Decentralized supervisory control of discrete event ... - Semantic Scholar
Automatica 43 (2007) 738 – 743 www.elsevier.com/locate/automatica
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Decentralized supervisory control of discrete event systems with communication delays based on conjunctive and permissive decision structures夡 Seong-Jin Park a , Kwang-Hyun Cho b,c,∗ a Department of Electrical and Computer Engineering, Ajou University, Suwon 443 749, Korea b College of Medicine, Seoul National University, Jongno-gu, Seoul 110 799, Korea c Bio-MAX Institute, Seoul National University, Gwanak-gu, Seoul 151 818, Korea
Received 11 October 2005; received in revised form 18 March 2006; accepted 27 October 2006
Abstract In many practical discrete event systems (DESs), some unexpected and uncontrollable events can subsequently occur before a proper control action is actually applied to a plant due to communication delays. For such DESs, this paper investigates necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor that can correctly achieve a given language specification when the decentralized supervisor is assumed to have a conjunctive and permissive decision structure. In particular, this paper presents a notion of delay-coobservability for a given language specification and shows that it is a key condition for the existence of such a decentralized supervisor. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete event systems; Decentralized supervisors; Communication delays; Delay-coobservability
1. Introduction Most decentralized supervisory control schemes for discrete event systems (DESs) including Rudie and Wonham (1992) and Yoo and Lafortune (2002, 2004) have been developed based on the assumption that the control action issued by a decentralized supervisor can be applied to a plant without any delay. In other words, it has been commonly assumed that if a certain event occurs in the plant then no further event occurs until the information of previous occurrence is processed and a proper control decision is made and delivered to the plant through a communication channel. In many practical situations, there are, however, non-negligible delays in sensing, communicating, and 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Karl Henrik Johansson under the direction of Editor André Tits. ∗ Corresponding author. Bio-MAX Institute, Seoul National University, 3rd Floor, IVI, San 4-8, Bongcheon 7-dong, Gwanak-gu, Seoul 151 818, Republic of Korea. Tel.: +82 2 887 2650; fax: +82 2 887 2692. E-mail addresses: [email protected] (S.-J. Park), [email protected] (K.-H. Cho).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.10.016
actuating that should be considered for decentralized supervisory control in particular. Supervisory control of DESs with communication delays has been investigated within several contexts. For instance, Balemi (1992) considered the supervisor synthesis problem of input/output DESs with communication delays when a partial specification is given by a language over an output event set. Debouk, Lafortune, and Teneketzis (2003) considered the problem of decentralized failure diagnosis under a communication delay. Ricker and Schuppen (2001) investigated the decentralized diagnosis problem of timed DESs with communication delays among the diagnosers. Recently, Park and Cho (2006) studied the problem of centralized supervisory control to achieve a given language specification under communication delays and presented the existence conditions. On the other hand, considering the bounded or unbounded communication delays due to the buffer queuing of messages transmitted among the local supervisors, Tripakis (2004) investigated the decentralized supervisory control problem when a simple specification called responsiveness is imposed on a DES. It has been, however, not considered in either Park and Cho (2006) or Tripakis (2004) what the necessary and sufficient conditions
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are for the existence of a decentralized nonblocking supervisor that can correctly achieve a given language specification under communication delays. In this paper, we consider the decentralized supervisory control problem for DESs with communication delays, in which a series of uncontrollable events can unexpectedly occur before a proper control action is actually applied to a plant. The delays can occur in data processing elements such as sensors and actuators, and also in communication channels between controllers and plants; however, the message delivery between local supervisors in Tripakis (2004) is not to be considered in this paper. In addition, to illuminate the main features of a decentralized supervisory control under delays in a simplest way, we adopt in this paper only the conjunctive and permissive decision structures (Rudie & Wonham, 1992; Yoo & Lafortune, 2002). Within this framework, we investigate the existence conditions of a nonblocking decentralized supervisor that can correctly achieve a given language specification. To this end, we further make some assumptions as follows: (i) every controllable event can occur only if it is enabled by a decentralized supervisor, (ii) some uncontrollable events can be unobservable to all local supervisors while all locally controllable events are locally observable, and (iii) the number of possible subsequent occurrence of uncontrollable events in a plant is limited within a finite bound. Based on these assumptions, we present a notion of delay-coobservability for a given language specification and show that it is a key condition for the existence of a decentralized supervisor that can achieve the specification. 2. Main results It has been assumed in the conventional supervisory control (Ramadge & Wonham, 1987; Rudie & Wonham, 1992; Yoo & Lafortune, 2002, 2004) that as an event occurs in a plant it is promptly observed by a supervisor and the control action for an observed event string is also directly applied to the plant without any delay. This framework is only realistic when a plant is tightly coupled with its supervisor such that there is no communication delay between them. Plants are, however, often controlled by remote supervisors through communication networks. In such environments, the messages delivering observation or supervision can experience non-negligible transmission delays through the communication networks. Moreover, data processing time in sensors or motor operation time in actuators can sometimes cause non-negligible delays as well. As a result, a series of subsequent uncontrollable events (e.g., unobservable faults, completion of certain tasks, etc.) may unexpectedly occur before a proper control action is transmitted and applied to the plant. In this paper, we consider the problem of decentralized supervisory control of DESs with such communication delays as shown in Fig. 1, where a1 , . . . , aD denote uncontrollable events. In Fig. 1, it is illustrated that a proper decentralized supervisory control action Sdec (t) for the string t is actually applied to the plant after the subsequent occurrences of ai , i = 1, . . . , D. The plant to be controlled is modeled by a finite state automaton G = (Q, , q0 , , Qm ) where Q is the set of states,
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Fig. 1. Decentralized supervisory control under delays.
