Sustainability of Automated Manufacturing Systems with Resources by Means of Their Deadlock Prevention

1. Introduction

It is very important to deal with deadlocks in industrial MAS (Multi Agent Systems). AMS (Automated Manufacturing Systems) where two or more production lines share robots, machine tools, conveyors, automatically controlled vehicles, etc., are a typical example of such systems. These shared entities represent resources. Such production systems are very important in social practice. They should fluently produce expected products without any obstacles. However, deadlocks occurring in them prevent this. In general, deadlock is a status when two (or more) processes are waiting to continue their activity, but are preventing each other from doing so. Namely, they are competing for the same resources. A deadlock is an undesirable state in the global system, because the system itself, or a part of it, stagnates. Sustainability of the manufacturing process continuity is strongly disturbed. Therefore, the original intention of such a system cannot be achieved. In order to sustain processes in AMS fluently work it is necessary to prevent deadlocks.

S³PR, i.e., Systems of Simple Sequential Processes with Resources, modeling the Resource Allocation Systems (RAS) on the base of PN were defined, analyzed and controlled in [1]. S³PR were modeled there by Ordinary PN (OPN) and controlled by siphons. Here, more complicated Extended S³PR (ES³PR) models of RAS are analyzed by Generalized PN (GPN) and controlled by their siphons.

1.1. Basic PN Terminology

A Petri net is a quadruplet N = (P, T, F, W) with P and T being finite nonempty sets, a set of places P = {p₁, p₂, …, p_n} (|P| = n) and a set of transitions T = {t₁, t₂, …, t_m} (|T| = m), where P ∪ T ≠ ∅ and P ∩ T = ∅, and ∅ means an empty set. The set F = (P × T) ∪ (T × P) is a flow relation of N. Such flow consists of directed arcs from places to transitions and vice versa. The mapping W: (P × T) ∪ (T × P) → ℕ assigns a weight to an arc: W(f) > 0 if f ∈ F and W(f) = 0 otherwise. Here ℕ = {0, 1, 2, …} is the set of natural numbers plus zero. The net N is ordinary net if ∀ f ∈ F, W(f) = 1. Such net is denoted as N = (P, T, F). If ∃f ∈ F, W(f) > 1, N = (P, T, F, W) is named as a generalized net GPN.

The set M expressing states of marking of particular PN places, simply said the PN marking M, is an (n × 1) vector named also as the state vector. Its evolution is performed by means of the matrix/vector equation M_k+1 = M_k + [N]. σ_k, k ∈ ℕ, where M₀ is an initial marking. This equation represents the mathematical model of the net N. The matrix [N] = [Post]^T − [Pre] is the (n × m) incidence matrix. It represents the matrix notation of the set F. It can also be expressed as [N] (p, t) = W (t, p) − W (p, t). After all, σ_k is a (m × 1) vector expressing firing of transitions named also as the control vector. Namely, it expresses the state of transitions (a transition may be firable—it is indicated by 1, or not—it is indicated by 0).

For a place p, M(p) is an integer indicating the number of tokens placed inside of this place. We say that the place p is marked by M iff M(p) > 0. The abbreviation iff means if and only if in the whole paper.

A subset D ⊆ P is marked by M iff at least one place p ∈ D is marked by M. Thus, M (D) = Σ_p∈D M (p) denotes the sum of tokens in all places in D.

^●x = {y ∈ P ∪ T | (y, x) ∈ F} is the preset of a node x ∈ P ∪ T, while x^● = {y ∈ P ∪ T | (x, y) ∈ F} is the postset of a node x ∈ P ∪ T.

A transition t such that |t^●| > 1 is named as a fork. A transition t such that |^●t| > 1 is named as a join. A place p such that |p^●| > 1 is named as a choice. A place p such that |^●p| > 1 is named as an attribution.

SP (x₁, x_n) means a simple path from x₁ to x_n. It is a path whose all nodes are different.

We say that a transition t ∈ T is enabled at a marking M iff ∀p ∈ ^●t, M(p) ≥ W (p, t). After firing t a new marking M′ is evolved such that ∀p ∈ P, M′ (p) = M(p) − W (p, t) + W (t. p). However, enabled transition t may be fired, but it may not be fired.

A non-empty subset of places S ⊆ P is a siphon iff ^●S ⊆ S^●. A non-empty subset of places S ⊆ P is a trap iff S^● ⊆ ^●S. Siphon S is minimal iff it contains no other siphons (as its proper subset). If a minimal siphon S does not contain a marked trap, it is named strict.

A column vector I: P → Z indexed by P, where Z is the set of integers, is named the P-vector. A column vector J: T → Z indexed by T is named the T-vector. For the economy of space, a P-vector I is denoted by Σ_p∈P I (p)p and T-vector J is denoted by Σ_{t∈ T} J(t)t.

Let I is a P-invariant of (N, M₀). Then ∀M ∈ R (N, M₀), I^T.M = I^T.M₀. Here R (N, M₀) is the state space of N. In other words, it is a set of all reachable markings of N.

