Controller synthesis with very simplified linear constraints in PN model

Controller synthesis with very simplifie d linear constraints in PN model Dideban A. * Zareiee M. * Alla H. ** * Semnan Uni versity, IRAN ( Tel: (+98)231-3354123 ; e-mail: adideb an@ Semnan.ac.ir). * Semnan Uni versity, IRAN ( Tel: (+98)231-3354123 ; e-mail:mzareiee@ Semnan.ac.ir). ** GIPSA lab, 38402 St Martin d'Heres Cedex FRANCE , Hassane.alla@ inpg.fr). Abstract: This paper address es the problem of forbidden s tates for safe P etri net m odeling discrete e vent systems. We present an efficient met hod to construct a control ler. A set of linear const raints allow forbidding the reacha bility of specific states. The number of these so-called forbidden states and consequently the number of constraints are larg e and lead to a large num ber of control places. A systemati c method fo r constructi ng very sim plified cont roller is o ffered. By using a m ethod based on Petri nets partial invariants, maximal permissive controllers are determined. Keywords : Discrete Event Systems (DES), Petri Nets, Supervi sory contr ol, Controll er synthesi s, Forbidden stat es 1. INTRODUCTION A unifying framework for the control of di screte event systems i s provided by Ram adge and Wonham (1987, 1989), and so far their supervisory control t heory is the m ost general and comprehensive theory presente d for logical DES. In Ramadge & Wonham (1987, 1989), it is shown that we can model the syst ems by using of form al languages and finite automata . Methods exist for designing cont rollers based on automata system models, however t hese methods often involve exhaustive searches or sim ulations of the system behaviour, making t hem im practical for system s with large numbers of states and transi tions causing event (Moody, Antsaklis 2000). For solving t hese problems, we can model systems with Petri nets (PNs). Petri nets are a very appropriat e and useful tool for the study of discrete event systems b ecause of their modelling power and their mathematical propertie s. In the last decade, the research in the field of controller sy nthesis of discrete event systems has becom e one of the most act ive domains (Achour et al. 2004; Giua A & Xie X., 2005). In Giua et al., (1992) it is shown that it is possible to represent the forbidden states with linear constrain ts. With the idea presented by Yam alidou et al. (1996) we can design a controller by adding a contro l place to the PN mode. Each control place corresponds to a c onstraint. However, when the number of forbidden states is large, then the num ber of constraints has the same size. In Gi ua (1992b) it is shown that we can simplify some constraints. Bu t there is no systematic method for simpli f ying these constraints in general case. This simplifi cation is done by using the PNs st ructural properties. In Dideban and Alla (2005) a sy stemati c me thod for reducing the size and the number of constrai nts was introduced. It was applicable for safe and conservative PNs. In Dideban and Alla (2008), an effi cient m ethod for constructing a controlle r based on safe PN is presented. A set of linear constraints allow forbidding the reachability of specific states. The number of these so-called forbi dden states, and consequently the num ber of constraints are large and lead to a large number of control places. A system atic method for reducing the size and t he number of constraints for safe PN was introduced there. With applying the idea in Dideban & Alla (2008), it is possible that som etimes the fi nal number of constraint s may remian large. In this paper o ur objective is to develop the id ea in Dideban & Alla (2008) and to present a syst ematic m ethod for safe PN which allows to do more simplification than in the previous approaches. In this paper, we introduce the concept of Partial Place