Non-blocking Supervisory Control of Nondeterministic Systems via Prioritized Synchronization 1

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1 Non-blocking Supervisory Control of Nondeterministic Systems vi Prioritized Synchroniztion 1 Rtnesh Kumr Deprtment of Electricl Engineering University of Kentucky Lexington, KY Emil: kumr@engr.uky.edu Mrk A. Shymn Deprtment of Electricl Engineering nd Institute for Systems Reserch University of Mrylnd College Prk, MD Emil: shymn@eng.umd.edu November 20, This reserch ws supported in prt by the Center for Robotics nd Mnufcturing, University of Kentucky, in prt by by the Ntionl Science Foundtion under the Grnts NSFD-CDR , NSF-ECS , NSF-ECS , the Mint Mrtin Fund for Aeronuticl Reserch, nd the Generl Reserch Bord t the University of Mrylnd.

2 Abstrct In previous pper we showed tht supervisory control of nondeterministic discrete event systems, in the presence of driven events, cn be chieved using prioritized synchronous composition s mechnism of control, nd trjectory models s modeling formlism, first introduced by Heymnn. The specifictions considered in this erlier work were given by prefix-closed lnguges. In this pper, we extend this work to include mrkings so tht non-closed specifictions nd issues such s blocking cn be ddressed. It is shown tht the usul notion of non-blocking, clled lnguge model non-blocking, my not be dequte in the setting of nondeterministic systems, nd stronger notion, clled trjectory model non-blocking, is introduced. Necessry nd sufficient conditions for the existence of lnguge model non-blocking s well s trjectory model non-blocking supervisors re obtined for nondeterministic systems in the presence of driven events in terms of extended controllbility nd reltive-closure conditions, nd new condition clled the trjectory-closure condition. Keywords: discrete event systems, supervisory control, nondeterministic utomt, driven events, prioritized synchroniztion, trjectory models, blocking

3 1 Introduction Discrete event systems involve quntities which tke discrete set of vlues which remin constnt except t discrete times when events occur in the system. Exmples include communiction networks, intelligent vehicle highwy systems, mnufcturing systems nd computer progrms. Supervisory control theory ws developed to provide mthemticl frmework for the design of controllers for such systems in order to meet vrious qulittive constrints. For more detils refer to [15, 17, 10]. The mjority of the reserch effort in this re hs focused on the supervisory control of deterministic systems, nd reltively little progress hs been mde towrds tht of nondeterministic systems systems in which knowledge of the current stte nd next event is insufficient to uniquely determine the next stte. Such nondeterminism rises due to unmodeled system dynmics nd/or prtil observtion. For exmple chnge-giving mchine my give different combintion of coins s chnge (for the sme input mount) depending on the sequence in which coins re loded in the mchine. However, for simplicity, this detil my be suppressed while obtining model for the mchine leding to nondeterministic model of it. Similrly, mchine in mnufcturing system my incur prtil undetectble filure while performing certin tsk. This cn be modeled by hving nondeterministic trnsition on the tsk completion event leding to two successor sttes depending on whether or not the filure occurred while completing the tsk. Also, in communiction network, user is only ble to observe the externl events such s trnsmission nd reception of messges, wheres the internl events such s loss or collision of messges, cknowledgments, etc., re not observed. Such internl events cn be represented s silent or ɛ-trnsitions leding to nondeterministic model of the communiction network. In the Rmdge-Wonhm pproch to supervisory control, every event is generted by the plnt nd synchronously executed by the supervisor [14] which cts pssively by disbling certin controllble events possible in the open-loop plnt. The disblement ction is ccomplished by control-input mp which specifies set of disbled events bsed on the current stte of the supervisor. Alterntively, in the work of Kumr-Grg-Mrcus [11], the disblement ction is ccomplished by removing certin trnsitions from the structure of the supervisor while continuing to require tht the plnt nd supervisor be connected by strict synchronous composition (SSC). In the work of Golszewski-Rmdge [3] nd, in the rel-time setting, the work of Brndin-Wonhm [2], the supervisor is ble to initite certin so-clled forcible events tht the plnt synchronously executes. In the work of Blemi nd coworkers [1], events cn originte in the supervisor (so-clled commnd events) or in the plnt (so-clled response events). The ssumption is mde tht the plnt nd supervisor re mutully receptive, mening tht neither the plnt nor the supervisor cn refuse to execute n event initited by the other. Common to ll of the bove pproches is the ssumption tht there re never events which my occur in the supervisor without the prticiption of the plnt. However, this ssumption my be unresonbly restrictive for nondeterministic systems. When the plnt is nondeterministic, there is generlly no wy to know priori whether commnd issued 1

4 by the supervisor cn be executed by the plnt in its current stte. For exmple, it my be impossible to know tht device is in fulted stte until fter it fils to respond to commnd from the controller. Heymnn hs introduced n interconnection opertor clled prioritized synchronous composition (PSC) [4], which relxes the synchroniztion requirements between the plnt nd supervisor. Ech process in PSC-interconnection is ssigned priority set of events. For n event to be enbled in the interconnected system, it must be enbled in ll processes whose priority sets contin tht event. Also, when n enbled event occurs, it occurs in ech subsystem in which the event is enbled. In the context of supervisory control, the priority set of the plnt contins the controllble nd uncontrollble events, while the priority set of the supervisor contins the controllble nd driven events. Thus, controllble events require the prticiption of both plnt nd supervisor; uncontrollble events require the prticiption of the plnt nd will occur synchronously in the supervisor whenever possible; driven events require the prticiption of the supervisor nd will occur synchronously in the plnt whenever possible. It is importnt to distinguish between PSC nd other types of prllel composition in the literture. For exmple, Hore [7] defines concurrent composition opertor in which ech process hs its own lphbet nd the processes synchronize on the events in the intersection of their lphbets. This is generlized to trce-dependent lphbets, clled event-control sets, by Inn-Vriy [9]. The key difference between concurrent composition nd PSC is tht in PSC, lthough process cnnot block events which re outside its priority set, it my be ble to execute these events nd, whenever possible, will execute these events synchronously when they occur in the other process 1. Lnguge models identify processes tht hve the sme set of trces. The filures model of Hore [7] identifies processes tht hve the sme set of so-clled filures. Filure equivlence refines lnguge equivlence. Heymnn showed tht filure equivlence is too corse to support the PSC opertor [4]. In other words, there exist two different plnts with the sme filures model (nd hence with the sme lnguge model) such tht their PSC s with common supervisor hve different lnguge models. Thus, neither the lnguge model nor even the filures model retins enough informtion bout process to do control design using the opertion of PSC. This hs led Heymnn to introduce the trjectory model, refinement of the filures model [4, 6]. The trjectory model is similr to the filure-trce model (lso clled the refusl-testing model) in concurrency theory [13], but differs from this model in its tretment of hidden trnsitions. The trjectory model trets hidden trnsitions in wy tht is consistent with the filures model. In previous pper [16], we proved tht the trjectory model retins sufficient process detil to permit PSC-bsed controller design. In [16], we showed tht supervisory control of nondeterministic discrete event systems, in 1 If pplied to so-clled improper processes, the prllel opertor defined by Inn [8] cn be viewed s generlized form of PSC, but only in the deterministic setting. However, when supervisory control is considered in this reference, the ssumption is mde tht the plnt is proper nd hs constnt event control set. This ssumption excludes driven events. 2

