Efficient Deadlock Prevention in Petri Nets through the Generation of Selected Siphons

Transcription

1 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrB12.3 Efficient Deadlock Prevention in Petri Nets through the Generation of Selected Siphons Luigi Piroddi, Member, IEEE, Roberto Cordone, and Ivano Fumagalli Abstract Siphon-based control methods are often employed for deadlock prevention in Petri net models of flexible manufacturing systems. Such methods generally require siphon enumeration, which is a computationally intensive task, whose complexity grows with the Petri net size. However, only a small fraction of minimal siphons needs to be controlled to prevent all deadlocks. his paper introduces an algorithm to compute the required siphons, based on a set covering approach that optimally matches emptiable siphons to critical markings. his greatly reduces the computational load of the deadlock prevention algorithm with respect to alternative methods with comparable performance in terms of permissivity. I I. INRODUCION N Petri net (PN) models, particular subsets of places called siphons are associated to deadlocks, since a deadlock occurs if a siphon gets empty of tokens [11]. For this reason, it is of interest to investigate control methods that forbid the complete emptying of siphons by appropriately constraining the net s evolution. Some notable examples of such methods are given, e.g., in [1, 3, 5, 7, 9, 10, 12, 13, 15]. Besides deadlock prevention, other objectives are also important for the evaluation and comparison of control methods, namely control permissivity, size of the control subnet and computational load. ypically, computationally simple and small size solutions sacrifice permissivity, using heuristic and conservative conditions (see, e.g., [3, 5, 9, 13]). Such conditions are generally aimed to avoid the generation of new siphons by way of the added control constraints [7, 12, 15]. High permissivity is achieved by the Iterative Siphon Control (ISC) scheme, that at each step enumerates siphons (on progressively bigger PNs), and adds the relative control places, until all siphons are controlled. his may lead to an explosion of the computational load and of the size of the control subnet. he latter problem can be tackled by reducing the controller s redundancy [9, 10, 12, 14]. Actually, only a fraction of the siphons can get empty and not even all such siphons need to be controlled, due to dominance relations. More precisely, suitable dominance concepts for siphons Manuscript received Sept. 15, L. Piroddi is with the Dip. di Elettronica e Informazione, Politecnico di Milano, 20133, Milano, Italy (phone: ; fax: ; piroddi@elet.polimi.it). R. Cordone is with the Dip. di ecnologie dell Informazione, Università degli Studi di Milano, Polo Didattico e di Ricerca di Crema, 26013, Crema, Italy ( cordone@dti.unimi.it). I. Fumagalli is with Automata S.p.A., 21042, Caronno Pertusella (VA), Italy ( fumagalli.ivano@gmail.com). and markings, based on the siphons emptiability properties, were introduced in [12] to characterize the relevant ones for control purposes. Based on these, a set covering problem was formulated, to provide a systematic method for finding a minimum cardinality set of siphons to control at each iteration of the ISC scheme. he set covering problem maps emptiable siphons to critical markings, so that each constraint implies that at least one siphon is controlled among those that get empty in the associated marking. In the present work, the approach proposed in [12] is modified in order to avoid full siphon enumeration, which accounts for most of the computational load. he dominance relations between siphons and markings are exploited to generate only a small fraction of the whole set of minimal siphons. In practice, the necessary siphons and markings for the set covering problem are generated in an integrated process. his modification can dramatically improve the overall computational cost of the ISC scheme. II. PERI NES: BASIC DEFINIIONS AND NOAION A marked PN is a 5-tuple N = (P,, F, W, M 0 ), where P and are the sets of places and transitions, F is the flow relation, W is the arc weight function and M 0 the initial marking [11]. A PN is