invariant and then, we present the sim plification method. We apply this method t o a concrete exampl e to highlights it s advantages. We use constraints which are equivalent t o forbidden states. These constrai nts can be calculated in two different ways. They can be given directly as specifications or they can be deduced thanks to the Kumar approach (Kumar & Holloway , 1996). By applying the idea in Dideban & All a (2008) and the new approach, it is possible to m ake a powerful simplificat ion. The advantage of the new idea is to reduce more the number of forbidden states. This paper is oraganized as follo ws: in section 2, some important definitions are introduce d. In section 3, the idea of passage from forbi dden states to linear const raints will be introduced. The idea in Dideb an & Alla (2008) and a practical example are presented in secti on 4. The basic original idea of simplification an d the calculation of control places are presented in section 5. Finally, the conclusion is given in t he last secti on. 2. PRELIMINARY PRESENTATION In this paper, it is supposed that the read er is familiar with the PNs basis ( David & Alla, 2005 ) and the theory of super visory control (Ramadge & W onham, 1987, 198 9). We briefly introduce the idea that was presented in (Dideban & Alla, 2008). For m ore detail, the reader can refer to this refere nce. . A PN is represente d by a quadru plet R ={ P , T , W , M 0 } where P is the set of places, T i s th e set of transition, W the incidence matrix and M 0 is the initial marking. Here we show the number of tokens in a place P i with m i . The PN is assumed to be safe; then the m arking of each place is Boolean. M R denotes the set of PN reachabl e markings. In M R , two subsets can be dist inguished; th e set of authorized states M A and the set of forbidden states M F . The set of forbidden stat es correspond to tw o groups: 1) The set of reachable states ( M F ′ ) which either don’t respect the specifications or are deadlock states. 2) The set of states for which t he occurrence of uncontrollable events leads t o states in M F ′ . The set of authorized states ar e the reachable states without the set of forbidden states: M A = M R \ M F Among the forbidden stat es, an important subset is constituted by the bo rder forbidden states d enoted as M B (Kumar & Holloway , 1996). Definition 1 . let M B be the set of border forbidden states: } , | { i j A j c F i B M M M and M → ∈ ∃ ∑ ∈ ∃ ∈ = σ σ M M M Where ∑ c is the set of controllable transitions.  In this pa per we use the concept of su pport instea d of marking as presented in definition 2. Definition 2 . the function Support( X ) of a vector N X } 1 , 0 { ∈ is: Support( X ) = The set of marked places i n X .  Definition 3 . let R b e a P N , P the set of its places , M ( M 0 ) the set of reachable markings from M 0 and k a constant. a place invariant is obtained if there is a sub-set of places P ′ ={ P 1 , P 2 ,…, P r } included in P and a weighing vector ( q 1 , q 2 ,…, q r ) for which all the weights q i are positive integers such that: ) ( , ... 0 2 2 1 1 M m k m q m q m q i r r M ∈ ∀ = + + + Then we say P ′ is a place invariant, (David & Alla 2005).  But if we remove some places from the set P ′ , what happens? It follows the definition of partial invariant: Definition 4 . let P ′ ={ P 1 , P 2 ,…, P r } be a pla ce invariant i n PN model R , P i 1 ={ P 1 , P 2 ,…, P L }for which{ 1,2,.. L } ⊂ {1,2,..r}, is a partial place invariant and it satisfy the following e quation: q 1 m 1 + q 2 m 2 +…+ q L m L ≤ k , ∀ m ∈ M ( M 0 )  3. FORBIDDEN STA TES TO LINEAR CONSTRAINTS Let [] ) ,..., , ( 2 1 N T i i m m m M M = Be a forbidden state in set M B and Support( M i )={ P i 1 P i 2 P i 3 … P in } the set of marked places of M i . From a forbidden state, a linear constraint can be constructed (Giua et al., 1992). The