Optimal Supervisory Control Synthesis

Optimal Supervisory Control Synthesis Hassane ALLA Hassane.Alla@inpg.fr Laboratoi re GIPSA-Lab de Grenobl e, BP 46, 38402 St Martin d'Heres Cedex France Abstract : The place inva riant method is well known as an elegant way to construct a Petri net controller. It is possible to use the constraint for preventing forbidden states. But in general case, the number fo rbidden st ates can be very large gi ving a great number of cont rol places. In thi s paper is prese nted a system atic method to reduce the size and the num ber of constraints. This method is applicable for safe and conservative Pe tri nets givi ng a ma ximally perm issive contr oller. Keywords : controller, Place invariant, Petri ne t, forbidde n states, linear constrai nts Résumé : La méthode des invariants est bien connus comme u n moyen élégant pour synthétiser un contrôleur à partir d’un rése au de Petri. Il est possible d’utiliser les contraintes qui permettent d’empêcher d’all er ers les états interdi ts. Cependa nt dans le cas général, le nombre d’états interdit s peut être t rès gran d donnant ai nsi un nombre très grand de places de contrôle . Cette méthode est applicable pour le s réseaux de Petri conservatifs, et saufs don nant un contr ôleur maxim al permi ssif. Mots Clés : contrôleu r, invariant de marquage, réseau de Petri, états interdits, contrainte linéaire. 1. Introduction Petri Nets (PN) is an app ropriat e and useful tool fo r the study of Discree t-Event Systems (DES ) because of t heir model ing po wer and their mathematical properties. One of the advantages of PN m odels in DES control synt hesis is the use of a PN marking as a di stributed re presentation of t he system state. For c ontrol goals represented as st ate avoidance pro blems, this o ften allows a dec ompositi on of the control synt hesis problem into a series of sm aller problem s which are m ore efficiently solved. In the last decade, the research in t he field of cont roller synthesis of DES became one of the m ost active domains [1 ],[2],[3],[4]. In this pape r we present an ef ficient me thod for synt hesizing a cont roller to sol ve forbidden st ate problem s for a particular class of DES' s. We consider control specification which can be stated in the PN m odel. Synchr onous composit ion 2 between model of system and model of s pecification by exi stence the un controll able transitions, produce the forbidden states. The forbidde n state problem for a DES was introduced in a paper by Ramadge and Wonhom in wh ich they demonstrated sev eral fundamental p roperties of the solution [?]. The main limitatio n of such appro ach is the lack of structure in controlled automata and the large nu mber of states of the related state transition structures. Li and Wonham [??] have presented an algorithm which calculates the opti mal solution fo r the system s modeled by Petri nets wh ose uncontr ollable sub net s are loop-free. The controller has to solve on- line at each step linear intege r programs. The approach will be difficult to ap ply to the gra nd systems because of computati onal complexi ty. A method f or the comput ation of a ma ximally perm issive controller using t heory of re gion is presented i n Ghaffa ri and Xie []. The main i dea is to reduce the reachability gr aph to the set of adm issible markings, i.e., markings that respect the specifications. Control places are adde d to the original network, so that the closed loop plant respecting th e specifications. The theory of region is used on the reachability graph for the com put ation of the control places. For the computation of control places , a system of linear inequalities m ust be resolved. The solution of the syste m must be an integer positiv e solution. The advantage of th is method is that gives a m aximal ly permi ssive solution of the cont rol proble m, but t he computati on method is difficult an d does not guara ntee a soluti on. In t his method, for each forbidden state, we must com put e a control place. T hen there is many control places and it inc reases the complexity of system. There is m any of research for solving t he proble m of forbi dden st ates [],[],[] ,[]. Each of t hem prop osed the