Decentralized Control of Petri Nets

Transcription

1 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Decenralized onrol of Peri Nes Technical Reor of he ISIS Grou a he Universiy of Nore Dame ISIS Ocober, 00 Marian V. Iordache and Panos J. Ansaklis Dearmen of Elecrical Engineering Universiy of Nore Dame Nore Dame, IN 6556, USA iordache.@nd.edu, ansaklis.@nd.edu Inerdiscilinary Sudies of Inelligen Sysems

2 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Decenralized onrol of Peri Nes Marian V. Iordache and Panos J. Ansaklis Absrac Suervision based on lace invarians (SBPI) is an efficien echnique for he suervisory conrol of Peri nes. In his aer we roose exensions of he SBPI o a decenralized conrol seing. In our seing, a decenralized suervisor consiss of local suervisors, each conrolling and observing a ar of he Peri ne. We consider boh versions of decenralized conrol, wih communicaion, and wih no communicaion. In he case of communicaion, a local suervisor may receive observaions of evens ha are no locally observable and send enabling decisions concerning evens ha are no locally conrollable. In he firs ar of he aer we roose efficien algorihms for he design of decenralized suervisors, based on he exension of he SBPI conce of admissibiliy ha we define. Then, in he second ar of he aer, we roose he design of decenralized suervisors based on ransformaions o admissible consrains. The feasibiliy of his roblem is demonsraed wih a simle ineger rogramming aroach. This aroach can incororae communicaion beween local suervisors as well as communicaion consrains. Inroducion The decenralized conrol of discree even sysems (DES) has received considerable aenion in he recen years []. The curren research effor has been focused on he auomaa seing, and has considered boh versions of decenralized conrol, wih communicaion and wih no communicaion. This aer considers he decenralized conrol of Peri nes by means of he suervision based on lace invarians (SBPI) [, 0, 9]. Peri nes are comac models of concurren sysems, as hey do no reresen exlicily he sae sace of he sysem. Peri ne models arise naurally in a variey of alicaions, such as manufacuring sysems and communicaion neworks. Peri ne mehods relying on he srucure of he ne raher han he sae sace are of secial ineres, as he size of he sae sace, when finie, can be exonenially relaed o he size of he ne. Among such mehods, he SBPI offers an efficien echnique for he design of suervisors enforcing on Peri nes a aricular class of sae redicaes, called generalized muual exclusion consrains. Noe ha he generalized muual exclusion consrains can reresen any sae redicae of a safe Peri ne [9]. Furhermore, Dearmen of Elecrical Engineering, Universiy of Nore Dame, IN 6556, USA. iordache., ansaklis.@nd.edu. A Peri ne is safe if for all reachable markings no lace has more han one oken.

3 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. wihou loss of any of is benefis, he SBPI has been exended in [5] o handle any consrains ha can be enforced by conrol (monior) laces. While SBPI has been considered so far in a cenralized seing, his aer rooses exensions of SBPI o a decenralized seing. Admissibiliy is a key conce in he SBPI of Peri nes wih unconrollable and unobservable ransiions. When dealing wih such Peri nes, he SBPI aroach classifies he secificaions as admissible and inadmissible, where he former can be direcly enforced, and he laer are firs ransformed o an admissible form and hen enforced. In he auomaa seing [], admissibiliy corresonds o conrollabiliy and observabiliy, and he ransformaion o an admissible form o he comuaion of a conrollable and observable sublanguage. The main conribuions of his aer are as follows. Firs, we define d-admissibiliy (decenralized admissibiliy), as an exension of admissibiliy o he decenralized seing. Our conce of d-admissibiliy exends he admissibiliy conce in he sense ha a se of consrains ha is d-admissible can be direcly enforced via SBPI (i.e., wihou comuaional overhead) in a decenralized seing. Since d-admissibiliy idenifies consrains for which he suervisors can be easily comued, raher han he class of consrains for which suervisors can be comued, i does no arallel conrollabiliy and coobservabiliy in he auomaa seing []. Second, we show how o enforce d-admissible consrains and show how o check wheher a consrain is d-admissible. Third, o deal wih consrains ha are no d-admissible, we rovide an algorihmic aroach o make he consrains d-admissible by enabling communicaion of evens (ransiion firings). Fourh, o deal wih he case in which he consrains are no d-admissible and communicaion is resriced or unavailable, we roose a simle linear ineger rogramming aroach for he design of he decenralized conrol. The design rocess generaes boh he local suervisors and he communicaion olicy. ommunicaion enables he local suervisors o observe evens ha are no locally observable and o conrol evens ha are no locally conrollable. The communicaion olicy secifies for each local suervisor he evens i remoely observes and he evens i remoely conrols. This aroach allows communicaion consrains o be incororaed in he design rocess and can be used o minimize he communicaion. Wih regard o our use of ineger rogramming, noe ha while he develomen of alernaive mehods ha are less comuaionally inensive are a direcion for fuure research, in he auomaa seing i was shown ha a decenralized soluion canno be found wih olynomial comlexiy []. Noe also ha he size of he ineger rogram deends on he size of he Peri ne srucure, and no on he size of is sae sace (i.e. he size of is equivalen auomaon), which may no be finie. To our knowledge, he decenralized suervisory conrol of Peri nes has no been ye considered in he lieraure. In he auomaa seing, he relaed work is as follows. Decenralized conrol when he secificaion is already given in a decomosed form is sudied in [9]. The aer rooses a coordinaor for he enforcemen of addiional secificaions no given in he decomosed form. The exisence of decenralized suervisors exacly imlemening a given language has been firs sudied in []. In [] coobservabiliy was defined, and i was shown ha a decenralized suervisor