linear constraint deduced from the forbidden state M i is given below. The state M i does not verify this rel ation. Therefore, by applying thi s relation, M i will be forbidden. ∑ = − ≤ n k ik n m 1 1 Where n=Card [support ( M i )], is the number of marked places of M i , and m ik is the marking of place P ik of state M i . Let M ( M T =[ m 1 , m 2 ,…, m N ]) be a general marki ng and M i be a forbidden state. The constraint (forbidden st ate M i ) is denoted by c i and can be rewritten as follows 1 ] ) support( [ Card . i − ≤ M M M T i For example if: 2 3 )] ( rt Card[suppo ] 1 , 0 , 0 , 0 , 1 , 1 , 0 [ 7 3 2 ≤ + + ⇒ = ⇒ = m m m M M i T i Verifying t he relation in above is equivalent to forbid ding state M i when the PN model i s conservative. However, in a safe PN not necessari ly conserva tive, this equivalence is not always true. In follow we recall the con cept of over state that was introduced in Dideban & Alla (2008). Definition 5 . let M 2 = P 21 P 22 … P 2 m be an accessible state, M 1 = P 11 P 12 … P 1 n will be an over-state of M 2 if: 2 1 M M ≤ . That means all of the places i n M 1 are in M 2 .  For example M 1 = P 1 P 3 is an over-state of M 2 = P 1 P 3 P 6 P 9 . The name over-state is used because the constraint corresponding to an over-state holds constraint. For exam ple, the constraint m 4 + m 6 ≤ 1 that is correspond to the over-sta te M 1 = P 4 P 6 holds both following const raints: m 1 + m 4 + m 6 + m 9 ≤ 3 , m 2 + m 4 + m 6 + m 9 ≤ 3 These two constraints forbid states M 6 = P 1 P 4 P 6 P 9 and M 7 = P 2 P 4 P 6 P 9 . P 4 P 6 is an over-state of both states P 1 P 4 P 6 P 9 and P 2 P 4 P 6 P 9 which coul d be verified by M 1 ≤ M 6 and M 1 ≤ M 7 . Thus by using onl y the constraint m 4 + m 6 ≤ 1, both states M 6 and M 7 will be forbidden. Remark 1 . W ith each over-state b i , we associate a constraint c i : b i =( P i 1 P i 2 P i 3 … P in ) → c i =( P i 1 P i 2 P I 3 … P in , n -1). That means: m i 1 + m i 1 +…+ m in ≤ n -1.  The concept of over-state could be used for the reduction t h e number of constraints deduced from states or over-states (Dideban & Alla 2008). In the next secti on we introduce these ideas and the problem to solve. 4. RECUCTION THE NUMBER OF CON STRAINTS Now we explain the idea in Dideban & Alla (2008). Suppose that in a system we have th e set of authorized states M A and the set of border forbidden states M B . In the first step of reduction, we must construct the sets B 1 (All the over-states of border forbidden states) and A 1 (All the over-states of authorized states). Then we remove from B 1 , the states or over-states that exist in A 1 and call this set B 2 . In the next step we remove the states which their over-states are in B 2 from it and we call it B 3 . Sometimes it is possible, to simplify the reduced set B 3 , by application of the idea that is presented in Dideban (2007). But sometim es it is not the simplest set. The rules t o choose the final over-states are similar to the rules of the Quine- McCluskey m ethod to simplify the logical expressions (Morris Mano 2001). 4.1. Introduction of a practi cal example Consider a system wi th three machines and two robots. The closed loop PN model of t h is system is presented in fig. 1. The start command of each m achine is accomplished by controllable transitions c 1 , c 2 , c 3 and the stop comm and is accomplished by uncontrollable transitions f 1 , f 2 , f 3 . Two Robots are used to free three machines. All of machines can be turned on or off independently. The end of process for each machine when at least one of the robots are not free, is forbidden. From constructing the reachability graph, we can calculate the set of border forbidden states. Fig. 1. close d loop PN m odel Fig.2, gives part of reachability graph of the PN model presented in fig 1. The cons truction of the reachability graph is