method for so lving this pro blem. Most relevant to our work are [5],[6 ]. In [5], auth or presented a method for construct a opti mal controller for the system modeled by Grafcet. The m ethod is based o n the equivalence bet ween the set of forbidden stat es and the set of linear constraints deduct ed from it. The use of the Yam alidou technique in [5],[6] allows const ructing a set of control places whic h represent the optimal controller. However t he number of cont rol places is equal to the number of constraints and can beco me very important. Intu itively, we realized that the PN solution is not unique from the point of view of the structure, bu t the reachable space (graph of the markings) is naturally unique. The problem comes from all the linear constraints wh ich can be sim plified by holding struct ural prope rties of the P N. In this paper is pr oposed a syst ematic m ethod of reduc tion of the num ber of constraints. Th e minimal solution is not uni que in the general case, but a method to choose the best solution is presen ted. This approach is based on the hypothesis that the PN models are safe and conser vative. This pa per is orga nized as bel ow: In the second section will be presented th e fundamental definitions then ho w we can construct the linear constraints from forbidd en states. Later, in the fourth section, the method to reduce th e number and the size of the constraints will be presented. At the end, th is method will be explain ed through an example. 3 2. Fundamental definitions In th is paper it is supposed th at reader knows the bases of the Petri n et paradigm and we present onl y the notions use d in this paper. For m ore details, see the b ook in (David and Al la, 2005 ). A PN is presented by a 4-uplet N = { P, T, Pre, P ost } where P is the set of places, T the set of transitions, Pre : P × T → N is the pre-incid ence function that defines weighted arcs from places to transitions . Pos t : T × P → N is the post-incidence function that defines wei ghted arcs from transitions to places. C = Post - Pre is the incidence matrix. The reachability graph is an automata = { M , Σ , δ , m 0 } where M is the states set, Σ is the even ts set, δ : M x Σ → M the evolution functi on, m 0 is the initial state. This graph corresponds to all th e possible evolutions of th e PN. The reachability graph consists of nodes which corr espond to the accessible mark ings, and of arcs t o the firing of transitions. In this paper, we use the word state instead of marking, represented by all the marked places. Note 1 : The corresponding state to the marking M i will be noted as following: M i = ( P i 1 … P i j … P in ) │ ∀ j ∈ {1, …, k } m ( P ij ) =1 & m ( P l ) = 0 for other places  For example the marking M 1 = [0, 1, 1, 0, 0, 1, 0, 0, 0] will be presented as state M 1 = P 2 P 3 P 6 . In the reachability graph, th ere are some of states: the authorized state, the forbidden st ate and the not reachable state. Among the fo rbidden states a particular and important subset is const ituted by the border forbi dden states, i t is denote d by the set M B . These states are such that all the input transitions are controllable. In this paper we use t he definit ion that prese nted in (Tom a 2005). Definition 1: The set of forbidde n states is not ed M F and is ex pressed: M F = { m ∈ M | ∃ w ∈ L R \ L Rd and m = δ ( m 0 , w )} i.e., a state is forbidden if it is a reac hable state and if it viol ates the specifications.  Definition 2: The set of dangerous states is: M D = { m ∈ M | ∃ w ∈ Σ u * and δ ( m , w ) ∈ M F } i.e., a dangerous state is a reachable m arki ng from which there is at least a sequence of uncontrollable transition s who leads to a forbidden state. We will consider, also, that a forbidden state is a dangerous state.  Definition 3: The set M A of admissible states is the gre atest set of reachable states so as: 4 • M A ∩ M D = ∅ • If m ∈ M A and δ ( m , e ) ∈ M F than e ∈ Σ c , i.e., the passage from an admissible state to a dangerous stat e is made by the firing of a controllable transition. Furthermore, M A is the most pe rmissive behavi or respecting t he specificat ions.  