5 the presence of driven events, cn be chieved using prioritized synchronous composition s mechnism of control, nd trjectory models s modeling formlism. The specifictions considered in [16] were given by prefix-closed lnguges. In this pper, we extend our erlier work to include the notion of mrkings by introducing the notion of recognized nd generted trjectory sets, so tht non-closed specifictions nd the issue of blocking cn be ddressed. It is importnt to understnd the difference between the bsence of dedlock nd the bsence of blocking. While the former cn be described using refusl sets, the lter needs the notion of mrking: The bsence of dedlock requires tht the controlled system should never rech stte with refusl set being the entire event set, wheres the bsence of blocking requires tht the controlled system should never rech stte from where it is not possible to complete its execution by reching mrked stte. If the refusl set of stte is the entire event set, then this mens tht no events re possible in tht stte nd the system would dedlock. However, tht stte my represent completion of tsk, in which cse the stte would be mrked nd not result in blocking. On the other hnd, the system could become trpped in cycle tht contins no sttes tht represent completion of tsks i.e., no mrked sttes. In this cse, the refusl set would not be the entire event set, i.e., system would not dedlock, nd yet the system is blocked since it is in livelock. Consequently, refusl sets re insufficient to distinguish blocking nd nonblocking sttes, nd the notion of mrking is required. The usul notion of non-blocking, referred to s lnguge model non-blocking in this pper, requires tht ech trce belonging to the generted lnguge of controlled system be extendble to trce belonging to the recognized lnguge. This property dequtely cptures the notion of non-blocking in deterministic setting. However, in nondeterministic setting, the execution of certin trce belonging to the generted behvior my led to more thn one stte. Lnguge model non-blocking only requires tht ech such trce be extendble to trce in the recognized behvior from t lest one such stte s opposed to ll such sttes. Thus, lnguge model non-blocking nondeterministic system cn get blocked, s illustrted by the exmple Section 6. Consequently, there is need for stronger type of non-blocking for nondeterministic systems. This leds us to introduce the property of trjectory model non-blocking, which requires tht ech refusl-trce belonging to the generted trjectory set of nondeterministic system be extendble to refusl-trce belonging to the recognized trjectory set. This stronger notion of non-blocking seems dequte for prcticl systems, lthough s explined in Remrk 4 it does not lwys gurntee the bsence of blocking. Another desirble property of supervisor is tht it should be non-mrking, i.e., certin trce (respectively, refusl-trce) of the controlled system should belong to the recognized lnguge (respectively, the recognized trjectory set) of the controlled system if nd only if mrked stte of the uncontrolled system is reched due to its execution regrdless of the type of stte reched in the supervisor. We first obtin necessry nd sufficient condition for the existence of non-mrking nd lnguge model non-blocking supervisor for given nondeterministic system in the presence of driven events. This result is then used to obtin necessry nd sufficient condition for the existence of non-mrking nd trjectory model 3

6 non-blocking supervisor in tht setting. 2 Nottion nd Preliminries Given finite event set Σ, Σ is used to denote the collection of ll trces, i.e., finite sequences of events, including the zero length sequence, denoted by ɛ. A subset of Σ is clled lnguge. Symbols H, K, etc. re used to denote lnguges. The set 2 Σ (Σ 2 Σ ) is used to denote the collection of ll refusl-trces, i.e., finite sequences of lternting refusls nd events [6, 16] of the type: Σ 0 (σ 1, Σ 1 )... (σ n, Σ n ), where n N. The sequence σ 1... σ n Σ is the trce, nd for ech i n, Σ i Σ is the set of events refused (if offered) t the indicted point. Symbols P, Q, R, S, etc. re used to denote sets of refusl-trces. Refusl-trces re lso referred to s trjectories. Given s Σ, we use s to denote the length of s, nd for ech k s, σ k (s) Σ is used to denote the kth event in s. If t Σ is nother trce such tht t s nd for ech k t, σ k (t) = σ k (s), then t is sid to be prefix of s, denoted t s. For ech k s, s k denotes the prefix of length k of s. The prefix-closure of s Σ, denoted pr(s) Σ, is defined s pr(s) := {t Σ t s}. The prefix-closure mp cn be defined for set of trces in nturl wy. Given e 2 Σ (Σ 2 Σ ), we use e to denote the length of e, nd for ech k e, Σ k (e) Σ is used to denote the kth refusl in e nd σ k (e) Σ is used to denote the kth event in e, i.e., e = Σ 0 (e)(σ 1 (e), Σ 1 (e))... (σ k (e), Σ k (e))... (σ e (e), Σ e (e)). If f 2 Σ (Σ 2 Σ ) is nother refusl-trce such tht f e nd for ech k f, Σ k (f) = Σ k (e) nd σ k (f) = σ k (e), then f is sid to be prefix of e, denoted f e. For ech k e, e k is used to denote the prefix of length k of e. If f 2 Σ (Σ 2 Σ ) is such tht f = e nd for ech k f, Σ k (f) Σ k (e) nd σ k (f) = σ k (e), then f is sid to be dominted by e, denoted f e. The prefix-closure of e 2 Σ (Σ 2 Σ ), denoted pr(e) 2 Σ (Σ 2 Σ ), is defined s pr(e) := {f 2 Σ (Σ 2 Σ ) f e}, nd the dominnce-closure of e, denoted dom(e) 2 Σ (Σ 2 Σ ), is defined s dom(e) := {f 2 Σ (Σ 2 Σ ) f e}. The prefix-closure nd dominnce-closure mps cn be defined for set of refusl-trces in nturl wy. Given refusl-trce e 2 Σ (Σ 2 Σ ), the trce of e, denoted tr(e) Σ, is defined s tr(e) := σ 1 (e)... σ e (e). The trce mp cn be extended to set of refusl-trces in nturl wy. Given set of refusl-trces P 2 Σ (Σ 2 Σ ), we use L(P ) := tr(p ) to denote its set of trces. Symbols P, Q, R, etc. re used to denote NSM s (with ɛ-moves). Let the 5-tuple P := (X P, Σ, δ P, x 0 P, X m P ) 4