4 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. exacly enforcing a language exiss iff he language is conrollable and coobservable. In [7], he roblem of finding a decenralized soluion wih he same erformance as a cenralized soluion is considered in a seing in which communicaion is allowed. The communicaion consiss of local suervisors sending o oher local suervisors observaion srings. The decenralized conrol roblem wih communicaion is sudied in []. In his roblem boh he communicaion olicy and he suervisors are designed. The communicaion seing consiss of suervisors broadcasing heir sae esimaes. Oher decenralized conrol work can be found in he survey [] and he references herein. Lieraure on SBPI or closely relaed o i is found in [, 9, 0, 7, 8,, 5] and he references herein. omared o he relaed work, he idea of informaion srucures in [7] is relaed o he clusering of subsysems in our aer. However, unlike [7, ], our communicaion seing involves sending observed evens raher han observaion srings or sae esimaes. This kind of communicaion has also been considered in []. Furhermore, noe ha in his aer our focus is on comuaionally efficien or racable mehods for he decenralized conrol of Peri nes. Therefore, while he oimaliy of he resul and he generaliy of he soluion are also maers of ineres, hey are no he rimary goals of our aroach. This differeniaes our work from he fundamenal resuls of [, ], concerning oimal soluions in he general DES framework. The vas majoriy of he decenralized conrol aers consider language secificaions. In his aer mos develomens are focused on a aricular class of sae redicae secificaions on Peri nes. In he auomaa seing, he exisence of a decenralized soluion enforcing sae redicaes is sudied in [6]. The relaion beween sae redicae secificaions and language secificaions is as follows: any language can be reresened as a sae redicae on a sysem consising of he lan and a memory DES [7]. The aer is organized as follows. Secion describes he noaion and oulines he SBPI. Secion describes he decenralized seing of our aroach. Secion defines he d-admissibiliy, shows how d-admissible consrains can be enforced, and resens he algorihm checking wheher a consrain is d-admissible. Then, d-admissibiliy is alied o he design of local suervisors wih communicaion in secion 5. The algorihm resened in secion 5 uses communicaion in order o reduce (when ossible) he enforcemen of consrains ha are no d-admissible o he enforcemen of d-admissible consrains. Secion 6 describes he suervisory aroach for he enforcemen of consrains ha are no d-admissible in he case in which he communicaion is resriced or no available. Secion 7 shows how he resuls obained in he revious secions exend o he generalized ye of consrains described in [5] and o he auomaa seing. Finally, secion 8 illusraes our aroach on a manufacuring examle from [9]. Preliminaries A Peri ne srucure is denoed by N =(P, T, F, W), where P is he se of laces, T he se of ransiions, F he se of ransiion arcs, and W he weigh funcion. The incidence marix of N is

5 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. denoed by D (laces corresond o rows and ransiions o columns). A lace (ransiion) denoed by j ( i ) is he lace (ransiion) corresonding o he j h (i h) row (column) of he incidence marix. The secificaion of he SBPI [, 0, 9] consiss of he sae consrains Lµ b () where L Z nc P, b Z nc,andµ is he marking of N. To disinguish beween he case n c = and n c >, we say ha () reresens a consrain when n c =, and ha () reresens aseof consrains when n c >. Noe ha N reresens he lan. The SBPI rovides a suervisor in he form of a Peri ne N s =(P s,t,f s,w s )wih D s = LD () µ 0,s = b Lµ 0 () where D s is he incidence marix of he suervisor, µ 0,s he iniial marking of he suervisor, and µ 0 is he iniial marking of N. The laces of he suervisor are called conrol laces. Thesuervised sysem, ha is he closed-loo sysem, is a Peri ne of incidence marix: [ ] D D c = () LD An examle is shown in Figure (b), in which he suervisor enforcing µ + µ andµ + µ consiss of he conrol laces and. Noe ha () imlies ha when he lan and he suervisor are in closed-loo, he iniial marking of he lan saisfies (). Le µ c be he marking of he closed-loo, and le µ c N denoe µ c resriced o he lan N.Le T be a ransiion. is closed-loo enabled if µ c enables. is lan-enabled, ifµ c N enables in N.Thesuervisordeecs if is closed-loo enabled a some reachable marking µ c and firing changes he marking of some conrol lace. The suervisor conrols if here is a reachable marking µ c such ha is lan-enabled bu no closed-loo enabled. Given µ c,hesuervisordisables if here is a conrol lace such ha (, ) F s and µ c () <W s (, ). In Peri nes wih unconrollable and unobservable ransiions, admissibiliy issues arise. Indeed, a suervisor generaed as shown above may include conrol laces revening lan-enabled unconrollable ransiions o fire, and may conain conrol laces wih marking varied by firings of closed-loo enabled unobservable ransiions. Such a suervisor is clearly no imlemenable. We say ha a suervisor is admissible, if i only conrols conrollable ransiions, and i only deecs observable ransiions. The consrains Lµ b are admissible if he suervisor defined by ( ) is admissible. When inadmissible, he consrains Lµ b are ransformed (if ossible) o an admissible form L a µ b a such ha L a µ b a Lµ b [0]. Then, he suervisor enforcing L a µ b a is admissible, and enforces Lµ b as well. Our discussion on admissibiliy is carried ou in more

6 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. conrollable and observable conrollable and unobservable unconrollable and observable unconrollable and unobservable Figure : Grahical reresenaion of he ransiion yes. Assembly area Pars bin Assembly area omuer omuer Nework onnecion Figure : Roboic manufacuring sysem. deail in secion. We will denoe N wih ses of unconrollable and unobservable ransiions T uc and T uo by (N,T uc,t uo ). Finally, Figure shows he grahical reresenaion of he unconrollable and/or unobservable ransiions ha is used in his aer. The Model We assume ha he sysem is given as a Peri ne srucure N =(P, T, F, W). A decenralized suervisor consiss of a se of local suervisors S, S,...S n, each acing uon individual ars of he sysem, called subsysems, where he simulaneous oeraion of he local suervisors achieves a global secificaion. A local suervisor S i observes he sysem hrough he se of locally observable ransiions T o,i, and conrols i hrough he se of locally conrollable ransiions T c,i. So, from he viewoin of S i, he ses of unconrollable and unobservable ransiions are T uc,i = T \ T c,i and T uo,i = T \ T o,i. This is he design roblem: Given a global secificaion and he ses of unconrollable and unobservable ransiions T uc,, T uc,,... T uc,n and T uo,, T uo,,... T uo,n, find a se of local suervisors S, S,...S n whose simulaneous oeraion guaranees ha he global secificaion is saisfied, where each S i can conrol T \ T uc,i and observe T \ T uo,i. A sysem N wih subsysems of unconrollable and unobservable ransiions T uc,i and T uo,i will be denoed by (N,T uc,,...t uc,n,t uo,,...t uo,n ). 5

