A General Architecture for Reliable Decentralized Supervisory Control of Discrete Event Systems

Transcription

1 Jont 48th IEEE Conference on Decson and Control and 28th Chnese Control Conference Shangha, P.R. Chna, December 16-18, 2009 WeA06.3 A General Archtecture for Relable Decentralzed Supervsory Control of Dscrete Event Systems Fuchun Lu and Ha Ln Abstract In ths paper, we study the relable decentralzed supervsory control of dscrete event systems (DESs) under the general archtecture, n whch the decson for controllable events employed s a combnaton of the conjunctve fuson and dsjunctve fuson rules. For a plant equpped wth n local supervsors, the noton of k-relable (1 k n) decentralzed supervsor s formalzed and nvestgated here. By k-relable decentralzed supervsor, we mean that the specfcaton can be acheved exactly even under possble falures of any no more than n k local supervsors. It s worth notng that the standard decentralzed supervsory control problem n the general archtecture [14] can be regarded as a specal case of k-relable decentralzed supervsory control wth k = n. The man contrbutons of the paper le on the proposed necessary and suffcent condtons for the exstences of k- relable decentralzed supervsor and nonblockng k-relable decentralzed supervsor n the context of the general archtecture, based on the notons of the Σuc-controllablty and k-relably Σc-coobservablty of a sublanguage. Ths represents a generalzaton of the results n [11]. Keywords: Dscrete event systems, supervsory control, relable decentralzed supervsor, conjunctve archtecture, dsjunctve archtecture. I. INTRODUCTION When a system consdered s physcally dstrbuted, the decentralzed supervsory control s commonly regarded as more effcent than the centralzed one [9], n whch there are a set of local supervsors, each makes control decsons based only on ts own drect observatons. The past decade has seen ncreasng research actvtes n the area of decentralzed supervsory control of dscrete event systems (DESs), such as [1], [3], [4], [7]-[10], [11]-[15]. In partcular, Yoo and Lafortune [14] presented a general archtecture for decentralzed supervsory control of DESs based on conjunctve and dsjunctve fuson rules for local decsons. Snce the general decentralzed archtecture was ntated by Yoo and Lafortune [14], t has been extensvely adopted n the lterature. Rohloff and Lafortune [9] presented a new approach of state estmaton and safe controllers synthess n supervsory control of DESs under the general archtecture. For the framework of [14], Park and Cho proposed the Ths work was supported by the Natonal Natural Scence Foundaton of Chna ( ), Natural Scence Foundaton of Guangdong Provnce ( ) of Chna, and Sngapore Mnstry of Educaton s AcRF Ter 1 fundng, TDSI, TL. F. Lu s wth Faculty of Computer, Guangdong Unversty of Technology, Guangzhou , Chna; and also wth Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore, , Sngapore lufch8@gmal.com H. Ln s wth Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore, , Sngapore elelh@nus.edu.sg exstence condton of a decentralzed supervsor for an uncertan DES modeled by a set of possble nondetermnstc automata wth nternal events [8]. Kumar and Taka [3] presented an nference-based ambguty management n decentralzed decson-makng n the general decentralzed archtecture. Reference [15] generalzed the archtecture of [14] to a condtonal archtecture, and studed the supervsor exstence of decentralzed supervsory control wth condtonal decsons. Park and Cho nvestgated the decentralzed supervsory control of DESs wth communcaton delays based on conjunctve and permssve decson structures [7]. Recently, the relable decentralzed