An Optimization Approach to Improved Petri Net ... - IEEE Xplore

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 10, NO. 3, JULY 2013

An Optimization Approach to Improved Petri Net Controller Design for Automated Manufacturing Systems Hesuan Hu, Member, IEEE, MengChu Zhou, Fellow, IEEE, Zhiwu Li, Senior Member, IEEE, and Ying Tang, Senior Member, IEEE

Abstract—Sensors and actuators are two indispensable parts in the paradigm of feedback control. Their implementation cost should be properly evaluated and constrained. In the previous work, a Petri net monitor with the least cost is synthesized through integer programming formulation. Despite its technical correctness, the existing method may lead to undesirable results when the net structure contains some shared or unshared resource places of a manufacturing-oriented net model. A necessary and sufficient condition is established to show that certain structures can lead to deadlock-prone supervisors. An efficient algorithm is developed to identify such structures. Furthermore, it is shown that if one can identify such structures at the initial stage, it is possible to achieve desirable controllers for the original systems. The theoretical correctness of the proposed algorithm is discussed. A manufacturing example is provided to illustrate the proposed approach. Note to Practitioners—A modern manufacturing system requires the use of sensors and actuators whose quantity is largely decided by its supervisory controller. Clearly, more complex controllers imply higher overall implementation cost and failure probability. This paper proposes an effective method to reduce size and thus the implementation cost of a supervisory controller. In the Petri net model, each transition is associated with observation and control cost that represents the implementation cost of a sensor and an actuator at certain physical positions. A mathematical programming approach is proposed to effectively determine the controller with the minimum cost. It is applied to automated manufacturing systems (AMS) with complex operations. The experimental results prove its effectiveness and efficiency. Manuscript received April 01, 2012; accepted May 13, 2012. Date of publication July 19, 2012; date of current version June 27, 2013. This paper was recommended for publication by Associate Editor B. Turchiano and Editor Y. Narahari upon evaluation of the reviewers’ comments. This work was supported in part by the Natural Science Foundation of China under Grants 60474018, 60773001, 61074035, 61034004, 61050110145, and 51105395, in part by the Fundamental Research Funds for the Central Universities under Grant JY10000904001, in part by the National Research Foundation for the Doctoral Program of Higher Education, the Ministry of Education, China, under Grant 20090203110009, in part by the Research Fellowship for International Young Scientists, and in part by the Alexander von Humboldt Foundation. H. Hu is with the School of Electro-Mechanical Engineering, Xidian University, Xi’an, Shaanxi 710071, China (e-mail: [email protected]). M. C. Zhou is with the The Key Laboratory of Embedded System and Service Computing, Ministry of Education, Tongji University Shanghai 201804, China, and also with the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]). Z. Li is with the School of Electro-Mechanical Engineering, Xidian University, Xi’an, Shaanxi 710071, China, and also with the Department of Computer Science, Martin-Luther University, Halle 06120, Germany (e-mail: [email protected]). Y. Tang is with the Department of Electrical and Computer Engineering, Rowan University, Glassboro, NJ 08028 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TASE.2012.2201714

Index Terms—Discrete event systems, integer programming, Petri nets, supervisory control.

I. INTRODUCTION

D

IFFERENT from most natural systems, many man-made systems cannot be described with differential or difference equations [1], [3], [9]–[12], [22], [23]. Typical examples are automated manufacturing systems (AMS), computer operating systems, and communication systems [2], [5], [13]–[17], [21], [25], [26], [39], [41]–[43], [45], [47]–[60]. In such systems, one event occurs largely owing to the triggering from other events. The occurrence of each event may lead the system from one state to others. Each event occurs abruptly at undetermined time instants. The state of such systems evolves in a discrete rather than continuous way. A trajectory of such systems is composed of these states along with the events connecting different states. While techniques are abundant to handle continuous systems, less are available to tackle discrete event ones. Currently, the most popular models for the latter are Petri nets and automata. Similar to the scenarios in continuous systems, it is of importance to consider the control issues arising in discrete event systems. The work is initiated in [46], where automata are explored to synthesize proper supervisory controllers. Formal languages are also involved so as to describe such systems and produce proper supervisors. To model a system with automata, one needs to divide the system into several subsystems. For each subsystem, an automaton model is established. The combination of these subsystem models can lead to the model for a whole system. Evidently, this can cause the well-known state explosion issue. This is infeasible in many cases since most discrete event systems are too complex to be described elaborately. Petri nets are a promising mathematical tool to palliate such a problem since it is proved that they can represent a system in a quite compact way. In terms of Petri nets, generalized mutual exclusion constraints (GMEC) are a significant concept initially proposed in [8] to limit the weighted sum of tokens contained in a subset of places. Through them, a linear supervisory specification can be implemented by a control place (monitor, for short) and its related flows to some controllable transitions [2]–[7], [13]–[20], [24]–[38], [40], [44]. Events are distinguished by controllable/observable and uncontrollable/unobservable ones such that more complex and practical cases can be handled.