is the set of events, q0 is the initial state, : Q × → Q is the state transition (partial) function, and Qm is the set of marked states. Let ∗ denote the set of all finite strings of elements in , including the empty string . For s, t ∈ ∗ , we denote st as a concatenation of the strings s and t. We note that the function can be extended to ∗ by defining (q, ) := q and (q, s) := ((q, s), ) for all s ∈ ∗ and ∈ . The prefix closure of a language L(⊂ ∗ ) is defined as pr(L) := {t ∈ ∗ |tu ∈ L for some u ∈ ∗ }, and L is said to be prefix-closed (or closed) if L = pr(L). The closed behavior and the marked behavior of G are defined by L(G) := {s ∈ ∗ |(q0 , s) (is defined)} and Lm (G) := {s ∈ ∗ |(q0 , s)! ∈ Qm }, respectively. For s ∈ ∗ and L ⊂ ∗ , L (s) := { ∈ |s ∈ pr(L)}. The plant G is controlled by joint actions of n local supervisors S1 , . . . , Sn according to the conjunctive fusion rule on the permissive local decision actions defined under communication delays. Without communication delays, the conjunctive fusion rule means that a controllable event is permitted to occur only if there is no disabling decision over that event by any of the local supervisors; the permissive local decision rule implies that the control action of a local supervisor under insufficient information is an enablement of the event (Rudie & Wonham, 1992; Yoo & Lafortune, 2002). However, in the presence of communication delays, these two decision rules should be redefined to describe the control actions in a proper way. In this paper, we explicitly define the conjunctive fusion rule and the permissive local decision rule under communication delays. Other decision structures such as disjunctive and anti-permissive rules (Yoo & Lafortune, 2002, 2004) are not considered in this paper, but the results obtained can be easily extended for such decision structures. Each local supervisor Si can observe the set of locally observable events denoted by o,i and can control the set of
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locally controllable events denoted by c,i . Each local supervisor is designed to achieve a given language specification K ⊆ L(G) with a priori information including the uncontrolled behavior L(G) and the desired behavior K. Each block Pi represents the usual projection mapping Pi : → o,i . We denote uo := \ o and uc := \c where o := and := o,i c i∈I i∈I c,i where I := {1, . . . , n}. A language K ⊆ L(G) is controllable with respect to (w.r.t.) G if pr(K)uc ∩ L(G) ⊆ pr(K), and it is Lm (G)-closed if pr(K) ∩ Lm (G) = K (Ramadge & Wonham, 1987). Before proceeding to the main results, we summarize in the following some assumptions that are made to develop our ideas: • Every controllable event is disabled by default and is permitted to occur only if it is enabled by a decentralized supervisor. Specifically, a controllable event in the supervised system can occur only if it is enabled by all local supervisors according to the conjunctive fusion rule. From the permissive decision rule, the enabling decision of each local supervisor implies that the controllable event is legal for at least one possible trajectory estimated by the local supervisor. In other words, a controllable event cannot occur if any of the local supervisors disables the event, and the disabling decision of the local supervisor implies that the controllable event must be illegal for all possible estimations of the local supervisor. • Every locally controllable event is locally observable, i.e. c,i ⊆ o,i . • The number of possible subsequent occurrence of uncontrollable events in a plant G is limited within a finite bound. • If an event a occurs before an event b in a plant, then a is observed by local supervisors always before b. In addition, control commands issued by a decentralized supervisor are applied to the plant without changing their order. The first assumption has been made to exclude any possible occurrence of subsequent controllable events prior to the application of a control action to the plant. Hence, only uncontrollable events can subsequently occur before a proper control action is applied to the plant. This implies that the plant does not suffer from any unexpected occurrence of events prior to the control action if only controllable events are defined in the plant. Following the framework of Ramadge and Wonham (1987), the supervisor adopted in this paper controls a plant through enabling controllable events without employing auxiliary forcing mechanisms and thereby the supervised system chooses which event to generate. However, the first assumption differs from that of Ramadge and Wonham in that controllable events do not occur spontaneously but occur as responses to enabling requests imposed by the supervisor. This assumption has been already employed to reflect the practical aspect of implementing supervisory control systems (Lauzon, Mills, & Benhabib, 1997). The other three assumptions have been made to deliver the main idea in a clear way without loss of generality. Let us present the design method of a decentralized supervisor and consider the behavior of a closed-loop system under
a
b a
b,c
u1 c
b
a u2
a,c
Fig. 2. An example plant for illustration of the design of local supervisors.
communication delays. To achieve a given language specification K ⊆ Lm (G) of a plant G, we adopt the local supervisors designed according to the following definition. Definition 1. For the occurrence of a string s ∈ pr(K), the set of enabled events for the observation Pi (s) of a local supervisor Si is defined as follows: Si (Pi (s)) := { ∈ c,i |s u ∈ pr(K) for some s ∈ ∗ and u ∈ ∗uc such that Pi (s ) = Pi (s) and K (s ) ∩ c,i = ∅} ∪ (c \c,i ), which is also called a local supervisory control action following the permissive local decision rule under communication delays. This design rule shows how the permissive decision rule in the absence of delays (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) can be extended in the situation when uncontrollable events can unexpectedly occur before a proper control action is applied to the plant due to communication delays. It states that, for each estimation s of the string s, if there are locally controllable legal events determined to occur immediately after the estimation (K (s )∩c,i = ∅) then every locally controllable legal event defined after a series of uncontrollable events u should be included in the set of enabled events for the observation Pi (s) of Si . For instance, let us consider the example shown in Fig. 2, where c,1 = {a, c}, o,1 = {a, c, u1 }, c,2 = {b, c}, o,2 = {b, c, u2 }, and uc = {u1 , u2 }. Let a language specification be K = {abu1 b, abu1 c, bau2 c, bau2 a, bac}. Then, for s = ab ∈ pr(K), the strings s ’s satisfying P1 (s )=P1 (s)=a and K (s )∩ c,1 = ∅ are ba and bau2 . Hence, S1 (P1 (s)) = S1 (a) = {a, c} ∪ {b} since bac, bau2 a, bau2 c ∈ pr(K) and b ∈ c \c,1 . On the other hand, the strings s ’s satisfying P2 (s ) = P2 (s) = b and K (s ) ∩ c,2 = ∅ are ba and abu1 . Hence, S2 (P2 (s)) = S2 (b) = {b, c} ∪ {a} since bac, abu1 b, abu1 c ∈ pr(K) and a ∈ c \c,2 . Based on the local decisions, let us define the decentralized supervisory control actions as follows: Definition 2. For the occurrence of a string s ∈ pr(K), the set of enabled events of a decentralized supervisor Sdec is defined
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as follows:
⎧ ∅ ⎪ ⎨
∗
∗uc \{},
and u ∈ if s = tu for some t ∈ and Sdec (t) = ∅ and Sdec (tu ) = ∅ Sdec (s) := ⎪ ⎩ ∀u ∈ pr(u)\{, u}, S (P (s)) := i i i i Si (Pi (s)) otherwise,
which is also called a decentralized supervisory control action following the conjunctive fusion rule under communication delays. This definition shows how the conjunctive fusion rule on local decisions in the absence of delays (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) can be extended in a decentralized supervision under communication delays. We note that if a conjunctive decision for the observed string is nonempty then supervisory actions for subsequently occurring uncontrollable events after the string are empty. The empty supervisory actions mean that no controllable events are permitted to occur in the closed-loop system. The above definition also implies that when the occurrence of a controllable event is observed by a local supervisor Si , the control action i Si (Pi (·)) is always computed and checked whether it is empty or not. A supervised system is denoted by Sdec /G. As a consequence of the previous definition on a decentralized supervisor, its closed-loop behavior L(Sdec /G) can be described as follows: ∈ L(Sdec /G), and for s ∈ L(Sdec /G) and ∈ with s ∈ L(G), s ∈ L(Sdec /G) ⇔ (i) ∈ uc , or (ii) ∈ Sdec (t) ∩ c where s = tu for some t ∈ ∗ , u ∈ ∗uc and Sdec (tv) = ∅ ∀v ∈ pr(u)\{}. Here (ii) implies that if the supervisor issues an enabling command for a controllable event after the occurrence of t, i.e. ∈ Sdec (t), then the same control action Sdec (t) is applied to the plant for any subsequent occurrence of uncontrollable events, i.e. v ∈ pr(u)\{}. It is important to note that the supervisory control action should be empty (i.e., Sdec (tv) = ∅) for the observation of such uncontrollable events, which prevents the delay effect from being propagated beyond the current observed string. Based on the aforementioned definitions, we can formulate the problem of supervisor existence to be addressed as follows: Given a specification K ⊆ Lm (G) for a plant G with communication delays, find necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor Sdec such that Lm (Sdec /G) = K. Note that here Lm (Sdec /G) := L(Sdec /G) ∩ Lm (G) and a decentralized supervisor Sdec is called nonblocking w.r.t. G if pr(Lm (Sdec /G)) = L(Sdec /G). We present a language property called delay-coobservability as a principal condition for the existence of a decentralized supervisor that can achieve a given language specification under communication delays.