stopped when a forbidden stat e is reached. The state after these forbidden states is presented by . Fig. 2. A part of reachability graph of the PN model As it is obvious in fi g. 2, the existence of uncontrollable events leads to the forbidden states. For example when the system is in state ( P 1 P 4 P 5 P 8 P 10 ), it is possible to fire the uncontrollable event f 1 , while it is n ot authorized by the specifications. This state is ca lled as a forbidden st ate. And the set of border forbidden states can be determined from these forbidden states. In this example t he set of border forbidden states is: } , , , , , , , , , , , , { 10 8 6 3 1 10 8 5 4 1 10 8 5 3 2 10 7 6 4 1 9 8 6 4 1 10 7 6 3 2 9 8 6 3 2 10 7 5 4 2 9 8 5 4 2 10 8 6 4 2 10 7 6 4 2 9 8 6 4 2 9 7 6 4 2 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p B = M And The set of authorized states is: } , , , , , , , , , , , , , , , { 9 7 6 4 1 9 7 6 3 2 9 7 5 4 2 10 7 6 3 1 9 8 6 3 1 9 7 6 3 1 10 7 5 4 1 9 8 5 4 1 9 7 5 4 1 10 7 5 3 2 9 8 5 3 2 9 7 5 3 2 10 8 5 3 1 10 7 5 3 1 9 8 5 3 1 9 7 5 3 1 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p A = M Now, the sets of over-states of M A and M B are constructed. The set of over-states of M B is shown with B 1 and M A with A 1 . Hence the set of o ver-states of M B is: } , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , { 10 8 5 4 10 7 5 4 9 8 5 4 10 8 6 4 10 7 6 4 9 8 6 4 10 8 6 3 10 8 5 3 10 7 6 3 9 8 6 3 9 8 6 2 9 8 4 2 8 6 4 2 10 8 5 2 10 8 3 2 10 5 3 2 8 5 3 2 10 7 3 2 10 6 3 2 7 6 3 2 10 7 5 2 10 5 4 2 7 5 4 2 9 8 3 2 9 6 3 2 8 6 3 2 8 5 4 2 10 8 6 2 10 8 4 2 10 7 6 2 10 7 4 2 9 8 5 2 9 5 4 2 10 6 4 2 9 7 6 2 9 7 4 2 9 6 4 2 7 6 4 2 10 7 6 1 10 7 4 1 10 6 4 1 7 6 4 1 10 7 6 1 10 8 5 1 10 8 4 1 10 5 4 1 8 5 4 1 10 8 6 1 10 8 3 1 10 6 3 1 8 6 3 1 10 8 6 10 7 6 9 7 6 9 8 6 10 7 5 10 8 5 9 8 5 10 5 4 7 5 4 9 5 4 8 5 4 10 8 4 10 7 4 9 7 4 9 6 4 7 6 4 9 8 4 8 6 4 10 5 3 8 5 3 10 8 3 10 7 3 10 6 3 7 6 3 9 8 3 9 6 3 8 6 3 5 3 2 10 3 2 7 3 2 9 3 2 8 3 2 6 3 2 10 5 2 7 5 2 9 5 2 8 5 2 5 4 2 10 8 2 10 6 4 10 7 2 10 6 2 10 4 2 9 7 2 9 6 2 7 6 2 9 4 2 7 4 2 6 4 2 9 8 2 8 6 2 8 4 2 10 4 1 7 4 1 10 6 1 7 6 1 10 4 1 7 4 1 6 4 1 10 8 1 10 5 1 8 5 1 5 4 1 10 7 1 10 6 1 7 6 1 10 8 1 10 3 1 8 3 1 6 3 1 10 8 9 8 10 7 9 7 10 6 9 6 8 6 7 6 7 5 10 5 9 5 8 5 5 4 10 4 9 4 8 4 7 4 6 4 5 3 10 3 7 3 9 3 8 3 6 3 3 1 5 1 6 1 4 1 10 1 7 1 9 1 8 1 5 2 3 2 10 2 9 2 8 2 7 2 6 2 4 2 10 9 8 7 6 5 4 3 2 1 1 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p = B After this step we remove from B 1 , the states that are common i n B 1 and A 1 , and the residual set is named B 2 . Remark 2 . we don’t write the set of over-states of authorized states A 1 since with having M A we can remove the com mon states from B 1 .  } , , , , , , , , , { 10 8 4 10 8 6 10 8 2 10 6 4 10 6 2 10 4 2 8 6 4 8 6 2 8 4 2 6 4 2 2 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P = B By the method pre sented in Dideban & Alla (2008), the controller ca n be synthesized by final select ion of simplified over-states. 4.2. Controller synthesis Final selection is sim ilar to the McCluskey met hod for simplify ing logical term s. In this method we construct a t able (table1) where the first row represents the set of border forbidden states M B , and the first column is the set of simplified over-states B 2 . The relation between border forbidden states and the simpli f ied over-states can be expressed by the following defini tion: Definition 6 . let B 3 ={ b 1 , b 2 ,…, b m } be the set of simplified over-states and M B ={ M 1 , M 2 ,…, M N } be the set of border forbidden states. The relation } 1 , 0 { : 3 → × B M R B is as: ⎩ ⎨ ⎧ − ≤ = . 0 ) ( 1 ) , ( not if M of state over is b M b b M R i j i j j i The covering of a marking is an integer numb er: ∑ = ) , ( ) ( j i i b M R M Cv Cv ( M i ) ≥ 1 means that forbidden state M i is covered by at least one over-state.  