7 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. (a) Global sysem (b) Lef subsysem (c) Righ subsysem Figure : A Peri ne model of he roboic manufacuring sysem (a) (b) (c) (d) Figure : Examles of c-admissible suervision. As an examle, we consider he manufacuring sysem of [6], shown in Figure. In his examle, wo robos access a common ars bin. The sysem can be modeled by he Peri ne of Figure (a), where µ =(µ = ) when he lef (righ) robo is in he assembly area, and µ =(µ =) when he lef (righ) robo is in he ars bin. The se of conrollable ransiions of he lef (righ) subsysem may be aken as T c, = {, } (T c, = {, }). Assume ha he subsysem of each robo knows when he oher robo eners or leaves he ars bin. Then each subsysem conains he conrollable ransiions of he oher subsysem as observable ransiions; a ossible grahical reresenaion of he subsysems is shown in Figure (b) and (c). Admissibiliy To disinguish beween admissibiliy in he cenralized case and admissibiliy in he decenralized case (o be defined laer), we denoe by c-admissibiliy he admissibiliy roery in he cenralized case. Therefore, c-admissibiliy is aken wih resec o a Peri ne (N,µ 0 ) of unconrollable ransiions T uc and unobservable ransiions T uo. The significance of c-admissibiliy is as follows. A c-admissible se of consrains () can be imlemened wih he simle consrucion of ( ), as in he fully conrollable and observable case. 6

8 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. I is essenial for he undersanding of his aer o see ha suervisors defined by ( ) may be admissible even when hey have conrol laces conneced o unobservable ransiions, and conrol laces conneced o unconrollable ransiions by lace-o-ransiion arcs. We firs illusrae his fac by wo examles, and hen, in he nex aragrah, we show how such admissible suervisors can be (hysically) imlemened. In he firs examle, he suervisor enforcing µ + µ and µ + µ in he Peri ne of Figure (a) is shown in Figure (b). By definiion, he suervisor is admissible, in sie of he fac ha i is conneced o he unconrollable and unobservable ransiions 5 and 6. The reason is ha, on one side, whenever he suervisor disables 5 (or 6 ), 5 ( 6 )is anyway disabled by he lan and, on he oher side, 5 and 6 are dead (hey require µ + µ and µ +µ, resecively, in order o be lan-enabled) and so heir observaion is no necessary. In he second examle, he suervisor enforcing µ + µ + µ andµ in he Peri ne of Figure (c) is shown in Figure (d). Again, he suervisor is admissible, in sie of he fac ha i may disable he unconrollable ransiion 5. Indeed, he suervisor never disables 5 when 5 is lan-enabled, and so is disablemen decision does no need o be hysically imlemened. In fac, he arc (, 5 ) can be seen as corresonding o an observaion acion only, as he suervisor decremens he marking of whenever 5 fires. The revious examles moivae he following inerreaion of he arcs beween he conrol laces of an admissible suervisor and he unconrollable and/or unobservable ransiions. Le be a conrol lace and a ransiion. If is unconrollable, an arc (, ) models observaion only, due o he fac ha an admissible suervisor never disables a lan-enabled ransiion; hysically, his means ha he suervisor has a sensor o monior bu no acuaor o conrol. If is unobservable and conrollable, an arc (, ) models conrol only, as he fac ha an admissible suervisor does no observe closed-loo enabled unobservable ransiions indicaes ha is dead in he closed-loo ; hysically, he suervisor has an acuaor o conrol bu no sensor o monior. If is unobservable and unconrollable, arcs beween and can be ignored, as he fac ha an admissible suervisor would never disable or observe if lan-enabled, imlies ha in he closedloo is never lan-enabled. A summary of he inerreaion of he arcs beween conrol laces and ransiions is found in Aendix A. In he decenralized case, we are ineresed o define admissibiliy wih resec o a Peri ne (N,µ 0 ), and he ses of unconrollable and unobservable ransiions of he subsysems: T uc,... T uc,n and T uo,... T uo,n. Admissibiliy in he decenralized case is called d-admissibiliy. As in he case of c-admissibiliy, we would like d-admissibiliy o guaranee ha we are able o consruc he (decenralized) suervisor wihou emloying consrain ransformaions. This is achieved by he following definiion. Definiion. A consrain is d-admissible wih resec o (N,µ 0,T uc,...t uc,n,t uo,...t uo,n ), if here is a collecion of subsysems {,,...n},, such ha he consrain is c-admissible Self-loos do no arise as long as we limi ourselves o he consrains of he ye (). 7

9 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Lef subsysem Righ subsysem enralized conrol Decenralized conrol Figure 5: enralized conrol versus decenralized conrol. wih resec o (N,µ 0,T uc,t uo ),wheret uc = T uc,i and T uo = T uo,i. A se of consrains is d-admissible if each of is consrains is d-admissible. To illusrae he definiion, assume ha we have a consrain ha is c-admissible only wih resec o he firs subsysem. Then, i is d-admissible, as we can selec =. An ineresing consequence is ha when each subsysem has full observabiliy of he ne and every ransiion is conrollable wih resec o some subsysem, any consrain is d-admissible. This consequence is formally saed nex. Proosiion. Any se of consrains is d-admissible if T uo,i T uc,i =. i=...n = for all i =...n and The consrucion of a decenralized suervisor, given a d-admissible se of consrains, is illusraed on he Peri ne of Figure. The muual exclusion consrain µ + µ (5) is o be enforced. The cenralized conrol soluion is shown in Figure 5. In he case of decenralized suervision, here are wo subsysems: he firs one has T uo, = and T uc, = {, },andhe oher has T uo, = and T uc, = {, }. Noe ha (5) is no c-admissible wih resec o any of (N,T uc,,t uo, )or(n,t uc,,t uo, ). However, i is d-admissible for = {, }. Given wo variables x,x N, a decenralized suervisor S S enforcing (5) can be defined by he following rules: The suervisor S : The suervisor S : iniialize x o 0. disable if x =0 incremen x if or fires. decremen x if or fires. iniialize x o 0. disable if x =0 incremen x if or fires. decremen x if or fires. 8

