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

Abstract

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.

Appendix: Basics of Petri nets

A Petri net is a four-tuple N = (P, T, F, W) where P and T are finite, nonempty, and disjoint sets. P is the set of places and T is the set of transitions with P ∪ T ≠ ∅ and P ∩ T = ∅. $F\subseteq (P\times T)\cup (T\times P)$ is called flow relation of the net, represented by arcs with arrows from places to transitions or from transitions to places. W: F → N ⁺ is a mapping that assigns a weight to any arc, where N ⁺={1, 2, ⋯ }. A net is self-loop free if $\nexists x, y\in P\cup T, f(x,y)\in F\wedge f(y,x)\in F$. A self-loop free net N = (P, T, F, W) can be alternatively represented by its incidence matrix [N], where [N] is a |P| × |T| integer matrix and [N](p, t)=W(t, p) − W(p, t).

A marking M of N is a mapping from P to IN, where IN = {0, 1, 2, ⋯ }. M(p) denotes the number of tokens contained in place p, which is marked by M iff M(p) > 0. Let $S\subseteq P$ be a set of places. M(S) denotes the sum of tokens contained in S at marking M, where M(S) = ∑  _p ∈ S M(p). (N,M ₀) is called a net system or marked net. For economy of space, we use ∑  _p ∈ P M(p)p to denote vector M.

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}. This notation can be extended to a set of nodes as follows: given $X\subseteq P\cup T$, $^\bullet X=\cup_{x \in X}{^\bullet x}$, and $X^\bullet=\cup_{x\in X}x^\bullet$.

A transition t ∈ T is enabled at a marking M if ∀ p ∈ ^∙ t, M(p) ≥ W(f(p, t)), which will be denoted as $M[t\rangle$; when fired in a usual way, it gives a new marking M′ such that ∀ p ∈ P, M′(p) = M(p) − W(p, t) + W(t, p), which will be denoted as $M[t\rangle M^\prime$. Marking M′ is said to be reachable from M if there exists a sequence of transitions σ=t ₀ t ₁ ⋯ t _n and markings M ₁, M ₂, ⋯, and M _n such that $M[t_0\rangle M_1[t_1\rangle$ $M_2\cdots M_n[t_n\rangle M^\prime$ holds. The set of markings reachable from M in N is denoted as R(N, M).

A transition t ∈ T is live under M ₀ if ∀ M ∈ R(N, M ₀), $\exists M^\prime\in R(N,M)$, $M^\prime[t\rangle$. N is dead under M ₀ if $\nexists t\in T$, $M_0[t\rangle$ holds. (N, M ₀) is deadlock-free if ∀ M ∈ R(N, M ₀), $\exists t\in T$, $M[t\rangle$ holds. (N, M ₀) is quasi-live if ∀ t ∈ T, $\exists M\in R(N,M_0)$, $M[t\rangle$ holds. (N,M ₀) is live if ∀ t ∈ T, t is live under M ₀. (N, M ₀) is bounded if $\exists k\in\mathbf{IN}$, ∀ M ∈ R(N, M ₀), ∀ p ∈ P, M(p) ≤ k holds. (N, M ₀) is said to be reversible, if for each marking M ∈ R(G,M ₀), M ₀ is reachable from M. A marking M′ is said to be a home state, if for each marking M ∈ R(G, M ₀), M′ is reachable from M. Reversibility is a special case of the home state property, i.e. if the home state $M^\prime=M_0$, then the net is reversible.

A P-vector is a column vector I : P → Z indexed by P and a T-vector is a column vector J : T → Z indexed by T, where Z is the set of integers. A P(T)-vector I(J) is denoted by ∑  _p ∈ P I(p)p (∑  _t ∈ T J(t)t) for economy of space. We denote column vectors where every entry equals 0(1) by $\textbf{0}(\textbf{1})$. I ^T and [N]^T are the transposed versions of a vector I and a matrix [N], respectively. P-vector I is a P-invariant (place invariant) iff $I\neq\textbf{0}$ and $I^T[N]=\textbf{0}^T$. P-invariant I is said to be a P-semiflow if no element of I is negative. ||I|| = {p ∈ P|I(p) ≠ 0} is called the support of I. ||I||⁺ = {p|I(p) > 0} denotes the positive support of P-invariant I, while ||I||− = {p|I(p) < 0} denotes the negative support of I. An invariant is called minimal when its support is not a strict superset of the support of any other, and the greatest common divisor of its elements is one. If I is a P-invariant of (N, M ₀) then ∀ M ∈ R(N, M ₀), $I^TM=I^TM_0$.

Let P be set of places of net N, I be a P-vector, and $S\subseteq P$ be a subset of places of N. $I\backslash{S}$ is defined to be $\sum_{p\in P\backslash{S}}I(p)p$. For example, I = 2p ₁ + 3p ₂ + p ₃ + 4p ₄ is a P-vector and S = {p ₁, p ₄} in some net. Then we have $I\backslash{S}=3p_2+p_3$. I ∩ S ≠ ∅ means that $\exists p\in{S}$, I(p) ≠ 0.

Let S be a non-empty subset of places. $S\subseteq P$ is a siphon (trap) if $^\bullet S\subseteq S^\bullet$ $(S^\bullet\subseteq{^\bullet S})$. Siphon S is said to be minimal if it contains no other siphons as its proper subset. A minimal siphon S is strict if it does not contain a trap.

The following notations of generalized Petri nets are from Barkaoui and Pradat-Peyre (1996).

Definition 13

Let (N, M ₀) be a marked net and S be a siphon of N. S is said to be max-marked at a marking M if $\exists p\in S$ such that $M(p)\geq \max_{p^\bullet}$, where $\max_{p^\bullet}=\max\{W(p, t)\mid t\in p^\bullet\}$. S is said to be max-controlled if S is max-marked at any reachable marking.

Definition 14

A net (N, M ₀) is said to satisfy the max cs-property (controlled-siphon property) if each minimal siphon of N is max-controlled.

The cs-property is an important concept in net theory on which our approaches rely. A siphon satisfying the cs-property can be always marked sufficiently to allow firing a transition once at least. In order to check and use the cs-property, Barkaoui and Pradat-Peyre (1996) proposed the conditions to determine whether a given siphon is max-controlled and establish the relationship of the cs-property and liveness property.