Definition 4 : The set of border states is: M B = { m ∈ M D | ∃ e ∈ Σ c ∧∃ m a ∈ M A , s.t. δ ( m a , e )= m } i.e., the set of dange rous states which ar e reached by the firing of a c ontrollable transition from an admissible state.  Definition 5: A controller is maximally permissive (MPC) if all the admissible markings of M A are reachable under supervision a nd all the firings of a transition, which cause th e evolution of the plant from an adm issible state to a non-admissible one, are in hibited.  In this paper, an important definition will be also used; it concerns the notio n of over-state. Definition 6 : Let be M i = P i 1 P i 2 … P in an admissible state. M j = P j 1 P j 2 … P jm is an over - state of M i if: 1) ∀ P jk ∈ M j ∃ P ik ∈ M i such as P jk = P ik = 1 2) ∀ P jl , l ∉ [1,…, m] M j ( P jl ) = 0 or 1  For example state M 1 = P 1 P 4 P 7 is an over – state of M 2 = P 1 P 4 P 7 P 10 . 3. From Forbidden States to Linear Constraints The modelling approach presented in this paper, is the classic approach developed by Rama dge and Wonham (1987a, b, 1988). The m odel of the pr ocess is synchronized wit h the specifications gi ving the closed l oop desired fu nctioning. T he existence of uncontrollable transitio n often leads to the existen ce of forbidden states including the bo rder forbidde n states. They are system atically determin ed by using the Kumar algorit hm (Kum ar and Holloway , 1996). We consi der here that t he PNs models of both process an d specifications ar e safe , but this does not mean that the marking of the control places is Boolean. L et f i be a border fo rbidden st ate from F and { P i1 P i2 P i3 … P in } be all the marked places whic h correspond to it. From the 5 forbidden st ates 1 , linear constr aints can be const ructed (Ka ttan, 2004; Tom a et al, 2003). The lin ear cons traint for the fo rbidden state f i is given by Equati on (1). ∑ = n k 1 m ik ≤ n – 1 (1) Where n is the num ber of marked place of f i , and m ik is the Boolean m arking of place P ik in state f i . For example: If the state ( P 2 P 5 P 7 ) is a forbidden state, it can be avoided by using Equat ion (2) m 2 + m 5 + m 7 ≤ 2 (2) In this paper the forbi dden state and i ts constrai nt is present ed as below: f i = { P i1 P i2 P i3 … P in } c i = { P i1 P i2 P i3 … P in , n -1} (3) The last term in c i corresponds to the number of ma rk ed places in the forbidden state minus 1, it is called the bound . This is true only at the beginning. 4. Reduction of the Size and Number of Constraints In real systems, the number of for bidden states can be ve ry large. Knowing that f or each forbidden state, one control place is ad ded, the complexity of the controller can become unmanagea ble. For this reason, in this pa per we pr opose a system atic method t o reduce the size and the number of the c onstraints. The PNs models are ass umed to be safe a nd conservative; then it is impossible fo r two places belonging to the s ame place invari ant, to be marked sim ultaneously. This basic idea will be used for the simplificatio n of the constraints. Property 1: Let be {( P 1 P i1 … P i(n-1) ),...,( P r P i1 … P i(n-1) )} r forbidden states in M F . m 1 + m 2 +…+ m r = 1 (4) The r constraints are equi valent to one constraint as below: m 1 + m i1 +…+ m i ( n -1) ≤ n -1 m 2 + m i1 +…+ m i ( n -1) ≤ n -1 … m r + m i1 +…+ m i( n -1) ≤ n -1 ⇔ m i1 +…+ m i( n -1) ≤ n -2 Where n is the num ber of ma rked place. 1 Where there is no ambiguity, the word border w ill be omitted . 6 Proof: necessary Condition : The sum of all constraints gives the constraint follow: ( m 1 + m i1 +…+ m i(n-1) ) + ( m 2 + m i1 +…+ m i(n-1) ) + … + ( m r + m i1 +…+ m i(n-1) ) ≤ r ( n -1) (5) (4), (5) ⇒ 1+ r ( m i1 +…+ m i(n-1) ) ≤ r ( n -1) ⇒ m i1 +…+ m i(n-1) ≤ n -1-1/ r ( m i : Integer number) ⇒ m i1 +…+ m i(n-1) ≤ n -2 Sufficient Condition : m i1 +…+ m i(n-1) ≤ n -2 ∀ k ∈ {1,…, r } m k ≤ 1 ⇒ m k + m i1 +…+ m i(n-1) ≤ n -1  Let be F = {( P 1 P 4 P 7 ), ( P 1 P 4 P 8 ), ( P 1 P 4 P 9 )} set of a forbidden states and m 7 + m 8 + m 9 = 1, thus the over-state ( P 1 P 4 ) is forbidden state, without attention to m arked place of the set { P 7 , P 8 , P 9 }, or it is possible to use instead of three constr aints only one constraint as below: m 1 + m 4 + m 7 ≤ 2 m 1 + m 4 + m 8 ≤ 2 ⎯ ⎯ ⎯ ⎯ ⎯→ ← = + + 1) m m (m 9 8 7 m 1 + m 4 ≤ 1 m 1 + m 4 + m 9 ≤ 2 NOTE : it is possible to use property 1 more than one time.  