10 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. A grahical reresenaion of S and S is ossible, as shown in Figure 5. Thus, S is reresened by and S by ; x is he marking of and x he marking of. Grahically, and are coies of he conrol lace of he cenralized suervisor. However, as discussed earlier in he secion, (, )and(, ) reresen observaion arcs. This corresonds o he fac ha S never disables and S never disables. As and havehesameiniialmarkingas, heir markings say equal a all imes. So, whenever should be disabled, he disablemen acion is imlemened by, and whenever is o be disabled, he disablemen acion is imlemened by. In he general case, he consrucion of a suervisor enforcing a d-admissible consrain lµ c (l N P and c N) is as follows: (Noe ha he noaion of Definiion. is used) Algorihm. Suervisor Design for a D-admissible onsrain. Le µ 0 he iniial marking of N, he conrol lace of he cenralized SBPI suervisor N s =(P s,t,f s,w s ) enforcing lµ c, and he se of Definiion... For all i,lex i N be a sae variable of S i.. Define S i, for all i, by he following rules: Iniialize x i o c lµ 0. If T c,i, and x i <W s (, ), hen S i disables. If fires, T o,i and, henx i = x i + W s (, ). If fires, T o,i and, henx i = x i W s (, ). To enforce a d-admissible se of consrains Lµ b, he consrucion above is reeaed for each consrain lµ c. Noe ha in he grahical reresenaion of he suervisors S i corresonds o coies of he conrol lace of he cenralized suervisor, where each coy has he same iniial marking as. Nex we rove ha he resuling decenralized suervisor is feasible (hysically imlemenable) and as erforman as he cenralized suervisor. Firs, we formalize he feasibiliy conce. The decenralized suervisor S i is said o be feasible if for all reachable markings µ c of he closedloo and for all ransiions : (i) for all i =...n,if is closed-loo enabled and / T o,i,firing does no change he sae (marking) of S i ; (ii) if is lan-enabled bu no closed-loo enabled, here is an S i disabling such ha T c,i. Theorem. The decenralized suervisor consruced in Algorihm. is feasible, enforces he desired consrain, and is as ermissive as he cenralized suervisor of (N,T uc,t uo ). Proof: Feasibiliy is an immediae consequence of he consrucion of Algorihm.. To rove he remaining ar of he heorem, we show ha a firing sequence σ is enabled by he cenralized 9

11 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. suervisor a he iniial marking iff enabled by he decenralized suervisor a he iniial marking. The roof uses he noaion of he Algorihm. and of Definiion.. In addiion, le S be he cenralized suervisor imlemened by he conrol lace, ands d he decenralized suervisor S i. Given a firing sequence σ = i i... ik enabled from µ 0 in he oen-loo (N,µ 0 ), we denoe i by µ j he markings reached while firing σ: µ 0 µ Firs, noe ha for all firing sequences σ = i i... ik have ha a all markings µ j reached while firing σ i µ i...µk. enabled by boh S and S d from µ 0,we x i = c lµ j i (6) This is roven by inducion. For i = 0, (6) is saisfied, due o he way he variables x i are iniialized. Assume (6) saisfied for j<k. According o he SBPI, when he lan has he marking µ j he marking of is c lµ j,hesameasx i i. In view of Definiion., he d-admissibiliy of lµ c imlies ha S is c-admissible wih resec o (N,µ 0,T uc,t uo ). Then, since ij is no dead, ij / T uc. Then, ij / T uc ( i ) ij / T uo,i. Hence ij is observable o all S i,andsoallx i are changed in he same way. Moreover, according o he SBPI, firing ij changes he marking of hesamewayasx i are changed. From he SBPI we know ha he new marking of is c lµ j+. I follows ha when µ j+ is reached, x i = c lµ j+ i. Finally, we rove by conradicion ha he firing sequences enabled by S from µ 0 are he firing sequences enabled by S d from µ 0. Assume he conrary, ha here is σ ha is enabled by one suervisor and no enabled by he oher. We decomose σ ino σ = σ x x σ y, x T,whereσ x is σ enabled by boh suervisors and σ x x is no. If µ x 0 µ x, hen (6) is saisfied a µ j = µ x ;he marking of is also c lµ x. There are wo cases: (a) x enabled by ; (b) x no enabled by. As in he revious ar of he roof, case (a) leads o he conclusion ha S d enables also x,which conradics he assumion ha no boh S and S d enable x. In case (b), according o he SBPI, we have ha W s (, x ) <c lµ x and x / T uc, by he d-admissibiliy of lµ c. I follows ha here is i such ha S i disables x, and hence ha S d does no enable x. This conradics he fac ha one of S and S d enables x. Nex we urn our aenion o checking wheher a consrain is d-admissible. Le S be he cenralized suervisor ha enforces he consrain in he fully conrollable and observable version of N. Le Tuo M be he se of ransiions ha are no deeced by S and Tuc M he se of ransiions ha are no conrolled by S. Algorihm.5 hecking wheher a onsrain is D-admissible. Find T M uo and T M uc.. Find he larges se of subsysems such ha i : T uo,i T M uo.. If =, declare ha he consrain is no d-admissible and exi. 0