As we can see, the invariant I is a P-vector. ||I|| = {p ∈ P | I(p) ≠ 0} means a support of P. ||I||⁺ = {p ∈ P | I(p) > 0} is the positive support of P and ||I||^- = {p ∈ P | I(p) < 0} is the negative support of P.

P-invariant of (N, M₀) is a natural solution of the equation I^T.[N] = 0. If all elements of P-invariant I are nonnegative this invariant is named as P-semiflow.

T-invariant of (N, M₀) is a natural solution of the equation [N]. J = 0. It gives information about a possible loop in the net, i.e., about a sequence of transitions which leads back to the marking it starts in. If all elements of T-invariant J are nonnegative this invariant is named as T-semiflow.

When a place p is given, then max _{t ∈ p●} {W (p, t)} is denoted by max_p●.

More details about PN are introduced in [1,2,3,4] and in basic papers about PN [17,18,19,20].

1.2. Structure of the Article

In the next section preliminaries concerning definitions of particular PN models of AMS are introduced.

The third section is devoted to control of ES³PR paradigm od AMS model. It represents one of two substantive parts of this paper. Since ES³PR model belongs to a subclass of S⁴PR paradigm of models, accordingly such paradigm is defined there. The basic influence of siphons to the control of PN models of AMS is emphasized there too. Finally, definitions concerning the siphon-based control as well as the procedure making possible to find parameters of the supervisor, ensuring the deadlock-freeness of the ES³PR model, are introduced. To illustrate ES³PR model, simple example is presented in Figure 1 and its siphons are computed. Unfortunately, these siphons, satisfying for S³PR models cannot be used for the supervisor synthesis of ES³PR model, Namely, they do not ensure the deadlock-freeness.

In the fourth section, representing the second of two substantive parts of the paper, controllability of siphons in GPN is analyzed and elementary and dependent siphons are checked as to max-cs property. Then the supervisor computed in the sense of such a procedure is set. The Figure 2 illustrates the interconnections between the original uncontrolled model and the supervisor ensuring the deadlock-freeness.

Fifth section represents conclusions, where the topic and contribution of the paper is evaluated and the further work in future is pointed out to.

2. Preliminaries

As to nomenclature of sets in PN used below, P_S is a set of operation places, P_R is a set of resource places, P₀ is a set of idle places.

Definition 1 [6]: An ordinary Petri net N = (P_S ∪ {p₀}, T, F) is a Simple Sequential Process (S²P) when the following hold true:

1. P_S ≠ ∅, p₀ ∉ P_S.

2. N is a strongly connected state machine (i.e., OPN where each transition has only one input place and only one output place).

3. Every circuit in N contains place p₀.

Definition 2 [6]: An extended simple sequential process with resources—ES²PR is a generalized pure Petri net N = (P_S ∪ {p₀} ∪ P_R, T, F, W), such that:

1. The subnet generated by the set X = P_S ∪ {p₀} × T is an S²P.

2. P_R ≠ ∅ and (P_S ∪ {p₀}) ∩ P_R = ∅.

3. ∀t ∈ T, ∀p ∈ ^●t, W (p, t) = 1.

4. ∀r ∈ P_R, ∃ a minimal P-semiflow I_r such that {r} = ||I_r|| ∩ P_R, p₀ ∉ ||I_r||, ||I_r|| ∩ P_S ≠ ∅ and I_r(r) = 1.

Definition 3 [6]: Let N = (P_S ∪ {p₀} ∪ P_R, T, F, W) be an ES²PR. An initial marking M₀ is acceptable for N iff:

1. M₀(p₀) > 0.

2. ∀p ∈ P_S, M₀(p) = 0.

3. ∀r ∈ P_R, M₀(r) > 0

4. Elements in the support of each minimal T-semiflow can be fired sequentially.

Definition 4 [6]: An ES³PR is the composition of a finite set of ES²PR via the fusion of resource places.

Definition 5 [6]: Let N_i = (P_i, T_i, F_i, W_i) (i = 1, 2) be two generalized Petri nets. Then:

1. N₁ and N₂ are composable (via fusion of places) iff T₁ ∩ T₂ = ∅, and P_C = P₁ ∩ P₂ ≠ ∅.

2. Net N = (P, T, F, W) is called the composed net of N₁ and N₂, denoted by N₁ ○ N₂, where P = P₁ ∪ P₂, T = T₁ ∪ T₂, W (p, t) = if t ∈ T₁ then W₁ (p, t) else W₂ (p, t).

Definition 6 [6]: Let I_m be a set of indices, N_i = (P_i, T_i, F_i, W_i) (i ∈ I_m) be a set of generalized Petri nets. We denote by N = ◯_{i ∈ Im} N_i the net obtained by the following operation: if card (I_m) = 1 then N = N₁ else N = N_k ○ ◯ _{i ∈ Im \ {k}} N_i.

Definition 7 [6]: An ES³PR is the composition of a finite set of ES²PR via the fusion of resource places.

