons is chosen from all strict minimal ones to be controlled and thus the structural complexity of a supervisor is greatly reduced. The concept of elementary siphons is originally proposed for ordinary Petri nets. When applied to generalized Petri nets, their selection and controllability require an additional study. In this work, the concept of augmented siphons is proposed to extend the application of the elementary ones to a class of generalized Petri nets, GLS³PR. Based on graph theory, a siphon extraction algorithm is developed to obtain all strict minimal siphons, from which augmented elementary ones are computed. In addition, the controllability conditions of dependent siphons are developed. Through fully investigating the net structure, especially weight information, the set of augmented elementary siphons is more compact and well suits for generalized Petri net models under consideration. Some examples are used to illustrate the proposed method.     

Liveness-enforcing supervisors synthesis for a class of generalised Petri nets based on two-stage deadlock control and mathematical programming

In this article we deal with deadlock prevention problems for S⁴PR, a class of generalised Petri nets, which can well model a large class of flexible manufacturing systems where deadlocks are caused by insufficiently marked siphons. We present a deadlock prevention methodology that is an iterative approach consisting of two stages. The first one is called siphon control, which is to add for each insufficiently marked minimal siphon a control place to the original net. Its objective is to prevent a minimal siphon from being insufficiently marked. The second one, called control-induced siphon control, is to add a control place to the augmented net with its output arcs connecting to the source transitions, which assures that there are no new insufficiently marked siphons generated. At each iteration, a mixed integer programming approach is adopted for generalised Petri nets to obtain an insufficiently marked minimal siphon from the maximal deadly siphon. This way complete siphon enumeration is avoided that is much more time-consuming for a sizeable plant model than the proposed method. The relation of the proposed method and the liveness and reversibility of the controlled net is obtained. Examples are presented to demonstrate the presented method.     

基于Petri网的柔性制造系统一种预防死锁方法

基于Petri网的结构特性分析,研究了FMS(柔性制造系统)一种预防死锁方法.提出了 Petri网的一种特殊拓扑结构--基本信标的概念.在Petri网中基本信标的集合是SMS(严格极 小信标)集合的一个真子集.尤其在大型Petri网系统中,基本信标的集合比SMS的集合要小得 多.对于Petri网的一个子类S3PR,只对每一个基本信标添加一个库所使其不被清空,就可实现 预防死锁,也就是说无须控制S3PR的所有SMS而达到无信标被清空的目的.此外,对于S3PR, 还提出了一种求取SMS和基本信标的方法.相对于现在普遍采用的控制所有SMS来预防死锁 的策略,其具三方面优势.1)只需控制少量的SMS即所谓的基本信标.相应地,添加少量的控制 库所和连接弧,就可得到无死锁或活的Petri网.2)不需要先行计算出极小信标的集合.3)明显 地,这种方法更适合大型Petri网系统.我们通过穿插在文中的一个例子来说明这些方法.     

Structure reduction of liveness‐enforcing Petri nets using mixed integer programming

Many deadlock prevention policies existing in the literature are to add control places (CPs) to cope with deadlocks in practical systems modeled with Petri nets. Since the number of CPs determined by these policies is not minimal under the condition that a controlled systems is live, this usually leads to a liveness‐enforcing Petri net supervisor with redundant CPs. Based on mixed integer programming (MIP) and the concept of implicit places (IPs), this paper develops a novel iterative algorithm of simplifying the structural complexity for a live Petri net. Under the condition that liveness is preserved in the iteration, this algorithm computes a feasible solution of an MIP for each CP to confirm whether redundant CPs exist in the live controlled system. Necessary and redundant CPs are then kept in or removed from the simplified live Petri net, respectively. As a result, a live controlled system with simpler structure is obtained, which directly reduces computational cost in further design and verification phases and possibly leads to more permissive behavior. Effectiveness of this algorithm is proved via a theoretic analysis and examples. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

基于混合整数规划的非阻塞监督控制器设计

提出一种新的死锁控制策略, 保证含有并发执行装配过程的一类柔性制造系统(Flexible manufacturing system, FMS) G-system的非阻塞性, 即在控制下, 受控系统从任意可达状态都可到达理想状态. 首先对Petri网模型运用混合整数规划算法求取一个最大的死信标, 然后从最大的死信标中求取一个需要受控的极小信标, 并对其添加控制库所, 从而保证所有信标的最大可控. 和现有方法相比, 该策略避免了求取所有的信标, 且添加较少的控制库所即可获得结构简单、许可行为趋于最优的控制器.