Table 1: Relation between over-st ates and border forbidden states P 2 P 4 P 6 P 7 P 9 P 2 P 4 P 6 P 8 P 9 P 2 P 4 P 6 P 7 P 10 P 2 P 4 P 6 P 8 P 10 P 2 P 4 P 5 P 8 P 9 P 2 P 4 P 5 P 7 P 10 P 2 P 3 P 6 P 8 P 9 P 2 P 3 P 6 P 7 P 10 P 1 P 4 P 6 P 8 P 9 P 1 P 4 P 6 P 7 P 10 P 2 P 3 P 5 P 8 P 10 P 1 P 4 P 5 P 8 P 10 P 1 P 3 P 6 P 8 P 10 P 2 P 4 P 6 1 1 1 1 00000 0 0 0 0 P 2 P 4 P 8 0 1 0 1 10000 0 0 0 0 P 2 P 6 P 8 0 1 0 1 00100 0 0 0 0 P 4 P 6 P 8 0 1 0 1 00001 0 0 0 0 P 2 P 4 P 10 0 0 1 1 01000 0 0 0 0 P 2 P 6 P 10 0 0 1 1 00010 0 0 0 0 P 4 P 6 P 10 0 0 1 1 00100 1 0 0 0 P 2 P 8 P 10 0 0 0 1 00000 0 1 0 0 P 4 P 8 P 10 0 0 0 1 00000 0 0 1 0 P 6 P 8 P 10 0 0 0 1 00000 0 0 0 1 C v (M j ) 1 4 41 0 11211 1 1 1 1 The final constraints m ust cover all the border forbidden states. This means that with applying the constraints equivalent to the final sets, al l the border forbidden states are forbidden. These constraints are as follow: } , , , , , , , , { 10 8 6 10 8 4 10 8 2 8 6 4 10 6 2 10 6 4 10 4 2 8 4 2 6 4 2 3 P P P P P P P P P P P P P P P P P P P P P P P P P P P = B Before using the method i n Dideban & Alla (2008), the number of constraint s and equivalently t h e number of border forbidden states for our example was 13, but wi th using this method, the number of constraints is 9. It is clear that the number of simpli fied constraints is l arge, and for designing a controller by the m ethod presented by Yamal idou et al. (1996), the number of control places is also large. However, is it possible to simplify these constraints more? The answer to this question is given by a m ethod based on the concept of partial place invariant that wa s defined in section 2. 5. INTRODUCTI ON OF THE N EW IDEA FO R SIMPLIFICATION We consider the partial invariant for safe and conservative PNs. From the invariant relations we can construct partial invariants. For example: m 1 + m 2 +…+ m n = k ⇒ m 1 + m 2 +…+ m n -1 ≤ k In safe PN in general case, it is not p ossible to construct this equation directly. In the followi ng, we present some properties based on partial invariant s that in many cases allow us to simplify the set of constrain ts representing the forbidden states. Property 1. ; suppose that m i 1 , m i 2 are the number of tokens in the places P i 1 , P i 2 respectively. If P i 1 P i 2 is not in the set of over-states of authorized states we have: m i 1 + m i 2 ≤ 1  Proof: Suppose that the constraint is not t r ue. Then we have: m i 1 + m i 2 > 1 So, for safe PNs we have: m i 1 + m i 2 =2 → m i 1 = m i 2 =1 This means P i 1 P i 2 is in the set of over-states o f authorized states that it is not true. Then: m i 1 + m i 2 ≤ 1  Property 2 (Extension of Property 1) . suppose that the constraint m i 1 +…+ m in ≤ 1 is true. If all of the over- states { P i 1 P i (n+1) ,…, P in P i ( n +1) } are not in the set of ov er-states of authorized states, we have: m i 1 +…+ m in + m i ( n +1) ≤ 1  Proof: The proof of this property is si mila r to the proof of property1. Suppose that this rel ation is not true, so we can write: m i 1 +…+ m in + m i ( n +1) > 1 hence we ha ve: m i 1 +…+ m in + m i ( n +1) = 2 m i ( n +1) = 1 m i ( n +1) ≤ 1 (for safe PN) m i 1 +…+ m in = 1 m i 1 +…+ m in ≤ 1 m i 1 +…+ m in = 1 ⇒ ∃ m ik = 1( k ∈ [1, n ]) Then P ik P i ( n +1) is an over-state of authorized states that is not true, then: m i 1 + …+ m in + m i ( n +1) ≤ 1  Remark 3 . A constraint m 1 + m 2 +…+ m k ≤ k can be presented by ( P 1 P 2 … P k , k ).  