Definition 8 [6]: (N, M₀), ◯_i∈Im Ni, N_i= (P_i, T_i, F_i, W_i), and (N_i, M_0i) being initially acceptably marked ES²PR, is an acceptably marked ES³PR iff:

1. ∀i ∈ I_m, ∀p ∈ P_Si ∪ {p_0i}, M₀(p) = M_0i (p).

2. ∀i ∈ I_m, ∀r ∈ P_R, M₀(r) = if card (I_m) = 1 then M_0i(r) else max_i∈Im M_0i(r).

In [10] the following property was proved:

Let N = (P_S ∪ P₀ ∪ P, T, F, W) be an ES³PR, where P₀ = ∪_{i ∈ Im} {p_0i}. Then, rank ([N]) = |P_S|.

In [11] the following Theorem was proved:

Theorem 1 [11]: Let S be a strict minimal siphon in an ES³PR N = (P_S ∪ P₀ ∪ P, T, F, W). Then S = S_R ∪ S_S satisfies S ∩ P₀ = ∅, S ∩ P_R = S_R ≠ ∅, and S ∩ P_S = S_S ≠ ∅.

3. Control of ES³PR

The paradigm ES³PR models a class of concurrently cyclic sequential processes sharing common resources. An approach to the deadlock prevention by controlling siphons in ES³PR will be presented here. Namely,

1. the control of elementary siphons

2. finding the relevant approach to finding conditions for control of dependent siphons;

3. finding the way of making the ES³PR deadlock-free.

With respect to [6] ES³PR is a subclass of S⁴R (Systems of Sequential Systems with Shared Resources) defined by the next definition.

Definition 8 [6]: An S⁴R is a generalized pure net N = ◯_i ∈ImN_i = (P, T, F, W), where

• N_i = (P_Si ∪ {p_0i} ∪ P_Ri, T_i, F_i, W_i), i ∈ I_m
• P = Ps ∪ P₀ ∪ P_R is a partition such that
  • P_S = ∪ _{i ∈ Im} P_Si, P_Si ≠ ∅ and P_Si ∩ P_Sj = ∅, ∀i ≠ j (i, j ∈ I_m)
  • P_R = ∪ _{i ∈ Im} P_Ri = {r₁, r_2, …, r_n}, n > 0
  • P₀ = ∪ _{i ∈ Im} {p_0i}
  • The elements in P₀, P_S, and P_R are called idle, operation, and resource places, respectively
  • The output transitions of an idle place are called source transitions.
• T = ∪ _{i ∈ Im} T_i, T_i ≠ ∅, T_i ∩ T_j = ∅ for all i ≠ j.
• ∀i ∈ I_m the subset N_i generated by P_Si ∪ {p_0i} ∪ T_i is a strongly connected state machine such that every cycle contains p_0i
• ∀r ∈ P_R, there exists a unique minimal P-semiflow I_r ∈ IN^|P| (here IN = {0, 1, 2, …}) such that {r} = ||I_r|| ∩ P_R, P₀ ∩ ||I_r|| = ∅, P_S ∩ ||I_r|| ≠ ∅ and I_r(r) = 1
• P_S = ∪ _{r ∈ PR} (||Ir||\{r})
• N is a strongly connected net.

In [11] the following Theorem was proved:

Theorem 2 [11]: Let (N, M₀) be a marked S⁴R net. N is live under M₀ iff it satisfies max-cs property.

Consider that r is a resource place, S is a strict minimal siphon and H(r) = I_r\{r} in a S⁴R net. We can define Th(S) = Σ_r∈SR H (r) \ S. It can be seen that |Th(S)| ⊆ P_S is true. Use Σ_p∈|Th(S)| h_S(p)p to denote Th(S). h_S(p) indicates that if the number of tokens in p increases by one, the siphon S loses h_S (p) tokens

In [9] the following Lemma was proved:

Lemma 1 [9]: Let (N, M₀) be a marked net and S be a siphon of N. S is max-controlled if there exists a P-invariant I such that ∀p ∈ (||I||^- ∩ S), max_p● = 1, ||I||⁺ ⊆ S, and Σ_{p∈ P} I (p)M₀(p) > Σ_{p∈ S} I(p) (max_p● − 1).

This Lemma will be applied below, within the context of Figure 2.

3.1. Siphons in Petri Nets Control

Siphons and traps were formally defined above at the end of the Section Introduction. Elementary and dependent siphons in OPN were proposed in [12]. Later, the terminology was more specified in [16]. In this section siphons and traps are defined in GPN.

With respect to [7] we have the following knowledge:

Definition 9 [7]: Let S ⊆ P be a subset of places of N = (P, T, F, W). P-vector λ_S is called the characteristic P-vector of S iff ∀p ∈ S, λ_S(p) = 1; otherwise λ_S(p) = 0.

Definition 10 [7]: Let S ⊆ P be a subset of places of N = (P, T, F, W). η_S is called the characteristic T-vector of S iff η^T_S = λ^T_S .[N] .

Definition 11 [7]: Let N = (P, T, F, W) be a net with |P| = n, which has k siphons S₁, …, S_k, m, k ∈ ℕ ⁺. We define [λ]_k×n = [λ_S1|λ_S2|… |λ_Sk]^T and [η]_k×m = [λ]_k×n.[N]_n×m = [η_S1|η_S2||…|η_Sk]^T is called the characteristic matrix of the siphons in N.

