Supervisor Reconfiguration for Deadlock Prevention by Resources Reallocation

Abstract

Analysis and control of deadlocks play an important role in the design and operation of automated flexible manufacturing systems (FMSs). In FMS, deadlocks are highly undesirable situations, which always cause unnecessary cost. The design problem of an optimal supervisor is in general NP-hard. A computationally efficient method often ends up with a suboptimal one. This paper develops a deadlock prevention policy based on resources reallocation and supervisor reconfiguration. First, given a plant model, we reallocate the marking of each resource place to be one, obtaining a net model whose reachable states are much less than that of the original one. In this case, we find a controlled system for it by using the theory of regions. Next, the markings of the resource places in the controlled system are restored to their original ones. Without changing the structure of the obtained controlled system, we compute the markings of the monitors gradually, which can be realized by two algorithms proposed in this paper. Finally, we decide a marking for each monitor such that it makes the controlled system live with nearly optimal permissive behavior. Two FMS examples are used to illustrate the application of the proposed method and show its superior efficiency.

1. Introduction

Traditional mass production systems can hardly cope with intensive competition in market and rapid variation in requirements. Hence, automated flexible manufacturing systems (FMSs) arise, aiming to offer a novel production mode with a small batch and multiple product types. The analysis and control of such systems have become the hot topics in the field of manufacturing systems. Petri nets [1] are a graph-based mathematical formalism suitable to describe, model, and analyze the behavior of automated flexible manufacturing systems.

Due to the existence of shared resources, an FMS may contain deadlocks. Based on Petri nets, researchers have developed many policies to deal with the deadlock problem [2–6] in FMS. Generally, there are mainly two analysis techniques to deal with deadlock prevention in FMS: structure [7–11] and reachability graph analysis [12–16]. The former always obtains a deadlock prevention policy through special structural objects of a Petri net such as siphons and resource-transition circuits. This method can usually obtain a computationally efficient liveness-enforcing supervisor in general but at the same time restrict a system such that a portion of permissive behavior is excluded. For the latter, the reachability graph can completely reflect the behavior of a system. Though a very highly or even maximally permissive liveness-enforcing supervisor can always be obtained, its computation is very expensive.

As stated, reachability graph analysis [17] is an important technique for deadlock controll; however, it always suffers from a state explosion problem. This is due to the fact that it is impossible to achieve the enumeration of all or a part of reachable markings in practice. Based on this technique, an optimal or suboptimal supervisor with highly behavioral permissiveness can always be obtained for a small system. In [18], Uzam and Zhou develop an iterative approach to design an optimal or suboptimal supervisor. This method is easy to use if the reachable space of a system is small but cannot guarantee the optimality of the supervisor.

The theory of regions developed in [19] can be used as an effective approach to find an optimal liveness-enforcing Petri net supervisor if such a supervisor exists. However, it suffers from computational and structural complexity problems. The work in [20] proposes a vector covering approach to improve the computation efficiency of the work in [19].

A siphon-based deadlock prevention policy is a typical application of structure analysis techniques of Petri nets. Although not optimal and even overly restrictive, this approach is computationally tractable and allows its supervisor to be reused when a system experiences new job instances [2, 21–28]. Recent effective and computationally efficient deadlock prevention policies are proposed by Piroddi et al. in [29, 30].

Nowadays, the distribution of resources in an FMS may change frequently and dynamically due to fluctuant customer demands. Such changes mean different production or service requirements for the providers. Therefore, the supervisory control system has to reconfigure rapidly in response to the changes in its physical entity and the control specifications by modifying or adjusting its plant model and controller. Once the resource configurations are changed, the supervisors are updated accordingly. Recently, the study in [31] proposes a novel deadlock prevention policy based on reconfiguration of Petri net supervisors. The method presented in [31] is near optimal, but its performance needs to be further improved.

Motivated by existing work, this paper presents an effective and computationally efficient method to design nearly optimal control places based on resources reallocation and supervisor reconfiguration. The main idea of the method is stated as follows. (i) The proposed method works on the premise of a class of ordinary Petri net (𝑁,𝑀₀) with 𝑁=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅,𝑇,𝐹), where for all  𝑖∈{1,2,…,𝑛}, 𝑃⁰=∪^𝑛_𝑖=1{𝑝⁰_𝑖} is called a set of idle process places, 𝑃_𝐴=∪^𝑛_𝑖=1𝑃_𝐴𝑖 is called a set of activity places, and 𝑃_𝑅=∪^𝑛_𝑖=1𝑃_𝑅𝑖 is called a set of resource places. (ii) We first reallocate the initial marking of each resource place in (𝑁,𝑀₀) to be one, obtaining a model (𝑁¹,𝑀¹) where 𝑁¹=𝑁, for all  𝑝∈𝑃⁰∪𝑃_𝐴, 𝑀¹(𝑝)=𝑀₀(𝑝), and forall𝑟∈𝑃_𝑅, 𝑀¹(𝑟)=1. In this case, we can design monitors by using the theory of regions and find a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) with 𝑁^1𝑐=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅∪𝑃_𝑉,𝑇^1𝑐,𝐹^1𝑐), where forall𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^1𝑐(𝑝)=𝑀¹(𝑝), for all  𝑖∈{1,2,…,𝑛}, 𝑃_𝑉=∪^𝑛_𝑖=1𝑃_𝑉𝑖 is called a set of control places.  (iii)Then, the markings of the resource places in (𝑁^1𝑐,𝑀^1𝑐) are restored to their original ones, obtaining a controlled system (𝑁^𝑐,𝑀^𝑐) with 𝑁^𝑐=𝑁^1𝑐, where for all  𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^𝑐(𝑝)=𝑀₀(𝑝), and for all  𝑣∈𝑃_𝑉, 𝑀^𝑐(𝑣)=𝑀^1𝑐(𝑣). (iv) Without changing the structure of the controlled system (𝑁^𝑐,𝑀^𝑐), we compute the markings of the monitors gradually. Finally, we decide a marking for each monitor such that forall𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^𝑐(𝑝)=𝑀₀(𝑝), and for all  𝑣∈𝑃_𝑉, 𝑀^𝑐(𝑣) makes the controlled system (𝑁^𝑐,𝑀^𝑐) live, where (𝑁^𝑐,𝑀^𝑐) is a controlled system for (𝑁,𝑀₀) with 𝑁^𝑐=𝑁^1𝑐.