Property3 . Let M = {( P 1 P k … P j , k ),...,( P r P k … P j , k )} be the subset of constraint s veri fied the authorized states. If the authorized states v erify the partial invariant m 1 + m 2 +... + m r ≤ 1 the r constraints are equivalent to one constrai nt as bellow: ( m 1 + m 2 +…+ m r ) + m k +…+ m j ≤ k  Proof: Necessary condition : ∀ q ∈ {1,…, r } m q + m i +…+ m j ≤ k ⇒ m i +…+ m j ≤ k And also from the constraint , we have: m 1 + m 2 +... + m r ≤ 1 By adding two constraint s, we have: ( m 1 + m 2 +…+ m r ) + m k +…+ m j ≤ k +1 We show th at the value k +1 is not any time accessible. Suppose that k +1 is accessible. We have: m 1 + m 2 +…+ m r =1 , m i +…+ m j = k From the r constraints and equation m i +…+ m j = k , we ha ve: m 1 = m 2 =…= m r =0 Then: ( m 1 + m 2 +…+ m r )=0 That is not t rue. Sufficient condition: ( m 1 +…+ m r ) + m i +…+ m j ≤ k ∀ q ∈ {1,…, r } m q ≥ 0 ⇒ ∀ q ∈ {1,…, r } m q + m i +…+ m j ≤ k  Generally, by applicat ion of these properties, it is possible to simplify the constraints. However, for our exam ple, it is not possible to do more sim plification using these properties. Suppose that the sets A ’ 1 and B ’ 1 be the sets of authorized and forbidden states, respectively: } , , , { 8 6 4 2 8 5 4 1 7 5 3 2 7 5 3 1 ' 1 P P P P P P P P P P P P P P P P A = } , , , { 7 5 4 7 5 2 5 4 2 5 4 1 ' 1 P P P P P P P P P P P P B = By applicati on of property 1 and 3 for st ates { P 1 P 4 P 5 , P 2 P 4 P 5 } we have: m 1 + m 2 + m 4 + m 5 ≤ 2. For states { P 2 P 5 P 7 , P 4 P 5 P 7 }, the constraint m 2 + m 4 ≤ 1 is not verifi ed, but is it necessary? We show that it is not necessary. When t he borne of constraints is m ore than 1, for having m 2 + m 4 + m 5 + m 7 ≤ 2, it is not necessary m 2 + m 4 ≤ 1, bu t also it is necessary that m 2 + m 4 + m 5 ≤ 2 and m 2 + m 4 + m 7 ≤ 2 be true. The following pro pertie generalizes this idea. Property 4 suppose that we have the set of constraints: C = {( P i 1 P 1 P 2 … P n , n ), ( P i2 P 1 P 2 … P n , n )}. Consider n over- states as follow: P i1 P i2 P 1 P 2 ... P j -1 P j +1 … P n for 2 ≤ j ≤ n -1 P i 1 P i 2 P 2 … P n , P i 1 P i 2 P 1 P 2 … P n -1 If none of these states are not in the set of over-states of authorized states, then we can reduce these constraints to one constraint as follow: P i 1 P i 2 P 1 P 2 … P n that means: m i 1 + m i 2 + m 1 + m 2 +…+ m n ≤ n  Proof : the proof of this property is clear. Suppose that t he inequality m i 1 + m i 2 + m 1 + m 2 +…+ m n ≤ n is not true then m i 1 + m i 2 + m 1 + m 2 +…+ m n = n +1,, that means ( n +1) places is marked in the set of P = ( P i 1 , P i 2 , P 1 , P 2 , …, P n ) then at least one of the condition in the property is vi olated. So our supposition is not true.  Property 5 . Suppose that we have the constraints C 1 = ( P i 1 P i 2 … P im P 1 P 2 … P n , n ) and C 2 = {( P i ( m+ 1) P 1 P 2 … P n , n ), that verify the authorized st ates. If all of the ov er-states content n +1 places from the sets P = ( P i ( m+ 1) , P i 1 P i 2 … P im P 1 P 2 … P n ,) are not exist in the sets of over-states of authorized states, the two constraints C 1 and C 2 can be replaced by one constraint as follow : P i 1 P i 2 … P im P i ( m+ 1) P 1 P 2 … P n , n  Proof: the proof of this property is sim ilar to the proof of property 4. Suppose that the inequality m i 1 + m i 2 +…+ m im + m i ( m+ 1) + m 1 + m 2 + … + m n ≤ n is not true then m i 1 + m i 2 +…+ m im + m i ( m+ 1) + m 1 + m 2 +…+ m n = n +1, that means ( n +1) places is marked in the set of P = ( P i 1 , P i 2 , …, P im ,P i ( m+ 1), P 1 , P 2 , …, P n ) then at least one of the condition in the property is violated. So our supposition is not true. 5.1. More simplification in our example by new idea From secti on 4.1 we ha ve: } , , , , , , , , { 10 8 6 10 8 4 10 8 2 8 6 4 10 6 2 10 6 4 10 4 2 8 4 2 6 4 2 3 P P P P P P P P P P P P P P P P P P P P P P P P P P P = B We can select the subset C 1 that contains two common places and one different place. C 1 = {( P 2 P 4 P 6 , 2), ( P 2 P 4 P 8 , 2). The over-state P 6 P 8 is in the set of over-states of au thorized states. So it is impossible to use the property 2, but the states P 2 P 6 P 8 , P 4 P 6 P 8 , are not in the set of over-states of authorized