Definition 12 [7]: Let η_Sα, η_Sβ, …, and η_Sγ ({α, β, …, γ} ⊆ {1, 2, ..., k}) be a linearly independent maximal set of the matrix [η]. Then Π_E = {S_α, S_β, …, S_γ} is called a set of elementary siphons in N.

Definition 13 [7]: S ∉ Π_E is called a strongly dependent siphon if η_s = Σ_{S ∈ ΠE} a_i.η_si, where a_i ≥ 0.

Definition 14 [7]: S ∉ Π_E is called a weakly dependent siphon if ∃ non-empty A, B ⸦ Π_E, such that A ∩ B = ∅ and η_S = Σ _Si∈A a_i.η_Si − Σ_{Si∈ B} a_i.η_Si, where a_i > 0.

Lemma 2 [7]: The number of elements in any set of elementary siphons in net N equals to rank ([η]).

Verbally said, this important Lemma shows that the rank ([η]) directly determines the number of elementary siphons, However, it simultaneously determines also the number of dependent siphons because Π = Π_E ∪ Π_D.

Theorem 3 [7]: Let N_ES be the number of elementary siphons in net N = (P, T, F, W). Then N_ES ≤ min {|P|, |T|}.

Corollary 1 [7]: Let N = ◯_{i ∈ Im} Ni, N_i= (P_i, T_i, F_i, W_i) be an ES³PR. Then, N_ES ≤ |P_S|.

Verbally said, this important Corollary indicates that the maximal number of elementary siphons in an ES³PR net is not greater than the number of its operation places p ∈ P_S.

3.2. Procedure of Setting the Supervisor for ES³PR

In [6,7] it can be found the following definition:

Defnition 15 [6,7]: Let S be a strict minimal siphon in an ES³PR net model of a plant (N_µ0, M_µ0) where N_µ0 = ◯_{i∈{1, …, n}} N_i = (P₀ ∪ P_S ∪ P_R, T, F_µ0, W_µ0). Let {α, β, …, γ} ⊆ {1, 2, …, n} such that i ∈ {α, β, …, γ}, |Th_S| ∩ p ∈ P_Si ≠ ∅ and ∀j ∈ {1, 2, …, n} \ {α, β, …, γ}, |Th_S| ∩ p ∈ P_Sj ≠ ∅. For S, a nonnegative P-vector k_S is constructed as follows:

Step 1: ∀p ∈ P₀ ∪ P_S ∪ P_R, k_S(p): = 0;

Step 2: ∀p ∈ |Th(S)|, k_S(p): = h_S(p), where Th(S) = Σ_p∈|Th(S)| h_S(p)p;

Step 3: ∀i ∈ {α, β, …, γ}, let p_s ∈ ||Th_S|| ∩ P_Si be such a place that ∀p_t ∈ SP (p_u, p_0i), p_u ∈ p_s^••, p_t ∉ |T h(S)|. Suppose that there are m such places p_1s, p_2s, …, p_ms. Assuredly, we have {p_is |i = 1, 2, …, m} ⊆ |Th(S)| ∩ P_Si. ∀p_is let p_iν ∈ SP (p_0i, p_is) be such a place that h_S(p_iν) > h_S(p_w), ∀p_w ∈ SP (p_0i, p_is). ∀p_x ∈ SP (p_0i, p_is), k_S(p_x): = h_S(p_iν).

Step 4: ∀I_m’ ⊆ {1, 2, …, m}. ∀p_y ∈ ∩ _i∈Im’ SP (p_0i, p_is), k_S(p_y): = h_S(p_iz), where p_iz ∈ |Th(S)| ∩ P_Si, and ∄p ∈ Th(S) ∩ P_Si, s.t. h_S(p) > h_S(p_iz).

In general, such procedure makes possible to finalize the supervisor synthesis, especially to set markings of the supervisor monitors V_Si, i = 1, 2, …

3.3. Illustrative Example of ES³PR—Siphons and P-Invariants

Let us apply the theory of siphon-based control, introduced above, on the example of ES³PR. Consider the GPN in Figure 1 modelling an AMS. Here, the set of operation places is P_S = {p₁, p₂, p₃, p₄, p₅, p₇, p₈, p₉}, the set of resource places is P_R = {p₁₁, p₁₂, p₁₃, p₁₄, p₁₅, p₁₆}, and the set of idle places is P₀ = {p₆, p₁₀}.

As we can see in Figure 1, there exists one arc with the weight equal to 2—notice the arc from t₆ to p₁₃. Consequently, in Figure 1 is GPN. The state space of the net N expressed by the reachability tree (RT) has 308 nodes (including the initial state). Because of such large amount of nodes, the RT cannot be displayed here. The minimal siphons are introduced in Table 1.