The rest of this paper is organized as follows. Section 2 briefly reviews preliminaries used in this paper. Section 3 formulates the considered problem through a motivation example. Section 4 presents a deadlock prevention policy that is formalized by Algorithm 2. Two FMS examples are given in Section 5, showing the superiority of the proposed method. A comparison between the proposed method and a previous one is made in Section 6. Finally, Section 7 concludes this paper.

2. Preliminaries

2.1. Basics of Petri Nets

A generalized Petri net (structure) [7] is a four-tuple 𝑁=(𝑃,𝑇,𝐹,𝑊), where 𝑃 and 𝑇 are finite, nonempty, and disjoint sets. 𝑃 is a set of places and 𝑇 is a set of transitions with 𝑃∪𝑇≠∅ and 𝑃∩𝑇=∅. 𝐹⊆(𝑃×𝑇)∪(𝑇×𝑃) is called a flow relation of a net, represented by arcs with arrows from places to transitions or from transitions to places. 𝑊∶(𝑃×𝑇)∪(𝑇×𝑃)→ℕ is a mapping that assigns a weight to an arc: 𝑊(𝑥,𝑦)>0 if (𝑥,𝑦)∈𝐹, and W(𝑥,𝑦)=0 otherwise, where 𝑥,𝑦∈𝑃∪𝑇 and ℕ={0,1,2,…}. If 𝑊(𝑥,𝑦)=1, for all  (𝑥,𝑦)∈𝐹, the net is called an ordinary Petri net.

A transition 𝑡∈𝑇 is enabled at a marking 𝑀 if for all  𝑝∈^•𝑡, 𝑀(𝑝)≥𝑊(𝑝,𝑡). This fact is denoted as 𝑀[𝑡⟩. Firing it yields a new marking 𝑀^′ such that forall𝑝∈𝑃, 𝑀^′(𝑝)=𝑀(𝑝)−𝑊(𝑝,𝑡)+𝑊(𝑡,𝑝), as denoted by 𝑀[𝑡⟩𝑀^′. 𝑀^′ is called an immediately reachable marking from 𝑀. Marking 𝑀^′′ is said to be reachable from 𝑀 if there exists a sequence of transitions 𝜎=𝑡₀𝑡₁…𝑡_𝑛 and markings 𝑀₁,𝑀₂,…, and 𝑀_𝑛 such that 𝑀[𝑡₀⟩𝑀₁[𝑡₁⟩𝑀₂…𝑀_𝑛[𝑡_𝑛⟩𝑀^′′ holds. The set of markings reachable from 𝑀 in 𝑁 is called the reachability set of Petri net (𝑁,𝑀) and denoted as 𝑅(𝑁,𝑀). [𝑁] is called the incidence matrix of 𝑁. It is a |𝑃|×|𝑇| integer matrix with [𝑁](𝑝,𝑡)=𝑊(𝑡,𝑝)−𝑊(𝑝,𝑡).

A 𝑃-vector is a column vector 𝐼∶𝑃→ℤ index by 𝑃 and a 𝑇-vector is a column vector 𝐽∶𝑇→ℤ index by 𝑇, where ℤ is the set of integers. 𝐼 is a 𝑃-invariant if 𝐼≠𝟎 and 𝐼^𝑇[𝑁]=𝟎^𝑇. A 𝑃-invariant 𝐼 is said to be a 𝑃-semiflow if every element of 𝐼 is nonnegative.

A nonempty set 𝑆⊆𝑃 is a siphon if ^•𝑆⊆𝑆^•. A siphon is minimal if there is no siphon contained in it as a proper subset. A minimal siphon that does not contain the support of any 𝑃-invariant is called a strict minimal siphon (SMS).

A siphon 𝑆 is said to be max-marked at 𝑀∈𝑅(𝑁,𝑀₀) if ∃𝑝∈𝑆 such that 𝑀(𝑝)≥max_𝑝•, where max_𝑝•=max{𝑊(𝑝,𝑡)∣𝑡∈𝑝^•}. 𝑆 is max-controlled if it is max-marked at any reachable marking. (𝑁,𝑀₀) satisfies the maximal controlled-siphon (cs) property if each minimal siphon of 𝑁 is max-controlled [33]. Siphon 𝑆 is called uncontrolled in (𝑁,𝑀₀) if ∃𝑀∈𝑅(𝑁,𝑀₀), for all  𝑡∈𝑆^•, 𝑡 is dead at 𝑀.

A marking 𝑀 of a Petri net 𝑁 is a mapping from 𝑃 to ℕ. 𝑀(𝑝) denotes the number of tokens in place 𝑝. A place 𝑝 is marked at a marking 𝑀 if 𝑀(𝑝)>0. A subnet 𝑆⊆𝑃 is marked at a marking 𝑀 if at least one place in 𝑆 is marked at 𝑀. The sum of tokens of all places in 𝑆 is denoted by 𝑀(𝑆), that is, 𝑀(𝑆)=∑_𝑝∈𝑆𝑀(𝑝). 𝑆 is said to be empty at 𝑀 if 𝑀(𝑆)=0. (𝑁,𝑀₀) is called a net system or marked net and 𝑀₀ is called an initial marking of 𝑁.

Markings and vectors are usually represented via using a multiset. As a result, vector 𝑀 is denoted by ∑_𝑝∈𝑃𝑀(𝑝)𝑝. For instance, a marking that puts two tokens in place 𝑝₁ and three tokens in place 𝑝₃ only in a net with 𝑃={𝑝₁−𝑝₆} is denoted as 𝑀=2𝑝₁+3𝑝₃ instead of (2,0,3,0,0,0)^𝑇.

Let 𝑥∈𝑃∪𝑇 be a node of net 𝑁=(𝑃,𝑇,𝐹,𝑊). The preset of 𝑥 is defined as ^•𝑥={𝑦∈𝑃∪𝑇∣(𝑦,𝑥)∈𝐹}. While the postset of 𝑥 is defined as 𝑥^•={𝑦∈𝑃∪𝑇∣(𝑥,𝑦)∈𝐹}. Furthermore, we have 𝑥^••=∪_{𝑦∈𝑥•}𝑦^• and ^••𝑥=∪_{𝑦∈•𝑥}^•𝑦.