For example, if the set F is as below: F = {( P 1 P 4 P 7 P 10 ) , ( P 1 P 4 P 7 P 11 ) , ( P 1 P 4 P 8 P 10 ) , ( P 1 P 4 P 8 P 11 ), ( P 1 P 4 P 9 P 10 ), ( P 1 P 4 P 9 P 11 )} and m 10 + m 11 = 1 m 7 + m 8 + m 9 = 1 After the first use of Property 1 for the invariant between P 10 and P 11 the result is: {( P 1 P 4 P 7 ), ( P 1 P 4 P 8 ), ( P 1 P 4 P 9 )}. Now it is possi ble to use Property 1 for the othe r invariant. The final resu lt will be the ov er–state ( P 1 P 4 ). Let us co nsider anot her set of f orbidden s tates as below; F ’ = {( P 1 P 4 P 7 P 10 ), ( P 1 P 4 P 7 P 11 ), ( P 1 P 4 P 8 P 10 ), ( P 1 P 4 P 8 P 11 )} At the end of first simplification the result is: {( P 1 P 4 P 7 ), ( P 1 P 4 P 8 )} and it is not possible to use Property 1, bu t is it possible t o reduce the number of c onstraint ? The answer is yes. Because of the inva riant between P 7 , P 8 and P 9 , the states P 7 and P 8 7 cannot be marke d in the same time. Thus bot h constraint s can decrease in a constraint as below: m 1 + m 4 + m 7 + m 8 ≤ 2 (6) The property discussed above, is for malized in property 2: Property 2: Let b e C = {( P 1 P i 1 …P i(n- 1 ) , k ), ( P 2 P i 1 … P i(n- 1 ) , k ), ..., ( P r P i 1 … P i(n- 1 ) , k ) } the equi valent const raints to f orbidden stat es of marking graph and m 1 + m 2 +…+ m r ≤ 1 (7) The r constraints are equi valent to one constraint as below: m 1 +m i 1 +…+m i(n- 1 ) ≤ k m 2 + m i 1 +…+m i(n- 1 ) ≤ k … m r + m i 1 +…+m i(n- 1 ) ≤ k ⇔ m 1 + m 2 + …+ m r + m i 1 +…+m i(n- 1 ) ≤ k Where n is the number of m arked place. Proof: necessary Condition : ∀ j ∈ {1,…, r } m j + m i 1 +…+ m i(n- 1 ) ≤ k (8) and m 1 + m 2 +…+ m r ≤ 1 ⇒ ( m 1 + m 2 + …+ m r ) +m i1 +…+m i(n- 1 ) ≤ k +1 We are going to show that limit k +1 is never reached. If not, it is necessary that m 1 + m 2 +…+ m r = 1 and m i1 +…+m i(n-1) = k . But if m i1 +…+m i(n- 1) = k , thus because of the constraints (8) it is necessary that for ∀ j ∈ {1,…, r } m j = 0. Then m 1 + m 2 + …+ m r = 0. Thus limit k +1 never is rea ched. Sufficient Condition : ( m 1 + m 2 + …+ m r ) + m i 1 +…+m i ( n-1 ) ≤ k and ∀ i ∈ {1,…, r } m i ≥ 0 Then ∀ i ∈ {1,…, r } m i + m i 1 +…+m i ( n-1 ) ≤ k  NOTE : While exploiting t his property , constraints are used instea d of states since the bound m ay change acc ording to the simpli fications.  8 4.1 Unreachabl e states The objective of this app roach is re duction the size and the numbers of the constraints. The final result is simpler if Pr operty 1 is u sed. For this reason, our idea is to add the unreachable states in orde r to increase the s pace of forbidden states. Definition 7 : The unreacha ble states are the st ates that are not accessible from the initial state of the PN model or are reachable from border forbidden states.  If these st ates help to reduce constr aints, it is possi ble to use these states as forbidden st ates. But at t he end, it is not nece ssary to choose a constraint whic h does not covere d any forbidden state. This will be explained later in Section 5. 4.2 Minimum of constr aints In order to have the min imum of constraints, it is necessary to ch oose the simplest constraints cont aining all the forbi dden stat es. To reach this goal it is necessary to write all the final const raints wi th the forbidde n states. A met hod similar with t he one propose d by M ac Cluskey f or the reduct ion of t he logic al expression s is used for determining t he best choi ce(Morri s Mano 200 1). Firstly , we must cho ose the simplified constraints which contain the fo rbidden states that are not in ot her constraints. Secondly for the forbidde n stat es which are in two or several simplified constraints, i t is needed t o choose t he constrai nt containi ng the greatest number of forbidden states which were not c hosen, or t he const raint whic h is sim pler. In generally, there is not only one answer. This will be more explained in the following section. 