supervsory control of DESs has been formulated n [11] [12]. For a system controlled by n local supervsors, a k-relable (1 k n) decentralzed supervsor requres that t exactly acheves the gven specfcaton under possble falures of any no more than n k local supervsors. The authors n [11] presented the condtons for the exstence of a k-relable decentralzed supervsor by means of the modfed controllablty and relable coobservablty. It s worth notng that the standard decentralzed supervsory control problem [2] can be regarded as a specal case wth k = n (.e., syntheszng a n-relable decentralzed supervsor). However, the decentralzed archtecture consdered n [11] [12] s the conjunctve archtecture. In ths paper, we nvestgate the followng relable decentralzed supervsory control of DESs n the general archtecture: For gven a system controlled by n local supervsors and a desred specfcaton, check whether there exsts a k-relable decentralzed supervsor n the general archtecture. If there exsts such decentralzed supervsor, how to desgn the local supervsors such that the decentralzed supervsor syntheszed s k-relable (1 k n). Frstly, we formalze the noton of k-relable decentralzed supervsor n the general archtecture. Roughly speakng, a decentralzed supervsor s sad to be k-relable f the local supervsors acheve exactly the specfcaton under possble falures of any no more than n k local supervsors, where the decsons for dsabled and enabled events are based on the conjunctve and dsjunctve fuson rules. Then the concepts of Σ uc -controllablty and k-relably Σ c -coobservablty of a language are ntroduced. In partcular, we present the necessary and suffcent condtons for the exstence of a k-relable decentralzed supervsor n the general archtecture by means of the Σ uc -controllablty and k-relably Σ c - coobservablty. In addton, we formulate the constructon of /09/$ IEEE 193

2 the local supervsors and the relable decentralzed supervsor based on the conjunctve and dsjunctve fuson rules. The approach proposed n ths paper s dfferent from those n the lterature. The framework of [11] s based on the conjunctve fuson rule for the decsons of dsabled events. Reference [12] extends the work of [11] to the case of the marked language specfcaton, whch s stll n the conjunctve archtecture. The archtecture adopted by ths paper s the general decentralzed archtecture proposed by Yoo and Lafortune [14]. The noton of k-relable decentralzed supervsor and the concepts of Σ uc -controllablty and k-relably Σ c -coobservablty ntroduced n ths paper are dfferent from those n [13]. As a result, the local supervsors and the decentralzed supervsor constructed n ths paper dffer from those of [13]. The rest of the paper s organzed as follows. Secton II recalls some prelmnares of DESs. In Secton III, we propose an approach to synthesze a part of local supervsors whch s used to deduce the necessary and suffcent condtons of the exstence of relable decentralzed supervsors. In Secton IV, we nvestgate the relable decentralzed supervsor n the general archtecture. In partcular, we present the necessary and suffcent condtons for the exstence of a (nonblockng) k-relable decentralzed supervsor. In order to llustrate the results proposed, an example s provded n Secton V. Fnally, n Secton VI, we summarze the man results of the paper and address some related ssues. II. PRELIMINARIES Consder a DES modeled by an automaton G = (Q, Σ, δ, q 0, Q m ), (1) where Q s the set of states, Σ s the fnte set of events, δ : Q Σ Q s the transton functon, q 0 Q s the ntal state, and Q m Q s the set of marked states. Let Σ denote the set of all fnte strngs over Σ, ncludng the empty strng ǫ. The transton