As we can see in Table 1, there exist 11 minimal siphons in this GPN model of AMS. They can be enumerated e.g., by the tool GPenSIM [13,14,15] or by another PN tools. Because the traps corresponding to the crossed siphons are marked, crossed siphons may be omitted from the list of siphons. Hence, only 4 of them are strict minimal siphons—S₃, S₇, S₈ and S₁₀. Other 7 siphons are not relevant because they are equal to corresponding traps.

Let us renumber siphons S₃, S₇, S₈ and S₁₀ to siphons S₃, S_1, S₂ and S₄, respectively. Consequently, S₁ = {p₅, p₈, p₁₅, p₁₆}, S₂ = {p₄, p₉, p₁₄, p₁₅}, S₃ = {p₄, p₁₃} and S₄ = {p₅, p₉, p₁₄, p₁₅, p₁₆}. In the form of row P-vectors they are as follows:

S₁ = (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1)

S₂ = (0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0)

S₃ = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0)

S₄ = (0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1)

Corresponding matrix [λ] = [S₁^T, S₂^T, S₃^T, S₄^T]^T can be utilized (analogically to [1] in the case of S³PR paradigm) for finding potential interconnections with the supervisor eliminating the deadlocks. Namely, when we multiply matrix [λ] by [N], being (16 × 11) dimensional incidence matrix of GPN in Figure 1, we obtain the matrix [η] = [λ]. [N] = [η₁^T, η₂^T, η₃^T, η₄^T]^Tconsisting of the following T-vectors

η₁ = (0, 0, 0, 0, −1, 1, 0, −1, 1, 0, 0)

η₂ = (0, 0, 0, −1, 1, 0, 0, 0, −1, 1, 0)

η₃ = (0, 0, 0, −1, 0, 1, 0, 0, 0, 0, 0)

η₄ = (0, 0, 0, −1, 0, 1, 0, −1, 0, 1, 0).

Hence, we have λ_S1 = p₅ + p₈ + p₁₅ + p₁₆, λ_S2 = p₄ + p₉ + p₁₄ + p₁₅, λ_S3 = p₄ + p₁₃, λ_S4 = p₅ + p₉ + p₁₄ + p₁₅ + p₁₆ and η_S1 = − t₅ + t₆ − t₈ + t₉, η_S2 = − t₄ + t₅ − t₉ + t₁₀, η_S3 = − t₄ + t₆, and η_S4 = − t₄ + t₆ − t₈ + t₁₀. It is easy to verify that η_S4 = η_S1 + η_S2. However, this set of vectors η_{Si ,} i = 1, …, 4 is not linearly independent, because η₄ = η₁ + η₂, i.e., rank([η]) = 3, and rank([η]) = N_ES = 3 < rank([N]) = 8. It means that there are three elementary siphons and one strongly dependent siphon. Thus, Π_E = {S_1, S₂, S₃} and Π_D= {S₄}.

In Figure 1 we have p₀₁ = p₆, p₀₂ = p₁₀, P_S1 = {p₁, …, p₅}, P_S2 = {p₇, …, p₉}, P_R1 = {p₁₁, …, p₁₆}, and P_R2 = {p₁₄, …, p₁₆}. Unfortunately, the system is deadlocked.

GPN in Figure 1 has eight minimal P-invariants (see Table 2). They may also be computed e.g., by PN tools [13,14,15]. Namely, I₁ = p₂ + p₁₁, I₂ = p₁ + p₁₂, I₃ = p₃ + 2p₄ + p₁₃, I₄ = p₃ + p₉ + p₁₄, I₅ = p₄ + p₈ + p₁₅, I₆ = p₅ + p₇ + p₁₆, I₇ = p₁ + p₂ + p₃ + p₄ + p₅ + p₆ and I₈ = p₇ + p₈ + p₉ + p₁₀. P-invariants in the form of row vectors we can see in Table 2. Negative elements in the vectors η_i in S³PR paradigm (see [1]) mean that directed arcs are emerging from the GPN model and enter through corresponding transition t_i (index i depends on the position of the transition in a vector η_j) to the supervisor, while positive elements represent directed arcs in opposite direction, i.e., from the supervisor to GPN model through corresponding transitions. However in case of ES³PR as well as S⁴PR paradigms the supervisor synthesis is not so simple like in S³PR paradigm. The structural properties presented here will be applied in the continuation of this Example introduced in the Section 4.3.

If we applied these parameters (which are suitable for S³PR) to ES³PR, we would find that the responding supervisor does not comply for such paradigm of the AMS model, because it does not prevent deadlocks.

4.1. Controllability of Elementary Siphons

Here, for elementary siphon control especially the knowledge from the Section 3.1 will be utilized. Moreover, the following definition for settings the markings to monitors V_Si creating the supervisor will be applied.

Definition 19 [6]: Let S be a strict minimal siphon in an ES³PR model (N_μ0, M_μ0), where N = ◯_{i ∈ {1.,…, n}} N_i = (P₀ ∪ P_S ∪ P_R, T, F_μ0,W_μ0). Let {α, β, ..., γ} ⊆ {1,2, ..., n} such that ∀i ∈ {α, β, ..., γ}, |Th(S)| ∩ P_Si ≠ ∅ and ∀j ∈ {1, 2, ..., n} \ {α, β, ..., γ}, |Th(S)| ∩ P_Sj ≠ ∅. For S, a nonnegative P-vector k_S is constructed by 4 steps introduced above in the Definition 15.