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HU et al.: AN OPTIMIZATION APPROACH TO IMPROVED PETRI NET CONTROLLER DESIGN FOR AUTOMATED MANUFACTURING SYSTEMS

A set of integer programming (IP) formulations is developed to derive monitors for systems with uncontrollable and/or unobservable events. This method always results in a more restrictive but safer monitor. To further generalize the approach in the same vein, each transition is associated with an observation and control cost in [1]. Consequently, a supervisor with the least implementation cost can be synthesized through a set of IP formulations. Owing to the lack of structural constraints, the results obtained via the method in [1] can be undesirable for some complex systems. In this paper, we show through a few examples that the approach in [1] leads to paradoxical results in some cases. As a result, despite its significance for supervisory control, its applicability is limited without proper modification. In some cases, the controlled system are dead at the initial state due to the over-restrictive control imposed by the synthesized monitors and their related flows. Finally, an improved approach is proposed so as to produce a more desirable monitor. In the consecutive sections, basic notations of Petri nets are given. The problem of the approach in [1] is illustrated through a few explanatory examples. The mechanisms that cause the mentioned problems are discussed. Effort is then devoted to the development of a proper mechanism to solve it. An example from automated manufacturing domain is used to show the effectiveness and significance of the proposed method. II. BASIC NOTATIONS OF PETRI NETS A Petri net is , where and are finite, nonempty, and disjoint sets, and . (resp., , ) denotes a finite set of places (resp., transitions, flow relations or arcs). . if and if so. A net is said to be an ordinary one if , . It is said to be a general net if , . The input and output incidence matrices are defined as and , respectively. The incidence matrix is defined as . (resp., , ) is the row of (resp., , ). A marking of is a mapping , where and . is called a net system or a marked net. is a net system with an initial marking . is enabled at , denoted by , if . is reachable from , denoted by , if there exists a firing sequence such that . The set of all markings reachable from is denoted by . A transition is live under if , , holds. is deadlock-free if , , . is dead at if such that holds. is live if , is live under . A -vector is a column vector indexed by , where is the set of integers. We denote a column vector where every entry equals 0 by . and are the transposed versions of vector and matrix , respectively. (resp., ) is a -invariant (resp., -invariant) if (resp., ) and hold. It is said to be a -semiflow (resp., -semiflow) if no component of (resp., ) is negative. is called the support of . A net is said to be conservative (resp., consis-

tent) if there exists a (resp., )

-semiflow (resp.,

so that ) postset

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-semiflow)

(resp.,

its preset

, its

. They can be extended to set, e.g. is a set and . The following concept is due to the early work that deals with the synthesis of Petri net models for automated manufacturing systems [15]–[17], [48]. A resource place (resource, for short) is a place such that , with , and . A resource is said to be shared if ; otherwise, is unshared. It and its related activity or operation places form the support of a -semiflow. The set of resources in is denoted by . III. BRIEF REVIEW OF THE PRIOR WORK The GMEC is initially proposed in [8] as a linear inequality , where means an arbitrary reachconstraint able marking, while and are an integer vector and scalar, respectively [8]. This constraint can be enforced with a monitor denoted by . A monitor should be superimposed on a net structure according to a row of an incidence row vector . The initial marking of the monitor, denoted , is determined by . Mathematically, by a negative initial marking is unacceptable for a monitor. This . According to the definition of incidence mameans trix, can be decomposed into a row of post-incidence matrix and a row of pre-incidence one such that , and . where The approach in [7] generalizes the one in [8] by distinguishing the transitions into controllable/observable and uncontrollable/unobservable ones. More importantly, a set of IP formulations is established to efficiently derive a more restrictive constraint such that no flow goes from a monitor to an uncontrollable transition while neither outgoing nor ingoing flow exists from a monitor to an unobservable transiis desired such that tion. Thus, another augmented pair , while other requirements are also satisfied. Without loss of generality, this transformation can be and , implemented by assuming -dimensional nonnegative vector and is a where is a positive scalar. The unknown values of and can be produced by solving the following IP problem [7]:

subject to (1) (2) (3) means the objective function while and denote the columns of an incidence matrix corresponding to the uncontrollable and unobservable transitions, where

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Fig. 1. A system in Example 1.

Fig. 2. Monitors for system shown in Fig. 1 according to three approaches. (a) Approach in [8]. (b) Approach in [7]. (c) Approach in [1].

i.e., and , respectively. The correctness of (1)–(3) is shown in the Appendix through several properties. Note, these properties and their proofs are already-known results in literature like [7]. We present them for the clarity of this paper. In [1], to evaluate the implementation cost of a monitor, each transition in a Petri net is associated with an observation and and control cost such that , where means the set of positive real numbers. The cost of a monitor is defined as the sum of observation and control costs of related transitions. To derive a monitor with the least cost, a set of IP formulations are presented in [1] as follows: (4)

Fig. 3. Net systems in Cases 1 to 3. (a) Net system in Case 1. (b) Net system in Case 2. (c) Net system in Case 3.