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Definition 3. A language K ⊆ Lm (G) is delay-coobservable w.r.t. G and (o,i , c,i )i∈I if for any su ∈ pr(K) and ∈ c satisfying u ∈ ∗uc and su ∈ / pr(K), there exists i ∈ {1, . . . , n} such that ∈ c,i and ∀s ∈ pr(K) satisfying Pi (s) = Pi (s ) and K (s ) ∩ c,i = ∅, the following is satisfied: / pr(K) ∀u ∈ ∗uc . s u ∈ This definition means that for any illegal controllable event there exists a local supervisor which carries information sufficient enough to make a disabling decision over the event in the case of subsequent occurrences of uncontrollable events. We note that if the terms ‘u’, ‘u ’, and ‘K (s ) ∩ c,i = ∅’ are removed from the definition, it becomes equivalent to the C&P (conjunctive & permissive) coobservability in Yoo and Lafortune (2002). The term ‘K (s ) ∩ c,i = ∅’ is required since a local supervisor should be designed to include enabled events only for an estimation s after which some legal local events are defined immediately. For instance, let us reconsider the example shown in Fig. 2. For a language specification K={abu1 b, abu1 c, bau2 c, bau2 a, bac}, we notice that K is not delay-coobservable w.r.t. (o,i , c,i )i∈{1,2} due to the following reason: For s = ab, / pr(K). For s = ba it holds that su1 ∈ pr(K) but su1 a ∈ and i = 1 with a ∈ c,1 , it holds that P1 (s) = P1 (s ) = a, K (s ) ∩ c,1 = {c} but s u2 a ∈ pr(K). Thus, the condition for s = ab in Definition 3 is not satisfied for i = 1. Since a ∈ / c,2 , checking the condition of Definition 3 is not required for i = 2. On the other hand, considering a local supervisor S3 with c,3 = o,3 = {a, b}, it turns out that K is delay-coobservable w.r.t. (o,i , c,i )i∈{1,2,3} from the following reason: For s = ab / pr(K), and i = 3 with su1 ∈ pr(K), a ∈ c,3 , and su1 a ∈ the unique string s satisfying P3 (s) = P3 (s ) = ab and K (s ) ∩ c,3 = ∅ is abu1 (K (s ) ∩ c,3 = {b}) which results in s a ∈ / pr(K). Therefore, the condition for s = ab is satisfied for i = 3. Let us discuss the computational complexity of verifying the delay-coobservability of a language K. The following analysis considers two local supervisors, but the result can be extended to any finite number of local supervisors in a straightforward way. Suppose that the number of uncontrollable events that can subsequently occur in a plant G is bounded by a finite integer value D. Let QK be the state space of a deterministic automaton that recognizes K. From the previous study (Rudie & Willems, 1995), the C&P coobservability is verifiable in polynomial time by constructing a special nondeterministic automaton M whose state space is QK × QK × QK × Q ∪ {d}. The dump state d indicates the case when C&P coobservability fails. Transitions to the dump state are checked during the construction of M without additional computations. Deciding the C&P coobservability is equivalent to verifying whether there is a path from the initial state to the dump state in M. From the verification algorithm of C&P coobservability, the delay-coobservability can also be verified by constructing M. However, the delay-coobservability requires to check its conditions for subsequently occurring uncontrollable events in the plant. Hence, checking the dump state transitions during the construction of M requires an additional |QK |3 |Q| · 4|uc |D times state space searching in a worst case
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(|uc |D transitions in the state space of G and the deterministic automaton recognizing K for each state of QK ×QK ×QK ×Q in M). Note that K is not delay-coobservable if and only if there is a path from the initial state to the dump state in M. Hence, the computational complexity of verifying the delaycoobservability of K is O(n2 ) where n is the size of the state space QK × QK × QK × Q ∪ {d}. Theorem 1. Given a language specification K ⊆ Lm (G) for a plant G with communication delays, there exists a nonblocking decentralized supervisor Sdec such that Lm (Sdec /G) =K if and only if (1) K is controllable w.r.t. G, (2) K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I , and (3) K is Lm (G)-closed. Proof. (If) For s ∈ ∗ and i ∈ I , let us consider the following local supervisors: Si (Pi (s)) := { ∈ c,i |s u ∈ pr(K) for some s ∈ ∗ and u ∈ ∗uc s.t. Pi (s ) = Pi (s) and K (s ) ∩ c,i = ∅}, and consider the following decentralized supervisor: ⎧ ∅ ⎪ ⎨
if s = tu for some t ∈ ∗ and u ∈ ∗uc \{}, and Sdec (t) = ∅ and Sdec (tu ) = ∅ Sdec (s) := ⎪ ⎩ ∀u ∈ pr(u)\{, u}, i Si (Pi (s)) otherwise. First, we prove that L(Sdec /G) = pr(K). The proof can be done by induction on the length of the strings. It holds that ∈ L(Sdec /G) ∩ pr(K). Let us assume that for any string s with |s| n, s ∈ L(Sdec /G) if and only if s ∈ pr(K) (|s| denotes the length of s). We first show that s ∈ L(Sdec /G) implies s ∈ pr(K). In case of ∈ uc , since K is controllable w.r.t. G, it naturally holds that s ∈ pr(K). In case of ∈ c , we assume s ∈ / pr(K) and use a contradiction method. Then, s ∈ L(Sdec /G) ⇒ ∈ Sdec (t) for some t ∈ ∗ and u ∈ ∗uc where s = tu and Sdec (tv) = ∅ ∀v ∈ pr(u)\{} (by the definition of L(Sdec /G)) ⇒ ∈ i Si (Pi (t)) (by the definition of Sdec ) ⇒ ∀i ∈ {1, . . . , n} satisfying ∈ c,i , ∃ t ∈ ∗ and u ∈ ∗uc such that (s.t.) t u ∈ pr(K), Pi (t ) = Pi (t), and K (t ) ∩ c,i = ∅ (by the definition of Si ) ⇒ i ∈ {1, . . . , n} s.t. ∀t ∈ pr(K) satisfying Pi (t) = Pi (t ) and K (t ) ∩ c,i = ∅, / pr(K)∀u ∈ ∗uc the following is satisfied: t u ∈ ⇒ contradiction to the delay-coobservability of K (by the assumption of s = tu ∈ / pr(K)). Thus, we conclude that s ∈ pr(K).