functon δ can be extended to doman Q Σ n the followng recursve manner: δ(q, ǫ) = q and δ(q, sσ) = δ(δ(q, s), σ) for all s Σ and σ Σ. A subset of Σ s called a language. The language generated by G, denoted by L(G), s defned by L(G) = {s Σ : δ(q 0, s) s defned}, (2) and the language marked by G s defned as L m (G) = {s L(G) : δ(q 0, s) Q m }. (3) For a language K Σ, we denote the set of all prefxes of strngs n K as K,.e., K = {s Σ : st K for some t Σ }. (4) K s called to be prefx-closed f K = K; and K s called to be L m (G)-closed f K = K L m (G). In the decentralzed control archtecture [2], a system G s jontly controlled by n local supervsors S P1, S P2,, S Pn accordng to the fuson rule on the local decson actons, and each local supervsor can observe the locally observable events and can control the controllable events. Denote Σ,c and Σ,uc as the sets of locally controllable and uncontrollable events, respectvely; Σ,o and Σ,uo as the sets of locally observable and unobservable events, respectvely, where I = {1, 2,, n}. The projecton P : Σ Σ,o s defned nductvely as P (ǫ) = ǫ, and for σ Σ and s Σ, { P (s)σ, f σ Σ P (sσ) =,o, (5) P (s), otherwse. The sets of globally controllable and observable events are respectvely defned as Σ c = I Σ,c, Σ o = I Σ,o, and the sets of globally uncontrollable and unobservable events are defned respectvely as Σ uc = Σ Σ c and Σ uo = Σ Σ o. In ths paper, we consder the relable decentralzed supervsory control problem based on the general archtecture proposed by Yoo and Lafortune [14], whch can be depcted as Fg.1. In the settng of general decentralzed control archtecture [14], the local supervsors make local enable decson and local dsable decson, and the decson fuson for global enable and dsable events s a fxed combnaton of the conjunctve and dsjunctve fusons. Formally, the set of controllable events Σ c s further parttoned nto Σ c,e and Σ c,d,.e., Σ c = Σ c,e Σ c,d, where the default settng for controllable events n Σ c,e s enablement and the default settng for controllable events n Σ c,d s dsablement. The local decsons over Σ c,e are processed by the conjunctve fuson rule whle the local decsons over Σ c,d are processed by the dsjunctve fuson rule. In addton, we denote Σ,c,e = Σ,c Σ c,e and Σ,c,d = Σ,c Σ c,d. Conjunctve Fuson Dsjunctve Fuson S P1 S P2 S Pn G P 1 P 2 P n Fg. 1. The general decentralzed control archtecture. III. AN APPROACH OF SYNTHESIS FOR A PART OF LOCAL SUPERVISORS IN GENERAL ARCHITECTURE In order to llustrate the relable decentralzed supervsory control of DESs n the general archtecture, n ths secton, we present an approach to synthesze a part of local supervsors based on the conjunctve and dsjunctve fuson rules, and then nvestgate some man propertes of the synthess, whch wll be used to deduce the condtons of the exstence of relable decentralzed supervsor. 194

3 For σ Σ c, denote In(σ) = { I : σ Σ,c }, where I = {1, 2,, n}. Let A 2 I, defne Σ A,c = A Σ,c and Σ A,uc = Σ Σ A,c. For I, the local supervsor s defned as a functon S P : P (Σ ) Γ, where Γ = {γ 2 Σ : Σ uc (Σ c,e Σ,c ) γ, (Σ c,d Σ,c ) γ = }. (6) Usually, only the observable events can be observed by the local supervsors. Therefore, the local supervsors are supposed to make the same decson for the behavors wth the same projecton. Ths property s formally descrbed as follows. Defnton 1: Let P be a projecton. The local supervsor S P s called to be feasble f S P (P (s)) = S P (P (s )) holds for any s, s Σ wth P (s) = P (s ). Defnton 2: For A 2 I, the A-decentralzed supervsor n the general archtecture, denoted by S A, s defned as S A (s) = P Σc,e ( A S P (P (s))) P Σc,d ( A S P (P (s))) Σ A,uc, where P