whether shared or unshared, are allowed in a net structure. This can be shown by the following two examples. — Case 1: Fig. 3(a) shows a Petri net with three unshared resources, i.e., , , and . Consider a GMEC with and , which is equivalent to an inequality . By assuming and , we can establish an IP formulation strictly according to (4)–(7) so as to obtain a more reasonable constraint in terms of its implementation cost

subject to

subject to (5) (6) (7) For a feasible original constraint , we claim that a solution for above formulations always exists. This is owing to and , which means that in the worst case, we have no other constraints can ensure a lower cost than the original one. 1) Example 1: In [1], [7], and [8], these three approaches are applied to quite simple systems structurally similar to the one shown in Fig. 1, where and are controllable and observable, is neither controllable nor observable, and is observable but uncontrollable. For clear presentation, and are depicted by drawing them as black boxes, by an empty one, while by a double-edge one. To further elaborate the controllability and observability of all transitions, each of them is associated with and a control and observation cost, e.g., . Applying the three approaches in [1], [7], and [8] to Fig. 1, we obtain the three monitors, as shown in Figs. 2(a)–(c), respectively. Their costs are 4, 4, and 3, respectively. IV. ANALYSIS THROUGH CASE STUDIES Although it is correct with solid theoretical support, the approach in [1] may lead to undesirable solutions when resources,

where , , and . After calculation with Lingo 8.0, which is a popular optimization software package, we obtain and . Remember the definition and . An augis obtained with mented GMEC and . In the form of inequality, this is equivalent to and can be imposed on the plant net with and . Observation shows that is the support of a -semiflow such that . Consequently, the actual constraint is equivalent to . Such a result is apparently undesirable since it prohibits the firing of any transition at . Thus, its corresponding controlled system is dead at the initial state. Figs. 4(a) and 5(a) demonstrate the monitors determined by and , respectively. Obviously, the latter cannot substitute the former one properly. — Case 2: Fig. 3(b) shows another Petri net, where places represent three shared resources. Consider a GMEC with and which


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Fig. 4. Monitors for Cases 1–3 according to the proposed approach [8].

Fig. 5. Monitors for Cases 1–3 according to the proposed approach in [1].

is equivalent to an inequality . By assuming and , we can establish an IP model in the same way as we do in Case 1 with , , and . We can find that and , which lead to an augmented GMEC with and . Its corresponding inequality, post-, and pre-incidence matrices are , , and , respectively. The fact that is the support of a -semiflow leads to . Thus, the actual constraint is equivalent to . This does not fulfill the prerequisite for such a transformation since it leads to an initially dead system. Figs. 4(b) and 5(b) demonstrate the monitors determined by and , respectively. Similar to Case 1, the latter cannot replace the former. However, not all the cases that involve resources can incur such an undesirable result. This is shown by another case, where resources are involved. The computational results according to [1] are verified to be true as shown below. — Case 3: Fig. 3(c) shows a Petri net modified from Fig. 3(a). Let us introduce the observation and control costs: and and consider a GMEC with and . Similar to the previous two cases, an IP formulation produces a monitor with the least cost. As a result, we obtain and which lead to a monitor with and , whose post- and pre-incidence matrices are and , respectively. The implementation cost is equal to 3. The controlled system is live. V. OVER-RESTRICTION ISSUES AND THEIR RESOLUTION The noticeable issue in Cases 1 and 2 is that the monitor with the minimum implementation cost may over-restrict the

Fig. 6. Illustrations using Case 1. (a) Net system in Case 1. (b) Net system controlled by the method in [1]. (c) Net system controlled by a very restrictive monitor.

Fig. 7. Over-restrictive monitors for Cases 2 and 3. (a) Case 2. (b) Case 3.

system behavior and sometimes lead to a deadlock-prone controlled system. This is quite undesirable and must be improved either in theory or practice. For clear presentation, we present the system of Case 1 in Fig. 6(a). Fig. 6(b) shows its controlled version by the method in [1]. We acknowledge that the constraint is not violated since the system actually cannot proceed such that . Despite its superficial reasonableness, such a result should definitely be improved. From the viewpoint of supervisory control, liveness is a primary and implicit requirement for any controlled system. Moreover, it is always expected that the behavior of a controllable system should be the least restricted if possible. Many paradoxical situations might result if one can derive a controlled system without considering its liveness and permissiveness. In the extreme case, one can introduce a monitor as shown in Fig. 6(c) without any calculation. Formally, this monitor implies . Since this monitor can inhibit the occurrence of any event from the initial state, any other constraints are also implicitly or explicitly implemented. Nevertheless, this result is not acceptable although it can realize any constraint without computation. Similarly, is unacceptable, either. Similar arguments apply to Case 2 too. Fig. 7 shows the other over-restrictive monitors that are derived without calculation. Although they can implement any linear constraint, they are unacceptable since a dead controlled system results. In fact, such a riddle can be cracked through the comparative examination of the augmented coefficients in the above three cases. In the sequel, for clear presentation, the augmented coefficient in Cases 1–3 are denoted by . We notice that in Cases

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1 and 2, both and tors such that ,