7 represent discrete event system modeled s n NSM, where X P is the stte set, Σ is the finite event set, δ P : X P (Σ {ɛ}) 2 X P denotes the nondeterministic trnsition function 2, x 0 P X P is the initil stte, nd XP m X P is the set of ccepting or mrked sttes. A triple (x 1, σ, x 2 ) X P (Σ {ɛ}) X P is sid to be trnsition if x 2 δ P (x 1, σ). A trnsition (x 1, ɛ, x 2 ) is referred to s silent or hidden trnsition. We ssume tht the plnt cnnot undergo n unbounded number of silent trnsitions, i.e., P does not contin ny cycle of silent trnsitions. The ɛ-closure of x X P, denoted ɛ P(x) X P, is defined recursively s: x ɛ P(x); [x ɛ P(x)] [δ P (x, ɛ) ɛ (x)], nd the set of refusl events t x X P, denoted R P (x) Σ, is defined s R P (x) := {σ Σ δ P (x, σ) =, x ɛ P(x)}. In other words, given x X P, ɛ P(x) is the set of sttes tht cn be reched from x on zero or more ɛ-moves, nd R P (x) is the set of events tht re undefined t ech stte in the ɛ-closure of x. The trnsition function δ P : X (Σ {ɛ}) 2 X P is extended to the set of trces s δ P : X Σ 2 X P, which is defined inductively s: x X P : { δ P (x, ɛ) := ɛ P(x), s Σ, σ Σ : δ P(x, sσ) := ɛ P(δ P (δ P(x, s), σ)), where in the lst equlity the trnsition function δ P : X (Σ {ɛ}) 2 X P hs been extended to δ P : 2 X P (Σ {ɛ}) 2 X P in nturl wy. The trnsition function is lso extended to the set of refusl-trces s δp T : X (2 Σ (Σ 2 Σ ) ) 2 X P, which is defined inductively s: Σ Σ : δp(x, T Σ ) := {x ɛ P(x) Σ R P (x )}, x X P : e 2 Σ (Σ 2 Σ ), σ Σ, Σ Σ : δp(x, T e(σ, Σ )) := {x ɛ P(δ P (δp(x, T e), σ)) Σ R P (x )}. In other words, stte x X P is reched by executing zero-length refusl-trce Σ Σ from stte x X P if x cn be reched in zero or more ɛ-moves from x, nd ech event in Σ is refused t x. A stte x X P is reched by executing refusl-trce e(σ, Σ ) 2 Σ (Σ 2 Σ ) from stte x X P if x cn be reched by executing the event σ followed by zero or more ɛ-moves from stte reched by executing the refusl-trce e from x, nd ech event in Σ is refused t stte x. The extended trnsition functions re then used to obtin the lnguge models nd the trjectory models of P s follows: L(P) := {s Σ δ P(x 0 P, s) }, L m (P) := {s L(P) δ P(x 0 P, s) X m P }, T (P) := {e 2 Σ (Σ 2 Σ ) δ T P(x 0 P, e) }, T m (P) := {e T (P) δ T P(x 0 P, e) X m P }. 2 ɛ represents both n internl or unobservble event nd n internl or nondeterministic choice [7, 12]. 5

8 L(P), L m (P), T (P), T m (P) re clled the generted lnguge, recognized or mrked lnguge, generted trjectory set, recognized or mrked trjectory set, respectively, of P. It is esily seen tht L(T m (P)) = L m (P) nd L(T (P )) = L(P). The pirs (L m (P), L(P)) nd (T m (P), T (P)) re clled the lnguge model nd the trjectory model, respectively, of P. Two lnguge models (K m 1, K 1 ), (K m 2, K 2 ) re sid to be equl, written (K m 1, K 1 ) = (K m 2, K 2 ), if K m 1 = K m 2, K 1 = K 2 ; equlity of two trjectory models is defined nlogously. 3 Trjectory Models Above we defined the trjectory model of NSM s. Trjectory models re importnt for our eventul gol of supervisory control of nondeterministic systems since we wish to exercise control by composing systems in prioritized synchrony, nd s shown in the next section trjectory models retin sufficient informtion bout the system to llow dequte supervisory design (lnguge model nd even the filures model re not dequte s they re not detiled enough). In order to gin further understnding into the structure of trjectory models, we next obtin necessry nd sufficient condition for given refusl-trce set pir to be trjectory model. This requires the definition of sturted refusl-trces. Given refusl-trce set P 2 Σ (Σ 2 Σ ), we define the sturtion mp on P by st P : P 2 Σ (Σ 2 Σ ) where st P (e) := Σ 0 (σ 1 (e), Σ 1 )... (σ e, Σ e ), where k e : Σ k := Σ k (e) {σ Σ e k (σ, ) dom(pr(p ))}. Thus if n event is not executble in P t certin point of its refusl-trce e, then it is dded to the refusl set of e t tht point to obtin st P (e). The sturted refusl-trces of P, denoted P st P, is defined to be the set of fixed points of st P ( ). Heymnn-Meyer used n xiomtic chrcteriztion for defining generted trjectory set. In contrst, we proved in [16, Theorem 1] tht these xioms re necessry nd sufficient for given refusl-trce set to be generted trjectory set of some NSM. This we recll here: Theorem 1 [16, Theorem 1] Given refusl-trce set P 2 Σ (Σ 2 Σ ), there exists n NSM P with generted trjectory set P, i.e., P = T (P), if nd only if 1. P 2. P = pr(p ) 3. P = dom(p ) 4. st P (P ) P 5. e P : σ k+1 (e) Σ k (e), k e 1 Corollry 1 If P is generted trjectory set, then 1. st P ( ) is idempotent, 2. P st = st P (P ). 6

9 Proof: Let e P, ê := st P (e), nd k e = ê. We need to show tht if σ Σ k (ê), then g := ê k (σ, ) P. Since σ Σ k (ê), f := e k (σ, ) P by definition of st P ( ). Since (st P (f)) k = st P (e k ) = ê k, it follows tht g st P (f) P by Theorem 1, Property 4. By Property 3, it follows tht g P, so st P (ê) = ê. The second clim is n immedite consequence of the first. Remrk 1 It follows esily from Corollry 1 tht Properties 3 nd 4 in Theorem 1 cn be replced by the single property P = dom(p st ). In order to see this first suppose P = dom(p st ). Then P is dominnce-closed, so we hve P = dom(p ), i.e., T3 holds. Also, from Corollry 1, st P (P ) = P st dom(p st ) = P, i.e., T4 holds. On the other hnd, if T3 nd T4 hold, then from Corollry 1 nd T4 we hve P st = st P (P ) P. Hence it follows from T3 tht dom(p st ) dom(p ) = P. The reverse continment P dom(p st ) = dom(st P (P )) follows from the definitions of the sturtion mp nd the dominnce-closure opertion. The following result generlizes Theorem 1 to chrcterize those refusl-trce set pirs tht re trjectory models of NSM s. If Σ 1,..., Σ n re subsets of Σ, then min(σ 1, Σ 2,..., Σ n ) denotes the set of miniml sets from mong the given subsets with respect to the inclusion prtil order. Theorem 2 Given pir of refusl-trce sets (P m, P ) with P m, P 2 Σ (Σ 2 Σ ), it is trjectory model if nd only if T1: P T2: P = pr(p ) T3: P = dom(p st ) T4: e P : σ k+1 (e) Σ k (e), k e 1 T5: P m = dom(p st P m ) Proof: First suppose tht (P m, P ) is trjectory model, i.e., there exists n NSM P := (X P, Σ, δ P, x 0 P, XP m ) such tht T m (P) = P m nd T (P) = P. Then it follows from Theorem 1 nd Remrk 1 tht T1 through T4 hold. In order to show tht T5 lso holds, we first show tht P m = dom(p m ), i.e., dom(p m ) P m. Pick e dom(p m ), then there exists f P m such tht e f. Hence δp(x T 0 P, e) δp(x T 0 P, f), which implies tht δp(x T 0 P, e) XP m δp(x T 0 P, f) XP m, which is nonempty since f P m. Thus e P m, so P m = dom(p m ). Hence, dom(p st P m ) dom(p m ) = P m. It remins to show tht P m dom(p st P m ). We show using induction on length of refusl-trces tht for ech e P nd x δp(x T 0 P, e), there exists f P st such tht e f nd x δp(x T 0 P, f). If e = 0, then e = Σ Σ. If x δp(x T 0 P, Σ ), then x ɛ P(x 0 P) nd Σ R P (x). Set f := R P (x); then clerly, f P st, e f, nd x δp(x T 0 P, f). This proves the bse step of induction. In order to prove the induction step, let e = ē(σ, Σ ) P, x δp(x T 0 P, e). Then x ɛ P(δ P ( x, σ)), where x δp(x T 0 P, ē) nd Σ R P (x). Since T2 holds, e P implies ē P. Hence from induction hypothesis, there exists f P st such tht ē f nd x δp(x T 0 P, f). Set f := f(σ, R P (x)); then f P st, e f, nd x δp(x T 0 P, f). This proves the induction step. Hence it follows tht given e P m, so tht there exists x XP m with x δp(x T 0 P, e), we cn select f P st such tht e f nd 7

