Decentralized Control of Discrete Event Systems with Multiple Local Specializations Shengbing Jiang, Vigyan Chandra, Ratnesh Kumar Department of Electrical Engineering University of Kentucky Lexington, KY 40506-0046 {sjian0, vigyan, kumar}@engr.uky.edu Abstract The decentralized supervisory control problem of discrete event systems with local specifications is studied in this paper. A necessary and sufficient condition is obtained for the existence of modular supervisors for ensuring that a group of local specifications is achieved by the controlled system. For illustration an example of a simple manufacturing system is given. This paper provides a modular approach for the design of supervisors for large systems. Keywords: Discrete event system, supervisory control, controllability, observability, decentralized control. Introduction The supervisory control theory for discrete event systems was first proposed by Ramadge and Wonham. In [7], they introduced the notion of controllability as a necessary and sufficient condition for the existence of a supervisor that achieves a desired controlled behavior for a given discrete event system under the complete observation of events. When the events are not completely observed by the supervisor but are filtered by an observation mask, an additional condition of observability introduced by Lin and Wonham [4], and Cieslak et al. [] is needed for the existence of the supervisor. In the more general case of decentralized control when there are more than one supervisors, the uncontrollable event set for the ith supervisor is Σ ui, and its observation is filtered through the mask M i, the condition of co-observability is required in place of observability, as shown in Cieslak et al. [] for the case when the controlled behavior is given as a prefix-closed language, which was later generalized to the non-prefixclosed case by Rudie and Wonham [8]. This research was supported by the National Science Foundation under the Grant NSF-ECS-9709796. In [5], Lin and Wonham considered a specialized version of the decentralized control, where the mask M i is a projection type mask, and the local controllable event set Σ ci is a subset of local observable event set Σ i := M i (Σ). The authors gave a sufficient condition in terms of the normality that guarantees that the decentralized control can achieve the optimal behavior achievable by a centralized supervisor. Later in [6], they extended their results to the systems with partial observation. They gave a sufficient condition for the existence of decentralized supervisors that ensure that the controlled behavior of the system lies in a given range expressed by conjunction of local specifications. In [9], Willner and Heymann also studied the problem of [5], except that the systems they considered were concurrent systems. They introduced the notion of separability and proved that under certain conditions the separability is a necessary and sufficient condition that guarantees that the decentralized control can achieve the optimal behavior achievable by a centralized supervisor. In [], we studied a version of decentralized control problem that is more general than those reported above, and provided a necessary and sufficient condition for the existence of decentralized supervisors for keeping the controlled behavior of the system in a given range. Results of [5, 6, 9] can be derived as special cases of our result. All the above work on decentralized control has to do with a global specification which may be specified through local specifications, i.e., given K L(G), find local supervisors {S i } such that L(( i S i)/g) = K. If K is given through local specifications {K i }, then K = i T i (K i ). Even if the global specification is given as a conjunction of local specifications, the satisfaction of global specification does not imply the satisfying of each local specification, i.e., [L(( i S i)/g) = ( i T i (K i )) L(G)] [T j [L(( i S i)/g)] = K j, j]
where K i T i (L(G)) is the local specification, and T i is the natural projection from the global event set to the local event set. But in practice, we may encounter the case that each local specification must be satisfied. In this paper we study this problem. We first examine the general decentralized control problem studied in [] and shown in Figure. Instead of con- σ M ( σ) T G S Σ disable G..... set union..... T n Gn n Σ disable S n Figure : Diagram of the system σ n M n( σ n) sidering a global specification which is the case in [], we consider a group of local specifications. We obtain a necessary and sufficient condition for the existence of decentralized supervisors ensuring that each local specification is satisfied. Next we consider a special case of the above problem, namely, the decentralized control of concurrent systems with partial observations where G = i G i, which implies L(G) = i T i [L(G i )]. A necessary and sufficient condition is also derived for