5. EXAMPLE To illustrate the ideas develop ed in this paper, let us cons ider the example which was presented in (K attan 2004). A manufactu ring system is composed of two independent m achines M 1 a nd M 2, two transfer robots of the parts and on e test bench where the final pr oducts are test ed (Fig. 1). Fig. 1 . Production line 9 Each machine has the following operatin g c ycle: By occurrence of the event c i , the machine starts working. When t he work is finishe d (occurrence o f event f i ) the produced part is transferred by the rob ot on the test bench, an d a new cy cle can be started again by occu rrence of event t i . There are two types of ev ents in this system: the controllable ev ents and unco ntrollable even ts, only events c 1 and c 2 are controllabl e: ∑ c = { c 1 , c 2 } et , ∑ u = { f 1 , t 1 , f 2 , t 2 }. The model PN for this sy stem is give n in Fig. 2. Fig. 2. PN process model . Fig. 3. PN specification model The model of the specification of this system is presen ted in the Fig. 3. It is asked that firstly, the robot transfers the produ ct of machine M 1 and then the product of machine M 2 . 5.1 Desired fu nctioning in cl osed loop The determination of the forbidden states is made by the study of the controllability of the languag e of specification with regard to the languag e generated by the model of desired func tioning i n closed loop. In this ex ample only transitions c 1 and c 2 are controllable a nd the othe r transiti ons are not cont rollable. The closed loop m odel of this system is illustrated in Fig. 4 and will be denoted as R d . In this pa per thi s model of PN is called Quasi-PN de fined below. 10 Fig. 4. Petri net model of closed loop system Definition 8: A Quasi -PN is a PN which respects the following rules of firing: 1) A controllable transition is firable in the same way as in a classical PN. 2) An uncontrollable transition is firable if all its input places belonging to the process are marke d  The difference of reachable markings bet ween these two m odels (PN and Quasi— PN) indicates the forbidden states. Definition 9 : If in from a marking, an uncontrollable transition is firable for the Quasi—PN model, but not for the real PN model, then this state is forbidden.  To determine the forbidde n states by this method, it is necessary to construct the reachability graph for both models. 5.2 Reachability graph of the functioning closed loop The reachability graph is an id entification tool of the beha viour of the PN. A node in the reachability Graph re presents the re achable m arkings, and a rcs characterize the possible passages betw een these ma rkings . A marki ng corresp onds to a combination of m arked places. There are 2 N possible states for a safe PN with N places. But for a conservative PN, the number of states is sm aller. Thanks to the method pre sented in (Da vid and Alla, 2005) , the set of possible st ates can be calculated. The number of stat e is determined as below: N = ∏ = m i 1 ( n i ) (9) Where n i is the num ber of places in each marking invaria nt and m is the num ber of invariant. From the reachability graph, the theory devel oped by Ram adge and Wonham allows to obtain t he maxim al permissive su pervisor . If the la nguage ge nerated by the reachability graph is controllable, then th e problem is resolved and the PN m odel R d constitutes the controller. But if it is not co ntrollable, it is necessary to determine all the forbidden states. The reachability graph of th e real – PN and quasi – PN are presen ted in Fig. 5 and 6. The comparison of the two graphs , gives the forbidden states. 11 Fig. 5. Reachability graph of real – PN Fig. 6. Reachability graph of quasi – PN By using Definition 3 it is possible to determine the set of forbidden states as below: E = {( P 7 P 6 P 1 ), ( P 7 P 6 P 2 ), ( P 7 P 6 P 3 ), ( P 8 P 4 P 3 ), ( P 8 P 5 P 3 ), ( P 8 P 6 P 3 )} (10) From these states, wea kly forbidden states are deduced. By the technique presente d in(Kattan 2004), it is possible to find all the forbidden an d weakly forbidden states that are called dangerous states. To preven t from reaching the da ngerous states, it is sufficient to forbid th e border states. A ll the input transitions of these states are controllabl e. For the example, the border st ates are: F = {( P 7 P 5 P 1 ), ( P 7 P 5 P 2 ), ( P 7 P 5 P 