10 x δ T P(x 0 P, f), i.e., f P st P m. Since e f, this implies tht e dom(p st P m ) s desired. Next ssume tht T1 through T5 hold. We need to show tht (P m, P ) is trjectory model, i.e., there exists n NSM P such tht T m (P) = P m nd T (P) = P. Consider the NSM P := (X P, Σ, δ P, x 0 P, X m P ) (refer to Remrk 2 for n explntion), where X P := P st, x 0 P := {σ Σ (σ, ) P }, XP m := P st P m, δ P : X P (Σ {ɛ}) 2 X P is defined s: 1. e P st, σ Σ : { e(σ, {σ δ P (e, σ) := Σ e(σ, )(σ, ) P }) if e(σ, ) P otherwise, 2 (). Σ Σ such tht Σ P st : δ P (Σ, ɛ) := min({σ Σ Σ P st, Σ Σ }), 2 (b). e 2 Σ (Σ 2 Σ ), σ Σ, Σ Σ such tht e(σ, Σ ) P st : δ P (e(σ, Σ ), ɛ) := {e(σ, Σ ) Σ min({ˆσ Σ e(σ, ˆΣ) P st, Σ ˆΣ})}. From [16, Lemm 1], it follows tht x 0 P P st nd for ech e P st, σ Σ, δ P (e, σ) P st whenever it is nonempty. Thus NSM P is well-defined. It follows from [16, Proposition 2] tht T (P) = P. It remins to show tht T m (P) = P m. By definition we hve T m (P) = {e T (P) δ T P(x 0 P, e) X m P }. Since T (P) = P, X m P = P st P m, nd for ech e P, δ T P(x 0 P, e) = {f P st e f} [16, Corollry 1], we hve T m (P) = {e P {f P st e f} (P st P m ) } = {e P {f P st P m e f} } = dom(p st P m ) = P m, where the lst equlity follows from T5. Remrk 2 In the proof of the sufficiency prt of Theorem 2 the NSM P is constructed from given refusl-trce set pir (P m, P ) stisfying T1-T5 s follows: The stte spce of P equls P st, the set of sturted refusl-trces of P ; the mrked sttes of P re those sturted refusl-trces which lso belong to P m ; nd the initil stte of P is the (unique) miniml zero-length sturted refusl-trce of P. The stte reched by executing nonepsilon event σ Σ from stte e P st equls the miniml sturted refusl-trce of the type e(σ, Σ ) dominting e(σ, ). The set of sttes reched by executing n epsilon 8

11 trnsition from zero-length refusl-trce Σ P st = X P equls the set of miniml zerolength sturted refusl-trces dominting Σ. Also, the set of sttes reched by executing n epsilon trnsition from refusl-trce e(σ, Σ ) P st = X P equls the set of miniml sturted refusl-trces of the type e(σ, Σ ) dominting e(σ, Σ ). This NSM construction is the sme s tht given in [16, Algorithm 1] except tht ccepting sttes re lso defined. A construction procedure somewht similr to the bove construction ws first given without ny proof in [6]. Our construction hs the dvntge tht it voids introduction of certin uxiliry sttes [16, Remrk 3]. The following result ws obtined in the course of the proof of Theorem 2. Corollry 2 Let P := (X P, Σ, δ P, x 0 P, X m P ) be n NSM. Then for ech e T (P) nd x δ T P(x 0 P, e), there exists f (T (P)) st such tht e f, x δ T P(x 0 P, f) nd Σ f (f) = R P (x). The result of Theorem 2 is not trivil generliztion of Theorem 1. In fct for lnguge model (K m, K), the prefix-closure of the recognized lnguge K m is the generted lnguge of n pproprite stte mchine provided K m is nonempty. The sitution is different for trjectory model (P m, P ). If P m is nonempty, then its prefix-closure, pr(p m ), stisfies properties T1,T2,T4. However, pr(p m ) need not stisfy T3 in which cse it cnnot be the generted trjectory set of ny NSM. (See Exmple 1 below.) The following result shows tht generted trjectory set cn be obtined by tking sturtion closure nd provides further insight into the structure of trjectory models. Proposition 1 Let Q be nonempty refusl-trce set stisfying pr(dom(q)) = Q nd T4. Then R := dom(st Q (Q)) is generted trjectory set. Proof: R is trivilly nonempty nd R = dom(r). Since Q is prefix-closed nd for ech e Q, k e, st Q (e k ) = (st Q (e)) k, R is prefix-closed. Also, from the definition of st Q ( ), T4 holds for R since it holds for Q. It remins only to show tht st R (R) R. Since st R ( ) is monotone (with respect to ) nd R is dominnce-closed, it suffices to show tht st R (st Q (Q)) R. Let e Q nd let ê := st Q (e). Then by definition, ê R. Thus in order to show tht st R (ê) R, it suffices to show st R (ê) = ê. We need to show tht if there exist k, σ such tht σ Σ k (ê), then ê k (σ, ) R. Since σ Σ k (ê), it follows tht f := e k (σ, ) Q. Since the mp st Q ( ) commutes with the opertion of tking the length-k prefix, we hve ê k (σ, ) = (st Q (e)) k (σ, ) = st Q (e k )(σ, ) st Q (f) R. This shows tht st R (ê) = ê, completing the proof. Corollry 3 Let P m be nonempty recognized trjectory set. Then dom(st pr(p m )(pr(p m )) is generted trjectory set. 9