can be decomposed into two positive vecand . Specifically, , , , , and . Since both and are -semiand , where flows, we have and denote the incidence matrices of the Petri nets in Cases . For 1 and 2, respectively. Consider the equation Case 1, we have . For Case 2, we have . This means that no flow is introduced by or in either Case 1 or 2. Because the objective function aims to minimize the cost induced by the additional ingoing or outgoing flows, any intends to be composed of a -semiflow with as many nonzero elements as possible. On the other hand, one may assume that itself can become a -semiflow in the extreme situation such that the value of the objective function is 0. However, we argue that this is impossible. Consider Case 3 where . By contradiction, we assume that this extreme situation exists such that with and . The implementation cost is zero since is a -semiflow and no flow is connected to any transition. However, this is not a feasible solution of the IP formulation owing to the requirement of (6). Once , we have and . As indicated in Fig. 3(c), . By substituting all these values to (6), its left hand is equal to 0, which is larger than on the right hand. Now, it is clear that the approach in [1] can lead to undesirable solutions in a Petri net involving a resource structure whose formal definition is introduced subsequently. Given a GMEC , a subset of all the places with can interact with resource places so as to constitute the support of a -semiflow. This results in an unacceptable solution from the viewpoint of practice. To avoid such a problem, we need to set when models a resource. In other words, the method in [1] is only adoptable to systems without taking into account the requirements of resources. Nevertheless, the concept of resources is indispensable in the cases their constraint upon a system is not negligible. Corresponding to resources, there is the concept of activities or operations. In the paradigm of Petri nets, a resource can be denoted by a resource place whose tokens represent the availability of such a resource. Its output and input transitions represent the allocation and deallocation of such a resource to and from certain activities, respectively. In the sequel, we present the concepts of activities and resources. Their combination produces a quite general class of Petri nets where activities and resources can be clearly distinguished. Definition 1: A general process net with is an ordinary Petri net that satisfies the following four conditions. 1) is connected and acyclic, i.e., no inner or local circuit is contained in . 2) There exist two sets of transitions, which are , , respectively. is called source (initialization) transitions such that , and . is called sink (end) transitions such that and .

3) 4)

, an activity place.

. For clarification,

is called

denotes the flows between and . In the framework of discrete event systems, a process net represents one of the process flows where the source and sink transitions mean the initialization and end of a process, respectively. Compared with the prior work, the process net in this paper can describe complex system structures. This is owing to the ignorance of some behavioral requirements such as consistency and liveness. Despite their meaningfulness in certain circumstances, these requirements always limit the considered systems to some subclasses of Petri nets, thus undesirably constraining the applicability of research results. In many real concurrent systems, the constraints from the resources must be taken into account. In the paradigm of Petri nets, such constraints can be implemented by resource places (or resources, for simplicity). Definition 2: Given process nets and their composition , there exists a set of resources where . For each , there exists one and only one process net , such that while . Moreover, a conservative subnet is constituted by , and the flows between them. , a unique minimal -semiflow such that , , and . Moreover, . For , . Hereby and in the sequel, we also use to represent either of and for simplicity. denotes the flows established by the interaction between and . 1) Example 2: Take the Petri nets in Fig. 3 as examples. In Fig. 3(a), places , transitions , and the flows among them constitute a process net, while places , are resources. In Fig. 3(b), there are two processes. One is constituted by , , and the flows among them. Another one is constituted by , , and the flows among them. Places , , and are resources. In Fig. 3(c), , , and the flows among them constitute a process net. Places , , and are resources. Clearly, the liveness of one or more process nets is largely dependent on the shared resources associated with them. On the basis of Definitions 1 and 2, we develop the formal definition for the Petri nets concerned throughout this paper. Definition 3: A general process net with resource (GPNR) with is a Petri net combined by the process nets and resources defined in Definitions 1 and 2 such that , , where , , , and compose a strongly connected net; , , . To facilitate further investigation, the following two important and evidently true structural properties of GPNR are clarified. We provide their proof since they are quite helpful to derive the mathematical formalism in the next section. 2) Property 1: Given a GPNR , a) is covered by -semiflows; and b) is conservative. Proof: Proof of part a). On one hand, thanks to its conservativeness, for each process net along with its related


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Fig. 8. GMEC involving resources. (a) Original net. (b) Controlled net.

resource

, we have a -semiflow such that , ; otherwise, . On the other hand, straightforward from Definition 2, , a unique minimal -semiflow such that , , , and . Therefore, for , there exists a -semiflow containing . So, is covered by -semiflows. Proof of part b). According to the proof of part a), we have a unique minimal -semiflow , where , . Thus, is conservative. 3) Property 2: Let be a GPNR and be a -semiflow. There must exist a resource such that . Proof: By contradiction, we assume that is a -semiflow such that and . Obviously, this is impossible since all the process nets are acyclic according to Definition 1. For clarification, we present the concept of an over-restrictive constraint as follows. Definition 4: Let be a GMEC enforced with a monitor . It is said to be over-restrictive if . and be the original Lemma 1: Let and augmented GMEC, respectively. We have . Proof: We know that . Thus, we have . Considering the requirement that , we have . Thanks to the requirement , we have . As a conclusion, . represents an arbitrary Theorem 1: Suppose that . augmented GMEC from an original one is a special case of such that can be decomposed into two and constraints, i.e., and , where . The constraint must lead to an over-restrictive constraint iff holds in the case that . Proof: It is obvious that corresponds to the output incidence vector, while corresponds to the input incidence vector. Thus, represents the cost to realize a monitor. To show the necessity part, suppose that there exists a consuch that straint . On the basis of the mathematical programming formulations (4)–(7), cannot be the optimal constraint in terms of implementation cost. leads To show the sufficiency part, suppose that to a constraint with the least implementation cost. According to Lemma 1, can be decomposed into and . Consider that