12 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00.. Define T uc = T uc,i. 5. Does T uc saisfy T uc T M uc? If yes, declare he consrain d-admissible. Oherwise, declare ha he consrain is no d-admissible. In he algorihm above, as long as a consrain is d-admissible, he consrain can be imlemened for a minimal se min conaining he minimal number of subsysems such ha Tuc M T uc,i. min Noe ha checking wheher a se of consrains is d-admissible involves checking each consrain individually. The reason he algorihm checks single consrains is ha checking ses of consrains as a whole may cause d-admissible ses of consrains o be declared no d-admissible. To see his, noe ha he overall se Tuo M of a d-admissible se of consrains may be emy, which would cause he algorihm o declare he consrains no d-admissible (see se ). Indeed, Tuo M = is ossible in sie of d-admissibiliy since Tuo M = Tuo i,m,wheretuo i,m is he Tuo M arameer for he i h consrain i of he se of consrains. Proosiion.6 The algorihm checking d-admissibiliy is correc. Proof: I is sufficien o rove ha he algorihm declares a consrain d-admissible only if i is d-admissible, and ha all d-admissible consrains are declared d-admissible. Le T uo = T uo,i. By consrucion, T uo Tuo M. A consrain is declared d-admissible if and T uc Tuc M. The definiion of Tuo M and Tuc M imlies ha he consrain is c-admissible wih resec o (N,T uc,t uo ). Then, in view of Definiion., he algorihm is righ o declare he consrain d-admissible. Nex, assume a d-admissible consrain. Then, here is a se of subsysems such ha he consrain is c-admissible wih resec o (N,T uc,t uo) (wheret uc = T uc,i and T uo = T uc,i ). Then T uo Tuo M ; T uo Tuo M T uc T uc T uc Tuc M. onsequenly, he algorihm declares he consrain o be d-admissible. In general, i may be difficul o comue he ses Tuc M and Tuo M. Then esimaes Tuc e Tuc M and Tuo e Tuo M can be used in he algorihm insead. In his case he algorihm only checks a sufficien condiion for d-admissibiliy, and so i can no longer deec consrains ha are no d- admissible. In he case of he SBPI, such esimaes can be found from he srucural admissibiliy es of [0], saing ha Lµ b is c-admissible if LD uc 0andLD uo =0,whereD uc and D uo are he resricions of D o he columns of T uc and T uo. Noe ha when i is ossible and convenien o communicae in a reliable fashion wih each subsysem of a decenralized sysem, a cenralized soluion wih T uc = T uc,i and T uo = i=...n T uo,i is ossible. Finally, noe ha in he imlemenaion of d-admissible consrains, each i=...n suervisor S i wih i relies on he roer oeraion of he oher suervisors S j wih j

13 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00.. By iself, a local suervisor may no be able o imlemen a d-admissible consrain or is imlemenaion may be overresricive. For insance, in he examle of Figure, he suervisor of he firs subsysem can only enforce µ +µ by iself by enforcing µ = 0. However, his soluion is overresricive. D-admissibiliy illusraes he fac ha more can be achieved when suervisors cooerae o achieve a given ask, raher han when a suervisor ries on is own o achieve i (cf. wo heads beer han one in []). 5 Suervision wih ommunicaion Obviously, communicaion can be used o change he aribues of oherwise inaccessible ransiions o observable or even conrollable. We begin wih an illusraion. 5. Illusraion As an illusraion, consider again he roboic sysem of Figure. We assume ha he comuers conrolling he wo robos are able o communicae hrough a nework connecion. The secificaion is ha he robos should no access a he same ime he ars bin. By requiring each of he comuers o signal any ransiion firing in he subsysem i conrols, he ses of observable ransiions become T o, = T o, = {,,, }. Then he decenralized suervisory soluion of Figure 5 can be alied. The realizaion of a rogram imlemening a local suervisor is illusraed on he lef subsysem. The marking of may be imlemened by a variable c. Each ime he righ subsysem signals ha fires, c is incremened, and each ime he righ subsysem announces ha fires, c is decremened. Furhermore, is he only ransiion conrolled by he lef subsysem; is allowed o fire only when c. When fires, he righ subsysem is announced and c is decremened. When fires, he righ subsysem is announced and c is incremened. 5. Decenralized Suervisors wih ommunicaion The urose of communicaion is o reduce he se of unobservable ransiions T uo,i such ha, if ossible, he given consrains are c-admissible wih resec o (N,T uc,t uo ). Noe ha communicaion canno reduce T uo below he aainable lower bound T uo,l T uo,wheret uo,l = T uo,i. T uc can i=...n be changed by selecing a differen se. However, i canno be reduced below T uc,l = T uc,i. i=...n Indeed, T uc,l (T uo,l ) is he se of ransiions unconrollable (unobservable) in all subsysems. Noe ha in his examle an arbiraion rocedure should be available, o ensure ha a ransiion in he lef subsysem does no fire a he same ime as one in he righ subsysem. Such an arbiraion mehod could be, for insance, ha ransiions in he lef subsysem may fire only a a ime 0 +kδ, while ransiions in he righ subsysem only a a ime 0 +(k +)δ. Furhermore, he communicaion beween he wo comuers is o be reliable (e.g. no los ransiion-firing messages).

14 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Algorihm 5. Decenralized Suervisor Design. Is he secificaion admissible wih resec o (N,T uc,l,t uo,l )? If no, ransform i o be admissible (an aroach of [0] could be used) or use he decenralized design aroach of secion 6.. Le S be he cenralized SBPI suervisor enforcing he secificaion. Le T c be he se of ransiions conrolled by S and T o he se of ransiions deeced by S.. Find a se such ha T uc = T uc,i T \ T c.. Design he decenralized suervisor by alying Algorihm. o N and. 5. The communicaion can be designed as follows: for all T o ( T uo,i ), a subsysem j such ha T o,j ransmis he firings of o all suervisors S k wih T uo,k and k. Noe he following. Firs, no communicaion arises when T o ( T uo,i )=. Second, he algorihm does no ake in accoun communicaion limiaions, such as bandwidh limiaions of he communicaion channel. Bandwidh limiaions can be considered in he aroach considered nex in secion 6. Third, in his soluion communicaion is used only o make some locally unobservable ransiions observable; here is no remoe conrol of locally unconrollable ransiions. Fourh, his soluion ends o require less communicaion han a cenralized soluion. Indeed, a cenral suervisor no only needs o send he conrol decisions o he local subsysems, bu also o remoely observe all ransiions in T o. Fifh, he main limiaion of he algorihm is ha in he case of inadmissible secificaions, he ransformaion a he se may resul in consrains ha are oo resricive. If so, he alernaive soluion we roose in secion 6 could be used. Finally, he only way he algorihm can fail is a se, when he secificaion is inadmissible and he ransformaions o an admissible form fail. Proosiion 5. The decenralized suervisor is feasible and equally ermissive o he cenralized suervisor S enforcing he secificaion on (N,T uc,t uo,l ). Proof: Since S is admissible, T c T uc = and T o T uo,l =. ommunicaion ensures ha he ses of locally unobservable ransiions become T uo,i = T uo,i \ T o. I follows ha he secificaion is d-admissible wih resec o (N,T uc,,...t uc,n,t uo,,...t uo,n ) and so he conclusion follows by Theorem.. A leas one soluion exiss, = {...n}. This can be seen from he fac ha S admissible w.r.. (N,T uc,l,t uo,l) imlies T uc,l T c =, andfromt uc,l = T uc,i. i=...n