Σc,e : Σ Σ c,e and P Σc,d : Σ Σ c,d are projecton mappngs [14]. Defnton 3: The language generated by S A, denoted by L(G, S A ), s defned recursvely n the usual manner: ǫ L(G, S A ) and for any s Σ and σ Σ, sσ L(G, S A ) s L(G, S A ), sσ L(G), σ S A (s). (8) The marked language L m (G, S A ) = L(G, S A ) L m (G). Proposton 1: Let A, B 2 I. For language K L(G), f L(G, S A ) = K = L(G, S B ), then L(G, S A B ) = K. Proof: It can be verfed drectly by nducton on the length of the strngs n L(G, S A ), L(G, S B ) and K. Next, we present the condtons of the exstence of A- decentralzed supervsor S A satsfyng L(G, S A ) = K by ntroducng the followng notons of the controllablty and coobservablty of K. Defnton 4: Let A 2 I. A language K L(G) s sad to be Σ A,uc -controllable f KΣ A,uc L(G) K. Defnton 5: Let A 2 I. A language K L(G) s sad to be Σ A,c -coobservable n the general archtecture, f for any s K and any σ Σ A,c, the followng condtons hold: (C1) If σ Σ c,e, then sσ L(G) K mples (7) ( A In(σ))P 1 P (s)σ K = ; (9) (C2) If σ Σ c,d, then sσ K mples ( A In(σ))(P 1 P (s) K)σ L(G) K. (10) Remark 1: The defnton of coobservablty under the conjunctve archtecture s ntroduced by Barrett [1] and Cassandras and Lafortune [2], n whch language K s called to be coobservable, f for any s K and any σ Σ c that sσ L(G) K, then there s I such that σ Σ,c and P 1 P (s)σ K =. For the dsjunctve archtecture, the noton of coobservablty of K n [14] s defned as: for any s K and any σ Σ c that sσ K, then there s I such that σ Σ,c and (P 1 P (s) K)σ L(G) K. Therefore, when A = I and Σ c = Σ c,e, the above Defnton 5 n the general archtecture degenerates nto the coobservablty n the conjunctve archtecture; and f A = I and Σ c = Σ c,d, then Defnton 5 s consstent wth the coobservablty n the dsjunctve archtecture. Now we present a necessary and suffcent condton for the exstence of A-decentralzed supervsor by usng the notons of Σ A,uc -controllablty and Σ A,c -coobservablty. Theorem 1: Let A 2 I. For a nonempty language K L(G), there s an A-decentralzed supervsor S A such that L(G, S A ) = K f and only f K s Σ A,uc -controllable and Σ A,c -coobservable. Proof: ( ) For any s K and any σ Σ A,uc that sσ L(G), by Eq. (7) and L(G, S A ) = K, we have s L(G, S A ) and σ S A (s). Accordng to the defnton of L(G, S A ), sσ L(G, S A ),.e., sσ K. Therefore, KΣ A,uc L(G) K, that s, K s Σ A,uc -controllable. Next, we further verfy that K s Σ A,c -coobservable from the followng two cases. Case 1.1: For any s K and any σ Σ A,c, f σ Σ c,e and sσ L(G) K, then σ S A (s) for L(G, S A ) = K. That s, there s 0 A such that σ S P0 (P 0 (s))). From Eq. (6), σ Σ c,e Σ 0,c,.e., 0 In(σ). Moreover, f P 1 0 P 0 (s)σ K, then there exsts s Σ such that P 0 (s ) = P 0 (s) and s σ K. From L(G, S A ) = K, we have s σ L(G, S A ), whch ndcates that σ S A (s ). Note that σ Σ c,e, we obtan σ A S P (P (s )), and then σ S P0 (P 0 (s )). Due to P 0 (s ) = P 0 (s), by the feasblty condton (see Defnton 1), σ S P0 (P 0 (s)), whch s n contradcton wth σ S P0 (P 0 (s))). Therefore, P 1 0 P 0 (s)σ K =. Case 1.2: If σ Σ c,d and sσ K, then from L(G, S A ) = K, we have sσ L(G, S A ), whch mples σ S A (s). So σ A S P (P (s)) for σ Σ c,d. That s, there s j 0 A such that σ S Pj0 (P j0 (s)). Notce that (Σ c,d Σ j0,c) S Pj0 (P j0 (s)) =, we have σ Σ j0,c,.e., j 0 In(σ). By the defnton of S A and L(G, S A ) = K, t can be proved drectly that (P 1 j 0 P j0 (s) K)σ L(G) K. ( ) Assume that K s Σ A,uc -controllable and Σ A,c - coobservable, where A 2 I. For s Σ and each I, defne the local supervsor S P (P (s)) as follows: S P (P (s)) = {σ Σ,c,d : (P 1 P (s) K)σ L(G) K} {σ Σ,c,e : P 1 P (s)σ K } (Σ c,e Σ,c ) Σ uc, (11) and the A-decentralzed supervsor S A s defned as Eq. (7). In order to prove L(G, S A ) = K,.e., s L(G, S A ) ff s K for all s Σ, we show t by nducton on the length s. If s = 0,.e., s = ǫ, the base case holds obvously. Suppose that s L(G, S A ) ff s K for any strng s wth 195