can be decomposed to be two constraints, i.e., and , where and . The existence of the second part implies that the augmented constraint can result in a monitor that introduces a -invariant involving other places. This definitely leads to an over-restrictive constraint. Corollary 1: Suppose that is an arbitrary augis its corresponding original mented GMEC and one. If , the constraint must lead to a place with zero initial marking. Proof: It is known that the control place for a constraint is determined by two equations, i.e., and . The augmentation of a constraint is and . through two equations, i.e., According to the assumption, we have . Therefore, . Consider . According to the assumption that , we have . Proposition 1: Suppose that is a GMEC. There must exist an equivalent GMEC such that . Proof: According to Definitions 1 and 2, each corresponds to a -semiflow such that and . . As a result, we have By substituting each resource in the same way, we can finally obtain a GMEC involving only the activity places. 4) Example 3: To illustrate the correctness of Proposition 1, we present an example as shown in Fig. 8, where Fig. 8(a) shows an uncontrolled Petri net. Evidently, and are resources. Let us consider a GMEC . It is obvious that . By substituting with , we have , which is equivalent to . Through the proper calculation with the two equations, i.e., and , we obtain and . The corresponding control place is shown in Fig. 8(b). On the basis of Proposition 1, we can always assume that a GMEC only involves operation places since the resource items can be substituted by the operation place items. Another interesting issue is that the converted version of GMEC does not ensure the positiveness of weighted integer vector, i.e., . This phenomenon is already shown in the above example. One may notice that the coefficient for is negative rather than positive. The involvement of negative coefficients is of significance to represent ratio assignment and synchronic distance. On this topic, detailed discussion is beyond this paper due to limited space. In the sequel, we always assume that the vector is nonnegative, .

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Remark 1: Let be a GMEC with as its resulting control place generated through two equations, i.e., and . Then, is equivalent to a resource. On the basis of the two equations, we have the row of incidence matrix corresponding to through the equation , which is equivalent to . In the form of a matrix, this can be written as . We can suppose . Also, it is obvious that is the incidence matrix of the controlled net after the introduction of . According to Proposition 1, . This is equivalent to say that , . Furthermore, , , and . From Definition 2, it is evident that is equivalent to a resource. Remark 2: Let be a Petri net system, be the initial marking, and be a resource place. Then, is not live if . According to the assumption, . According to Definitions 1 and 2, a -semiflow corresponding to such that . Consider that . We have . According to the basic property of -semiflow, we have . Then, no output transition of can be enabled any more. Therefore, the net is not live. Remark 3: Given a net system . It is not if it leads to a monitor live under the constraint of with zero initial marking. According to Remark 1, the control place generated by the constraint is equivalent to a resource. According to Remark 2, a resource with the zero initial marking results in a nonlive net. Therefore, we can conclude that is not live under the constraint of if it leads to a monitor with zero initial marking. VI. ALGORITHM MODIFICATION From the above discussion, it is evident that the approach in [1] can cause certain problems if some resources are involved in the augmented constraints. Thus, our solution to such a problem is to properly avoid such involvement. To do so, we need to identify the resource structures according to the subsequent algorithm. According to the above examples and comparative analysis, we can improve the IP formulation defined by (5)–(7) as follows such that this model can cope with the Petri nets with resource places: (8) subject to (5)–(7) and (9) Theorem 2: The computational complexity of Algorithm 1 is . Proof: Algorithm 1 involves cycles. Each solves a -dimensional linear homogeneous equations. To solve such equations, we employ Gauss elimination method. Its complexity is well-known to be , where

. Evidently, the computational complexity of the entire algorithm is . Algorithm 1: Identification of Resource Structures Input: A GPNR

with incidence matrix

Output: A set of resources , 1) , ; 2) while (new strict minimal -semiflow can be found) do begin 3) An unknown vector is introduced to obtain new strict minimal -semiflow through the equation ; 4) if such that and ; 5) ; end Based on such formulations, we can further develop the method to obtain desired monitors with the least implementation cost. The corresponding algorithm is presented as follows. Algorithm 2: Generation of Monitors With the Least Cost Input: A set of GMEC

, where

Output: A set of monitors with the least cost 1) Identify all the resources using Algorithm 1; 2) ; 3) while do begin 4) ; 5) Convert to with the mathematical programming formulations specified as in (5)–(9); 6) Generate monitor such that and ; end Theorem 3: Let be an original GMEC. is the resulting augmented one using Algorithm 2. Then, does not lead to a monitor with zero initial marking. Proof: In , and are obtained through two equations and , respectively. Since , is always nonzero. To obtain the control place, we have . does not contain resource places. According to Algorithm 2, Thus, , according to Definition 1. As a . Therefore, . result, we have Algorithm 2 involves the solving of a set of integer programming formulations, whose complexity is exponential; however, it is verified to be efficient in practice. Regarding this point, further research is needed. Let us now apply the above theoretical results to the three problems in Cases 1–3. The produced solutions are shown in Table I, while the corresponding monitors are illustrated in Fig. 9. Evidently, all of them are correct. It is easily verified that


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TABLE I SOLUTIONS FOR CASES 1–3 ACCORDING TO THE IMPROVED APPROACH

Fig. 9. Monitors for Cases 1–3 according to the improved approach.

Fig. 11. The Petri net model of the AMS in Fig. 10.