15 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober onsrain Transformaions for Suervisor Design Given a d-admissible se of consrains, a suervisor enforcing i can be easily consruced, as shown in Algorihm.. This secion considers ransformaions of ses of consrains ha are no d-admissible. These ransformaions aim o obain (more resricive) d-admissible consrains, in order o reduce he roblem o he enforcemen of d-admissible consrains. Two aroaches are roosed: ransformaions o single ses of consrains and ransformaions o mulile ses of consrains. The former is a aricular case of he laer, and can be done using echniques from he lieraure [0, 5]. As he ransformaion o a single se of consrains canno deal effecively wih some ineresing roblems, we will focus on he ransformaion o mulile ses of consrains. This aroach will be resened in boh suervisory frameworks, wih communicaion and wih no communicaion. 6. Transformaion o a single se of consrains A ossible aroach o ransform a se of consrains o a d-admissible se of consrains is:. Selec a nonemy subse of {,,...n}.. Transform 5 he se of consrains o a c-admissible se of consrains wih resec o (N,T uc,t uo ), for T uc = T uc,i and T uo = T uo,i. In racice, i may no be rivial o selec he bes se. However, for some aricular cases he choice of is more obvious: If T uo, = T uo, =...T uo,n (in aricular, his is rue when full observabiliy is available in each subsysem: T uo,i = i =...n), hen canbechosenas = {,,...n}, o minimize he number of ransiions in T uc. If T o,i T o,j = for all disinc i, j =...n, hen we could aem o se o each of {}, {},... {n}, do in each case he ransformaion o admissible consrains, and hen selec he one yielding he leas resricive consrains. The main drawback of his aroach is ha i fails for many ineresing sysems and consrains. For insance, i fails o rovide a soluion for he sysem of Figure 6, wih T uc, = T uo, = {, }, T uc, = T uo, = {, }, iniial marking as shown in figure, and secificaion µ + µ (7) Indeed, no maer how is chosen, no d-admissible inequaliy imlying (7) is saisfied by he iniial 5 Techniques ha can be used o erform his ransformaion aear in [0, 5].

16 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Global sysem Lef subsysem Righ subsysem Lef subsysem Righ subsysem enralized soluion Decenralized soluion Figure 6: Decenralized conrol examle. marking. However, i is ossible o enforce (7) wih wo d-admissible inequaliies µ (8) µ (9) as shown in Figure 6. To see ha (8) and (9) are d-admissible, noe ha (8) saisfies Definiion. for = {}, and (9) saisfies Definiion. for = {}. Noe also ha none of (8) and (9), by iself, imlies (7). This examle moivaes he ransformaion o mulile consrains, which is resened nex. 6. Transformaion o mulile ses of consrains The roblem can be saed as follows: Given a se of consrains Lµ b ha is no d-admissible, find d-admissible ses of consrains L µ b... L m µ b m such ha (L µ b L µ b...l m µ b m ) Lµ b (0) omared o he revious aroach, we now use several ses,,..., m o design each of he L µ b, L µ b,..., L m µ b m, insead of a single se. For insance, if T o,i T o,j = for all i j, henwemayake i = {i}, for all i. Furhermore, noe ha his framework includes he case when no all consrains L i µ b i are necessary o imlemen Lµ b, by allowing L i =0and b i =0. In general, (0) may have many soluions, no all equally ineresing. In order o have a more ineresing soluion, we can use a se of markings of ineres M I, and consrain each L i and b i o 5

17 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. saisfy L i µ b i µ M I. This condiion can be wrien as L i M b i T () where means ha each elemen of L i M is less or equal o he elemen of he same indices in b i T, M is a marix whose columns are he markings of ineres, and T is a row vecor of aroriae dimension in which all elemens are. The roblem is more racable if we relace (0) wih he sronger condiion below: [(α L + α L +...α m L m )µ (α b + α b +...α m b m )] Lµ b () where α i are nonnegaive scalars. 6 Wihou loss of generaliy, () assumes ha L... L m have he same number of rows. Again, wihou loss of generaliy, () can be relaced by We furher simlify our roblem o [(L + L +...L m )µ (b + b +...b m )] Lµ b () L + L +...L m = R + R L () b + b +...b m = R (b + ) (5) for R wih nonnegaive ineger elemens and R diagonal wih osiive inegers on he diagonal. Noe ha [(R + R L)µ R (b + ) ] Lµ b has been roved in [0]. I is known ha a sufficien condiion for he c-admissibiliy of a se of consrains Lµ b is ha LD uc 0andLD uo =0,whereD uc and D uo are he resricions of he incidence marix D o he ses of unconrollable and unobservable ransiions [0]. The admissibiliy requiremens in our seing can hen be wrien as L i D uc (i) 0 (6) L i D uo (i) = 0 (7) where D uc (i) and D uo (i) are he resricions of D o he ses T uc (i) = T uc,i and T uo (i) = T uo,i.then i i our roblem becomes: find a feasible soluion of () and ( 7). The unknowns are R, R, L i,andb i, and ineger rogramming can be used o find hem. The nex resul is an immediae consequence of our consideraions above. Proosiion 6. Any ses of consrains L i µ b i saisfying () and ( 7) are d-admissible and [L i µ b i ] Lµ b. i=...n 6 In he lieraure, a relaxaion of a hard roblem ha is similar o he relaxaion from (0) o () is he S- rocedure menioned in [8] a age 6. 6