Given a Petri net (𝑁,𝑀₀), 𝑡∈𝑇 is live at 𝑀₀ if for all  𝑀∈𝑅(𝑁,𝑀₀), ∃𝑀^′∈𝑅(𝑁,𝑀), 𝑀^′[𝑡⟩. (𝑁,𝑀₀) is live if forall𝑡∈𝑇, 𝑡 is live at 𝑀₀. (𝑁,𝑀₀) is dead at 𝑀₀ if ∄𝑡∈𝑇, 𝑀₀[𝑡⟩. (𝑁,𝑀₀) is deadlock-free if for all  𝑀∈𝑅(𝑁,𝑀₀), ∃𝑡∈𝑇, 𝑀[𝑡⟩.

With respect to the concepts of supervisors and controlled systems, the reader is referred to [14]. A supervisor is said to be optimal if its resulting controlled system covers all safe states of a plant and every reachable state in the controlled system is a safe state of the plant. Such a controlled system is said to be optimal.

2.2. M-Nets

This paper considers a class of manufacturing-oriented Petri nets, M-nets [31]. It is a generalization of the existing net classes that can model FMS. In this paper, we just focus on the research of ordinary M-nets, a type of M-nets.

Definition 1. An M-net denoted by (𝑁,𝑀₀) satisfies the following statements:(1)𝑁=○^𝑛_𝑖=1𝑁_𝑖=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅,𝑇,𝐹,𝑊) is composed of 𝑛 nets 𝑁₁,𝑁₂,…, and 𝑁_𝑛, where for all  𝑖∈ℕ_𝑛, ℕ_𝑛={1,2,…,𝑛}, 𝑁_𝑖=({𝑝⁰_𝑖}∪𝑃_𝐴𝑖∪𝑃_𝑅𝑖,𝑇_𝑖,𝐹_𝑖,𝑊_𝑖) is called a subnet of 𝑁.(2)𝑃⁰=∪^𝑛_𝑖=1{𝑝⁰_𝑖} is called a set of idle process places with 𝑝⁰_𝑖≠𝑝⁰_𝑗, for all  𝑖,𝑗∈ℕ_𝑛, 𝑖≠𝑗; 𝑃_𝐴=∪^𝑛_𝑖=1𝑃_𝐴𝑖 is called a set of activity places with 𝑃_𝐴𝑖∩𝑃_𝐴𝑗=∅, for all 𝑖,𝑗∈ℕ_𝑛, 𝑖≠𝑗; and 𝑃_𝑅=∪^𝑛_𝑖=1𝑃_𝑅𝑖 is called a set of activity places.(3) For all  𝑖,𝑗∈ℕ_𝑛, 𝑖≠𝑗, 𝑇_𝑖∩𝑇_𝑗=∅.(4) For all  𝑟∈𝑃_𝑅, it is associated with a minimal 𝑃-semiflow 𝐼_𝑟 such that 𝐼_𝑟(𝑟)=1, for all  𝑝∈𝑃_𝐴, 𝐼_𝑟(𝑝)≥0, and forall𝑝∈𝑃⁰, 𝐼_𝑟(𝑝)=0.(5) For all  𝑝∈𝑃_𝐴, 𝑝 is associated with a minimal 𝑃-semiflow 𝐼_𝑝, where ‖𝐼_𝑝‖⊆𝑃_𝐴.(6)(𝑁_𝑖,𝑀_0𝑖) is quasi-live, bounded, and conservative.(7)(𝑁^′_𝑖,𝑀^′_0𝑖) with 𝑁^′_𝑖=({𝑝⁰_𝑖}∪𝑃_𝐴𝑖,𝑇_𝑖,𝐹^′_𝑖,𝑊^′_𝑖) is live, bounded, and reversible, where 𝑁^′_𝑖 is the resulting net from removing resource places in (𝑁_𝑖,𝑀_0𝑖).(8) Let (𝑁_𝑖,𝑀_0𝑖)(𝑖=1,2) be two subnets with 𝑁_𝑖=({𝑝⁰_𝑖}∪𝑃_𝐴𝑖∪𝑃_𝑅𝑖,𝑇_𝑖,𝐹_𝑖,𝑊_𝑖). Their composition denoted by (𝑁₁₂,𝑀₁₂) with 𝑁₁₂=𝑁₁∘𝑁₂=(𝑃⁰₁₂∪𝑃_𝐴12∪𝑃_𝑅12,𝑇₁₂,𝐹₁₂,𝑊₁₂) is defined as follows:(i)  𝑃⁰₁₂={𝑝⁰₁}∪{𝑝⁰₂}={𝑝⁰₁,𝑝⁰₂}, 𝑃_𝐴12=𝑃_𝐴1∪𝑃_𝐴2, and 𝑃_𝑅12=𝑃_𝑅1∪𝑃_𝑅2,(ii)  𝑇₁₂=𝑇₁∪𝑇₂,(iii) 𝐹₁₂=𝐹₁∪𝐹₂,(iv) forall𝑓∈𝐹₁, 𝑊(𝑓)=𝑊₁(𝑓) and forall𝑓∈𝐹₂, 𝑊(𝑓)=𝑊₂(𝑓),(v)  for all  𝑝∈{𝑝⁰₁}∪𝑃_𝐴1, 𝑀₁₂(𝑝)=𝑀₀₁(𝑝); forall𝑝∈{𝑝⁰₂}∪𝑃_𝐴2, 𝑀₁₂(𝑝)=𝑀₀₂(𝑝); forall𝑟∈𝑃_𝑅1⧵𝑃_𝑅2, M₁₂(𝑟)=𝑀₀₁(𝑟); forall𝑟∈𝑃_𝑅2⧵𝑃_𝑅1, 𝑀₁₂(𝑟)=𝑀₀₂(𝑟); and for all  𝑟∈𝑃_𝑅1∩𝑃_𝑅2, 𝑀₁₂(𝑟)=max{𝑀₀₁(𝑟),𝑀₀₂(𝑟)}.(9) The net 𝑁 resulting from the composition of 𝑛 subnets 𝑁₁,𝑁₂,…, and 𝑁_𝑛 is defined as follows: if 𝑛=1, then 𝑁=𝑁₁; if 𝑛>1, then 𝑁=○^𝑛_𝑖=1𝑁_𝑖=(○^𝑛−1_𝑖=1𝑁_𝑖)∘𝑁_𝑛.(10)Forall𝑝∈𝑃⁰, 𝑀₀(𝑝)>0; forall𝑝∈𝑃_𝐴, 𝑀₀(𝑝)=0; and forall𝑟∈𝑃_𝑅, 𝑀₀(𝑟)≥max{𝐼_𝑟(𝑝)∣𝑝∈‖𝐼_𝑟‖}. Such a marking is said to be an admissible initial marking.(11) An uncontrolled siphon in (𝑁,𝑀₀) contains at least one resource place and one activity place but no idle process place.(12)(𝑁,𝑀₀) is live if no siphon is uncontrolled.(13) If (𝑁,𝑀₀) is not live, liveness can be enforced by adding monitors whose addition leads to a controlled system.(14) Let (𝑁^𝑐,𝑀^𝑐₀) be a controlled system for (𝑁,𝑀₀). (𝑁^𝑐,𝑀^𝑐₀) is live if it is ordinary and no siphon is unmarked. (𝑁^𝑐,𝑀^𝑐₀) is live if it is generalized and satisfies the controlled-siphon (cs) property.(15) Let 𝑃_𝑉 be the set of monitors in (𝑁^𝑐,𝑀^𝑐₀). For all  𝑣∈𝑃_𝑉, there exists a minimal 𝑃-semiflow 𝐼_𝑣 such that 𝐼_𝑣(𝑣)=1 and for all  𝑝∈‖𝐼_𝑣‖⧵{𝑣}, 𝑝∈𝑃_𝐴.
In order to make Definition 1 clear, an example is given in Appendix A. It is easy to find that M-nets are more general than almost all manufacturing-oriented Petri nets, for example, the ones in [2, 22, 24, 32].