Fig. 10. An example of AMS.

they are all live. For Case 3, the method in [1] and the modified one produce the same result. VII. ILLUSTRATIVE EXAMPLE Fig. 10 shows the layout of an AMS. It produces two product types, i.e., . It is composed of three robots and three machines . Each of them can hold or process three products at a time. There are two loading ones and two unloading ones to load and unload AMS. Products can be concurrently manufactured. According to the predefined routings, a raw product is taken from by and moved to . After being processed by , it is moved to by . After being processed by , it is moved to directly by . A raw product is taken from by and moved to . After being processed by . It can be moved to either or by . After being processed, the finished product of is moved to by . Fig. 11 shows the net model of this AMS. The meanings of the states are as follows. A token in and means that one unit of a type of product is being held by a robot or processed by a machine. A token in means the availability of one copy of either a robot or machine. Under the initial state, we assume that no product is being processed. Thus, the initial marking of each place in represents its capacity. For places and , their initial markings represent the maximum number of products that are allowed to be processed for either or . The meanings of transitions are omitted since they simply denote the start

Fig. 12. Monitors for the Petri net in Fig. 11 without considering cost issues.

and/or end of operations. To evaluate their controllability and observability, each transition is associated with a control and observation cost such that and . Since the initial markings are appropriately assigned, this net proves to be deadlock-free and live since no transition can be dead during the evolution. Owing to certain technical reasons, the behavior of such system must satisfy three inequalities, i.e., , , and . Without considering the implementation issue, one can easily derive the three monitors, as shown in Fig. 12. Specifically, Figs. 12(a)–(c) correspond to the first, second, and third inequality, respectively. It is verified that their costs are 16, 10, and 12, respectively. This is not satisfactory since the low cost is always expected in practice. The approach in [1] is applied to the monitors with the least cost first. Fig. 13 shows the resultant ones whose implementation costs are 3, 3, and 3, respectively. However, these monitors can only result in a dead system since no transition can fire. With the aid of our proposed approach, such a defect can be eliminated. Our obtained controllers, as shown in Fig. 14, can satisfy the required inequalities without influencing the system

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Proof: Since

Fig. 13. Monitors for the Petri net in Fig. 11 using approach in [1] to reduce cost.

, we have , which corresponds a monitor whose incidence matrix is . Obviously, the components in with respect to these uncontrollable transitions are either zero or negative. This means that there are no arcs from the monitor to the uncontrollable transitions. Property A.4: Given a plant Petri net whose unobservable transitions are denoted by incidence matrix . implies a correct constraint. , we have Proof: Since , which corresponds a monitor whose incidence matrix is . Obviously, the components in with respect to these unobservable transitions are all zeros. This means that there are no arcs between the monitor to the uncontrollable transitions.

Fig. 14. Monitors for the Petri net in Fig. 11 using our improved approach to reduce cost.

REFERENCES liveness. Further analysis shows the their costs are 10, 8, and 8, respectively. VIII. CONCLUSION A key point in [1] is to minimize the implementation cost of a supervisor in the framework of an integer programming problem, which might unnecessarily lead to a more restrictive monitor when resource places are involved in some Petri nets. The root cause of this problem is analyzed. Consequently, a modification is proposed such that the method can deal with arbitrary resource place structures. The proposed solution is of significance for the optimal monitor design. In the future, our research will be focused on more complex manufacturing systems and complicated resource requirements. APPENDIX Property A.1: Let be a -dimensional nonnegative vector and a positive scalar. If where and , then . implies Proof: which is equivalent to . Dividing both sides by , we have . Considering the nonnegativity of and positivity of , we have . Thus, obviously holds. This says . Property A.2: If , then there exists a feasible transformation. Proof: For a GMEC , we know . For a feasible transformation, holds. This implies which is equivalent to . Since and , we have such that holds. Property A.3: Given a plant Petri net whose uncontrollable transitions are denoted by incidence matrix . implies a correct constraint.