P-ordinary iff the arcs from places to transitions are unitary [7]. he net topology can also be specified through the input (I) and output (O), or incidence (C = O I) matrices. A transition t j is said to be enabled in a marking M if M I(,j), where A(,j) indicates the jth column of a generic matrix A. Firing a sequence of enabled transitions produces a new marking M * = M + Cσ, where the jth element of the firing vector σ represents the number of times that transition t j fires in the sequence. he set of reachable markings by way of enabled transition sequences is denoted R(N). An extended marking space R * (N) = {M 0 σ N s.t. M = M 0 + Cσ} [1, 5], where N is the set of nonnegative integers and X denotes the cardinality of set X, will also be employed in the following. A marking M R(N), such that t j enabled in M, is called a dead marking and represents a (total) deadlock state. A PN such that no marking in R(N) is dead is called deadlock-free. Given S P, the characteristic P-vector of S is a column binary vector λ S such that λ S (i) = 1 iff p i S and 0 otherwise [3, 9]. A P-invariant is a column vector x such that x C = 0. A P-invariant satisfies the following marking relation: x M = x M 0, M R(N). A siphon is a set S of places such that S S, where /09/$ AACC 5006

2 X = {y P x X s.t. (y,x) F} denotes the pre-set of a set of nodes X P, and X = {y P x X s.t. (x,y) F} its post-set of X. When a siphon gets empty of tokens, all the transitions in S are permanently disabled and a partial or total deadlock results. A siphon is minimal if it does not strictly contain other siphons. Siphon control-based deadlock prevention methods employ generalized mutual exclusion constraints (GMEC) of the type λ Sj M ξj to avoid the emptying of siphons, where ξ j 1 (ξ j = 1 for least restrictive methods). he P-invariant based control approach (PIBC) [15] implements GMECs, by adding for each scalar constraint a control place (or monitor), suitably marked and connected to the existing net s transitions. Let Π be the set of minimal siphons of a PN. A siphon that is marked at any reachable marking is termed a controlled siphon [6]. Let E Sj = {M R(N) λ Sj M = 0} be the set of markings where a siphon gets empty, and E Π* = Sj Π* E Sj for a generic subset of siphons Π *. hen, Π u = {S j Π E Sj } is the set of uncontrolled siphons, and the set of critical markings E Πu groups the markings where at least one siphon is empty [12]. A covering set of uncontrolled siphons (CSUS) is a subset of siphons Π c Π u, such that E Πc = E Πu [12]. Forbidding all the markings where any of the siphons of a CSUS Π c gets unmarked controls all siphons in Π. A siphon S j Π u is essential iff M E Sj such that λ Sk M > 0, Sk Π u \{S j } [12], i.e., there exists at least one reachable marking where only S j gets empty. A siphon S j Π u is dominated by a subset of siphons Π * Π u \{S j } iff E Sj E Π* [12]. A critical marking M i is dominated by M j iff λ Sk Mi = 0, S k Π u such that λ Sk Mj = 0 [12]. Whatever siphon is controlled to forbid marking M j, it also forbids M i. hus, dominated markings can be neglected for control purposes. Notice that for an essential siphon S j all markings in E Sj such that λ Sk M = 0 for some Sk Π u \{S j } are dominated. III. HE SE COVERING-BASED SIPHON CONROL MEHOD Selecting a minimum cardinality CSUS to forbid the critical markings of a PN (at a given iteration of the ISC scheme) clearly provides the smallest possible control subnet (since Π c min may be significantly smaller than Π), while achieving maximal permissivity. Also, avoiding redundancy indirectly affects the overall computational effort in the iterative scheme, since fewer constraints imply fewer new siphons, so that siphon enumeration is faster and easier and possibly the overall number of iterations is also reduced. A minimum cardinality CSUS can be found as the optimal solution of the following set covering problem: Definition 1 Set covering problem (SCP) [12] Let N = (P,, F, W, M 0 ) be a P-ordinary PN, Π the set of n S minimal siphons of N, and E Πu the set of n M critical markings. Let A = (a ij ) be a binary n M n S matrix, such that a ij = 1 if λ Sj Mi = 0, and 0 otherwise. Denote by SCP the following set covering