In the next, we show that s ∈ pr(K) implies s ∈ L(Sdec /G). In case of ∈ uc , it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). Let us consider the case of ∈ c . For t ∈ ∗ and u ∈ ∗uc , let s = tu and then we can consider the following two cases: Case (1): K (t) ∩ c = ∅; Case (2): K (tv) ∩ c = ∅ ∀v ∈ pr(u)\{u}. In Case (1), it follows from the definition of Si that ∈ Si (Pi (t)) for any Si with ∈ c,i . Then, it leads to ∈ i Si (Pi (t)) and hence it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). In Case (2), it follows from s ∈ pr(K) that ∈ S i (Pi (s)) for any Si with ∈ c,i . Then, it leads to ∈ i Si (Pi (s)) and it turns out that s ∈ L(Sdec /G). From the above two cases, s ∈ L(Sdec /G). Therefore, we conclude that L(Sdec /G) = pr(K). Furthermore, since L(Sdec /G) = pr(K) and K is Lm (G)closed, it holds that Lm (Sdec /G) = L(Sdec /G) ∩ Lm (G) = pr(K) ∩ Lm (G) = K. So, pr(Lm (Sdec /G)) = pr(K) = L(Sdec /G). Finally, the decentralized supervisor Sdec turns out to be nonblocking. (Only if) Assume that Sdec is a nonblocking decentralized supervisor satisfying Lm (Sdec /G) = K. Let s ∈ pr(K) and ∈ uc satisfying s ∈ L(G). Then, s ∈ L(Sdec /G) since L(Sdec /G) = pr(K), and it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). As a result, s ∈ pr(K) since L(Sdec /G) = pr(K), and we conclude therefore that K is controllable w.r.t. G. In addition, since Sdec is nonblocking, it is true that pr(Lm (Sdec /G)) = L(Sdec /G) = pr(K), and Lm (Sdec /G) = L(Sdec /G) ∩ Lm (G) = pr(K) ∩ Lm (G) = K. Thus K is Lm (G)-closed. Finally, let us show that K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I using a contradiction method. For this purpose, assume that for some su ∈ pr(K) and ∈ c satisfying u ∈ / pr(K), there is no i ∈ {1, . . . , n} such that ∈ ∗uc and su ∈ c,i and ∀s ∈ pr(K) satisfying Pi (s) = Pi (s ) and K (s ) ∩ c,i = ∅, the following is satisfied: s u ∈ / pr(K) ∀u ∈ ∗uc . Then, the above assumption implies that
⇒ ⇒ ⇒ ⇒
∀ i ∈ {1, . . . , n} s.t. ∈ c,i , ∃ some s ∈ ∗ and u ∈ ∗uc satisfying Pi (s) = Pi (s ), K (s ) ∩ c,i = ∅, and s u ∈ pr(K) ∈ i Si (Pi (s)) = Sdec (s) su ∈ L(Sdec /G) (by the definition of L(Sdec /G) together with s ∈ pr(K) = L(Sdec /G) and u ∈ ∗uc ) su ∈ pr(K) since L(Sdec /G) = pr(K) contradiction to the assumption su ∈ / pr(K).
Therefore, we conclude that K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I .
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The delay-coobservability is a stronger notion than the C&P coobservability of Rudie and Wonham (1992) and Yoo and Lafortune (2002). It is obvious from Definition 3 that if a local supervisor makes a disabling decision on an illegal event for a given language K, it also makes a disabling decision on the event for any sublanguage of K in the presence of communication delays. In other words, the class of delay-coobservable languages is closed under intersection. Hence, we can infer that there exists an infimal prefix closed and delay-coobservable superlanguage for a given language. We can obtain this language by extending the formula of an infimal prefix closed and C&P coobservable superlanguage (Takai, Kumar, & Ushio, 2005). Moreover, the prefix closed and delay-coobservable sublanguage can also be obtained by extending the formula of a prefix closed and C&P coobservable sublanguage based on the class of locally observable languages (Takai et al., 2005). 3. Conclusions In this paper, we have introduced the notion of delaycoobservability for a given language specification as an extension of the coobservability (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) to the case of decentralized supervisory control under communication delays. Based on this notion, we have presented the necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor that can achieve a given language specification. The main results have been developed based on the conjunctive and permissive decentralized supervisory control structures, but these can be easily extended to any other decentralized control structures such as disjunctive and anti-permissive ones (Yoo & Lafortune, 2002, 2004). Acknowledgments This work was supported from Korea Ministry of Science and Technology through the Korean Systems Biology Research Grant (M10503010001-05N030100111) and the 21C Frontier Microbial Genomics and Application Center Program
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(Grant MG05-0204-3-0), and in part by 2005-B0000002 from Korea Bio-Hub Program of Korea Ministry of Commerce, Industry & Energy. K.-H. Cho was supported by the second stage Brain Korea 21 Project in 2006. References Balemi, S. (1992). Communication delays in connections of input/output discrete event processes. In Proceedings of the 31st IEEE conference on decision control 1992 (pp. 3374–3379). Berlin: Springer. Debouk, R., Lafortune, S., & Teneketzis, D. (2003). On the effect of communication delays in failure diagnosis of decentralized discrete event systems. Discrete Event Dynamic Systems: Theory and Applications, 13, 263–289. Lauzon, S. C., Mills, J. K., & Benhabib, B. (1997). An implementation methodology for the supervisory controller for flexible manufacturing workcells. SME Journal of Manufacturing Systems, 16(1), 91–101. Park, S.-J., & Cho, K.-H. (2006). Delay-robust supervisory control of discrete event systems with bounded communication delays. IEEE Transactions on Automatic Control, 51(5), 911–915. Ramadge, P. J., & Wonham, W. M. (1987). Supervisory control of a class of discrete event processes. SIAM Journal on Control and Optimization, 25(1), 206–230. Ricker, L., & Schuppen, J. V. (2001). Decentralized failure diagnosis with asynchronous communication between supervisors. In Proceedings of the European control conference. Rudie, K., & Willems, J. C. (1995). The computational complexity of decentralized discrete-event control problems. IEEE Transactions on Automatic Control, 40(7), 1313–1319. Rudie, K., & Wonham, W. M. (1992). Think gloablly, act locally: Decentralized supervisory control. IEEE Transactions on Automatic Control, 37, 1692–1708. Takai, S., Kumar, R., & Ushio, T. (2005). Characterization of co-observable languages and formulas for their super/sublanguages. IEEE Transactions on Automatic Control, 50(4), 434–447. Tripakis, S. (2004). Decentralized control of discrete-event systems with bounded or unbounded delay communication. IEEE Transactions on Automatic Control, 49(9), 1489–1501. Yoo, T.-S., & Lafortune, S. (2002). A general architecture for decentralized supervisory control of discrete-event systems. Discrete Event Dynamic Systems: Theory and Applications, 12(3), 335–377. Yoo, T.-S., & Lafortune, S. (2004). Decentralized supervisory control with conditional decisions: Supervisor existence. IEEE Transactions on Automatic Control, 49(11), 1886–1904.