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[42] N. Q. Wu, F. Chu, C. B. Chu, and M. C. Zhou, “Petri net-based scheduling of single-arm cluster tools with reentrant atomic layer deposition processes,” IEEE Trans. Autom. Sci. Eng., vol. 8, no. 1, pp. 42–55, Jan. 2011. [43] N. Q. Wu, F. Chu, C. B. Chu, and M. C. Zhou, “Schedulability analysis of short-term scheduling for crude oil operations in refinery with oil residency time and charging-tank-switch-overlap constraints,” IEEE Trans. Autom. Sci. Eng., vol. 8, no. 1, pp. 190–204, Jan. 2011. [44] N. Q. Wu and M. C. Zhou, System Modeling and Control With Resource-Oriented Petri Nets. New York: CRC Press, Taylor & Francis Group, 2009. [45] S. G. Wang, C. Y. Wang, M. C. Zhou, and Z. W. Li, “A method to compute strict minimal siphons in a class of Petri nets based on loop resource subsets,” IEEE Trans. Syst., Man, Cybern. A. Syst., Humans, vol. 42, no. 1, pp. 226–237, Jan. 2012. [46] P. J. Ramadge and W. M. Wonham, “Supervisory control of a class of discrete event processes,” SIAM J. Contr. Opt., vol. 25, no. 1, pp. 206–230, Jan. 1987. [47] K. Y. Xing, M. C. Zhou, F. Wang, H. X. Liu, and F. Tian, “Resourcetransition circuits and siphons for deadlock control of automated manufacturing systems,” IEEE Trans. Syst., Man, Cybern. A. Syst., Humans, vol. 41, no. 1, pp. 74–84, Jan. 2011. [48] M. C. Zhou and F. DiCesare, Petri Net Synthesis for Discrete Event Control of Manufacturing Systems. Norwell, MA: Kluwer, 1993. [49] G. J. Liu, C. J. Jiang, and M. C. Zhou, “Two simple deadlock prevention policies for based on key-resource/operation-place pairs,” IEEE Trans. Autom. Sci. Eng., vol. 7, no. 1, pp. 945–957, Oct. 2012. [50] W. Wang, S. Lafortune, F. Lin, and A. R. Girard, “Minimization of dynamic sensor activation in discrete event systems for the purpose of control,” IEEE Trans. Autom. Contr., vol. 55, no. 11, pp. 2447–2461, Nov. 2010. [51] M. C. Zhou and F. DiCesare, “Parallel and sequential mutual exclusions for Petri net modeling of manufacturing systems with shared resources,” IEEE Trans. Robot. Autom., vol. 7, no. 4, pp. 515–527, Aug. 1991. [52] M. C. Zhou and F. DiCesare, “Adaptive design of Petri net controllers for error recovery in automated manufacturing systems,” IEEE Trans. Syst. Man, Cybern., vol. 19, no. 5, pp. 963–973, Sep./Oct. 1989. [53] M. C. Zhou, F. DiCesare, and A. Desrochers, “A hybrid methodology for synthesis of Petri nets for manufacturing systems,” IEEE Trans. Robot. Autom., vol. 8, no. 3, pp. 350–361, Jun. 1992. [54] M. C. Zhou, F. DiCesare, and D. Rudolph, “Design and implementation of a Petri net based supervisor for a flexible manufacturing system,” IFAC J. Automatica, vol. 28, no. 6, pp. 1199–1208, 1992. [55] M. C. Zhou and F. DiCesare, “Petri net modeling of buffers in automated manufacturing systems,” IEEE Trans. Syst. Man, Cybern. - Part B: Cybern., vol. 26, no. 1, pp. 157–164, Feb. 1996. [56] M. C. Zhou and M. D. Jeng, “Modeling, analysis, simulation, scheduling, and control of semiconductor manufacturing systems: A Petri net approach,” IEEE Trans. Semiconductor Manuf., vol. 11, no. 3, pp. 333–357, Aug. 1998. [57] M. C. Zhou and M. C. Leu, “Modeling and performance analysis of a flexible PCB assembly station using Petri nets,” Trans. ASME J. Electron. Packag., vol. 113, no. 4, pp. 410–416, 1991. [58] M. C. Zhou, K. McDermott, and P. A. Patel, “Petri net synthesis and analysis of a flexible manufacturing system cell,” IEEE Trans. Syst. Man, Cybern., vol. 23, no. 2, pp. 523–531, Mar./Apr. 1993. [59] M. C. Zhou and K. Venkatesh, Modeling, Simulation and Control of Flexible Manufacturing Systems: A Petri Net Approach. Singapore: World Scientific, 1998. [60] R. Zurawski and M. C. Zhou, “Petri nets and industrial applications: A tutorial,” IEEE Trans. Ind. Electron., vol. 41, no. 6, pp. 567–583, Dec. 1994. Hesuan Hu (M’11) received the B.S. degree in computer engineering and the M.S. and Ph.D. degrees in electromechanical engineering from Xidian University, Xi’an, China, in 2003, 2005, and 2010, respectively. From 1997 to 2000, he was an Assistant Engineer with the Metrology Department, Xi’an Aircraft Engine Control Engineering Company, Ltd., where he served as the Director of the Electrical and Electronic Laboratory from 2001 to 2002 and the Leader for several innovation projects. From 2008 to 2011, he worked as a Research Scholar in the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark. At the same institute, he also served as an Adjunct Instructor in charge of several courses in

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the field of electrical engineering at the graduate level. He is currently a Faculty Member at Xidian University. His research interests include Petri nets, discrete-event systems, automated manufacturing systems, multimedia streaming systems, and artificial intelligence. In the aforementioned areas, he has 50 publications in journals, book chapters, and conference proceedings. Dr. Hu was the recipient of the Franklin V. Taylor Outstanding Paper Award from the IEEE SMC Society in 2010. He was a Program Committee member for the following conferences: the 2008 International Conference on Industrial Engineering and Other Applications of Applied Intelligent Systems, the 2008 IEEE International Conference on Automation Science and Engineering, the 2008–2011 IEEE International Conference on Networking, Sensing and Control, the 2010 IEEE International Conference on Systems, Man, and Cybernetics (SMC), and the 2010 IEEE International Conference on Mechatronics and Automation. He serves as a frequent Reviewer for more than 20 international journals, including the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART A: SYSTEMS AND HUMANS; the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS; the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING; the IEEE COMMUNICATIONS LETTERS; the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY; the IEEE TRANSACTIONS ON COMPUTERS; the ACM Transactions on Embedded Computing Systems; and Discrete Event Dynamic Systems. He is on the Editorial Board of the Journal of Control Engineering and Technology. He is listed in Who’s Who in America.