problem: n S min xj j=1 n S aij x j 1, i = 1,, n M j=1 x j {0, 1}, j = 1,, n S All feasible solutions of the SCP are CSUS, and the optimal ones have minimum cardinality. Notice that, although set covering is known to be an NP-hard problem, available integer linear programming (ILP) solvers are efficient enough to exactly solve set covering problems of thousands of rows and columns in very small time. he SCP of Def. 1 requires the complete knowledge of the minimal siphons and the critical markings of the PN. However, the same result can be achieved using a reduced SCP that generates only a few selected and (preferably) non dominated markings by means of an ILP method [12]. In detail, for every (not yet explored) siphon S * the method repeatedly generates markings in which the siphon is empty, by means of a suitable ILP problem denoted Marking Generation Problem (MGP, see below). Each generated marking implicitly provides a row of the SCP, since at least one of the siphons that get empty in that marking must be controlled, but redundant rows corresponding to markings dominated by previously generated ones are avoided by means of suitable constraints. Also, if a marking is found for which the current siphon S * is essential, then all the previously generated rows associated to S * can be safely removed and the search carried on to the next siphon. Definition 2 Marking generation problem (MGP) [12] Let N be a P-ordinary PN with incidence matrix C and initial marking M 0. Let Π be the set of minimal siphons, and assume that Π old Π is the set of previously analyzed siphons and S * Π is the current one. Let also M k, k = 1,..., k * 1, be the markings previously generated for siphon S *, and Π k = {S Π\{S * } λ S Mk = 0}. Let Σ k = Πk S be the siphon defined as the union of the siphons that get empty in M k (except S * ). Denote by MGP the following ILP problem: min f = M(p) p P M = M 0 + Cσ, M 0, σ N λ S M 1, S Πold λ S * M = 0 λ Σk M 1, k = 1,..., k * 1 (1.a) (1.b) (1.c) (1.d) (1.e) In Def. 2, constraints (1.b) account for the (extended) reachability problem, constraints (1.c) rule out markings where any of the previously analyzed siphons get empty, constraint (1.d) imposes that the current siphon is empty, and constraints (1.e) exclude dominated markings. Notice, in fact, that λ Σk M 1 is a synthetic form to express the condition that λ S M 1 for at least one siphon in Πk. he objective function (1.a) maximizes the probability of finding 5007

3 a marking for which S * is essential, if it exists. Although the extended reachability space R * (N) is used instead of R(N) [1, 5], experimental practice has shown negligible effects due to spurious markings. he integrality constraint on σ can be initially relaxed, and reconsidered only if an admissible solution exists (and is non integer). IV. COMBINING SIPHON AND MARKING GENERAION O AVOID FULL SIPHON ENUMERAION he siphon and dominance relations can be further exploited to relax the computationally critical requirement that all minimal siphons are enumerated in advance. o this aim, siphon enumeration is combined with the marking generation, exploiting the constraints derived from the latter to avoid the computation of already controlled or dominated siphons. he basic siphon enumeration scheme [2] employs a combinatorial formulation and a node branching approach, where each node represents a specific siphon enumeration subproblem P = {N, P in, P out }, N being the current PN, and P in and P out the sets of places constrained to be in and out of the sought solutions, respectively. Initially, P in = P out =. Several structural properties are used to extend the place constraints at each node, thereby reducing the search space, and suitable conditions are applied to find a minimal siphon abiding by the place constraints, if any exists. When a minimal siphon S = {p S1,, p Sn } is found, the current node is branched into n subnodes, associated to other potential solutions not containing the found siphon. Precisely, the kth subnode inherits the P in and P out sets of the father node, and augments them with {p S1,, p S,k-1 } and {p Sk }, respectively. Child nodes with inconsistent constraints are discarded. wo slightly different siphon