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Decentralized supervisory control of discrete event systems with communication delays based on conjunctive and permissive decision structures夡 Seong-Jin Park a , Kwang-Hyun Cho b,c,∗ a Department of Electrical and Computer Engineering, Ajou University, Suwon 443 749, Korea b College of Medicine, Seoul National University, Jongno-gu, Seoul 110 799, Korea c Bio-MAX Institute, Seoul National University, Gwanak-gu, Seoul 151 818, Korea
Received 11 October 2005; received in revised form 18 March 2006; accepted 27 October 2006
Abstract In many practical discrete event systems (DESs), some unexpected and uncontrollable events can subsequently occur before a proper control action is actually applied to a plant due to communication delays. For such DESs, this paper investigates necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor that can correctly achieve a given language specification when the decentralized supervisor is assumed to have a conjunctive and permissive decision structure. In particular, this paper presents a notion of delay-coobservability for a given language specification and shows that it is a key condition for the existence of such a decentralized supervisor. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete event systems; Decentralized supervisors; Communication delays; Delay-coobservability
1. Introduction Most decentralized supervisory control schemes for discrete event systems (DESs) including Rudie and Wonham (1992) and Yoo and Lafortune (2002, 2004) have been developed based on the assumption that the control action issued by a decentralized supervisor can be applied to a plant without any delay. In other words, it has been commonly assumed that if a certain event occurs in the plant then no further event occurs until the information of previous occurrence is processed and a proper control decision is made and delivered to the plant through a communication channel. In many practical situations, there are, however, non-negligible delays in sensing, communicating, and 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Karl Henrik Johansson under the direction of Editor André Tits. ∗ Corresponding author. Bio-MAX Institute, Seoul National University, 3rd Floor, IVI, San 4-8, Bongcheon 7-dong, Gwanak-gu, Seoul 151 818, Republic of Korea. Tel.: +82 2 887 2650; fax: +82 2 887 2692. E-mail addresses: [email protected] (S.-J. Park), [email protected] (K.-H. Cho).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.10.016
actuating that should be considered for decentralized supervisory control in particular. Supervisory control of DESs with communication delays has been investigated within several contexts. For instance, Balemi (1992) considered the supervisor synthesis problem of input/output DESs with communication delays when a partial specification is given by a language over an output event set. Debouk, Lafortune, and Teneketzis (2003) considered the problem of decentralized failure diagnosis under a communication delay. Ricker and Schuppen (2001) investigated the decentralized diagnosis problem of timed DESs with communication delays among the diagnosers. Recently, Park and Cho (2006) studied the problem of centralized supervisory control to achieve a given language specification under communication delays and presented the existence conditions. On the other hand, considering the bounded or unbounded communication delays due to the buffer queuing of messages transmitted among the local supervisors, Tripakis (2004) investigated the decentralized supervisory control problem when a simple specification called responsiveness is imposed on a DES. It has been, however, not considered in either Park and Cho (2006) or Tripakis (2004) what the necessary and sufficient conditions
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are for the existence of a decentralized nonblocking supervisor that can correctly achieve a given language specification under communication delays. In this paper, we consider the decentralized supervisory control problem for DESs with communication delays, in which a series of uncontrollable events can unexpectedly occur before a proper control action is actually applied to a plant. The delays can occur in data processing elements such as sensors and actuators, and also in communication channels between controllers and plants; however, the message delivery between local supervisors in Tripakis (2004) is not to be considered in this paper. In addition, to illuminate the main features of a decentralized supervisory control under delays in a simplest way, we adopt in this paper only the conjunctive and permissive decision structures (Rudie & Wonham, 1992; Yoo & Lafortune, 2002). Within this framework, we investigate the existence conditions of a nonblocking decentralized supervisor that can correctly achieve a given language specification. To this end, we further make some assumptions as follows: (i) every controllable event can occur only if it is enabled by a decentralized supervisor, (ii) some uncontrollable events can be unobservable to all local supervisors while all locally controllable events are locally observable, and (iii) the number of possible subsequent occurrence of uncontrollable events in a plant is limited within a finite bound. Based on these assumptions, we present a notion of delay-coobservability for a given language specification and show that it is a key condition for the existence of a decentralized supervisor that can achieve the specification. 2. Main results It has been assumed in the conventional supervisory control (Ramadge & Wonham, 1987; Rudie & Wonham, 1992; Yoo & Lafortune, 2002, 2004) that as an event occurs in a plant it is promptly observed by a supervisor and the control action for an observed event string is also directly applied to the plant without any delay. This framework is only realistic when a plant is tightly coupled with its supervisor such that there is no communication delay between them. Plants are, however, often controlled by remote supervisors through communication networks. In such environments, the messages delivering observation or supervision can experience non-negligible transmission delays through the communication networks. Moreover, data processing time in sensors or motor operation time in actuators can sometimes cause non-negligible delays as well. As a result, a series of subsequent uncontrollable events (e.g., unobservable faults, completion of certain tasks, etc.) may unexpectedly occur before a proper control action is transmitted and applied to the plant. In this paper, we consider the problem of decentralized supervisory control of DESs with such communication delays as shown in Fig. 1, where a1 , . . . , aD denote uncontrollable events. In Fig. 1, it is illustrated that a proper decentralized supervisory control action Sdec (t) for the string t is actually applied to the plant after the subsequent occurrences of ai , i = 1, . . . , D. The plant to be controlled is modeled by a finite state automaton G = (Q, , q0 , , Qm ) where Q is the set of states,
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Fig. 1. Decentralized supervisory control under delays.