2.3. An MIP-Based Deadlock Detection Method

In this paper, by using a technique that is called the mixed integer programming (MIP) approach proposed in [32], siphons that cause deadlocks can be detected. Let (𝑁,𝑀₀) be an ordinary net with 𝑁=(𝑃,𝑇,𝐹) and 𝑆 the maximal empty siphon at 𝑀, that is, forall𝑝∉𝑆, 𝑀(𝑝)>0. Finding 𝑆 in 𝑁 is the solution of a mixed integer programming problem. For all  𝑝∉𝑆, let 𝑣_𝑝=1 and for all  𝑡∉𝑆^•, let 𝑧_𝑡=1.

It is easy to see that any 𝑝 with 𝑣_𝑝=1 and any 𝑡 with 𝑧_𝑡=1 are removed from the net. Since 𝑆 is a siphon, we have that for all  𝑡∈𝑝^•, 𝑣_𝑝=0 implies 𝑧_𝑡=0 and for all  𝑝∈𝑡^•, 𝑧_𝑡=1 implies the truth of 𝑣_𝑝=1. This leads to 𝑧_𝑡≥_{𝑝∈•𝑡}𝑣_𝑝−^•𝑡𝑣+1,∀𝑡∈𝑇_𝑝≥𝑧_𝑡𝑣,∀𝑡,𝑝∈𝐹_𝑝,𝑧_𝑡∈{0,1}. (1)(2)(3)

For a structurally bounded net, we have 𝑣_𝑝≥𝑀𝑝SB𝑝,∀𝑝∈𝑃, (4) where SB(𝑝)=max{𝑀(𝑝)∣𝑀=𝑀₀+[𝑁]𝑌,𝑀≥0,𝑌≥0} is the structural bound of place 𝑝. Therefore, the maximal siphon unmarked at a given marking 𝑀 can be determined by the following MIP problem and there exist siphons unmarked at 𝑀 iff 𝐺^MIP<|𝑃| [32]: 𝐺^MIP=Minimize_𝑝∈𝑃𝑣_𝑝 (5) s.t. constraints (1)–(4) and 𝑀=𝑀₀+[𝑁]𝑌,𝑀≥0,𝑌≥0, (6) where [𝑁] is the incidence matrix and 𝑀 and 𝑌 are vectors of real numbers. Relation 𝑀=𝑀₀+[𝑁]𝑌 is usually called the state equation.

Theorem 2 (see [32]). Let (𝑁,𝑀₀) be a Petri net with 𝑁=(𝑃,𝑇,𝐹). There is no emptiable siphon if 𝐺^MIP=|𝑃|.

Corollary 3. Let (𝑁,𝑀₀) be an ordinary M-net with 𝑁=(𝑃,𝑇,𝐹). There is no emptiable siphon if 𝐺^𝑀𝐼𝑃=|𝑃|.

The proof of Corollary 3 is given in Appendix B.

Theorem 4. Let (𝑁,𝑀₀) be an ordinary M-net with 𝑁=(𝑃,𝑇,𝐹). Then (𝑁,𝑀₀) is live if 𝐺^MIP=|𝑃|.

The proof of Theorem 4 is given in Appendix B.

3. Structure Design of a Petri Net Supervisor

Now, let us go briefly through the processes by using the theory of regions to design a supervisor for a Petri net model. One first needs to generate its reachability graph. Then, the set of marking/transition separation instances should be found, whose number is in theory exponential with respect to the net size and the initial marking. Finally, for each instance, a monitor should be found by solving a linear programming problem in which the number of constraints is approximately equal to that of nodes in the reachability graph. For such a method, the size of a reachability graph is rather sensitive to the size and the initial marking of a net. These facts make it infeasible for the theory of regions to be applied to real-world problems.

With the purpose of formulating the proposed method even more clearly, we design Algorithm 1 to find a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) that can be obtained by reallocating the marking of each resource place in (𝑁,𝑀₀) to be one, with 𝑁^1𝑐=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅∪𝑃_𝑉,𝑇,𝐹^1𝑐,𝑊^1𝑐).