Besides, almost every fact introduced below in the Section 4.2 touch both elementary siphons and dependent ones.

4.2. Controllability of Dependent Siphons

In addition to the control of elementary siphons, the control of dependent siphons is also very important.

Lemma 4 [8]: Let (N, M₀) be a marked net and S be a siphon of N. S is max-controlled if there exists a P-invariant I such that ∀p ∈ (||I||⁻ ∩ S), max_p• = 1, ||I||⁺ ⊆ S, and Σ_p∈P I(p)M₀(p) > Σ_p∈S I(p) (max_p• − 1).

When in an S⁴R net r is a resource place, S is a strict minimal siphon and H(r) = I_r\{r}. Let Th(S) = Σ_r∈ SR H(r)\S. We can see that |Th(S)| ⊆ P_S is true. Let us use Σ_p∈|Th(s)| h_S(p)p to denote Th(S). With respect to the Theorem 2 in the beginning of the Section 3, h_S(p) shows that siphon S loses just h_S(p) tokens if the number of tokens in p increases by one.

In [6] the following Theorem (such Theorem is also introduced in [7]) dealing with strongly dependent siphons is introduced:

Theorem 4 [6,7]: Let (N, M₀), N = (P, T, F, W), be a marked net and S be a strongly dependent siphon with η_S = Σ_i∈{1,n} a_iη_Si, where S₁, …, S_n are elementary siphons of S. S is max-controlled if

1. ∀i ∈ {1, 2, …, n}, I_i is a P-invariant of N, ||I_i||⁺ = S_i, and ∀p ∈ S_i, I_i(p) = 1;

2. M₀(S) > Σ_i∈{1,n} Σ_p∈||Ii||⁻ (a_i|I_i(p)|M₀(p)) + Σ_{p∈ S} (max_p• − 1).

In our Example 3.3 of ES³PR the siphon S₄ = {p₅, p₉, p₁₄, p₁₅, p₁₆} is, due to η_S4 = η_S1 + η_S2, a strongly dependent siphon and S₁ and S₂ are its elementary siphons. Using previous Theorem 4 we can verify the controllability of S₄. As to Figure 2, h₁ and h₂ are P-invariants satisfying the condition 1 of the Theorem 4. Now, we only have to check the condition 2 of the Theorem 4: Σ_i∈{1,2} Σ_{p∈||hi||−} (a_i|h_i(p)|M_µ1(p)) + Σ_p∈S4 (max_p•−1). After computing the supervisor consisting of monitors V_Si , i = 1, …, 3, we may denote the controlled net as (N_µ1 , M_µ1).

4.3. Illustrative Example of ES³PR—Control

Let us continue in the illustrative example introduced in the Section 3.3. While there rather the GPN model and its structural properties were presented, here the supervisor synthesis will be performed.

In our Example of ES³PR we have P₀ = {p₆, p₁₀} where p₀₁ = p₆, p₀₂ = p₁₀, P_S1 = {p₁, …, p₅}, P_S2 = {p₇, …, p₉}, P_R1 = {p₁₁, …, p₁₆}, P_R2 = {p₁₄, …, p₁₆}.

Ip₁₅ = p₄ + p₈ + p₁₅ and I_p16 = p₅ + p₇ + p₁₆ are minimal P-semiflows associated with resource places p₁₅, p₁₆, respectively. Hence, Th(S₁) = (H(p₁₅) + H(p₁₆)) \S₁ = ((p₄ + p₈) + (p₅ + p₇)) \ {p₅, p₈} = p₄ + p₇. I_p14 = p₃ + p₉ + p₁₄ and I_p15 = p₄ + p₈ + p₁₅ are minimal P-semiflows associated with resource places p₁₄, p₁₅, respectively. Hence, Th(S₂) Σ_{p ∈ P} h₂(p)M_µ1 (p) = M_µ1 (p₁₄) + M_µ1 (p₁₅) − M_µ1 (V_S2) = 2 + 1 − 2 = 1 > Σ_{p ∈ S2} h₂(p)(max_p• − 1) = 0. Thus, we can say that S₂ is max-controlled by P-invariant h₂.

P-invariants h_i, i = 1, 2, 3 indicate how the structure of interconnections of monitors V_Si (in Figure 2 right) with the net model representing the original uncontrolled plant will look like. Marking of particular monitors V_Si needs to be computed as it can be seen below: h₁ and h₂ are P-invariants satisfying the condition 1 of the above introduced Theorem 4.