12 {} {} {b} ε ε ε ε b {b} {} b {,b} {b} {} b {,b} {,b} {,b} {,b} {,b} {,b} () NSM P (b) NSM Q (c) NSM R Figure 1: Digrm illustrting Exmple 1 Exmple 1 Consider the NSM P with Σ = {, b} nd unspecified mrking shown in Figure 1(); ech stte is lbeled with the set of events tht re refused t tht stte, i.e., the set of events tht re not executble following zero or more silent trnsitions. Thus, P is obtined from the nondeterministic choice between two deterministic subsystems one tht executes nd dedlocks, the other tht executes b nd dedlocks. Then the generted trjectory set of P is given by P := T (P) = dom(pr({e 1, e 2 })), where e 1 := {b}(, {, b}), e 2 := {}(b, {, b}). The sturted refusl-trces of P re given by P st = {, {}, {b}, e 1, e 2, e 3, e 4 }, where e 3 := (, {, b}), e 4 := (b, {, b}). Let P m := dom({e 3 }). Then dom(p m P st ) = dom({e 3 }) = P m, so tht the refusl-trce set pir (P m, P ) stisfies T1-T5. Hence it follows from Theorem 2 tht there exists NSM Q such tht T (Q) = P, T m (Q) = P m. One choice for Q is the cnonicl NSM described in the proof of Theorem 2, which is shown in Figure 1(b). However, there is no mrking of the sttes of P for which T m (P) = P m. Thus specifiction of the mrking informtion results in refinement of the ssocited stte mchine. Furthermore, if we consider P := pr(p m ) = pr(dom({e 3 })) = { } dom({e 3 }), then we hve st P (P ) = pr(e 1 ). Clerly, dom(st P (P )) = dom(pr(e 1 )) P, i.e., T3 does not hold for P ; consequently, it cnnot be the generted trjectory set of ny NSM. However, it follows from Corollry 3 tht dom(st P (P )) = dom(pr(e 1 )) is generted trjectory set. The NSM R shown in Figure 1(c) genertes this trjectory set. Next we identify the trjectory models of deterministic stte mchines. This is used lter for designing supervisors which hve deterministic stte mchine representtion. Definition 1 A trjectory model (P m, P ) is sid to be deterministic if there exists deterministic stte mchine P := (X P, Σ, δ P, x 0 P, X m P ) such tht T m (P) = P m nd T (P) = P. An equivlent definition of deterministic generted trjectory set ws first given in [6, definition 12.4] (refer to Remrk 3 below). Note tht given trjectory model, the trce mp 10

13 cn be used to obtin the ssocited lnguge model. Conversely, given lnguge model (K m, K), the trjectory mp trj K : K 2 Σ (Σ 2 Σ ) cn be used to obtin the ssocited deterministic trjectory model: trj K (s) := Σ 0 (s)(σ 1 (s), Σ 1 (s))... (σ s (s), Σ s (s)), where Σ k (s) := {σ Σ s k σ K}, k s. Thus the kth refusl-set in the refusl-trce trj K (s) is the set of events tht re unexecutble in K fter the prefix of length k of s. Let det(k) := dom(trj K (K)) nd det m (K m, K) := dom(trj K (K m )). Proposition 2 Given lnguge model (K m, K), (det m (K m, K), det(k)) is the unique deterministic trjectory model with lnguge model (K m, K). Proof: From stndrd result, there exists deterministic stte mchine P such tht L m (P) = K m nd L(P) = K. Let (P m, P ) be the trjectory model of P. By [16, Proposition 3], det(k) is the unique deterministic generted trjectory model with generted lnguge K, so P = det(k). It is cler from the definition of trj K tht P st = trj K (K). By T5, it follows tht P m = dom(trj K (K) P m ). Thus, K m = L(P m ) = L(trj K (K) P m ), which implies tht trj K (K) P m = trj K (K m ), so P m = det m (K m, K). Remrk 3 It follows from Proposition 2 nd [6, Proposition 12.5] tht the definition of deterministic generted trjectory set given in [6, Definition 12.4] nd the definition given bove re in fct equivlent. 4 Prioritized Synchroniztion nd Augmenttion In this section we formlly introduce prioritized synchronous composition (PSC) which is used for interconnecting vrious systems including plnt nd supervisor. We show tht trjectory model retins sufficient informtion to infer the dequte behvior of the interconnected system for the behvior of the component systems. For PSC, ech system is ssigned priority set of events. When systems re interconnected vi PSC, n event cn occur in the composite system only if it cn occur in ech subsystem which hs priority over it. In this wy, subsystem cn prevent the occurrence of certin events, thereby implementing type of supervisory control. The following definition of PSC of two NSM s trivilly extends the one given by us in [16, Definition 9] by incorporting the notion of mrkings. The initil definition given by Heymnn [4] ws limited to NSM s without epsilon-moves. Definition 2 Let P := (X P, Σ, δ P, x 0 P, X m P ), Q := (X Q, Σ, δ Q, x 0 Q, X m Q ) be two NSM s hving priority sets A, B Σ respectively. The PSC of P nd Q is nother NSM which is denoted by P A B Q := R := (X R, Σ, δ R, x 0 R, X m R ), 11

14 where X R = X P X Q, x 0 R = (x 0 P, x 0 Q), XR m = XP m δ R : X R (Σ {ɛ}) 2 X R is defined s: X m Q, nd the stte trnsition function x r = (x p, x q ) X R : σ Σ : δ R (x r, σ) := δ P (x p, σ) δ Q (x q, σ) if δ P (x p, σ), δ Q (x q, σ) δ P (x p, σ) {x q } if δ P (x p, σ), σ R Q (x q ), σ B {x p } δ Q (x q, σ) if δ Q (x q, σ), σ R P (x p ), σ A otherwise, δ R (x r, ɛ) := [δ P (x p, ɛ) {x p }] [δ Q (x q, ɛ) {x q }] {(x p, x q )}. Thus n event is executed synchronously whenever both systems cn prticipte; however, it cn occur synchronously whenever one of the systems cn prticipte nd the second system refuses it but hs no priority over it. The bove definition gives the NSM resulting by composing the given NSM s. However, when only the trjectory models of the two NSM s re vilble, in order to obtin the trjectory model of the composed system we need to define the composition of the trjectory models, nd show tht this definition is consistent with the definition of composition for NSM s. This is wht we do next. For nottionl convenience, given Σ, Σ 1, Σ 2, Σ Σ, we define Σ Σ 1 Σ 2 Σ := (Σ Σ ) (Σ Σ 1 ) (Σ Σ 2 ). Given generted trjectory sets P, Q, the PSC of pir of trjectories e p P, e q Q ws first given in [6, Definition 13.1], which ws mde precise in [16, Definition 10] s follows: Definition 3 Let P, Q be generted trjectory sets with e p P, e q Q. Then the PSC of e p nd e q (with respect to P nd Q), denoted e p A B e q, is defined inductively on e p + e q s follows: Σ p, Σ q Σ such tht Σ p P, Σ q Q : Σ p A B Σ q := {Σ Σ p A B Σ q }, e p P, e q Q with e p + e q 1: (Let σ p := σ ep (e p ), σ q := σ eq (e q ), Σ p := Σ ep (e p ), Σ q := Σ eq (e q )) e p A B e q := T 1 T 2 T 3, where T 1 := {e(σ p, Σ ) e e ep 1 p A B e q ; Σ Σ p A B Σ q } if e p 1, σ p B nd e q (σ p, ) Q otherwise 12