Input: an ordinary M-net (𝑁,𝑀0) with 𝑁=(𝑃0∪𝑃𝐴∪𝑃𝑅,𝑇,𝐹,𝑊) Output: (𝑁1𝑐,𝑀1𝑐) begin { 𝑁1∶=𝑁 ∀𝑝∈𝑃0∪𝑃𝐴, 𝑀1(𝑝)∶=𝑀0(𝑝) ∀𝑟∈𝑃𝑅, 𝑀1(𝑟)∶=1 if  {there exists an optimal controlled system for (𝑁1,𝑀1)}   then  design a controlled system (𝑁1𝑐,𝑀1𝑐) for (𝑁1,𝑀1) by the theory of regions else  design a controlled system (𝑁1𝑐,𝑀1𝑐) for (𝑁1,𝑀1) by the method in [18] end if ∀𝑝∈𝑃0∪𝑃𝐴∪𝑃𝑅, 𝑀1𝑐(𝑝)∶=𝑀1(𝑝) 𝑁1𝑐∶=(𝑃0∪𝑃𝐴∪𝑃𝑅∪𝑃𝑉,𝑇,𝐹1𝑐,𝑊1𝑐) if   {(𝑁1𝑐,𝑀1𝑐) is an ordinary controlled system}   then  output (𝑁1𝑐,𝑀1𝑐) else  exit and stop the algorithm end if output (𝑁1𝑐,𝑀1𝑐) } end of the algorithm

Algorithm 1 

Structure design of a controlled system for 𝑁,𝑀₀.

Input: an ordinary M-net (𝑁,𝑀0) with 𝑁=(𝑃0∪𝑃𝐴∪𝑃𝑅,𝑇,𝐹,𝑊) Output: controlled system (𝑁𝑐,𝑀𝑐) begin { design a controlled system (𝑁1𝑐,𝑀1𝑐) for (𝑁1,𝑀1) by Algorithm 1, then 𝑁𝑐∶=𝑁1𝑐 ∀𝑣∈𝑃𝑉, 𝑀𝑐(𝑣)∶=𝑀1𝑐(𝑣) ∀𝑝∈𝑃0∪𝑃𝐴∪𝑃𝑅, 𝑀𝑐(𝑝)∶=𝑀0(𝑝) 𝑃𝑖∶=(••𝐻(𝑣𝑖))∩𝑃𝑅, 𝑖∈{1,2,…,𝑛} ∀𝑝∈𝑃𝑖, 𝑀𝑐(𝑣𝑖)∶=∑𝑝∈𝑃𝑖𝑀(𝑝)−1, 𝑖∈{1,2,…,𝑛} →𝜇∶=[𝑀𝑐(𝑣1),𝑀𝑐(𝑣2),…,𝑀𝑐(𝑣𝑗),…,𝑀𝑐(𝑣𝑛)]𝑇, 𝑗∈{1,2,…,𝑛} While   {→𝜇 makes (𝑁𝑐,𝑀𝑐) not live, which can be decided by the MIP-based deadlock detection method in [32]}  do  for  𝑗=1 to 𝑛  do   𝑀𝑐(𝑣𝑗)−−;  end for end while →𝑙∶=→𝜇      /∗ →𝑙 denotes the marking vector [𝑀𝑐𝑙(𝑣1),𝑀𝑐𝑙(𝑣2),…,𝑀𝑐𝑙(𝑣𝑗),…,𝑀𝑐𝑙(𝑣𝑛)]𝑇,𝑗∈{1,2,…,𝑛}∗/ for   𝑗=1 to 𝑛   do  𝑀𝑐(𝑣𝑗)++; end for →ℎ∶=→𝜇      /∗ →ℎ denotes the marking vector [𝑀𝑐ℎ(𝑣1),𝑀𝑐ℎ(𝑣2),…,𝑀𝑐ℎ(𝑣𝑗),…,𝑀𝑐ℎ(𝑣𝑛)]𝑇,𝑗∈{1,2,…,𝑛}∗/ while (1) do    for   𝑗=1 to 𝑛   do   if   (𝑀𝑐ℎ(𝑣𝑗)==𝑀𝑐𝑙(𝑣𝑗))    𝑀𝑐ℎ(𝑣𝑗)∶=𝑀𝑐(𝑣𝑗);   else    𝑀𝑐ℎ(𝑣𝑗)−−;    break;   end if  end for  if   {→ℎ makes (𝑁𝑐,𝑀𝑐) live, which can be decided by the MIP-based deadlock detection  method in [32]}do    break;  end if end while →𝜇∶=→ℎ ∀𝑝∈𝑃0∪𝑃𝐴∪𝑃𝑅, 𝑀𝑐(𝑝)∶=𝑀0(𝑝) 𝑁𝑐∶=𝑁1𝑐 output (𝑁𝑐,𝑀𝑐) } end of the algorithm

Algorithm 2 

Controlled system design for (𝑁,𝑀₀).

Proposition 5. Let (𝑁^1𝑐,𝑀^1𝑐) be the resulting net from adding monitors to an ordinary M-net (𝑁¹,𝑀¹) by using Algorithm 1. Then (𝑁^1𝑐,𝑀^1𝑐) is ordinary and live.

The proof of Proposition 5 is given in Appendix B.

In order to illustrate Algorithm 1, consider a small example from [31]. Figures 1(a) and 1(b) show an ordinary M-net (𝑁,𝑀₀) and its corresponding reachability graph with eight reachable states, respectively. Now by applying Algorithm 1 to (𝑁,𝑀₀), a plant mode (𝑁¹,𝑀¹) can be obtained, as shown in Figure 1(c). It has the same topology structure as (𝑁,𝑀₀) in Figure 1(a) but its resource places have a small initial marking with 𝑀(𝑝₅)=1 and 𝑀(𝑝₆)=1. Its reachability graph is shown in Figure 1(d) with five reachable states. Figure 1(e) shows a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹), which can be obtained by using the theory of regions [15, 19].

(a)

(b)

(c)

(d)

(e)

Figure 1 

(a) A plant model (𝑁,𝑀0), (b) the reachability graph of (𝑁,𝑀0), (c) a modified model (𝑁1,𝑀1), (d) the reachability graph of (𝑁1,𝑀1), and (e) a controlled system (𝑁1𝑐,𝑀1𝑐) for (𝑁1,𝑀1).

The principal objective of reallocating the marking of each resource place to be one is that it is more tractable by using the theory of regions to design a controlled system for (𝑁¹,𝑀¹) than that for (𝑁,𝑀₀). We can see that the reachable states of (𝑁¹,𝑀¹) are five that are less than that of (𝑁,𝑀₀) whose reachable states are eight. One can image the heavy computation if the theory of regions is applied to such a net shown in Figure 1(a), with an initial marking 𝑀₀=100𝑝₁+80𝑝₂+50𝑝₃ having more than 8×10⁴ states. Algorithm 1 considers (𝑁¹,𝑀¹), as shown in Figure 1(c), which has five reachable markings only. Therefore, it is easier for us to find a supervisor for (𝑁¹,𝑀¹) than that for (𝑁,𝑀₀). Then, a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) can be obtained by using the theory of regions, as shown in Figure 1(e).