MengChu Zhou (S’88–M’90–SM’93–F’03) received the B.S. degree in electrical engineering from the Nanjing University of Science and Technology, Nanjing, China, in 1983, the M.S. degree in automatic control from the Beijing Institute of Technology, Beijing, China, in 1986, and the Ph.D. degree in computer and systems engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1990. In 1990, he joined the New Jersey Institute of Technology (NJIT), Newark, where he is currently a Professor with the Department of Electrical and Computer Engineering and the Director of the Discrete-Event Systems Laboratory. He is also a Professor with the MoE Key Laboratory of Embedded System and Service Computing, Tongji University, Shanghai, China. He has more than 440 publications including ten books, more than 200 journal papers (majority in IEEE TRANSACTIONS), and 18 book chapters. His research interests include intelligent automation, lifecycle engineering and sustainability evaluation, Petri nets, wireless ad hoc and sensor networks, semiconductor manufacturing, and energy systems. Dr. Zhou is a Life Member of the Chinese Association for Science and Technology-USA and served as its President in 1999. He was recently elevated to Fellow of the American Association for the Advancement of Science.

Zhiwu Li (M’06–SM’07) received the B.S., M.S., and Ph.D. degrees in mechanical engineering, automatic control, and manufacturing engineering, respectively, from Xidian University, Xi’an, China, in 1989, 1992, and 1995, respectively. In 1992, he joined Xidian University, where he is currently a Professor with the School of Electro-Mechanical Engineering and the Director of Systems Control and Automation Group. From June 2002 to July 2003, he was a Visiting Professor with the Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. From February 2007 to February 2008, he was a Visiting Scientist with the Laboratory for Computer-Aided Design and Lifecycle Engineering, Department of Mechanical Engineering, Technion-Israel Institute of Technology, Technion City, Haifa, Israel. Since November 2008, he has been a Visiting Professor with the Automation Technology Laboratory, Institute of Computer Science, Martin-Luther University of Halle-Wittenburg, Halle (Saale), Germany. He was a Senior Visiting Scientist with Conservatoire National des Arts et Mtiers,

Paris, France, supported by the program Research in Paris in 2010. He was a Lecture Professor with Taiwan Chengchi University in 2011 and serves as a Host Professor of Research Fellowship for International Young Scientists, National Natural Science of Foundation of China. He is the author or coauthor of more than 120 publications. His current research interests include Petri net theory and application, supervisory control of discrete event systems, workflow modeling and analysis, and systems integration. Dr. Li is the recipient of the Alexander von Humboldt Research Grant, Alexander von Humboldt Foundation, Germany. He is a member of the Discrete Event Systems Technical Committee of the IEEE Systems, Man, and Cybernetics Society, and a member of the IFAC Technical Committee on Discrete Event and Hybrid Systems (2011–2014). He was the General Co-Chair of the IEEE International Conference on Automation Science and Engineering, Washington, DC, August 23–26, 2008. He was a Financial Co-Chair of the IEEE International Conference on Networking, Sensing, and Control, March 26–29, 2009, a member of the International Advisory Committee of the Tenth International Conference on Automation Technology, June 27–29, 2009, the Co-Chair of the Program Committee of the IEEE International Conference on Mechatronics and Automation, August 24–27, 2010, and a member of the program committees of many international conferences. He serves as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, PART A: SYSTEMS AND HUMAN BEINGS, the International Journal of Discrete Event Control Systems, IST Transactions of Robotics, Automation and Mechatronics—Theory and Applications, and the IST Transactions of Control Engineering—Theory and Applications. He is the Founding Chair of Xi’an Chapter of IEEE Systems, Man, and Cybernetics Society.

Ying Tang (S’99–M’02–SM’07) received the B.S. and M.S. degrees from the Northeastern University, Shenyang, China, in 1996 and 1998, respectively, and the Ph.D degree from the New Jersey Institute of Technology, Newark, in 2001. She is an Associate Professor of Electrical and Computer Engineering at Rowan University, Glassboro, NJ. She has led and participated in several research and education projects funded by National Science Foundation, the U.S. Department of Transportation, the U.S. Navy, the Charles A. and Anne Morrow Lindbergh Foundation, the Christian R. and Mary F. Lindback Foundation, and industry firms. Her work has resulted in 4 book chapters, and over 80 journal and conference proceedings articles. The majority of her work has been published in the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, the IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, and the International Journal of Production Research, etc. Her current research interests include discrete event systems and visualization, Petri nets and applications, virtual/augmented reality, green manufacturing design and optimization, intelligent learning game systems and artificial intelligence. Dr. Tang is currently an Associate Editor of the IEEE TRANSACTION ON AUTOMATION SCIENCE AND ENGINEERING, the International Journal of Intelligent Control and Systems, and an Editorial Board Member of the International Journal of Remanufacturing. She has chaired several technical sessions and served on program committees for many conferences. She was Program Co-Chair of the 2010 IEEE International Conference on Mechatronics and Automation; Publication Co-Chair of the 2012 IEEE International Conference on Networking, Sensor and Control, and the 2011 IEEE International Conference on Systems, Man, and Cybernetics; Publicity Chair of the 2011 IEEE Conference on Automation Science and Engineering; Special Session Chair of the 2008 IEEE Conference on Automation Science and Engineering, of the 2007 International Conference on Flexible Automation and Intelligent Manufacturing, and of the 2006 International Conference on Service Operations and Logistics, and Informatics; Finance Chair of the 2006 IEEE International Conference on Networking, Sensor and Control; Student Activity Chair of the 2005 IEEE International Conference on Networking, Senor and Control; Publication Chair of the 2005 International Conference on Service Operations and Logistics, and Informatics; and Award Committee Chair of the 2003 International Conference on Information Technology: Research and Education.