4 s n. The followng s to prove t for sσ where s = n and σ Σ. Let sσ L(G, S A ). By the defnton of L(G, S A ) and nducton hypothess, we have s K, sσ L(G) and σ S A (s). We verfy sσ K from the followng cases. Case 2.1: If σ Σ A,uc, then sσ K because of the Σ A,uc -controllablty of K. Case 2.2: If σ Σ A,c Σ c,e, then we show sσ K by contradcton. Snce K s Σ A,c -coobservable, f sσ K, then there s A In(σ) such that P 1 P (s)σ K =. From Eq. (11), σ S P (P (s)) and then σ A S P (P (s)), whch s n contradcton wth σ S A (s). Case 2.3: If σ Σ A,c Σ c,d, then σ S A (s), whch ndcates that σ A S P (P (s)). As a result, there s A such that σ S P (P (s)). From Eq. (11), we have (P 1 P (s) K)σ L(G) K. Notce that sσ (P 1 P (s) K)σ L(G), so sσ K. Conversely, let sσ K. We check that sσ L(G, S A ) from the followng cases. Case 3.1: If σ Σ A,uc, then σ S A (s) from the defnton of S A, whch mples sσ L(G, S A ) snce sσ L(G) and the nducton hypothess s L(G, S A ). Case 3.2: If σ Σ A,c Σ c,e, t s not dffcult to prove sσ L(G, S A ) wth the smlar process of Case 2.2. Case 3.3: If σ Σ A,c Σ c,d, then from sσ K and the Σ A,c -coobservablty of K, there s A In(σ) such that (P 1 P (s) K)σ L(G) K. Accordng to the defnton of S P,.e., Eq. (11), σ S P (P (s)) and then σ A S P (P (s)). Consequently, σ S A (s). By Defnton 3, we have sσ L(G, S A ). Defnton 6: Let A 2 I. The A-decentralzed supervsor S A s called to be nonblockng f L m (G, S A ) = L(G, S A ). Theorem 2: Let A 2 I. For a nonempty language K L(G), there s a nonblockng A-decentralzed supervsor S A such that L(G, S A ) = K and L m (G, S A ) = K f and only f K s Σ A,uc -controllable, Σ A,c -coobservable and L m (G)- closed. Proof: ( ) The Σ A,uc -controllablty and Σ A,c - coobservablty of K have been proved n Theorem 1. From L(G, S A ) = K and L m (G, S A ) = K, t s easy to show that K s L m (G)-closed snce K = L m (G, S A ) = L(G, S A ) L m (G) = K L m (G). ( ) We defne the local supervsors and A-decentralzed supervsor as Eq. (11) and Eq. (7), respectvely. By Theorem 1, we have L(G, S A ) = K. On the other hand, snce K s L m (G)-closed, we have K = K L m (G) = L(G, S A ) L m (G) = L m (G, S A ). As a result, L m (G, S A ) = K = L(G, S A ), that s, S A s nonblockng. IV. RELIABLE DECENTRALIZED SUPERVISORY CONTROL IN GENERAL ARCHITECTURE Based on the results presented n Secton 3, we are ready to nvestgate the relable decentralzed supervsor under the general archtecture, n whch a system G s jontly controlled by n local supervsors S P1, S P2,, S Pn accordng to a fxed combnaton of the conjunctve and dsjunctve fusons on the local decson actons. Defnton 7: Let K L(G) be a nonempty language and 1 k n. The decentralzed supervsor S dec s sad to be k-relable n the general archtecture, f L(G, S A ) = K for any A 2 I wth A k, where A represents the number of elements of A, and S A s the A-decentralzed supervsor n the general archtecture. For I, denote Σ,uc = Σ Σ,c and Σ,c = {σ Σ,c : In(σ) n k + 1}. Let A 2 I, defne Σ A,c = A Σ,c and Σ A,uc = Σ Σ A,c. For the sake of smplcty, when A = I, we denote Σ c = Σ I,c, Σ uc = Σ I,uc, and S dec = S I. Defnton 8: A language K L(G) s sad to be Σ uc - controllable f K Σ uc L(G) K. Defnton 9: Let 1 k n. A language K L(G) s sad to be k-relably Σ c -coobservable n the general archtecture, f A s,σ n k + 1 for any s K and any σ Σ c, where A s,σ = A s,σ,1 A s,σ,2, and A s,σ,1 = { In(σ) : P 