To check the condition 2 of the Theorem 4, Σ_{i ∈ (1,n)} Σ_{p ∈ ||hi||−} (a_i|h_i(p)|M_µ1(p)) + Σ_{p ∈ S4} (max_p• −1) = M_µ1(p₁) + M_µ1(p₃) + M_µ1(p₇) + M_µ1(V_S1) + M_µ1(V_S2) + (1−1) = (0 + 0 + 0 + 2 + 2 + 0) = 4. While M_µ1(S₄) = M_µ1(p₁₄) + M_µ1(p₁₅) + M_µ1(p₁₆) = 5. Thus, S₄ is max-controlled. When we consider V_S1, V_S2 we see that based on them we have Σ_{i ∈ (1,2)}Σ_{p ∈ ||hi||−} (a_i|h_i(p)|M_µ1 (p)) + Σ_{p ∈ S4} (max_p• −1) = M_µ1(p₁) + M_µ1(p₃) + M_µ1(p₇) + M_µ1(V_S1) + M_µ1(V_S2) + (1−1) = (0 + 0 + 0 + 2 + 2 + 0) = 4. While M_µ1(S₄) = M_µ0(S₄) = M_µ1(p₁₄) + M_µ1 (p₁₅) + M_µ1 (p₁₆) = 5. Thus, S₄ is max-controlled. Because g_S = k_S + V_S is a P-invariant of the resultant net system (N_µ1, M_µ1), a monitor V_S is added to the original net model (N_µ0, M_µ0). Thus, N_µ1 = (P₀ ∪ P_S ∪ P_R ∪ V_S, T, F_µ1, W_µ1), ∀p ∈ P₀ ∪ P_S ∪ P_R, M_µ1(p) = M_µ0(p). Let h_S = Σ_r∈SR I_r − g_S and N_µ1(V_S) = M_µ0(S) − ξ_S (ξ_S ∈ ℕ). Then S is max-controlled if ξ_S > Σ_{p ∈ S} h_S(p) (max_p• − 1).

As we can see in Figure 2, (N_µ0, M_µ0) is an ES³PR being the model of plant net, (N_µ1, M_µ1) is its liveness-enforcing supervisor. Monitors are added to control elementary siphons S₁–S₃ only. They enforce liveness for (N_µ0, M_µ0),

The number N_ES of elementary siphons is bounded—see Theorem 3 in the Section 3.1 (it is smaller than minimum of the pair the place count and the transition count). Therefore, dependent siphons have to be implicitly controlled by fittingly setting the initial number of tokens in monitors. Thus, we can get a structurally simple liveness-enforcing PN supervisor for an ES³PR paradigm of the plant net model.

In Figure 1 we have P_S1 = {p₁, …, p₅}, P_S2 = {p₇, …, p₉}. For S₁ = {p₅, p₈, p₁₅, p₁₆} we have Th(S₁) = p₄ + p₇. With respect to the procedure described in the Section 3.1: For P_S1 we have p¹_s = p₄, p¹_ν = p₄, p¹_z = p₄. For P_S2 we have p²_s = p²_ν = p²_z = 7. Finally, k_S1 (p₁) = k_S1 (p₃) = k_S1 (p₄) = k_S1 (p₇) = 1. ∀p ∈ P_S \ {p₁, p₃, p₄, p₇}, k_S1 (p) = 0. Naturally, we obtain k_S1 (p₆) = k_S1(p₁₀) = k_S1 (p₁₁) = k_S1(p₁₂) = k_S1(p₁₃) = k_S1(p₁₄) = k_S1 (p₁₅) = k_S1 (p₁₆) = 0 since p₆ and p₁₀ are idle places and p₁₁, …, p₁₆ are resource places. Let K_S = {p | k_S (p) ≠ 0, p ∉ |Th(S)|}. In Figure 1 we have K_S1 = {p₁, p₃}.

In Figure 2 we have the following results:

1. g₁ = p₁ + p₃ + p₄ + p₇ + V_S1 and l₁ = I_p15 + I_p16 = p₄ + p₈ + p₁₅ + p₅ + p₇ + p₁₆ are P-semiflows. Let h₁ = l₁ − g₁. h₁ = p₅ + p₈ + p₁₅ + p₁₆ − p₁ − p₃ − V_S1 is a P-invariant. When we notice that ||h₁||⁻ ∩ S₁ = ∅, ||h₁||⁺ = S₁, and Σ_{p ∈ P} h₁(p)M_µ1 (p) = M_µ1 (p₁₅) + M_µ1 (p₁₆) − M_µ1 (V_S1) = 1 + 2 − 2 = 1 > Σ_{p ∈ S1} h₁(p)(max_p• − 1) = 0 we can say that S₁ is max-controlled by P-invariant h₁.

2. g₂ = p₁ + p₃ + p₇ + p₈ + V_S2 and l₂ = I_p14 + I_p15 = p₃ + p₉ + p₁₄ + p₄ + p₈ + p₁₅ are P-semiflows of the controlled net. Let h₂ = l₂ − g₂. h₂ = p₄ + p₉ + p₁₄ + p₁₅ − p₁ − p₇ − V_S2 is hence a P-invariant. When we notice that ||h₂||⁻ ∩ S₂ = ∅, ||h₂||⁺ = S₂, and Σ_{p ∈ P} h₂(p) M_µ1(p) == M_µ1 (p₁₄) + M_µ1 (p₁₅) − M_µ1 (V_S1) = 2 + 1 − 2 = 1 > Σ_{p ∈ S2} h₁(p)(max_p• − 1) = 0, we can say that S₂ is max-controlled by P-invariant h₂.