generation problems must be solved for combined usage with the MGP. Either we are looking for a new emptiable siphon without reference to a specific marking or we need to enumerate all the siphons that get empty in a specific marking M *. Both problems require a modification of the basic siphon enumeration scheme of [2]. A. Generation of a new emptiable minimal siphon he first problem is trivial if there are already generated but not yet examined emptiable siphons. However, this is not the case, e.g., at the beginning of the siphon enumeration process, when no siphons or markings have been processed yet, so that an ad hoc procedure must be set up to find an emptiable siphon. For this purpose, the siphon generation and the reachability analysis can be combined in a single problem that selects a pair (S *, M * ) such that λ S * M * = 0. A variant of the algorithm proposed in [1] for the selection of one emptiable siphon can be used for this purpose, adding constraints to reduce the siphon-marking space as new siphons or markings are explored. Also a different objective function is here used, designed to maximize the probability of finding an essential siphon. Definition 3 Emptiable siphon generation problem (ESGP) Let N = (P,, F, W, M 0 ) be a generalized PN with incidence matrix C. Assume that each place p P is k p - bounded. Let Π be the set of already found siphons. Let also s(p) = 1 iff p S * and w(t) = 1 iff t S *. Denote ESGP the following ILP problem: min f = s(p) M(p) (2.a) p P p P s(p) w(t), (t, p) F (2.b) w(t) s(p), t (2.c) p t M = M 0 + Cσ, M 0, σ N (2.d) M(p) k p (1 s(p)), p P (2.e) s(p) 1 (2.f) p P s(p) S 1, S Π (2.g) p S λ S M 1, S Π (2.h) Constraints (2.b-c) translate the siphon definition property, S S. Constraints (2.d) characterize the extended set of markings R * (N), as usual. Constraints (2.e) impose that the obtained siphon/marking couple (S *, M * ) actually satisfies the emptiability condition λ S * M * = 0. With respect to [1] the following new constraints have been added. Constraint (2.f) excludes the null siphon solution. Constraint (2.g) excludes all the already found siphons (collected in set Π) from the search, as well as any (non minimal) siphon containing them, in analogy with the efficient siphon enumeration algorithm of [2]. Constraint (2.h) excludes all the markings where any of the already found siphons is empty. he objective function (2.a) includes both siphon (with reversed sign with respect to the objective function used in [1]) and marking variables, with the intent of finding siphons with few places, that are more likely to be minimal, and big markings, since this maximizes the chances to recognize an essential or, at least, non dominated siphon (the more places are marked in the selected marking, the less likely that another siphon will get empty in the same marking). Since the ESGP is used occasionally alongside the basic siphon enumeration scheme, mutual consistency must be enforced, i.e. the siphon found by the ESGP must be removed from the solution space of the current siphon enumeration branching tree. Constraints (2.g) guarantee that the siphon S * obtained with the ESGP has not yet been found in the enumeration and that, therefore, it is necessarily a solution of one (and only one) open node of the siphon enumeration branching tree. o continue the enumeration process, this node must then be found and partitioned with respect to S *. B. Enumeration of all minimal siphons that are empty in a specific marking he second siphon generation problem can be solved as 5008

4 follows. In the siphon enumeration branching tree, add the temporary constraint λ S M * = 0, by extending the P out of every node as follows: P out = P out {p P M * (p) > 0}. Now, perform an ordinary siphon enumeration, by exploring all open nodes of the branching tree. For each node, a solution is sought with respect to place constraints P in and P out. If a siphon is found, the node is partitioned with respect to it augmenting the original place constraints P in and P out. Otherwise, the node is retained with the original place constraints P in and P out (the corresponding problem could still admit other emptiable siphons). In this way, only siphons that are empty