15 T 2 := T 3 := {e(σ q, Σ ) e e p ; A B e eq 1 q ; Σ Σ p A B Σ q } if e q 1, σ q A nd e p (σ q, ) P {e(σ, Σ ) e e ep 1 p otherwise A B e eq 1 q ; Σ Σ p A B Σ q } if e p, e q 1 nd σ p = σ q := σ otherwise It should be noted tht e p A B e q is set of refusl-trces tht depends on the generted trjectory sets P, Q s well s on the prticulr trjectories e p, e q. The dependence on P, Q is not explicitly indicted in the nottion. The PSC of two zero-length refusl-trces Σ p P nd Σ q Q is obtined by computing Σ p A B Σ q = (Σ p Σ q ) (Σ p A) (Σ q B), i.e., n event is refused in the composed system if either it is refused in both the systems, or it is refused in system which hs priority over tht event. Next the PSC of two refusl-trces e p P, e q Q with t lest one of them of length more thn zero (so tht t lest one of them hs the form: e p = e ep 1 p (σ p, Σ p ), e q = e eq 1 q (σ q, Σ q )), is obtined by considering these three possible cses: (i) e p is of length more thn zero nd refusl-trce belonging to e ep 1 p A B e q hs lredy been executed in the composed system, nd t this point, the occurrence of the lst event of e p cnnot be blocked by Q (indicted by σ p B), nd Q cnnot prticipte in the occurrence of this event (indicted by e q (σ p, ) Q); (ii) e p is of length more thn zero nd refusl-trce belonging to e p A B e eq 1 q hs lredy been executed in the composed system, nd t this point, P cn neither block the occurrence of the lst event of e q, nor it cn prticipte in its occurrence; (iii) Both e p nd e q re of length more thn zero nd their finl events re sme (indicted by σ p = σ q := σ); refusl-trce belonging to e ep 1 p A B e eq 1 q hs lredy been executed in the composed system, nd t this point, σ is executble in both P nd Q. The definition of PSC of refusl-trces is extended to obtin the definition of the PSC of trjectory models: Definition 4 Let (P m, P ), (Q m, Q) be trjectory models with priority sets A, B Σ respectively. The PSC of (P m, P ) nd (Q m, Q), denoted (P m, P ) A B (Q m, Q), is the pir of refusl-trce sets (R m, R), where R m := e p A B e q, R := e p P m,e q Q m e p P,e q Q e p A B e q, where e p A B e q is with respect to P, Q in the definitions of both R m nd R. We will use the nottion (P m A B Q m, P A B Q) for (P m, P ) A B (Q m, Q). However, it must be kept in mind tht P m A B Q m implicitly depends on P nd Q. The following theorem proves tht the trjectory model retins sufficient system detil to support prioritized 13

16 synchronous composition. This result justifies the need for using trjectory models for representing nondeterministic systems tht re intercting vi prioritized synchroniztion such s plnt nd supervisor in the setting of supervisory control. (Heymnn [4] showed tht corresponding result does not hold for less detiled models such s lnguge or filures model.) Theorem 3 Let P, Q be NSM s nd A, B Σ. T m (P) A B T m (Q) = T m (P A B Q) nd T (P) A B T (Q) = T (P A B Q). Proof: The fct tht T (P) A B T (Q) = T (P A B Q) is proved in [16, Theorem 2]. For nottionl convenience, let R := P A B Q; we first prove tht T m (R) T m (P) A B T m (Q). Pick e T m (R). Then e T (R), nd there exists x r = (x p, x q ) δr(x T 0 R, e) XR m = δr(x T 0 R, e) (XP m XQ m ). It follows from [16, Corollry 5] tht given e nd x r, there exists e p T (P) nd e q T (Q) such tht e e p A B e q nd x r δ T P(x 0 P, e p ) δ T Q(x 0 Q, e q ). (1) It follows tht δp(x T 0 P, e p ) XP m nd δq(x T 0 Q, e q ) XQ m. This implies tht e p T m (P) nd e q T m (Q), proving tht e T m (P) A B T m (Q). Next we prove tht T m (P) A B T m (Q) T m (R). Pick e T m (P) A B T m (Q). Then there exist e p T m (P) nd e q T m (Q) such tht e e p A B e q. Since e p T (P) nd e q T (Q), it follows from [16, Corollry 5] tht δp(x T 0 P, e p ) δq(x T 0 Q, e q ) δ R (x 0 R, e). This implies tht [δp(x T 0 P, e p ) XP m ] [δq(x T 0 Q, e q ) XQ m ] δr(x T 0 R, e) XR m. (2) Since e p T m (P) nd e q T m (Q), we hve δ T P(x 0 P, e p ) X m P nd δ T Q(x 0 Q, e q ) X m Q. Thus, Eqution 2 implies tht δ T R(x 0 R, e) X m R, so e T m (R). Next we prove the ssocitivity property of PSC. This result is quite useful in the setting of supervisory control since plnt or supervisor my be composed of severl sub-plnts or sub-supervisors tht re intercting vi prioritized synchroniztion, nd their behvior should not depend on the order in which they re composed. First note from Definition 2 it is immedite tht for NSM s P, Q with priority sets A, B Σ respectively, the following holds for ech stte x r = (x p, x q ) X P X Q of the composed system R := P A B Q: R R (x r ) = R P (x p ) A B R Q (x q ) = [R P (x p ) R Q (x q )] [R P (x p ) A] [R Q (x q ) B]. Theorem 4 Let P, Q, R be NSM s nd A, B, C Σ. Then (P A B Q) A B C R = P A B C (Q B C R). Proof: Let S 1 := (P A B Q) A B C R, nd S 2 := P A B C (Q B C R). Then it follows from Definition 2 tht X S1 = X S2 = X P X Q X R := X S, x 0 S 1 = x 0 S 2 = (x 0 P, x 0 Q, x 0 R), X m S 1 = X m S 2 = X m P X m Q X m R, nd for ech x s = (x p, x q, x r ) X S : δ S1 (x s, ɛ) = δ S2 (x s, ɛ) = [δ P (x p, ɛ) {x p }] [δ Q (x q, ɛ) {x q }] [δ R (x r, ɛ) {x q }] {x s }. 14