3. g₃ = p₁ + p₃ + V_S3 and l₃ = I_p13 = p₃ + p₄ + p₁₃ are P-semiflows. Let h₃ = l₃ − g₃. h₃ = p₄ + p₁₃ − p₁ − V_S3 is a P-invariant. When we notice that ||h₃||⁻ ∩ S₃ = ∅, ||h₃||⁺ = S₃ and Σ_{p ∈ P} h₃(p)M_µ1 (p) = M_µ1(p₁₆) − M_µ1 (V_S3) = 2 − 1 = 1 > Σ_{p ∈ S3} h₃(p)(max_p• − 1) = 0 which implies that S₃ is max-controlled by P-invariant h₃.

The supervised ES³PR is deadlock free. Left the original GPN model of plant is displayed, while right its supervisor is placed. Both pictures are shown separately to avoid complicated and non-transparent interconnections between both the plant and the supervisor. Namely, their cooperation is brighter when one observes insinuated transitions of the supervisor and corresponding real ones in the plant.

As we can see in Figure 2, 0 < ξ_S1 < 3, 0 < ξ_S2 < 3, 0 < ξ_S3 < 2, and ξ_S1 + ξ_S2 > 1. Let ξ_S1 = ξ_S2 = ξ_S3 = 1. Hence, we have M_µ1(V_S1) = 2, M_µ1 (V_S2) = 2, and M_µ1 (V_S3) = 1. It is easy to verify that all strict minimal siphons are max-controlled—see results 1–3 introduced above in this Section 3.5. Consequently, the net system in Figure 2 is live, i.e., deadlock free.

5. Conclusions

Liveness (deadlock-freeness) is a very important property of PN from behavioural point of view. Siphons, traps and invariants are structural objects of PN. Especially siphons are closely related to the PN liveness. There are many deadlock prevention control policies for AMS (alternatively named also flexible manufacturing systems—FMS) based on siphons. Namely, AMS are frequently modelled by PN possessing siphons. There are many paradigms of PN models of AMS like S³PR, ES³PR, S⁴R, GS³PR and many others.

To prevent deadlocks in AMS liveness-enforcing supervisors are used. Most of existing methods of their design are based on adding monitors (control places) for siphons. Monitors are based on exact controllability conditions for siphons. In such a way a liveness-enforcing supervisor with permissive behaviour is obtained. However, as we have found, conditions of max controllability of siphons, are overly restrictive and in general only sufficient.

The subject of this paper was to deal with the problem of deadlocks in ES³PR paradigm of GPN models of AMS. This paper may be seen as a free continuation of the author’s paper [1], where similar problem was resolved for the S³PR paradigm of ordinary PN (OPN) models of AMS. A difference between OPN and GPN consists in the weights of directed arcs. While in OPN weights of directed arcs are only 1, in GPN may be greater than 1, i.e., there may occur multiplex arcs. Therefore, the approach used in [1] to deal with deadlocks in S³PR is insufficient for dealing with deadlocks in ES³PR and it was necessary to find another one.

An empty siphon in an OPN can cause some transitions to be disabled forever. Therefore, it is necessary to prevent emptying siphons. Such a case in GPN is much more complicated. In consequence of arcs with greater weights, an insufficiently marked siphon can lead to the occurrence of other deadlocks. Therefore, setting the marking of monitors V_Si (which create the supervisor), is very important as well as the selection of the control depth variables ξ_Si of elementary siphons S_i.

ES³PN models of APN can be understood to be a subset of S⁴PR models, which are even much more complicated.

The approach presented here is based on verification of the max cs-property of a net (N, M₀). When such net N satisfies this property, we can say that it is deadlock-free. In case of GPN, a marked S⁴PR net is live if it satisfies the cs-property. This means that the cs-property is a sufficient but not necessary condition (unlike ES³PR) for the liveness of an S⁴R.

It was presented here, that the deadlock prevention in GPN models of AMS is much more complicated as that in OPN models of AMS. Simultaneously, the deadlock prevention of ES³PR (as a subclass of GPN models of AMS) was performed and illustrated on the example.

More complex analysis of the deadlock prevention problem depends on specific paradigms of GPN. Namely, there are many paradigms of GPN with specific properties and corresponding specific polices how to deal with deadlocks in them. For example, the specific paradigm GS³PR (generalised systems of simple sequential processes with resources) was analysed in [22] as concerns the problem of the deadlock prevention.

In future it will be necessary to concern in more and more complicated structures of AMS modelled by GPN.

A serious review of siphon-based approaches to control GPN are presented in [21]. An application on a specific paradigm GS³PR was presented in [22].

In this paper was shown that prevention of deadlocks in AMS with resources ensures a correct allocation of resources and thereby ensure the continuous and fluent operation of AMS. Thus, by means of controlling siphons of their PN-based models, the fluent and continuous operation of constituents in industrial production can be sustainable what is very important in social practice.