is the set of events, q0 is the initial state, : Q × → Q is the state transition (partial) function, and Qm is the set of marked states. Let ∗ denote the set of all finite strings of elements in , including the empty string . For s, t ∈ ∗ , we denote st as a concatenation of the strings s and t. We note that the function can be extended to ∗ by defining (q, ) := q and (q, s) := ((q, s), ) for all s ∈ ∗ and ∈ . The prefix closure of a language L(⊂ ∗ ) is defined as pr(L) := {t ∈ ∗ |tu ∈ L for some u ∈ ∗ }, and L is said to be prefix-closed (or closed) if L = pr(L). The closed behavior and the marked behavior of G are defined by L(G) := {s ∈ ∗ |(q0 , s) (is defined)} and Lm (G) := {s ∈ ∗ |(q0 , s)! ∈ Qm }, respectively. For s ∈ ∗ and L ⊂ ∗ , L (s) := { ∈ |s ∈ pr(L)}. The plant G is controlled by joint actions of n local supervisors S1 , . . . , Sn according to the conjunctive fusion rule on the permissive local decision actions defined under communication delays. Without communication delays, the conjunctive fusion rule means that a controllable event is permitted to occur only if there is no disabling decision over that event by any of the local supervisors; the permissive local decision rule implies that the control action of a local supervisor under insufficient information is an enablement of the event (Rudie & Wonham, 1992; Yoo & Lafortune, 2002). However, in the presence of communication delays, these two decision rules should be redefined to describe the control actions in a proper way. In this paper, we explicitly define the conjunctive fusion rule and the permissive local decision rule under communication delays. Other decision structures such as disjunctive and anti-permissive rules (Yoo & Lafortune, 2002, 2004) are not considered in this paper, but the results obtained can be easily extended for such decision structures. Each local supervisor Si can observe the set of locally observable events denoted by o,i and can control the set of
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locally controllable events denoted by c,i . Each local supervisor is designed to achieve a given language specification K ⊆ L(G) with a priori information including the uncontrolled behavior L(G) and the desired behavior K. Each block Pi represents the usual projection mapping Pi : → o,i . We denote uo := \ o and uc := \c where o := and := o,i c i∈I i∈I c,i where I := {1, . . . , n}. A language K ⊆ L(G) is controllable with respect to (w.r.t.) G if pr(K)uc ∩ L(G) ⊆ pr(K), and it is Lm (G)-closed if pr(K) ∩ Lm (G) = K (Ramadge & Wonham, 1987). Before proceeding to the main results, we summarize in the following some assumptions that are made to develop our ideas: • Every controllable event is disabled by default and is permitted to occur only if it is enabled by a decentralized supervisor. Specifically, a controllable event in the supervised system can occur only if it is enabled by all local supervisors according to the conjunctive fusion rule. From the permissive decision rule, the enabling decision of each local supervisor implies that the controllable event is legal for at least one possible trajectory estimated by the local supervisor. In other words, a controllable event cannot occur if any of the local supervisors disables the event, and the disabling decision of the local supervisor implies that the controllable event must be illegal for all possible estimations of the local supervisor. • Every locally controllable event is locally observable, i.e. c,i ⊆ o,i . • The number of possible subsequent occurrence of uncontrollable events in a plant G is limited within a finite bound. • If an event a occurs before an event b in a plant, then a is observed by local supervisors always before b. In addition, control commands issued by a decentralized supervisor are applied to the plant without changing their order. The first assumption has been made to exclude any possible occurrence of subsequent controllable events prior to the application of a control action to the plant. Hence, only uncontrollable events can subsequently occur before a proper control action is applied to the plant. This implies that the plant does not suffer from any unexpected occurrence of events prior to the control action if only controllable events are defined in the plant. Following the framework of Ramadge and Wonham (1987), the supervisor adopted in this paper controls a plant through enabling controllable events without employing auxiliary forcing mechanisms and thereby the supervised system chooses which event to generate. However, the first assumption differs from that of Ramadge and Wonham in that controllable events do not occur spontaneously but occur as responses to enabling requests imposed by the supervisor. This assumption has been already employed to reflect the practical aspect of implementing supervisory control systems (Lauzon, Mills, & Benhabib, 1997). The other three assumptions have been made to deliver the main idea in a clear way without loss of generality. Let us present the design method of a decentralized supervisor and consider the behavior of a closed-loop system under
a
b a
b,c
u1 c
b
a u2
a,c
Fig. 2. An example plant for illustration of the design of local supervisors.
communication delays. To achieve a given language specification K ⊆ Lm (G) of a plant G, we adopt the local supervisors designed according to the following definition. Definition 1. For the occurrence of a string s ∈ pr(K), the set of enabled events for the observation Pi (s) of a local supervisor Si is defined as follows: Si (Pi (s)) := { ∈ c,i |s u ∈ pr(K) for some s ∈ ∗ and u ∈ ∗uc such that Pi (s ) = Pi (s) and K (s ) ∩ c,i = ∅} ∪ (c \c,i ), which is also called a local supervisory control action following the permissive local decision rule under communication delays. This design rule shows how the permissive decision rule in the absence of delays (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) can be extended in the situation when uncontrollable events can unexpectedly occur before a proper control action is applied to the plant due to communication delays. It states that, for each estimation s of the string s, if there are locally controllable legal events determined to occur immediately after the estimation (K (s )∩c,i = ∅) then every locally controllable legal event defined after a series of uncontrollable events u should be included in the set of enabled events for the observation Pi (s) of Si . For instance, let us consider the example shown in Fig. 2, where c,1 = {a, c}, o,1 = {a, c, u1 }, c,2 = {b, c}, o,2 = {b, c, u2 }, and uc = {u1 , u2 }. Let a language specification be K = {abu1 b, abu1 c, bau2 c, bau2 a, bac}. Then, for s = ab ∈ pr(K), the strings s ’s satisfying P1 (s )=P1 (s)=a and K (s )∩ c,1 = ∅ are ba and bau2 . Hence, S1 (P1 (s)) = S1 (a) = {a, c} ∪ {b} since bac, bau2 a, bau2 c ∈ pr(K) and b ∈ c \c,1 . On the other hand, the strings s ’s satisfying P2 (s ) = P2 (s) = b and K (s ) ∩ c,2 = ∅ are ba and abu1 . Hence, S2 (P2 (s)) = S2 (b) = {b, c} ∪ {a} since bac, abu1 b, abu1 c ∈ pr(K) and a ∈ c \c,2 . Based on the local decisions, let us define the decentralized supervisory control actions as follows: Definition 2. For the occurrence of a string s ∈ pr(K), the set of enabled events of a decentralized supervisor Sdec is defined
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as follows:
⎧ ∅ ⎪ ⎨
∗
∗uc \{},
and u ∈ if s = tu for some t ∈ and Sdec (t) = ∅ and Sdec (tu ) = ∅ Sdec (s) := ⎪ ⎩ ∀u ∈ pr(u)\{, u}, S (P (s)) := i i i i Si (Pi (s)) otherwise,
which is also called a decentralized supervisory control action following the conjunctive fusion rule under communication delays. This definition shows how the conjunctive fusion rule on local decisions in the absence of delays (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) can be extended in a decentralized supervision under communication delays. We note that if a conjunctive decision for the observed string is nonempty then supervisory actions for subsequently occurring uncontrollable events after the string are empty. The empty supervisory actions mean that no controllable events are permitted to occur in the closed-loop system. The above definition also implies that when the occurrence of a controllable event is observed by a local supervisor Si , the control action i Si (Pi (·)) is always computed and checked whether it is empty or not. A supervised system is denoted by Sdec /G. As a consequence of the previous definition on a decentralized supervisor, its closed-loop behavior L(Sdec /G) can be described as follows: ∈ L(Sdec /G), and for s ∈ L(Sdec /G) and ∈ with s ∈ L(G), s ∈ L(Sdec /G) ⇔ (i) ∈ uc , or (ii) ∈ Sdec (t) ∩ c where s = tu for some t ∈ ∗ , u ∈ ∗uc and Sdec (tv) = ∅ ∀v ∈ pr(u)\{}. Here (ii) implies that if the supervisor issues an enabling command for a controllable event after the occurrence of t, i.e. ∈ Sdec (t), then the same control action Sdec (t) is applied to the plant for any subsequent occurrence of uncontrollable events, i.e. v ∈ pr(u)\{}. It is important to note that the supervisory control action should be empty (i.e., Sdec (tv) = ∅) for the observation of such uncontrollable events, which prevents the delay effect from being propagated beyond the current observed string. Based on the aforementioned definitions, we can formulate the problem of supervisor existence to be addressed as follows: Given a specification K ⊆ Lm (G) for a plant G with communication delays, find necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor Sdec such that Lm (Sdec /G) = K. Note that here Lm (Sdec /G) := L(Sdec /G) ∩ Lm (G) and a decentralized supervisor Sdec is called nonblocking w.r.t. G if pr(Lm (Sdec /G)) = L(Sdec /G). We present a language property called delay-coobservability as a principal condition for the existence of a decentralized supervisor that can achieve a given language specification under communication delays.