in M * are found, but the siphon enumeration branching tree is still valid, in the sense that it provides the complete set of emptiable siphons. V. ELIMINAION OF REDUNDAN CONROL PLACES he adopted set covering approach guarantees the minimality of the set of siphons to control only with respect to the current PN. However, some residual redundancy may occur among constraints introduced at different iterations of the ISC scheme. A heuristic strategy has been adopted to address this problem, which, along the lines of [14], consists in checking for every control place that there does not exist a reachable marking that is prevented only by it. Definition 4 Redundancy check problem (RCP) Let N = (P,, F, W, M 0 ) be a generalized PN with incidence matrix C, and let p c P. Denote by RCP(N, p c ) the following ILP problem: min f = M(p c ) M = M 0 + Cσ M(p) 0, p p c σ N Property 1 Let N = (P,, F, W, M 0 ) be a generalized PN with incidence matrix C, and let p c be an additional control place, designed with the PIBC method, so as to realize the GMECs LM b. Let N = (P,, F, W, M 0 ) be the resulting controlled PN. hen, if the optimum of RCP(N, p c ) is nonnegative, the control place p c is redundant. Proof Omitted for brevity. A control place that is redundant according to Prop. 1 can be safely removed without enlarging the reachability space. Algorithm RC Input: N = (P,, F, W, M 0 ), a (sub)set of control places P c = {p c1,, p cn } P. Output: Reduced PN 0) Set i := 0. 1) Set i := i+1. Solve RCP(N, p ci ). If the optimal solution f of the RCP is such that f 0 then remove p ci (as well as any arc connected to p ci ) from N. If i = n then exit, else go to (1). By Prop. 1, the RC algorithm ensures that the reduced PN has exactly the same reachable states as the initial one. Notice that the obtained solution depends on the order of examination of the control place. In this regard, the forward order (i.e., the order in which the control places have been added by the ISC method) seems more appropriate, since we are interested in assessing if a previously obtained constraint is dominated by the set of constraints computed at the current iteration. Notice that the integrality constraint can be initially dropped, and must be reinforced only if f < 0 for the relaxed problem. VI. HE PROPOSED IERAIVE SIPHON CONROL SCHEME he combination of the general ISC approach with the SCP, the ESGP and RCP results in the scheme described next. Algorithm SC-ISC (Set Covering-based ISC) Input: N = (P,, F, W, M 0 ), N P-ordinary. Output: Controlled PN 0) Set i := 0 and Π (0) :=. Denote N (0) := N. 1.0) Set i := i+1. Set Π (i) =. Let A be an empty matrix. Set Π old =. Initialize the MGP as (1.a-b). 1.1) If Π (i) then extract one minimal emptiable siphon of N (i 1) from Π (i) else find a new one by solving the. ESGP for a siphon/marking couple (S *, M * ). If none exists then go to (1.5), else add constraint (1.d) to problem MGP and a constraint of type (2.g) and one of type (2.h) to the ESGP. 1.2) Solve problem MGP to find a reachable and (possibly) non dominated marking M where the current siphon gets empty, and none of the previously examined ones does (use the M * obtained from the ESGP if available). If no solution exists, set Π old = Π old {S * }, i.e. turn the current constraint (1.d) to one of type (1.c), and go to (1.1). 1.3) Find all other siphons of N (i 1) that get empty in M (check also the siphons already included in Π (i) and not yet extracted). Add the new ones to Π (i). 1.4) Add a binary row of length n S to A, with 1 in the positions associated to the siphons empty in M (extend the rows with 0s as the number of siphons grows). If S * is essential with respect to M, remove from A all previously added rows with a 1 in the column associated to S *, set Π old = Π old {S * }, i.e. turn the current constraint (1.d) to one of type (1.c), and go to (1.1), else add a constraint of type (1.e) to the MGP to exclude markings dominated by M from further search and go to (1.2). 1.5) If A = then exit, else formulate the SCP associated to matrix A and let Π c (i) = Πx, where x is an optimal solution of the SCP. 2) Use the PIBC method to implement on N (i 1) the set of GMECs {λ S M 1, S Πc (i) }. Apply RC to the place constraints derived at previous iterations. Denote the resulting net N (i). Go to (1). Property 2 Let N be a generalized PN. he SCP formulated at step (1) of algorithm SC-ISC on N admits the 5009