3 ), ( P 8 P 4 P 2 ), ( P 8 P 5 P 2 ), ( P 8 P 6 P 2 )} (11) From these 6 forbi dden sta tes, 6 constraint s are deduced. We can now use the presented properties to reduce th e size and th e number of these constraints. For this, it is necessary to construct a set co ntaining all the possible states and to indicate the type of each on e. In this example the m arking invariants a re obvious . For this ex ample the num ber of possible states is calculated as below: N = ∏ m i ( n i )= 3*3*2= 18 (12) States S States S States S P 7 P 4 P 1 1 P 7 P 4 P 2 1 P 7 P 4 P 3 1 P 7 P 5 P 1 0 P 7 P 5 P 2 0 P 7 P 5 P 3 0 12 P 7 P 6 P 1 Ф P 7 P 6 P 2 Ф P 7 P 6 P 3 Ф P 8 P 4 P 1 1 P 8 P 4 P 2 0 P 8 P 4 P 3 Ф P 8 P 5 P 1 1 P 8 P 5 P 2 0 P 8 P 5 P 3 Ф P 8 P 6 P 1 1 P 8 P 6 P 2 0 P 8 P 6 P 3 Ф 1: admissible state 0: forbidden state Ф : not reachable state Fig. 7. All possible states 5.3 Use of Propertie s 1 and 2 In this example there are three invarian ts and Property 1 is used for all. I: m 1 + m 2 + m 3 =1: Finding this invariant in Fig. 7 is easy, because in this case, all of the states are in one row. At the end it i s obtained two over- states as belo w: ( P 5 P 7 ), it covers the states: { P 7 P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 } ( P 6 P 7 ), it covers the states: { P 7 P 6 P 1 , P 7 P 6 P 2 , P 7 P 6 P 3 } (13) I І . m 4 + m 5 + m 6 =1: By applying Property 1 for th is invariant, the result is as belo w: ( P 8 P 2 ), covering { P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 } ( P 8 P 3 ), covering { P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 } (14) II І . m 7 + m 8 =1 ( P 5 P 2 ), cove ring { P 7 P 5 P 2 , P 8 P 5 P 2 } ( P 6 P 2 ), covering { P 7 P 6 P 2 , P 8 P 6 P 2 } ( P 5 P 3 ) , coveri ng { P 7 P 5 P 3 , P 8 P 5 P 3 } ( P 6 P 3 ), covering { P 7 P 6 P 3 , P 8 P 6 P 3 } ( 15) The following table contains all the over stat es and indicates all the forbi dden states which are c overed by t hem. If may happen t hat a state is not covered, then it must be added in the final selection. It is not the case in this example (Fig. 8). Over- The states which are covered S 13 states P 5 P 7 P 7 P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 0 P 6 P 7 P 7 P 6 P 1 , P 7 P 6 P 2 , P 7 P 6 P 3 Ф P 8 P 2 P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 0 P 8 P 3 P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 Ф P 5 P 2 P 7 P 5 P 2 , P 8 P 5 P 2 0 P 6 P 2 P 7 P 6 P 2 , P 8 P 6 P 2 0 P 5 P 3 P 7 P 5 P 3 , P 8 P 5 P 3 0 P 6 P 3 P 7 P 6 P 3 , P 8 P 6 P 3 Ф . Fig. 8. All states after first simplification Here, Property 1 cannot be used longer, but in general case, it is possible to examine the table for new sim plifications. Pr operty 2 can now be used. From now, we use the const raints instead of st ates or ov er-states since t he bound may change according to the simplifications. There is two over-states or two constraint {( P 5 P 7 , 1), ( P 6 P 7 , 1)} which can be simplifi ed by using Pro perty 2. It i s possible to u se instead of t wo constraint s only one co nstraint as bel low: {( P 5 P 7 , 1), ( P 6 P 7 , 1)} ⇒ ( P 5 P 6 P 7 , 1) ( 16) The result after using Property 2 for all the constraints of Fig. 8 is illustrated in Fig. 9. 14 Constraint The states which are covered S P 5 P 6 P 7 , 1 P 7 P 5 P 1 , P 7 P 5 P 2 , P 7 P 5 P 3 , P 7 P 6 P 1 , P 7 P 6 P 2 , P 7 P 6 P 3 0 P 8 P 2 P 3 ,1 P 8 P 4 P 2 , P 8 P 5 P 2 , P 8 P 6 P 2 , P 8 P 4 P 3 , P 8 P 5 P 3 , P 8 P 6 P 3 0 P 5 P 6 P 2 P 3 ,1 P 7 P 5 P 2 , P 8 P 5 P 2 , P 7 P 6 P 2 , P 8 P 6 P 2 , P 7 P 5 P 3 , P 8 P 5 P 3 , P 7 P 6 P 3 , P 8 P 6 P 3 0 Fig. 9. All states after the last simplifica tion 5.4 Minimum of constr ai nt for this example At the end of this level, it is necessary to cho ose the minimum and simplest constraints, containing all th e forbidden stat es. In general case, it is possi ble to ha ve some choice. The difference between two choi ces is the number of arc of the control places. The best choice is dete rmined afte r t he calculation of the control places. P 7 P 5 P 1 P 7 P 5 P 2 P 7 P 5 P 3 P 8 P 4 P 2 P 8 P 5 P 2 P 8 P 6 P 2 choice P 5 P 6 P 7 , 1 √ √ √ √ P 8 P 2 P 3 ,1 √ √ √ √ P 5 P 6 P 2 P 3 ,1 √ √ √ √ √ √ √ √ √ √ Fig. 10. Minimum of constraints