In this section, we propose a method to realize the structure design of a controlled system which can be obtained by Algorithm 1. That is to say, the structure of the supervisors has been found, based on which a deadlock prevention policy will be presented in the next section.

4. Deadlock Prevention Policy

This section proposes a deadlock prevention policy that can be carried out through the following Algorithm 2. In the previous section, a controlled system (𝑁^1𝑐,𝑀^1𝑐) can be obtained for (𝑁¹,𝑀¹) with 𝑁^1𝑐=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅∪𝑃_𝑉,𝑇,𝐹^1𝑐,𝑊^1𝑐) and 𝑁¹=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅,𝑇,𝐹,𝑊). In this section, we can find a controlled system (𝑁^𝑐,𝑀^𝑐) for (𝑁,𝑀₀) by restoring the markings of the resource places in (𝑁^1𝑐,𝑀^1𝑐) to their original ones and then computing a marking for each monitor. That is to say, even if the initial marking of the plant model changes, the structure of the controlled system obtained previously can be reused. This implies that we only need to compute the marking of each monitor in the controlled system when the markings of the resource places change.

Now, let us consider the relationship between the activity places, the monitors, and the resource places. First, we can find that the tokens that will flow into the activity places can not be greater than that of their holding resource places. Similarly, the tokens in the monitor should be less than that of the resource places which the monitor's activity places hold. That can be expressed by the following: for all  𝑖∈{1,2,…,𝑛}, 𝑣_𝑖∈𝑃_𝑉, 𝐻(𝑣_𝑖) is the set of the activity places controlled by  𝑣_𝑖, and for all  𝑝∈𝐻(𝑣_𝑖), 𝑝 is called the monitor's activity place. 𝑃_𝑖=(^••𝐻(𝑣_𝑖))∩𝑃_𝑅, for all  𝑝∈𝑃_𝑖, 𝑀^𝑐(𝑣_𝑖)≤∑_{𝑝∈𝑃𝑖}𝑀(𝑝)−1. Then, an upper limit value for each marking with respect to the monitors 𝑣₁,𝑣₂,…, and 𝑣_𝑛 can be obtained. Finally, we decide a marking for each monitor such that it makes the controlled system live which can be decided by the MIP-based deadlock detection method in [32]. Consequently, Algorithm 2 is designed to formulate the proposed method. Let (𝑁^𝑐,𝑀^𝑐) denote a controlled system for (𝑁,𝑀₀), which has the same net structure as (𝑁^1𝑐,𝑀^1𝑐) with 𝑁^𝑐=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅∪𝑃_𝑉,𝑇,𝐹^1𝑐,𝑊^1𝑐).

Theorem 6. Let (𝑁^𝑐,𝑀^𝑐) be a controlled system for an ordinary M-net (𝑁,𝑀₀) by using Algorithm 2. Then (𝑁^𝑐,𝑀^𝑐) is live.

The proof of Theorem 6 is given in Appendix B.

We briefly explain Algorithm 2 as follows. First, a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) can be obtained by using Algorithm 1 in the previous section. Next, the markings of the resource places in (𝑁^1𝑐,𝑀^1𝑐) are restored to their original ones. Then, we compute an upper limit value for the marking of each monitor and denote it by ^→𝜇∶=[𝑀^𝑐(𝑣₁),𝑀^𝑐(𝑣₂),…,𝑀^𝑐(𝑣_𝑗),…,𝑀^𝑐(𝑣_𝑛)]^𝑇, 𝑗∈{1,2,…,𝑛}. If the marking vector ^→𝜇 makes the controlled system live, we stop the algorithm and the marking vector ^→𝜇 is the result we want to obtain. If the marking vector ^→𝜇 makes the controlled system not live, then each marking in ^→𝜇 decreases by one; repeat this step until it makes the controlled system live and denote it by ^→𝑙=[𝑀^𝑐_𝑙(𝑣₁),𝑀^𝑐_𝑙(𝑣₂),…,𝑀^𝑐_𝑙(𝑣_𝑗),…,𝑀^𝑐_𝑙(𝑣_𝑛)]^𝑇,𝑗∈{1,2,…,𝑛}. Then, each marking in ^→𝑙 increases by one that can be denoted by ^→ℎ=[𝑀^𝑐_ℎ(𝑣₁),𝑀^𝑐_ℎ(𝑣₂),…,𝑀^ℎ_𝑙(𝑣_𝑗),…,𝑀^𝑐_ℎ(𝑣_𝑛)]^𝑇,𝑗∈{1,2,…,𝑛}. That is to say, we obtain a new higher limit value vector ^→ℎ and a lower limit value vector ^→𝑙 for the marking of each monitor. Finally, by using Algorithm 2, we decide a marking for each monitor such that for all  𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^𝑐(𝑝)=𝑀₀(𝑝), and forall𝑣∈𝑃_𝑉, 𝑀^𝑐(𝑣) makes the controlled system live which can be decided by the MIP-based deadlock detection method proposed in [32].

For example, a controlled system (𝑁^𝑐,𝑀^𝑐) can be obtained for the net in Figure 1(a) by utilizing Algorithm 2, as shown in Figure 2. In the previous section, a controlled system (𝑁^1𝑐,𝑀^1𝑐) has been obtained, as shown in Figure 1(e). Then, we can find a controlled system (𝑁^𝑐,𝑀^𝑐) for (𝑁,𝑀₀) by restoring the markings of the resource places in (𝑁^1𝑐,𝑀^1𝑐) to their original ones with 𝑀(𝑝₅)=2 and 𝑀(𝑝₆)=1, and then an upper limit value for the marking of the monitor 𝑝_𝑐 can be computed. We can find that the activity places controlled by 𝑝_𝑐 are 𝑝₂ and 𝑝₃. Therefore, 𝐻(𝑝_𝑐)={𝑝₂,𝑝₃}, {𝑝₅,𝑝₆}=(^••𝐻(𝑝_𝑐))∩𝑃_R. Then, we can obtain that 𝑀(𝑝_𝑐)≤𝑀(𝑝₅)+𝑀(𝑝₆)−1 implies that 𝑀(𝑝_𝑐)≤2. As shown in Figure 2, the controlled system is live with 𝑀(𝑝_𝑐)=2, which can be decided by the MIP-based deadlock detection method in [32]. Consequently, the marking 𝑀(𝑝_𝑐)=2 is the one that we want to obtain.