5 same set of optimal solutions of the full SCP of Def. 1. Proof he SC-ISC algorithm constructs a reduced SCP which employs fewer variables and constraints than the full one of Def. 1. However, the eliminated constraints are associated to dominated markings, and are therefore redundant. On the other hand, the eliminated variables correspond either to non-emptiable siphons or to siphons that get empty only in removed critical markings. Any solution containing a siphon of this type can be simplified by removing it, since the markings covered by that siphon are forbidden for free by controlling the other siphons of the solution, which by construction eliminate all the dominating markings. As a result, the reduced SCP has fewer solutions than the original problem, but excludes only non optimal ones. Property 3 Let N be a P-ordinary PN and N the controlled PN resulting from the application of the SC-ISC algorithm. If N is P-ordinary, then it is deadlock-free. Proof he algorithm is terminated if no more siphons are found (step 1). Since this implies that N does not have emptiable siphons, the thesis trivially follows. Notice that, except for pathological cases in which a siphon is non dominated only with respect to spurious markings in R * (N)\R(N), the control subnet is also nonredundant. If at the end of the SC-ISC the controlled PN is not P-ordinary, a simple test can be set up to ascertain its deadlock-freeness by searching the extended marking set R * (N) for dead markings. Definition 5 Deadlock generation problem (DGP) Let N = (P,, F, W, M 0 ) be a generalized PN with input matrix I and incidence matrix C. Let also δ ij = 1 iff p i enables t j and 0 otherwise. Denote by DGP(N) the following ILP problem: M = M 0 + Cσ, M(p) 0, σ N (3.a) pi t j δ ij t j 1, j = 1,..., (3.b) M(p i ) k ij δ ij + I(i,j) 1, i = 1,..., P, j = 1,..., (3.c) M(p i ) I(i,j)δ ij, i = 1,..., P, j = 1,..., (3.d) δ ij {0, 1}, i = 1,..., P, j = 1,..., (3.e) where k ij = max (M(p i) I(i,j)+1). M Constraints (3.c-e) formalize the transition enabling conditions. On the other hand, constraints (3.b) require that for each transition there exists at least one place that does not enable it. Finally, constraints (3.a) account for the (extended) reachability problem. Property 4 Let N be a generalized PN. hen if DGP(N) has no feasible solution, N is deadlock-free. Proof By hypothesis, there is no marking in R * (N) that does not enable at least one transition. Since R * (N) R(N), the thesis follows. he DGP can be first solved relaxing the integrality and binarity constraints: if no admissible solution exists in that case, none will exist in the non-relaxed setting either. In case the DGP admits feasible solutions, a net transformation approach can be applied to complete the control design, as explained in [8, 12]. Briefly, the generalized PN is transformed to a P-ordinary one, whose emptiable siphons are controlled with the PIBC method, and finally the resulting PN is back-transformed. VII. SIMULAION EXAMPLE AND PERFORMANCE ANALYSIS Consider the PN of Fig. 1 [3], which is a widely used benchmark for deadlock avoidance methods (see [12] and references listed there). he PN has reachable states, of which are safe, while 4226 lead to deadlock and should be forbidden. he set covering approach introduced in [12] computes a maximally permissive control that achieves a live PN with states, using 13 control places. Compared to the other solutions proposed in the literature, this one provides the best compromise in terms of permissivity and control size, as argued in [12]. p 2 t 11 t 14 t 2 t 12 t 3 t 8 t 18 p 21 p1 p p 3 p 8 p 12 p 17 7 p 5 t 13 t 4 t 9 t 17 p 4 p 23 p 24 p 7 p 9 t 5 t 1 t 6 t 7 t 20 p 6 p 15 p 10 Fig. 1. Benchmark PN example drawn from [3]. p 11 p 20 p 25 p 13 p 26 t 19 t 10 t 16 able I compares the set covering approach of [12] and the upgraded version introduced in this work. Each row reports the iteration number, the characteristics of the current net (P-ordinarity, number of places, nodes and