18 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober Decenralized conrol wih communicaion So far, we have ignored he ossibiliy ha he local suervisors S i may have he abiliy o communicae. We now consider he case in which he suervisors are able o communicae he firings of cerain ransiions. ommunicaion is useful, as i relaxes he admissibiliy consrains (6) and (7) by reducing he number of unconrollable and unobservable ransiions. However, communicaion consrains may be resen, and bandwidh limiaions may encourage he minimizaion of he communicaion over he nework. The analysis of his secion, wihou being comrehensive, serves as an illusraion of he fac ha such roblems can be aroached in his framework. For each se i and ransiion j,leα ij be a binary variable, where α ij =ifhefiringof j is made known o he subsysems in i. Noe ha we have he following consrains: where T uo,l = i=...n j T uo,l : α ij = 0 (8) T uo,i is he se of ransiions ha canno be observed anywhere in he sysem. (T uo,l is he se of ransiions whose firing canno be communicaed.) Le BL i and Bi U be lower and uer bounds of L id and A =[α ij ] be he marix of elemens α ij. Given a vecor v and a marix M, lediag(v) denoe he diagonal marix of diagonal v, M(k, ) he k h row of M, andm (i) T he resricion of M o he ransiions of T uo (i) (i.e. M (i) uo T conains uo he columns M(,j) such ha j T uo (i) ). We require L i D uo (i) [BUdiag(A(i, i ))] (i) T uo L i D uo (i) [BLdiag(A(i, i ))] (i) T uo (9) (0) insead of L i D uo (i) = 0. In his way, he admissibiliy requiremen L i D uo (i) = 0 is relaxed by eliminaing he consrains corresonding o he ransiions of T uo (i) ha have heir firings communicaed o he subsysems of i. Similarly, (6) can also be relaxed by communicaing enabling decisions of suervisors. Naurally, for each ransiion i conrols, each suervisor S i has wo enabling decisions: enable and disable. They deend on wheher all conrol laces of S i ha are conneced o saisfy µ c () W s (, ) or no. A communicaion olicy may be ha a suervisor announces a remoe acuaor each ime is enabling decision changes. Then he acuaor can deermine is enabling by aking he conjuncion of he decisions corresonding o all suervisors conrolling i. In our seing, d-admissibiliy imlies ha he suervisors wihin a cluser i have always he same enabling decisions, and so only communicaion beween clusers needs o be considered. Similarly o α ij,we can inroduce binary variables ε ij describing he communicaion of enabling decisions eraining o j. Thus, ε ij = if a suervisor from i communicaes is enabling decisions o j. As in he case of α ij,wehave j T uc,l : ε ij = 0 () 7

19 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. for T uc,l = i=...n T uc,i. Furhermore, if E =[ε ij ], (6) becomes: L i D (i) uc [Bi U diag(e(i, ))] T (i) uc ommunicaion consrains saing ha cerain ransiions canno be observed by communicaion or ha cerain ransiions canno be remoely conrolled by communicaion, can be incororaed by seing coefficiens α ij and ε ij o zero. onsrains limiing he average nework raffic can be incororaed as consrains of he form: A(i, )g i + E(i, )h i () i i where g i and h i are vecors of aroriae dimensions and is a scalar. As an examle, he elemens of g i could reflec average firing couns of he ransiions over he oeraion of he sysem. Noe ha () can be wrien more comacly as () Tr(AG + EH) () where G and H are he marices of columns g i and h i,andtr(m) denoes he race of a marix M. We may also choose o minimize he amoun of communicaion involved in he sysem. Then we can formulae our roblem as min L i,b i,a,e,r,r Tr(A + EF) (5) where he weigh marices and F are given, and he minimizaion is subjec o he consrains (), ( 5), (8 ), and α ij,ε ij {0, } T. This roblem can be solved using linear ineger rogramming. 6. Liveness onsrains One of he difficulies encounered wih his aroach is ha he ermissiviy of he generaed consrains is hard or imossible o be exressed in he cos funcion. Moreover, he generaed consrains may cause ars of he sysem o unavoidably deadlock. This siuaion can be revened by using a secial kind of consrains, ha we call liveness consrains. A liveness consrain consiss of a vecor x such ha for all i: L i x 0. A ossible way o obain such consrains is described nex. Given a finie firing sequence σ, lex σ be a vecor such ha x σ (i) is he number of occurrences of he ransiion i in σ. Given he Peri ne of incidence marix D and he consrains Lµ b, ley be a nonnegaive ineger vecor such ha Dy 0 and LDy 0. A vecor y saisfying hese inequaliies has he following roery. If σ is a firing sequence such ha (a) σ can be fired wihou violaing Lµ b and (b) x σ = y, henσ can be fired infiniely ofen wihou violaing Lµ b. However, if he decenralized conrol algorihm generaes aconsrainl i µ b i such ha L i Dy 0, hen any firing sequence σ having x σ = y canno be infiniely ofen fired in he closed-loo. If such a siuaion is undesirable, he marices L i can be required o saisfy L i x 0forx = Dy. An illusraion will be given in secion 8. 8

20 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober Reducing he omuaional omlexiy Obviously, an ineger rogramming aroach limis he size of he roblems ha can be solved. A ossible way o reduce he amoun of comuaion is o solve several smaller ineger rograms insead of a large ineger rogram. This aroach is oulined nex. Algorihm 6.. Le L 0 µ b 0 be he given se of consrains.. Le I = J = and se (L 0,i,b 0,i ) as emy ses of consrains for all i, i =...m.. For all inequaliies lµ β of he se of inequaliies L 0 µ b 0 do (a) Se L = l, b = β, and solve (5) subjec o (), ( 5), (8 ), α ij,ε ij {0, } T, α ij = (i, j) Iand ε ij = (i, j) J. (b) If he ineger rogram is feasible, include (L i,b i )in(l 0,i,b 0,i )andsei = {(i, j) :α ij 0} and J = {(i, j) :ε ij 0}. (c) If he ineger rogram is infeasible, exi, and declare failure.. The ouu are he consrains (L 0,i,b 0,i ), while communicaion is o ensure ha he firing of all ransiions j wih (i, j) Iis announced o i,andhaforall(i, j) J,asuervisor from i sends is enabling decisions o j. Oher comuaional savings can be obained by aking advanage of knowledge on he reachable markings. Indeed, u o now our aroach has relied exclusively on srucural roeries of he ne. However, knowledge on he reachable markings can be used o reduce he number of ransiions ha need o be considered conrollable and observable, based on reachabiliy informaion and he c-admissibiliy definiion. This is shown in Aendix B. Reducing he number of conrollable or observable ransiions resuls in a smaller number of consrains in he condiions (6) and (7). Significan comuaional savings can be achieved when only a ar of he elemens of each L i need o be calculaed. For insance, we may choose o se he j h column of L i o zero for all laces j ha are no ar of he subsysem i. The number of variables α ij and ε ij is significanly reduced when he communicaion olicy is changed o broadcas. Then observed ransiions and enabling decisions ha are o be communicaed are broadcased. In his case α ij = α j and ε ij = ε j,whereα j = means ha a suervisor observing j broadcass he firings of j,andε j = means ha any suervisor ha remoely conrols j broadcass is enabling decisions for j. 7 Exensions The main resuls resened so far in he aer consis of he definiion of d-admissibiliy, algorihms for decenralized conrol, and roofs of correcness. I can be noiced ha while he resuls of sec- 9