Figure 2 

A controlled system (𝑁𝑐,𝑀𝑐) for (𝑁,𝑀0).

Given a plant model (𝑁,𝑀₀), we reallocate the marking of each resource place to be one, obtaining a net model (𝑁¹,𝑀¹). By using Algorithm 1, a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) can be obtained. Then, the markings of the resource places in (𝑁^1𝑐,𝑀^1𝑐) are restored to their original ones. On the premise of not changing the structure of the controlled system, we compute the marking of each monitor by Algorithm 2. That is to say, even if the initial markings of the plant model change, the structure of the controlled system obtained previously can be reused. This implies that we only need to compute the marking of each monitor in the controlled system without changing the supervisor's structure. Figure 3 shows the flowchart of the proposed deadlock control strategy.

Figure 3 

Flowchart of the deadlock prevention policy.

5. Experimental Studies

In order to show the advantage of the proposed method, this section provides two typical examples that are taken from [31]. The computational results indicate that the proposed deadlock prevention policy is nearly optimal and superior to the one in [31].

An FMS consists of two robots R1 and R2 and three machines M1–M3. Its model is shown in Figure 4(a). It is an ordinary M-net, where 𝑝₁ and 𝑝₁₀ are idle places, 𝑝₁₁−𝑝₁₅ are resource places, and the others are activity places. As shown in Figure 4(b), (𝑁^1𝑐,𝑀^1𝑐) is the controlled system for the net (𝑁¹,𝑀¹) with the initial marking of each resource place being one.

(a)

(b)

Figure 4 

(a) An ordinary M-net (𝑁,𝑀0) and (b) controlled system (𝑁1𝑐,𝑀1𝑐).

To illustrate Algorithm 2, consider (𝑁^1𝑐,𝑀^1𝑐) shown in Figure 4(b). The activity places 𝑝₄ and 𝑝₉ are controlled by 𝑣₁ and hold the resource places 𝑝₁₂ and 𝑝₁₄, respectively. That can be expressed by the following: 𝐻(𝑣₁)={𝑝₄,𝑝₉}, {𝑝₁₂,𝑝₁₄}=(^••𝐻(𝑣₁))∩{𝑝₁₁,𝑝₁₂,𝑝₁₃,𝑝₁₄,𝑝₁₅}. Based on Algorithm 2, the marking of the monitor 𝑣₁ should be less than the total markings of the resource places 𝑝₁₂ and 𝑝₁₄. That can be expressed by an inequality: 𝑀𝑣₁≤𝑀𝑝₁₂+𝑀𝑝₁₄−1. (7) Much the same can be applied to monitors 𝑣₂ and 𝑣₃. The inequalities can be obtained as follows: 𝑀𝑣₂≤𝑀𝑝₁₂+𝑀𝑝₁₃𝑀𝑣−1,₃≤𝑀𝑝₁₃+𝑀𝑝₁₄−1. (8)

Now, the net under initial marking 4𝑝₁+4𝑝₁₀+2𝑝₁₁+2𝑝₁₂+2𝑝₁₃+2𝑝₁₄+2𝑝₁₅ is used to demonstrate the proposed method. From (7) and (8), the results can be obtained as follows: 𝑀𝑣₁𝑀𝑣≤2+2−1=3,₂𝑀𝑣≤2+2−1=3,₃≤2+2−1=3. (9)

Therefore, we can obtain an upper limit value for the marking of each monitor and denote it by ^→𝜇=[𝑀^𝑐(𝑣₁),𝑀^𝑐(𝑣₂),𝑀^𝑐(𝑣₃)]^𝑇=[3,3,3]^𝑇. Then, we find that the marking vector ^→𝜇 makes the controlled system live which can be decided by the MIP-based deadlock detection method proposed in [32]. Therefore, the marking vector [𝑀^𝑐(𝑣₁),𝑀^𝑐(𝑣₂),𝑀^𝑐(𝑣₃)]^𝑇=[3,3,3]^𝑇 is the result we want to obtain. It can be verified that the controlled model in Figure 4(b) under initial marking 4𝑝₁+4𝑝₁₀+2𝑝₁₁+2𝑝₁₂+2𝑝₁₃+2𝑝₁₄+2𝑝₁₅ with 𝑀(𝑣₁)=3,𝑀(𝑣₂)=3, and 𝑀(𝑣₃)=3 obtained by the proposed method is live with 1032 reachable states. Compared the proposed method with the one in [31] with 941 states, it can be clearly seen that we have achieved a better result, what we call a near-optimal result.

Compared with the method in [31], the superiority of the proposed policy can be verified. Table 1 shows the permissive behavior of the controlled systems under different initial markings, where the markings of the monitors are decided by Algorithm 2. In this table, 𝐵_𝑝 is the number of reachable states of (𝑁,𝑀₀), 𝐵_𝐿 represents the number of states that an optimal controlled system for (𝑁,𝑀₀) has, 𝐵_𝑐 indicates the number of states of the controlled system (𝑁^𝑐,𝑀^𝑐), and 𝐵_𝑐/𝐵_𝐿 implies the optimality degree. In order to make a comparative analysis of the proposed method and the one in [31], let Li’s denotes the results in [31]. For economy of space, the detailed computational steps are omitted. From this table, we conclude that the proposed method for this example is near optimal and superior to the one in [31].

Table 1 

Behavior permissiveness of the proposed deadlock prevention policy.

The second FMS is shown in Figure 5(a). It has two robots R1 and R2, each of which can hold one product at a time. The cell also contains four machines M1–M4, and each of them can hold one part. Parts enter FMS through two automatic loading buffers I1 and I2 and leave it through two unloading ones O1 and O2. The robots deal with the movements of parts. Two part types P1 and P2 are produced. Their respective production routes are shown in Figure 5(b).

(a)

(b)

Figure 5 

(a) Layout of an FMS and (b) routes of part types P1 and P2.

Figure 6(a) shows its net model that is an ordinary M-net in which 𝑃⁰={𝑝₁,𝑝₈}, 𝑃_𝑅={𝑝₁₅,𝑝₁₆,𝑝₁₇,𝑝₁₈,𝑝₁₉}, and the others are activity places. The controlled system of such a plant model is shown in Figure 6(b), which can be obtained by the theory of regions [15].