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Definition 3. A language K ⊆ Lm (G) is delay-coobservable w.r.t. G and (o,i , c,i )i∈I if for any su ∈ pr(K) and ∈ c satisfying u ∈ ∗uc and su ∈ / pr(K), there exists i ∈ {1, . . . , n} such that ∈ c,i and ∀s ∈ pr(K) satisfying Pi (s) = Pi (s ) and K (s ) ∩ c,i = ∅, the following is satisfied: / pr(K) ∀u ∈ ∗uc . s u ∈ This definition means that for any illegal controllable event there exists a local supervisor which carries information sufficient enough to make a disabling decision over the event in the case of subsequent occurrences of uncontrollable events. We note that if the terms ‘u’, ‘u ’, and ‘K (s ) ∩ c,i = ∅’ are removed from the definition, it becomes equivalent to the C&P (conjunctive & permissive) coobservability in Yoo and Lafortune (2002). The term ‘K (s ) ∩ c,i = ∅’ is required since a local supervisor should be designed to include enabled events only for an estimation s after which some legal local events are defined immediately. For instance, let us reconsider the example shown in Fig. 2. For a language specification K={abu1 b, abu1 c, bau2 c, bau2 a, bac}, we notice that K is not delay-coobservable w.r.t. (o,i , c,i )i∈{1,2} due to the following reason: For s = ab, / pr(K). For s = ba it holds that su1 ∈ pr(K) but su1 a ∈ and i = 1 with a ∈ c,1 , it holds that P1 (s) = P1 (s ) = a, K (s ) ∩ c,1 = {c} but s u2 a ∈ pr(K). Thus, the condition for s = ab in Definition 3 is not satisfied for i = 1. Since a ∈ / c,2 , checking the condition of Definition 3 is not required for i = 2. On the other hand, considering a local supervisor S3 with c,3 = o,3 = {a, b}, it turns out that K is delay-coobservable w.r.t. (o,i , c,i )i∈{1,2,3} from the following reason: For s = ab / pr(K), and i = 3 with su1 ∈ pr(K), a ∈ c,3 , and su1 a ∈ the unique string s satisfying P3 (s) = P3 (s ) = ab and K (s ) ∩ c,3 = ∅ is abu1 (K (s ) ∩ c,3 = {b}) which results in s a ∈ / pr(K). Therefore, the condition for s = ab is satisfied for i = 3. Let us discuss the computational complexity of verifying the delay-coobservability of a language K. The following analysis considers two local supervisors, but the result can be extended to any finite number of local supervisors in a straightforward way. Suppose that the number of uncontrollable events that can subsequently occur in a plant G is bounded by a finite integer value D. Let QK be the state space of a deterministic automaton that recognizes K. From the previous study (Rudie & Willems, 1995), the C&P coobservability is verifiable in polynomial time by constructing a special nondeterministic automaton M whose state space is QK × QK × QK × Q ∪ {d}. The dump state d indicates the case when C&P coobservability fails. Transitions to the dump state are checked during the construction of M without additional computations. Deciding the C&P coobservability is equivalent to verifying whether there is a path from the initial state to the dump state in M. From the verification algorithm of C&P coobservability, the delay-coobservability can also be verified by constructing M. However, the delay-coobservability requires to check its conditions for subsequently occurring uncontrollable events in the plant. Hence, checking the dump state transitions during the construction of M requires an additional |QK |3 |Q| · 4|uc |D times state space searching in a worst case
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(|uc |D transitions in the state space of G and the deterministic automaton recognizing K for each state of QK ×QK ×QK ×Q in M). Note that K is not delay-coobservable if and only if there is a path from the initial state to the dump state in M. Hence, the computational complexity of verifying the delaycoobservability of K is O(n2 ) where n is the size of the state space QK × QK × QK × Q ∪ {d}. Theorem 1. Given a language specification K ⊆ Lm (G) for a plant G with communication delays, there exists a nonblocking decentralized supervisor Sdec such that Lm (Sdec /G) =K if and only if (1) K is controllable w.r.t. G, (2) K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I , and (3) K is Lm (G)-closed. Proof. (If) For s ∈ ∗ and i ∈ I , let us consider the following local supervisors: Si (Pi (s)) := { ∈ c,i |s u ∈ pr(K) for some s ∈ ∗ and u ∈ ∗uc s.t. Pi (s ) = Pi (s) and K (s ) ∩ c,i = ∅}, and consider the following decentralized supervisor: ⎧ ∅ ⎪ ⎨
if s = tu for some t ∈ ∗ and u ∈ ∗uc \{}, and Sdec (t) = ∅ and Sdec (tu ) = ∅ Sdec (s) := ⎪ ⎩ ∀u ∈ pr(u)\{, u}, i Si (Pi (s)) otherwise. First, we prove that L(Sdec /G) = pr(K). The proof can be done by induction on the length of the strings. It holds that ∈ L(Sdec /G) ∩ pr(K). Let us assume that for any string s with |s| n, s ∈ L(Sdec /G) if and only if s ∈ pr(K) (|s| denotes the length of s). We first show that s ∈ L(Sdec /G) implies s ∈ pr(K). In case of ∈ uc , since K is controllable w.r.t. G, it naturally holds that s ∈ pr(K). In case of ∈ c , we assume s ∈ / pr(K) and use a contradiction method. Then, s ∈ L(Sdec /G) ⇒ ∈ Sdec (t) for some t ∈ ∗ and u ∈ ∗uc where s = tu and Sdec (tv) = ∅ ∀v ∈ pr(u)\{} (by the definition of L(Sdec /G)) ⇒ ∈ i Si (Pi (t)) (by the definition of Sdec ) ⇒ ∀i ∈ {1, . . . , n} satisfying ∈ c,i , ∃ t ∈ ∗ and u ∈ ∗uc such that (s.t.) t u ∈ pr(K), Pi (t ) = Pi (t), and K (t ) ∩ c,i = ∅ (by the definition of Si ) ⇒ i ∈ {1, . . . , n} s.t. ∀t ∈ pr(K) satisfying Pi (t) = Pi (t ) and K (t ) ∩ c,i = ∅, / pr(K)∀u ∈ ∗uc the following is satisfied: t u ∈ ⇒ contradiction to the delay-coobservability of K (by the assumption of s = tu ∈ / pr(K)). Thus, we conclude that s ∈ pr(K).