states, then it is possible to use the property 4. C 1 = {( P 2 P 4 P 6 , 2), ( P 2 P 4 P 8 , 2) ⇒ C ’ 1 = {( P 2 P 4 P 6 P 8 , 2) For applying the property 5 on the sets of C ’ 1 and C 2 = {( P 2 P 4 P 10 , 2)}, it is clear that the over-states P 2 P 8 P 10 , P 4 P 8 P 10 , P 2 P 6 P 10 , P 4 P 6 P 10 are not in the set of ove r-states of authorized states, then, we arriv e to the simplified constraint: C ’ 2 ={( P 2 P 4 P 6 P 8 P 10 , 2)} By the same way for the sets C 3 and C 4 we can ar rive to the simplified constraints as bellow: C 3 = {( P 2 P 6 P 10 , 2), ( P 4 P 6 P 10 , 2), ( P 6 P 8 P 10 , 2)} ⇒ C ’ 3 = ( P 2 P 4 P 6 P 8 P 10 , 2) C 4 = {( P 2 P 8 P 10 , 2), ( P 4 P 8 P 10 , 2), ( P 4 P 6 P 8 ,2)} ⇒ C ’ 4 = ( P 2 P 4 P 6 P 8 P 10 , 2) The simplified sets C ’ 2 and C ’ 3 and C ’ 4 are the same. Then the final set is: C ’ 4 = ( P 2 P 4 P 6 P 8 P 10 , 2) So it is obvious that with using t hese properties the 9 constraints have reduced to one constraint that is good reason for utilization of these p roperties. In this example, there is not the final selection, but generally by the method as the sam e was presented in last section, we can select the final result. 5.2. Calcul ation of control places To calculate the control pl aces corresponding for each linear constraint, we will use the me thod developed in Yam alidou et al. (2006). Now we briefly present t h is met hod that is based on concept of P-invariant. Suppose that we have the set of constraints as L . m p ≤ b that m p is the marki ng vector of system, L is a n × n matrix, b is a n c ×1 vector, n c is the number of constraints and n is the number of places. In this m ethod for each constraint, we add a place to the PN model of the system. In Yamali dou et al. (2006), it is proved that this places guarantees that these constraints are respected. Suppose W p be the incidence matrix of the system. For each constraint we add a row t o W p and we show this row as W c , corresponds to the incidence mat r ix of the controller. and W c is calculated as follows: W c = - L . W p W c is added to W p as follow: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = c p W W W If the initial marking of system is m p 0 , the initial marking fo r the added places is calculated as follow: m s 0 = b - L . m p 0 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 0 0 s p m m m Now we calculate the control places for our exam ple. The simplified constraint was ( P 2 P 4 P 6 P 8 P 10 , 2), Then we have: L =[0 1 0 1 0 1 0 1 0 1] ⇒ W c =[-1 0 0 -1 0 0 -1 0 0 1 1], M s 0 =2 The PN model of the final controller aft er adding the control place is indicated in fig 3. The control place and corresponding arcs are shown with gray color. Fig. 3. Co ntrolled P N model As it is obvious from fig. 3, with adding P c 1 , forbidden states can not be reached.. 6. CONCLUS ION In this paper, we have presented a syst ematic method to reduce the number of linear cons t r aints corresp onding to the forbidden st ates for a safe PN. This is real ized by using non- reachable states and by build ing the constraints using a systematic method. Th e important concept of over-state has been used; it corresp onds to a set of markings which keep the same prope rty (forbi dden or a uthorized). From the forbidden states, the set of over-states is calculated. The u tilization of non-reachable markings all ows great simplification of the constraints. In this paper, we have extended the idea presented in Dideban & Alla (2008). We have used the partial i nvariant for m ore simplifications. Finally, we have shown th at sometimes it is possible to sim p lify without invariant or partial invariant. By this met hod we can arrive to very si mplified constraints deduced from the forbidden stat es. REFERENCES Achour Z., & N.Rezg, X. Xie . (2004). 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