(a)

(b)

Figure 6 

(a) Petri net model of an FMS and (b) structure of the controlled system.

Consider the model shown in Figure 6(b). The monitor 𝑣₁ controls the activity places 𝑝₃, 𝑝₁₁, and 𝑝₁₂. The set of their corresponding resource places is {𝑝₁₅,𝑝₁₈}. Therefore, the markings of the monitor 𝑣₁ should be less than the total markings of the resource places 𝑝₁₅ and 𝑝₁₈. That can be expressed by an inequality: 𝑀𝑣₁≤𝑀𝑝₁₅+𝑀𝑝₁₈−1. (10)

The same situation can be applied to monitors 𝑣₂, 𝑣₃, 𝑣₄, 𝑣₅, and 𝑣₆. The inequalities can be obtained as follows: 𝑀𝑣₂≤𝑀𝑝₁₄+𝑀𝑝₁₅+𝑀𝑝₁₈𝑀𝑣−1,₃≤𝑀𝑝₁₆+𝑀𝑝₁₇+𝑀𝑝₁₈+𝑀𝑝₁₉𝑀𝑣−1,₄≤𝑀𝑝₁₅+𝑀𝑝₁₇+𝑀𝑝₁₈+𝑀𝑝₁₉𝑀𝑣−1₅≤𝑀𝑝₁₅+𝑀𝑝₁₆+𝑀𝑝₁₇+𝑀𝑝₁₈+𝑀𝑝₁₉𝑀𝑣−1,₆≤𝑀𝑝₁₄+𝑀𝑝₁₅+𝑀𝑝₁₆+𝑀𝑝₁₇+𝑀𝑝₁₈+𝑀𝑝₁₉−1. (11)

By exploiting Algorithm 2, the markings for the monitors can be obtained, as shown in Table 2. For economy of space, the computational steps for the system are not shown in detail. Table 2 shows the performance of the controlled systems under different initial markings. From this table, it is verified that the proposed method for this example is nearly optimal and may even achieve optimality. It is obviously superior to the method in [31].

Table 2 

Behavior permissiveness of the proposed deadlock prevention policy.

6. Comparision of Computational Efficiency

The deadlock prevention method proposed in [31] needs to calculate all SMS in the controlled system and infer algebraic inequalities. An improved method of avoiding this problem is presented in this paper, which is simple and practicable. To illustrate the application of the proposed method, two FMS examples are used in this paper. A comparison between the proposed method and the one in [31] is shown in Tables 1 and 2. The superiority of the proposed one is obvious. For a class of FMS considering resource allocation, this paper proposes a deadlock prevention policy by resources reallocation and supervisor reconfiguration, which can make a good tradeoff between optimality and computational tractability for a class of ordinary Petri nets.

7. Conclusion

The deadlock prevention policy is a static strategy that imposes restrictions on the interactions among resources and processes such that resource requests that may lead to deadlocks are prevented. Behavioral permissiveness is very important in designing a liveness-enforcing supervisor for a system to be controlled. An optimal liveness-enforcing supervisor can lead to high utilization of system resources. This paper proposes a deadlock prevention policy by resources reallocation and supervisor reconfiguration. Given a plant model, we first reallocate the marking of each resource place to be one, and then find a controlled system by using Algorithm 1. Next, the markings of the resource places in the controlled system are restored to their original ones. Without changing the structure of the controlled system, we compute the markings of the monitors. Finally, we decide a marking for each monitor such that it makes the controlled system live which can be realized by Algorithms 2 proposed in this paper. For a fixed net structure with different initial marking, the theory of regions is used once only. That is to say, the supervisory control system can be reconfigured rapidly in response to the changes in the initial markings of the plant model. Two FMS examples are used to illustrate the application of the proposed method and show its superior efficiency.

However, the proposed method suffers from the computational complexity problem due to the theory of regions. In theory, it suffers from the exponential complexity. Future efforts will be made to a near-optimal supervisor with low computational costs. In addition, the proposed method is applicable to ordinary M-nets only. Therefore, our future work will extend this method to more general classes of Petri nets, for example, the ones in [34, 35].

Appendices

A. An Example for Definition 1

As the net shown in Figure 7, it is an M-net, where 𝑝₁ is an idle process place, 𝑝₂,𝑝₃, and 𝑝₄ are activity places, and 𝑝₅ and 𝑝₆ are resource places. It is quasi-live, bounded, and conservative. It is live if no siphon is uncontrolled.

Figure 7 

An M-net model.

B. Proofs for Corollary 3, Theorems 4, 6, and Proposition 5

The proof of Corollary 3.

Proof. It follows immediately from the definition of an ordinary M-net that it is a class of ordinary Petri nets. According to Theorem 2, the result is true.

The proof of Theorem 4.

Proof. Let (𝑁,𝑀₀) be an ordinary M-net. There is no emptiable siphon if 𝐺^MIP=|𝑃| from Corollary 3. By the definition of M-nets, if (𝑁,𝑀₀) is ordinary and no emptiable siphon, that is, uncontrolled siphon in (𝑁,𝑀₀), (𝑁,𝑀₀) is live.

The proof of Proposition 5.

Proof. It follows immediately from the theory of regions that (𝑁^1𝑐,𝑀^1𝑐) is live. (𝑁^1𝑐,𝑀^1𝑐) is excluded by Algorithm 1 if it is a generalized net. Therefore, the result is true.

The proof of Theorem 6.

Proof. According to Proposition 5, (𝑁^1𝑐,𝑀^1𝑐) is ordinary and live. The structure of the controlled system (𝑁^𝑐,𝑀^𝑐) is the same as that of the (𝑁^1𝑐,𝑀^1𝑐). That is to say, (𝑁^𝑐,𝑀^𝑐) is an ordinary controlled system as well as an M-net according to Definition 1. Then, from Theorem 4, (𝑁^𝑐,𝑀^𝑐) is an ordinary M-net, (𝑁^𝑐,𝑀^𝑐) is live if 𝐺^MIP=|𝑃|. Therefore, the result is true.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants no. 61074035 and 61100056, the Fundamental Research Funds for the Central Universities under Grant no. 72103326, the Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12F03020, the Zhejiang Provincial Education Department Foundation under Grant no. Y201018216, and the Opening Project of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing, under Grant no. MCCSE2012A05.