In the next, we show that s ∈ pr(K) implies s ∈ L(Sdec /G). In case of ∈ uc , it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). Let us consider the case of ∈ c . For t ∈ ∗ and u ∈ ∗uc , let s = tu and then we can consider the following two cases: Case (1): K (t) ∩ c = ∅; Case (2): K (tv) ∩ c = ∅ ∀v ∈ pr(u)\{u}. In Case (1), it follows from the definition of Si that ∈ Si (Pi (t)) for any Si with ∈ c,i . Then, it leads to ∈ i Si (Pi (t)) and hence it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). In Case (2), it follows from s ∈ pr(K) that ∈ S i (Pi (s)) for any Si with ∈ c,i . Then, it leads to ∈ i Si (Pi (s)) and it turns out that s ∈ L(Sdec /G). From the above two cases, s ∈ L(Sdec /G). Therefore, we conclude that L(Sdec /G) = pr(K). Furthermore, since L(Sdec /G) = pr(K) and K is Lm (G)closed, it holds that Lm (Sdec /G) = L(Sdec /G) ∩ Lm (G) = pr(K) ∩ Lm (G) = K. So, pr(Lm (Sdec /G)) = pr(K) = L(Sdec /G). Finally, the decentralized supervisor Sdec turns out to be nonblocking. (Only if) Assume that Sdec is a nonblocking decentralized supervisor satisfying Lm (Sdec /G) = K. Let s ∈ pr(K) and ∈ uc satisfying s ∈ L(G). Then, s ∈ L(Sdec /G) since L(Sdec /G) = pr(K), and it follows from the definition of L(Sdec /G) that s ∈ L(Sdec /G). As a result, s ∈ pr(K) since L(Sdec /G) = pr(K), and we conclude therefore that K is controllable w.r.t. G. In addition, since Sdec is nonblocking, it is true that pr(Lm (Sdec /G)) = L(Sdec /G) = pr(K), and Lm (Sdec /G) = L(Sdec /G) ∩ Lm (G) = pr(K) ∩ Lm (G) = K. Thus K is Lm (G)-closed. Finally, let us show that K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I using a contradiction method. For this purpose, assume that for some su ∈ pr(K) and ∈ c satisfying u ∈ / pr(K), there is no i ∈ {1, . . . , n} such that ∈ ∗uc and su ∈ c,i and ∀s ∈ pr(K) satisfying Pi (s) = Pi (s ) and K (s ) ∩ c,i = ∅, the following is satisfied: s u ∈ / pr(K) ∀u ∈ ∗uc . Then, the above assumption implies that
⇒ ⇒ ⇒ ⇒
∀ i ∈ {1, . . . , n} s.t. ∈ c,i , ∃ some s ∈ ∗ and u ∈ ∗uc satisfying Pi (s) = Pi (s ), K (s ) ∩ c,i = ∅, and s u ∈ pr(K) ∈ i Si (Pi (s)) = Sdec (s) su ∈ L(Sdec /G) (by the definition of L(Sdec /G) together with s ∈ pr(K) = L(Sdec /G) and u ∈ ∗uc ) su ∈ pr(K) since L(Sdec /G) = pr(K) contradiction to the assumption su ∈ / pr(K).
Therefore, we conclude that K is delay-coobservable w.r.t. G and (o,i , c,i )i∈I .
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The delay-coobservability is a stronger notion than the C&P coobservability of Rudie and Wonham (1992) and Yoo and Lafortune (2002). It is obvious from Definition 3 that if a local supervisor makes a disabling decision on an illegal event for a given language K, it also makes a disabling decision on the event for any sublanguage of K in the presence of communication delays. In other words, the class of delay-coobservable languages is closed under intersection. Hence, we can infer that there exists an infimal prefix closed and delay-coobservable superlanguage for a given language. We can obtain this language by extending the formula of an infimal prefix closed and C&P coobservable superlanguage (Takai, Kumar, & Ushio, 2005). Moreover, the prefix closed and delay-coobservable sublanguage can also be obtained by extending the formula of a prefix closed and C&P coobservable sublanguage based on the class of locally observable languages (Takai et al., 2005). 3. Conclusions In this paper, we have introduced the notion of delaycoobservability for a given language specification as an extension of the coobservability (Rudie & Wonham, 1992; Yoo & Lafortune, 2002) to the case of decentralized supervisory control under communication delays. Based on this notion, we have presented the necessary and sufficient conditions for the existence of a nonblocking decentralized supervisor that can achieve a given language specification. The main results have been developed based on the conjunctive and permissive decentralized supervisory control structures, but these can be easily extended to any other decentralized control structures such as disjunctive and anti-permissive ones (Yoo & Lafortune, 2002, 2004). Acknowledgments This work was supported from Korea Ministry of Science and Technology through the Korean Systems Biology Research Grant (M10503010001-05N030100111) and the 21C Frontier Microbial Genomics and Application Center Program
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(Grant MG05-0204-3-0), and in part by 2005-B0000002 from Korea Bio-Hub Program of Korea Ministry of Commerce, Industry & Energy. K.-H. Cho was supported by the second stage Brain Korea 21 Project in 2006. References Balemi, S. (1992). Communication delays in connections of input/output discrete event processes. In Proceedings of the 31st IEEE conference on decision control 1992 (pp. 3374–3379). Berlin: Springer. Debouk, R., Lafortune, S., & Teneketzis, D. (2003). On the effect of communication delays in failure diagnosis of decentralized discrete event systems. Discrete Event Dynamic Systems: Theory and Applications, 13, 263–289. Lauzon, S. C., Mills, J. K., & Benhabib, B. (1997). An implementation methodology for the supervisory controller for flexible manufacturing workcells. SME Journal of Manufacturing Systems, 16(1), 91–101. Park, S.-J., & Cho, K.-H. (2006). Delay-robust supervisory control of discrete event systems with bounded communication delays. IEEE Transactions on Automatic Control, 51(5), 911–915. Ramadge, P. J., & Wonham, W. M. (1987). Supervisory control of a class of discrete event processes. SIAM Journal on Control and Optimization, 25(1), 206–230. Ricker, L., & Schuppen, J. V. (2001). Decentralized failure diagnosis with asynchronous communication between supervisors. In Proceedings of the European control conference. Rudie, K., & Willems, J. C. (1995). The computational complexity of decentralized discrete-event control problems. IEEE Transactions on Automatic Control, 40(7), 1313–1319. Rudie, K., & Wonham, W. M. (1992). Think gloablly, act locally: Decentralized supervisory control. IEEE Transactions on Automatic Control, 37, 1692–1708. Takai, S., Kumar, R., & Ushio, T. (2005). Characterization of co-observable languages and formulas for their super/sublanguages. IEEE Transactions on Automatic Control, 50(4), 434–447. Tripakis, S. (2004). Decentralized control of discrete-event systems with bounded or unbounded delay communication. IEEE Transactions on Automatic Control, 49(9), 1489–1501. Yoo, T.-S., & Lafortune, S. (2002). A general architecture for decentralized supervisory control of discrete-event systems. Discrete Event Dynamic Systems: Theory and Applications, 12(3), 335–377. Yoo, T.-S., & Lafortune, S. (2004). Decentralized supervisory control with conditional decisions: Supervisor existence. IEEE Transactions on Automatic Control, 49(11), 1886–1904.