1 P (s)σ K = }, (12) A s,σ,2 = { In(σ) : (P 1 P (s) K)σ L(G) K}. (13) Remark 2: The above defnton extends the correspondng noton n [11] to the general archtecture. In partcular, when Σ c = Σ c,e, t degenerates nto the noton of relable ( Σ c, k)-coobservablty n the conjunctve archtecture [11]. Lemma 1: Let 1 k n and K L(G). There s a k-relable decentralzed supervsor S dec n the general archtecture, f and only f, K s Σ A,uc -controllable and Σ A,c -coobservable for any A I k, where I k = {A 2 I : A = k}. Proof: ( ) Assume that there s a k-relable decentralzed supervsor S dec, then L(G, S A ) = K for any A 2 I wth A k. So L(G, S A ) = K for any A I k. By Theorem 1, we know that K s Σ A,uc -controllable and Σ A,c - coobservable. ( ) Assume that K s Σ A,uc -controllable and Σ A,c - coobservable for any A I k. We defne the local supervsor S P (P (s)) as Eq. (11) and the decentralzed supervsor S dec s the same as Eq. (7) wth A = I. Next, we prove that S dec s k-relable,.e., L(G, S B ) = K for any B 2 I wth B k. If B = k, then from the assumpton, K s Σ B,uc -controllable and Σ B,c -coobservable. By Theorem 1, we have L(G, S B ) = K. If B > k, then there are B 1, B 2,,B m such that B = B 1 B 2 B m and B I k for each B. By the assumpton, K s Σ B,uccontrollable and Σ B,c-coobservable for each B. Accordng to Theorem 1, we have L(G, S B1 ) = L(G, S B2 ) = = L(G, S Bm ) = K, 196

5 whch mples L(G, S B ) = K accordng to Proposton 1. So S dec s k-relable. Lemma 2: Let 1 k n and K L(G). K s Σ uc - controllable and k-relably Σ c -coobservable, f and only f, K s Σ A,uc -controllable and Σ A,c -coobservable for any A I k, where I k = {A 2 I : A = k}. Proof: ( ) We frst prove that K s Σ uc -controllable,.e., K Σ uc L(G) K. Denote then Σ c (k) = {σ Σ c : In(σ) n k}, K Σ uc L(G) = (KΣ c (k) L(G)) (KΣ uc L(G)). For any sσ KΣ c (k) L(G), we have σ Σ c and In(σ) n k, whch shows that there s B I k such that σ Σ B,uc. Notce that K s Σ B,uc -controllable,.e., KΣ B,uc L(G) K, so sσ K. That s, KΣ c (k) L(G) K. On the other hand, KΣ uc L(G) K snce KΣ uc L(G) KΣ B,uc L(G). Therefore, K Σ uc L(G) K. Next, we verfy that K s k-relably Σ c -coobservable by contradcton. Suppose that there s s K and σ Σ c satsfyng A s,σ n k, then In(σ) n k + 1 and In(σ) A s,σ 1. Therefore, there s j In(σ) A s,σ and B I k such that A s,σ B = and j B, whch mples σ Σ B,c. Note that K s Σ B,c -coobservable, for the above s and σ, f σ Σ c,e and sσ L(G) K, then there exsts l B satsfyng l In(σ) and P 1 l P l (s)σ K =,.e., l A s,σ,1 A s,σ. Hence l A s,σ B, whch s n contradcton wth A s,σ B =. On the other sde, f σ Σ c,d and sσ K, then there exsts h B satsfyng h In(σ) and (P 1 h P h(s) K)σ L(G) K,.e., h A s,σ,2 A s,σ. Hence h A s,σ B, whch s also n contradcton wth A s,σ B =. ( ) We frst prove that K s Σ A,uc -controllable for any A I k. Snce Σ A,uc = Σ uc {σ Σ c : σ Σ A,c } Σ uc {σ Σ c : In(σ) n k} = Σ uc, (14) where A I k. Therefore, KΣ A,uc L(G) K Σ uc L(G) K by the Σ uc -controllablty of K. Next, we prove that K s Σ A,c -coobservable for any A I k. For any s K and any σ Σ A,c, the proof s completed by the followng two cases. Case 1: If σ Σ A,c Σ c,.e., In(σ) n k + 1, then A s,σ n k + 1 snce K s k-relably Σ c -coobservable. Notce that A = k, we have A A s,σ, that s, there s A such that In(σ) and A s,σ,1 A s,σ,2. When σ Σ c,e and sσ L(G) K, by Eq. (13), t s obtaned that A s,σ,2. Therefore, A s,σ,1,.e., P 1 P (s)σ K =. On the other hand, when σ Σ c,d and sσ K, by Eq. (12), we have A s,σ,1 for sσ P 1 P (s)σ K. So, A s,σ,2,.e., (P 1 P (s) K)σ L(G) K. We complete the proof of Σ A,c -coobservablty of K n the frst case. Case 2: If σ Σ A,c (Σ A,c Σ c ),.e., σ Σ A,c and