17 It remins to show tht for ech x s = (x p, x q, x r ) X S nd σ Σ: It follows from Definition 2 tht δ P (x p, σ) δ Q (x q, σ) δ R (x r, σ) δ S1 (x s, σ) = δ S1 (x s, σ) = δ S2 (x s, σ). (3) if δ P (x p, σ), δ Q (x q, σ), δ R (x r, σ) δ P (x p, σ) {x q } δ R (x r, σ) if δ P (x p, σ), δ R (x r, σ), σ R Q (x q ), σ B {x p } δ Q (x q, σ) δ R (x r, σ) if δ Q (x q, σ), δ R (x r, σ), σ R P (x p ), σ A δ P (x p, σ) δ Q (x q, σ) {x r } if δ P (x p, σ), δ Q (x q, σ), σ R R (x r ), σ C δ P (x p, σ) {x q } {x r } if δ P (x p, σ), σ R Q (x q ) R R (x r ), σ B C {x p } δ Q (x q, σ) {x r } if δ Q (x q, σ), σ R P (x p ) R R (x r ), σ A C {x p } {x q } δ R (x r, σ) if δ R (x r, σ), σ R P (x p ) R Q (x q ), σ A B otherwise An expression for δ S2 (x s, σ) cn be nlogously obtined. Note tht in the fifth cluse of the expression for δ S1 (x s, σ), the condition σ R Q (x q ) R R (x r ) nd σ B C is equivlent to σ R Q (x q ) B C R R (x r ) nd σ B C. Similrly, in the sixth cluse of the expression for δ S1 (x s, σ), the condition σ R P (x p ) R R (x r ) nd σ A C is equivlent to σ R P (x p ) A C R R (x r ) nd σ A C. If similr simplifictions in the cluses of the expression for δ S2 (x s, σ) re performed, then Eqution 3 is esily proved. Appendix A lists some of the corollries of Theorems 3 nd 4. We conclude this section by extending the notion of ugmenttion first introduced in [16, Subsection 5.2]. The notion of ugmenttion is quite useful s under certin mild conditions tht re met in the setting of supervisory control, the PSC of systems cn be reduced to SSC (strict synchronous composition) of ppropritely ugmented systems. The underlying ide of ugmenttion is quite simple: Since n event tht belongs to the priority set of single system cn occur synchronously, if we ugment the other system by dding self-loops on such events, then the opertion of PSC reduces to tht of SSC of ugmented systems over those events tht belong to the priority set of t lest one system. Formlly, ugmenttion of n NSM with given event set D Σ is its PSC with the NSM D := ({x 0 D}, Σ, δ D, x 0 D, {x 0 D}), the trnsition function of which is defined s: { {x σ Σ {ɛ}, δ D (x 0 0 D, σ) := D } if σ D otherwise. In other words, D is deterministic stte mchine consisting of single mrked stte hving self-loops on events in D Σ. It is cler tht L m (D) = L(D) = D nd T m (D) = T (D) = det(d ). 15

18 Definition 5 Given n NSM P := (X P, Σ, δ P, x 0 P, X m P ) nd D Σ, the ugmented NSM with respect to D, denoted P D, is defined to be P D := P D. Given trjectory model (P m, P ), the ugmented trjectory model with respect to D, denoted (P m, P ) D, is defined to be (P m, P ) D := (P m, P ) (det(d ), det(d )). It follows from the bove definition tht the ugmented NSM P D is obtined by dding self-loops t ech stte of P on those events in D tht re refused t tht stte, i.e., P D := (X P, Σ, δ P D, x 0 P, XP m ), where the trnsition function is defined s: δ P (x, σ) if δ P (x, σ) x X P, σ Σ {ɛ} : δ P D(x, σ) := {x} if σ D R P (x) otherwise Using ((P m ) D, P D ) to denote (P m, P ) D, we hve (P m ) D = P m det(d ) nd P D = P det(d ). As is lwys the cse with the PSC of recognized trjectory sets, the expression for (P m ) D implicitly depends on the corresponding generted trjectory sets P nd det(d ). Clerly the trjectory model (T m (P D ), T (P D )) of the ugmented NSM P D is equl to ((T m (P)) D, (T (P)) D ). Also, since det(d ) cn lwys execute every event in D nd cn never execute ny event in Σ D, it follows tht given ny trjectory model (P m, P ), ny A Σ D, nd ny B D (P m ) D := P m det(d ) = P m A B det(d ), P D := P det(d ) = P A B det(d ). The following result shows tht the PSC of two systems is equivlent to the SSC (over the union of the two priority sets) of the ssocited ugmented systems. Proposition 3 Let (P m, P ), (Q m, Q) be trjectory models, nd A, B Σ. Then 1. P m A B Q m = (P m ) B A A B B Q m = (P m ) B A A B A B (Q m ) A B, 2. P A B Q = P B A A B B Q = P B A A B A B Q A B. Proof: We only prove the first prt; the second prt cn be proved nlogously. Agin, we only prove the first equlity s the second equlity follows from symmetry nd second ppliction of the first equlity. From the definition of ugmenttion nd ssocitivity of PSC (Corollry 8), we hve (P m ) B A A B B Q m = (P m A B A det((b A) )) A B B Q m = det((b A) ) B A A B (P m A B Q m ) = P m A B Q m, where the lst equlity follows from the two fcts: (i) The priority set of P m A B Q m is A B, nd B A A B, so det((b A) ) cnnot execute n event tht P m A B Q m does not execute. (ii) det((b A) ) cn lwys execute ech event in its priority set, so tht it cnnot block ny event in P m A B Q m. Note tht when A B = Σ s in the setting of supervisory control, then PSC cn be trnsformed into SSC of ugmented systems using the result of the bove proposition. 16