21 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. ion 6 are more secialized, aking advanage of he fac ha he consrains have he aricular form Lµ b, he resuls of secions and 5 are general. Indeed, in order o aly he laer resuls o oher yes of consrains, we only need o relace he SBPI wih he corresonding suervision echniques, and hen change accordingly he c-admissibiliy definiion. In his secion wo exensions of our resuls are discussed. Firs he exension o he suervision of generalized linear consrains [5] is discussed. Then, he exension o he auomaa seing is discussed. Noe ha while he firs exension effecively enhances he racical use of our resuls, he second exension has an illusraive urose. The racical significance of he secializaion of our aroach o he auomaa seing is a maer of furher invesigaion. 7. Exensions o Generalized Linear onsrains This secion alies he resuls derived so far in he aer o generalized linear consrains [5]. This ye of consrains generalizes he consrains Lµ b o he form Lµ + Hq + v b (6) where q, he firing vecor, and v, he Parikh vecor, are defined as follows. The firing vecor q idenifies a ransiion ha is o fire by he osiion of is nonzero elemen; namely, if i is he ransiion o fire, q j =forj = i and q j =0 j i. The Parikh vecor v consiss of elemens v i ha coun he number of firings of each ransiion i since he iniializaion of he sysem. A suervisor enforcing (6) ensures ha (i) all reachable saes (µ, v) of he lan saisfy Lµ + v b and (ii) a ransiion i can be fired from he sae (µ, v) onlyiflµ + Hq (i) + v b and Lµ + v b, whereq (i) j =ifj = i and q (i) j = 0 oherwise, µ is he marking reached by firing i, and v = v + q (i). As shown in [5], in he fully conrollable and observable case, a cenralized suervisor enforcing (6) is defined by he inu and ouu marices where The iniial marking of he suervisor is 7 7 Noe ha he iniial v is zero. D + s = D + lc +max(0,h D lc ) (7) Ds = max(d lc,h) (8) D + lc = max(0, LD ) (9) D lc = max(0,ld+ ) (0) µ s0 = b Lµ 0 () 0

22 Marian Iordache, Panos J.Ansaklis, Decenralized onrol of Peri Nes, ISIS Technical Reor ISIS , Ocober 00. Similarly o (), he closed-loo sysem has he incidence marix [ ] D D c = D s () where D s = D s + Ds. Noe ha as in he case of he SBPI, he suervisor consiss of conrol laces conneced o he ransiions of he lan. Therefore, he definiions for deecion and conrol in secion are sill valid. So, he c-admissibiliy definiions can be adaed as follows. A suervisor enforcing (6) is c-admissible if i only conrols conrollable ransiions and i only deecs observable ransiions. The se of consrains (6) is c-admissible if he suervisor defined by (7), (8) and () is c-admissible. The resuls of he secions, 5 and 6 aly o secificaions (6) as follows:. All resuls in he secions and 5 aly once he references o he SBPI are relaced by references o he suervision mehod of (7), (8) and ().. The maerial of secion 6 can be alied o secificaions in he form of consrains (6) afer alying he - andh-ransformaions defined in [5]. These ransformaions ransform he Peri ne and he consrains o reduce he roblem o he enforcemen of consrains L e µ b. 7. Exension o Auomaa In his secion he resuls of secions and 5 are secialized o he auomaa seing. Noe ha he resuls of secion 6 are harder o exend, as hey rely on a aricular ye of secificaions. Firs, he noaion is inroduced. An auomaon is denoed by he ule G =(Σ,Q,δ,q 0,Q m ), where Σ is he se of evens, Q is he se of saes, δ : Q Σ Q is a (arial) funcion reresening he ransiion funcion, q 0 Q is he iniial sae, and Q m Q is he se of marked saes. The emy symbol is denoed by ε. Le Σ denoe he se of all srings over he alhabe Σ. Le δ : Q Σ Q be he (arial) funcion recursively defined as follows: q Q: δ (q,ε) =q and q Q α Σ σ Σ : δ (q, σα) =δ(δ (q,σ),α)whenδ (q,σ) andδ(δ (q,σ),α) are defined. The fac ha δ (q,σ) is defined is denoed as δ (q,σ)!. The language acceed by G is L(G) ={σ Σ : δ (q 0,σ)!}, and he language marked by G is L m (G) ={σ L(G) :δ (q 0,σ) Q m }. onsider a refix-closed secificaion K L(G). The conrollabiliy and observabiliy of K are defined wih resec o he se of unconrollable evens Σ uc Σ and he se of unobservable evens Σ uo Σ. K is conrollable if KΣ uc L(G) K. K is observable if ( α Σ σ,σ K) [P (σ )=P(σ ) σ α, σ α L(G)] [σ α, σ α K σ α, σ α/ K], where P :Σ (Σ \ Σ uo ) is he rojecion removing he elemens of Σ uo from any sring. A suervisor S of G is defined as a ma S :(Σ\ Σ uo ) Σ. To imlemen K, hesuervisor S can be defined as follows: σ (Σ \ Σ uo ), S(σ) ={α Σ: σ x P (σ), σ x α K}. Then, if K is observable, he closed-loo language is L(S/G) = K. Noe ha he conrol acion aken afer he occurrence of σ L(G) ing is S(P (σ)).