σ Σ uc, by the Σ uc -controllablty of K, t can be verfed straght that the condtons of Defnton 5 hold. From Lemma 1 and Lemma 2, we have the followng necessary and suffcent condtons for the exstence of a k- relable decentralzed supervsor n the general archtecture. Theorem 3: Let 1 k n and K L(G). There s a k- relable decentralzed supervsor n the general archtecture, f and only f, K s Σ uc -controllable and k-relably Σ c - coobservable. Proof: It s a combnaton of Lemma 1 and Lemma 2. Remark 3: The exstence condtons of Theorem 3 generalze the results of [11] to the general archtecture. When Σ c = Σ c,e, Theorem 3 degenerates nto the necessary and suffcent condtons of k-relable decentralzed supervsor n conjunctve archtecture [11]. Theorem 4: Let 1 k n and K L(G). There s a nonblockng k-relable decentralzed supervsor S dec n the general archtecture such that L m (G, S A ) = K for any A 2 I wth A k, f and only f, K s Σ uc -controllable, k-relably Σ c -coobservable and L m (G)-closed. Proof: ( ) Due to Theorem 3, we only need to prove that K s L m (G)-closed. From Defnton 7 and L m (G, S A ) = K, we have K = L m (G, S A ) = L(G, S A ) L m (G) = K L m (G). ( ) By Theorem 3, there s a k-relable decentralzed supervsor S dec such that L(G, S A ) = K for any A 2 I wth A k. On the other hand, snce K s L m (G)-closed, K = K L m (G) = L(G, S A ) L m (G) = L m (G, S A ). Therefore, K = L m (G, S dec ) and L m (G, S dec ) = K = L(G, S dec ), that s, S dec s nonblockng. Remark 4: Theorem 4 s a generalzaton of one of the man results presented n [14]. In fact, the necessary and suffcent condtons for the exstence of nonblockng decentralzed supervsor n [14] (Theorem 1 on page 343 of [14]) are exactly consstent wth those of nonblockng n- relable decentralzed supervsor of Theorem 4 (.e., k = n n Theorem 4). V. AN ILLUSTRATIVE EXAMPLE Theorem 3 shows that we can check the exstence of the k-relable decentralzed supervsor by means of testng the Σ uc -controllablty and k-relably Σ c -coobservablty of K. Next, an llustratve example s gven n ths secton. Example 1: Consder a DES G modeled by an automaton shown n Fg. 2. Let n = 3 (.e., I = {1, 2, 3}) and Σ 1,o = {σ 1, σ 2, σ 5 }; Σ 1,c,e = {σ 1 }, Σ 1,c,d = {σ 2 }; Σ 2,o = {σ 1, σ 4 }; Σ 2,c,e = {σ 1 }, Σ 2,c,d = {σ 4 }; Σ 3,o = {σ 2, σ 3 }; Σ 3,c,e = {σ 3 }, Σ 3,c,d = {σ 2 }. 197

6 q 1 q σ 1 σ 3 0 σ σ 4 2 q 5 q 2 q 6 q 3 σ 5 σ 2 σ 5 σ 1 q 7 q 4 q 8 Fg. 2. A DES G. Consder language K = σ 1 + σ 2 + σ 4 σ 5 + σ 3 σ 5, then K L(G). In the followng, we verfy that there s a 2- relable decentralzed supervsor by checkng that K s Σ uc - controllable and 2-relably Σ c -coobservable, where Σ c = {σ Σ c : In(σ) 2} = {σ 1, σ 2 }, and Σ uc = Σ Σ c = {σ 3, σ 4, σ 5 }. (1) K s Σ uc -controllable because K Σ uc L(G) = {σ 3, σ 4, σ 3 σ 5, σ 4 σ 5 } K. (2) Next we prove that K s 2-relably Σ c -coobservable,.e., for any s K and any σ Σ c, A s,σ 2 holds. In fact, on the one hand, only σ = σ 1 and s = σ 4 σ 5 satsfes s K, σ Σ c, σ Σ c,e and sσ L(G) K, and then and A s,σ,1 = { In(σ) : P 1 P (s)σ K = } = {1, 2} A s,σ,2 = { In(σ) : (P 1 P (s) K)σ L(G) K} =. Therefore, A s,σ = A s,σ,1 A s,σ,2 = 2. On the other hand, only σ = σ 2 and s = ǫ satsfes s K, σ Σ c, σ Σ c,d and sσ K, n ths case, we can calculate A s,σ,1 = and A s,σ,2 = {1, 3}. Therefore, A s,σ = 2. By Defnton 9, K s 2-relably Σ c -coobservable. Therefore, accordng to Theorem 3, there s a 2-relable decentralzed supervsor n the general archtecture. In fact, from Eq. (11), the local supervsors can be desgned as follows: {σ 1, σ 2, σ 3, σ 5 }, f P 1 (s) = ǫ, S P1 (P 1 (s)) = {σ 3, σ 5 }, f P 1 (s) = σ 5, {σ 2, σ 3, σ 5 }, otherwse. { {σ1, σ S P2 (P 2 (s)) = 3, σ 4, σ 5 }, f P 2 (s) = ǫ, {σ 3, σ 4, σ 5 }, otherwse. S P3 (P 3 (s)) = {σ 1, σ 2, σ 3, σ 5 }, f P 3 (s) = ǫ, {σ 1, σ 5 }, f P 3 (s) = σ 3, {σ 1, σ 2, σ 5 }, otherwse. Consequently, we may check straght that the decentralzed supervsor s 2-relable, snce L(G, S A ) = {ǫ, σ 1, σ 2, σ 3, σ 4, σ 3 σ 5, σ 4 σ 5 } = K for any A 2 I wth A 2. to synthesze part of local supervsors, and formulated the noton of k-relable decentralzed supervsor n the general archtecture. Then the concepts of the Σ uc -controllablty and k-relably Σ c -coobservablty of a sublanguage were ntroduced. Based on these new concepts, necessary and suffcent condtons for the exstence of a (nonblockng) k-relable decentralzed supervsor were presented n the context of the general archtecture. Wth the results obtaned n ths paper, we wll consder the relable robust nonblockng supervsory control of DESs wth communcaton [1] and contnue our prevous work of [5], [6] to nvestgate the relable control for stochastc dscrete event systems n the subsequent work. REFERENCES [1] G. Barrett and S. Lafortune, Decentralzed Supervsory Control wth Communcatng Controllers, IEEE Trans. Automat. Contr., vol. 45, pp , Oct [2] C.G. Cassandras and S. Lafortune, Introducton to Dscrete Event Systems. Boston, MA: Kluwer, [3] R. Kumar and S. Taka, Inference-Based Ambguty Management n Decentralzed Decson-Makng: Decentralzed Control of Dscrete Event Systems, IEEE Trans. Automat. Contr., vol. 52, no. 10, pp , [4] F. Ln and W.M. Wonham, Decentralzed Control and Coordnaton of Dscrete Event Systems wth Partal Observaton, IEEE Trans. Automat. Contr., vol. 35, pp , Dec [5] F.C. Lu, D.W. Qu, H. Xng, and Z. Fan, Decentralzed Dagnoss of Stochastc Dscrete Event Systems, IEEE Trans. Automat. Contr., vol. 53, no. 2, pp , [6] F.C. Lu and D.W. Qu, Safe Dagnosablty of Stochastc Dscrete Event Systems, IEEE Trans. Automat. Contr., vol. 53, no. 5, pp , [7] S.J. Park and K.H. Cho, Decentralzed Supervsory Control of Dscrete Event Systems wth Communcaton Delays Based on Conjunctve and Permssve Decson Structures, Automata, vol. 43, pp , [8] S.J. Park and K. H. Cho, Decentralzed Supervsory Control of Nondetermnstc Dscrete Event Systems: The Exstence Condton of a Robust and Nonblockng Supervsor, Automatca, vol. 43, pp , [9] K. Rohloff and S. Lafortune, On the Synthess of Safe Control Polces n Decentralzed Control of Dscrete Event Systems, IEEE Trans. Automat. Contr., vol. 48, no. 6, pp , [10] K. Rude and W. M. Wonham, Thnk Globally, Act Locally: Decentralzed Supervsory Control, IEEE Trans. Automat. Contr., vol. 37, no. 11, pp , Nov [11] S. Taka and T. Usho, Relable Decentralzed Supervsory Control of Dscrete Event Systems, IEEE Trans. Syst., Man, Cubern.-B: Cubern., vol. 30, no. 5, pp , [12] S. Taka and T. Usho, Relable Decentralzed Supervsory Control for Marked Language Specfcatons, Asan J. Contr., vol. 5, no. 1, [13] S. Taka and T. Usho, Relable Decentralzed Supervsory Control of Dscrete Event Systems wth the Conjunctve and Dsjunctve Fuson Rules, n Proc Amer. Contr. Conf., June 2003, pp [14] T.-S. Yoo and S. Lafortune, A General Archtecture for Decentralzed Supervsory Control of Dscrete-Event Systems, Dscrete Event Dynamc Systems: Theory and Applcatons, 12(3), , [15] T.-S. Yoo and S. Lafortune, Decentralzed Supervsory Control wth Condtonal Decsons: Supervsor Exstence, IEEE Trans. Automat. Contr., 49(11), , VI. CONCLUSION In ths paper, we have nvestgated the relable decentralzed supervsory control problem wthout communcaton based on a combnaton of the conjunctve and dsjunctve fuson rules, whch has generalzed the results n the conjunctve archtecture [11]. Frstly, we have presented an approach 198