At the end, t he best solutio n contains t wo linear c onstraints(Fig. 10). m 5 + m 6 + m 7 ≤ 1 15 m 2 + m 3 + m 8 ≤ 1 (17) 5.5 Calcul ation of co ntrol pl aces To calculate control places for every linear const raint, there is a systematic method given in (Yam alidou et al 1996). The inci dence matri x of the PN R d , corresp onding to the model of functioning in clo sed loop is W p . The synthesis appr oach consists in adding to the i ncidence m atrix W p a matrix W c corresponding to the control places. The incidence matrix of contr olled system W will be constructed by adding a line to the matrix of incidence of co ntrolled pr ocess W p for every control place. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = c p W W W (18) It is then possible to calculate the matrix W c and its initial marking as bellow: W c = - L.W P (1 9) And M cinit = b - L . M pinit (20) The controller i s known by th e determinat ion of his matrix of inci dence W c and his initial marking M cinit . By applying this result to th e example, the result is: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 0 0 1 1 0 0 1 0 0 0 0 1 Wc Fig. 11. The con trol matrix W c The initial marking of the control place is calculated: m c 1 = 1; m c 2 = 0; The PN of the final controller is repr esented in Fig.12. Fig. 12. Petri net model of closed loop system with control places 16 6. Problem of state explosion Fig. 13. T he controlle d Automa ted Guided Vehicle 17 In the large system, the number of possible states is enormous and by this method we need to large memory and long time for calculatin g the simpler controller. By example for a system AVG by two in puts and two o utputs and 4 worksta tions and 4 critical zones, we ha ve 30,965, 760 states possible a nd com puting the cont roller is very complicated. But it is obvious that th e situation of output system don’t effect on the situation o f Wagon In zo ne 1. Therefore we can c ompute the co ntroller for the zone 1 without attention to mark in the model of outpu t system. The Petri Net model for controlled AVG is shown in Fig. 13. In this exam ple, supervis or is presente d by the const raints. By Yam alidou’s method, the supervisor is com puted. Bu t because of uncontrollable tr ansition this supervisor doesn’t work correctly. For re solve the problem of fi rst zone we ca n neglect the m odel of workst ation 1, 2, 3 and t he Wagons that move o ut of zone 1. This model is presented in Fig. 14. Fig. 14. The depende nce invariants to zone 1 Now, we can com pute the fo rbidden st ates and borde r forbidden st ates for this system and compute the controller by the method th at presented in this paper. Remark: The controller isn’t always simpler co ntroller, because s ome times we can construct one control place for two or more critical zone. 18 Remark: In supervisory idea the critical zone produced when we have an uncontrollable tran sition common between model of system and model of supervisor. We must find all of this type of t ransitions a nd dependence in variants . Finding the invariants that depend to each critical zone is possible intu itively for the simple system, but in the large syste m we need to an algorith m to computing the dependence invariant for eac h critical zone. The result for this example is shown in Fig. 15. Fig. 15. The controll er for zone 1 In this example we can apply the constraints for supervisory as the fo rbidden states in constructing the controller. Th is result is presented in Fig 16. 6. Conclusion In this paper we have presented a sy stematic met hod to reduce the num ber of forbidden st ates or the n umber of equivale nt linear constraints. The basic idea is to use the marki ngs invari ants of the P N to si m plify these constrain ts. This is realized by using the not reacha ble states. We obtain a structural solution easy to implement. The solution obtained by our approach gives the o ptimal controller. This op timality comes from the equivalence between all the adm issible marki ngs and all the linear constrai nts. The me thod that 19 is presented in this p aper is applicable for the systems modelled by Grafcet if the place of control is safe. 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