arcs of the reachability graph, minimal siphons, emptiable siphons), the set covering features (SCP dimension, number of computed control places and redundant ones), and computational times (emptiable siphon enumeration, control algorithm, stopping criterion, and total control design time). Such times refer to a 3.2 GHz Pentium 4 with 1 GB RAM, where ILOG CPLEX software has been used to solve ILPs. he rows marked as Final report the main characteristics of the final solution, i.e., the net s type, dimension and number of reachable states, the total number of control places computed and discarded (the difference is the net number of control places added), and the overall control computational time at the end of the iterative method. Although the same solutions are found at every iteration by the two approaches, the proposed algorithm is very p 22 t 15 p 16 p 18 p

6 Method iter. P- # reach. graph ord. places # nodes # arcs efficient in cutting down the size of the SCP, and avoids the siphon enumeration task which grows rapidly with size and amounts to more than 90% of the computational time required by the approach of [12]. For this reason that approach turns out to be inapplicable to large nets, while the proposed approach scales more smoothly with net size. Notice that, at the 4 th iteration, since no more emptiable siphons are found (out of 1113 minimal ones), the stopping criterion based on the DGP is tested. Since no deadlock is possible in the controlled net, the algorithm stops. For reference purposes, the following list reports the connectivity and initial marking of the 13 control places found by the SC-ISC method (non unitary arc weights are indicated by the number preceding the associated transition): p c1 = {t 10, t 16 }, p c1 = {t 9, t 15 }, M(p c1 ) = 2, p c2 = {t 4, t 13 }, p c2 = {t 3, t 11 }, M(p c2 ) = 2, p c3 = {t 8, t 18 }, p c3 = {t 7, t 17 }, M(p c3 ) = 2, p c4 = {t 9, t 17 }, p c4 = {t 8, t 16 }, M(p c4 ) = 2, p c5 = {t 10, t 17 }, p c5 = {t 8, t 15 }, M(p c5 ) = 3. p c6 = {t 5, t 10, t 13, t 17 }, p c6 = {t 3, t 8, t 11, t 15 }, M(p c6 ) = 5, p c7 = {t 3, t 8, t 19 }, p c7 = {t 1, t 17 }, M(p c7 ) = 5, p c8 = {t 8, t 17 }, p c8 = {t 7, t 16 }, M(p c8 ) = 3, p c9 = {t 3, t 5, t 8, 2t 10, t 17, 2t 19 }, p c9 = {2t 1, t 9, 2t 15, t 18 }, M(p c9 ) = 17. p c10 = {t 5, t 8, t 13, t 17 }, p c10 = {t 3, t 7, t 11, t 15 }, M(p c10 ) = 6, p c11 = {t 3, 2t 5, t 9, 2t 10, t 13, t 17, 3t 19 }, p c11 = {3t 1, t 11, 3t 15, t 18 }, M(p c11 ) = 27, p c12 = {2t 5, 3t 10, t 13, t 17, t 18, t 19 }, p c12 = {2t 1, t 9, t 11, 3t 15 }, M(p c12 ) = 22, p c13 = {t 8, t 10, 2t 17 }, p c13 = {2t 7, 2t 15 }, M(p c13 ) = 8. VIII. CONCLUSIONS Exploiting the results of a previous research work, where a non-redundant highly permissive siphon control method was developed, a new version of the method is here suggested, that avoids the main computational bottleneck of the original approach, i.e., the full siphon enumeration. Instead, the new method employs a combined siphon and marking generation strategy, that thanks to structural properties related to siphon emptiability picks out only the siphons and markings strictly relevant for the control design. As in the previous method, a set covering problem is set up for the optimal selection of ABLE I SIMULAION EXAMPLE: MEHOD COMPARISON # siphons (emptiable) Set cov.: rows cols # c. places (redundant) Computational times [s] empt. s. control stopping control enum. alg. criterion total Set covering 1 Y (18) (0) approach [12] 2 Y (32) (0) N (89) (1) N (0) (0) Final N (1) Proposed 1 Y (18) (0) approach 2 Y (32) (0) N (89) (1) N (0) (0) Final N (1) the set of siphons to control. he proposed method has been tested on a benchmark PN, proving the practical applicability of the approach. REFERENCES [1] F. Chu and X.-L. Xie, Deadlock analysis of Petri nets using siphons and mathematical programming, IEEE rans. on Robotics and Automation, vol. 13, n. 6, pp , [2] R. Cordone, L. Ferrarini, and L. Piroddi, Enumeration algorithms for minimal siphons in Petri nets based on place constraints, IEEE rans. on Systems, Man and Cybernetics, Part A, vol. 35, n. 6, pp , [3] J. Ezpeleta, J.-M. Colom, and J. 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