19 5 Supervisory Control Using PSC Supervisory control theory for nondeterministic systems in the setting of trjectory models nd PSC proposed by Heymnn [4] nd Heymnn-Meyer [6] ws rigorously developed in our previous work [16]. In this work, ll refusl-trces of the plnt were considered mrked nd the desired behvior ws specified by prefix-closed lnguge; thus the issue of blocking ws not investigted. In this section, we generlize the results in [16] to include non-closed specifictions nd rbitrry mrking of the plnt refusl-trces. Since trjectory models contin sufficient detil to support the opertion of prioritized synchroniztion, we use trjectory models, rther thn NSM s, to represent discrete event system. Unless otherwise specified, the trjectory model of the plnt nd tht of the supervisor re denoted by (P m, P ) nd (S m, S), nd the priority set of the plnt nd tht of the supervisor re denoted by A nd B respectively. The controlled (or closed-loop) system is (P m, P ) A B (S m, S). In certin situtions, plnt my consist of severl sub-plnts operting in prioritized synchrony. In such cse, if (Pi m, P i ) is the trjectory model of the ith sub-plnt, nd A i Σ is its priority set, then the trjectory model of the composed plnt is given by (P m, P ) := ( Ai Pi m, Ai P i ), where the nottion ( Ai Pi m, Ai P i ) is used to denote the PSC of sub-plnts {(Pi m, P i )}. (Since PSC is ssocitive (Corollry 8), ( Ai Pi m, Ai P i ) is well defined, nd it follows from Corollry 5 tht it is trjectory model.) The priority set of the composed plnt is given by A := i A i. In the setting of supervisory control, A = Σ u Σ c, where Σ u, Σ c Σ denote the sets of uncontrollble nd controllble events respectively [14, 15]; nd B = Σ c Σ d, where Σ d Σ denotes the set of so-clled driven [4] or forcible [3, 2] or commnd events [1]. By definition Σ u, Σ c nd Σ d re pirwise disjoint nd exhust the entire event set, i.e., A B = Σ. Since controllble events belong to the common priority set, they cnnot occur solely in the plnt or in the supervisor. This is consistent with the definition of controllble events in Rmdge- Wonhm theory [14] where lthough they originte in the plnt, their execution requires enblement by the supervisor. Only the plnt hs priority over the uncontrollble events, so they cn occur whenever the plnt cn prticipte. Agin this is consistent with the definition of uncontrollble events in Rmdge-Wonhm theory where uncontrollble events re lwys enbled by the supervisor. Driven events re dul to uncontrollble events. Only the supervisor hs priority over them, so they cn occur whenever the supervisor is ble to execute them, regrdless of whether the plnt cn prticipte. We begin by defining the notions of non-mrkingness nd non-blockingness which re desirble properties of supervisors. Informlly, supervisor is sid to be non-mrking if it does not ffect the mrking sttus of certin refusl-trce, i.e., if the execution of certin refusl-trce of controlled plnt leds to mrked stte of the plnt, then regrdless of wht stte is reched in the supervisor tht refusl-trce should be mrked refusl-trce of the controlled plnt. On the other hnd, supervisor is sid to be non-blocking if it is lwys possible to rech mrked stte of the controlled plnt from ny of its other sttes, i.e., the controlled system does not get blocked in one of its sttes tht is not mrked. 17

20 Definition 6 Given plnt (P m, P ) with priority set A Σ, supervisor, with trjectory model (S m, S) nd priority set B Σ, is sid to be non-mrking if S m = S; it is sid to be lnguge model non-blocking if pr(l(p m A B S m )) = L(P A B S); nd it is sid to be trjectory model non-blocking if pr(p m A B S m ) = P A B S. Note tht if supervisor is trjectory model non-blocking, then it is lso lnguge model non-blocking. On the other hnd, if plnt s well s supervisor re deterministic, nd the supervisor is lnguge model non-blocking, then it is lso trjectory model non-blocking. Remrk 4 It must be kept in mind tht lthough trjectory model non-blocking is stronger notion thn lnguge model non-blocking, it is possible to hve system which is trjectory model non-blocking yet it blocks. Consider for exmple simple NSM with event set Σ = {} nd consisting of three sttes, in which there is nondeterministic trnsition on event from the initil mrked stte to the other two sttes, only one of which is mrked; no other trnsition is defined. Then the recognized s well s generted trjectory set of this NSM is given by pr(dom( (, {})). Consequently, it is trjectory model non-blocking. However, fter executing the event in its initil stte, the system cn rech n unmrked stte from which no mrked stte is rechble. Thus the notion of trjectory model non-blocking is not dequte when blocking s well s non-blocking stte is rechble by the execution of the sme set of refusltrces. However, in prcticl setting this is unlikely, s whenever blocking nd non-blocking stte is reched by the execution of the sme trce due to the presence of nondeterminism, we expect the refusl-sets t the two sttes or the intermedite sttes to be different (s they re physiclly different sttes), so it is not possible to rech the two sttes by the sme set of refusl-trces. Next we obtin necessry nd sufficient condition for the existence of non-mrking nd lnguge model non-blocking supervisor. This result is then used to obtin necessry nd sufficient condition for the existence of non-mrking nd trjectory model non-blocking supervisor. Theorem 5 Let (P m, P ) be the trjectory model of plnt, A, B Σ with A B = Σ, nd K m L((P m ) Σ A ) with K m. Then there exists non-mrking nd lnguge model non-blocking supervisor with trjectory model (S, S) such tht L(P m A B S) = K m if nd only if Reltive-closure: pr(k m ) L((P m ) Σ A ) = K m Controllbility: pr(k m )(A B) L(P Σ A ) pr(k m ). In this cse, S cn be chosen to be det(pr(k m )). Proof: We begin with the proof for necessity. Suppose there exists non-mrking nd lnguge model non-blocking supervisor with trjectory model (S, S) such tht L(P m A B S) = K m. Then pr(k m ) = pr(l(p m A B S)) = L(P A B S), (4) 18

21 where the lst equlity follows from the supervisor being lnguge model non-blocking. It follows from Definitions 4 nd 5 tht (P m ) Σ A P Σ A, which implies tht pr(k m ) pr(l((p m ) Σ A )) L(P Σ A ). Hence it follows from the necessity prt of [16, Theorem 4] tht pr(k m )(A B) L(P Σ A ) pr(k m ) i.e., the controllbility property is stisfied. It remins to show tht the reltive-closure property is lso stisfied. We hve the following series of equlities: pr(k m ) L((P m ) Σ A ) = L(P A B S) L((P m ) Σ A ) = L((P A B S) Σ A P m ) = L((P A A P m ) A B S) = L((P A A P m ) Σ A ) L(S Σ B ), (5) where the first equlity follows from Eqution 4, the second equlity follows from Corollry 7 nd Proposition 3, the third equlity follows from ssocitivity of PSC (Corollry 8), nd the finl equlity follows from Corollry 7. On the other hnd, we hve K m = L(P m A B S) = L((P m ) Σ A ) L(S Σ B ), (6) where the lst equlity follows from Proposition 3 nd Corollry 7. It follows from Equtions 5 nd 6 tht in order to show tht the reltive-closure property is stisfied, it suffices to show L((P A A P m ) Σ A ) = L((P m ) Σ A ). We hve L((P A A P m ) Σ A ) = L((P A A P m ) A Σ A det((σ A) )) = L(P A Σ (P m A Σ A det((σ A) ))) = L(P A Σ (P m ) Σ A ) = L(P Σ A ) L((P m ) Σ A ) = L((P m ) Σ A ), where the first nd third equlities follow from the definition of ugmenttion, the second equlity follows from the ssocitivity of PSC (Corollry 8), nd the fourth equlity follows from Proposition 3. This completes the proof of necessity. Next we prove sufficiency. Suppose the reltive-closure nd controllbility properties re stisfied. Then it follows from the sufficiency prt of [16, Theorem 4] tht the non-mrking deterministic supervisor with trjectory model (S, S), where S := det(pr(k m )), yields L(P A B S) = pr(k m ), (7) nd S m is rbitrry. We select S m = S, so the supervisor is non-mrking. It follows from Eqution 7 tht pr(k m ) = L(P Σ A ) L(S Σ B ). (8) Using the reltive-closure property gives the following series of equlities: K m = pr(k m ) L((P m ) Σ A ) = [L(P Σ A ) L(S Σ B )] L((P m ) Σ A ) = L((P m ) Σ A ) L(S Σ B ) = L(P m A B S). 19

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