the existence of decentralized supervisors in this special case. Finally, an illustrative example based on a simple manufacturing system is provided. Notation and Preliminaries This paper is set in the supervisory control framework for discrete-event systems developed by Ramadge and Wonham [7]. For the readers convenience, some background results from the cited references are first provided in this section. For a detailed introduction of the theory, readers may refer to [3]. An uncontrolled discrete event system is modeled as an automaton G = (Q, Σ, δ, q 0, Q m ), where Q is a set of states, Σ is a set of event labels, q 0 Q is the initial state, Q m Q is the set of marked states, and δ : Σ Q Q, the transition function, is a partial function defined at each state in Q for a subset of Σ. Let Σ denote the set of all finite strings over Σ including the null string ɛ. Then δ : Σ Q Q can be extended in the obvious way to δ : Σ Q Q. The language generated by G is given by, L(G) := {s Σ δ(s, q 0 ) is defined}, and the language marked by G is given by, L m (G) := {s Σ δ(s, q 0 ) Q m }. In general, a language over Σ is any subset of Σ. The prefix-closure pr(k) of a language K Σ is the set of all prefixes of strings in K. K is called prefix-closed if pr(k) = K. To study control of a system, its event set Σ is partitioned into two subsets Σ c and Σ u of controllable and uncontrollable events respectively. A supervisor S is a pair (R, ψ) where R is an automaton which recognizes a language over the same event set as G, and ψ is the feedback map from the event set and the states of R to the set {enable, disable}. If X denotes the set of states of R, then ψ satisfies: ψ(σ, ) =enable if σ Σ u. R is considered to be driven by the strings generated by G, and in turn, at each state x of R, the control rule ψ(σ, x) dictates the occurrence of σ at the corresponding state of G. The behavior of the supervised system is represented by an automaton S/G. The language generated by the supervised system is denoted by L(S/G), and its marked language is L m (S/G) := L(S/G) L m (G). A supervisor S = ((X, Σ, ξ, x 0 ), ψ) is complete for G if for every string s in Σ and every event σ in Σ, the conditions s L(S/G), sσ L(G), and ψ(σ, ξ(s, x 0 )) =enable together imply that sσ L(S/G). It will always be assumed that S is complete. Given a nonempty prefix-closed sublanguage K of L(G), there exists a supervisor S such that L(S/G) = K if and only if K is (L(G), Σ u )-controllable [7], i.e., pr(k)σ u L(G) pr(k). If K is not controllable, then a supervisor is synthesized for achieving the supremal prefixed closed and controllable sublanguage of K, denoted by supp C(K). When the supervisor can not observe all the events, the concept of observation mask is introduced. An observation mask is a function M : Σ {ɛ}, where ɛ, and is called the set of observed events. The mask function can be extended to the set of strings in a natural way. A supervisor S for G is said to be mask-compatible [3] if it observes only M(L(G)). In this paper, it is assumed that all supervisors for partially observed systems must be mask-compatible supervisors. A sublanguage K of L(G) is said to be (L(G), M)- observable [4] if the conditions s, s pr(k), M(s) = M(s ), sσ pr(k), and s σ L(G) together imply
s σ pr(k). If K is not observable, then the infimal prefix closed and observable superlanguage of K denoted inf P O(K) exists [4]. Given a nonempty prefixclosed sublanguage K of L(G) and an observation mask M, there exists a supervisor S such that L(S/G) = K if and only if K is (L(G), Σ u )-controllable and (L(G), M)- observable [4]. Let A, E Σ be two languages with A E L m (G), M be an observation mask, then there exists a supervisor S such that A L(S/G) E if and only if infp CO(A) E, where infp CO(A) is the infimal prefix-closed, controllable, and observable superlanguage of A [4]. 3 Decentralized Control In this section we study the decentralized supervisory control problem of discrete event systems with local specifications for ensuring the satisfaction of a group of local specifications. We consider the general setting of decentralized control problem studied in []. Let G be a plant, Σ be the global event set, Σ i Σ, i I, be the local event sets. Let T i : Σ Σ i {ɛ} denote the natural projection from the global event set to the local event set. We use G i to denote the automaton with L(G i ) = T i [L(G)], i I, called the local system; here L(G i ) represents the ith local behavior of the system G. Let Σ ci Σ i be the ith local controllable event set, then the global controllable event set is given by Σ c := i Σ ci, the local uncontrollable event set is given by Σ ui := Σ Σ ci, the global uncontrollable event set is given by Σ u := Σ Σ c = Σ i (Σ Σ ui ) = i Σ ui. Finally, let M i : Σ i i {ɛ} be the local observation mask. If S i is a local supervisor for G i, then we can extend S i to a global supervisor S i for G, by letting S i not observe all events in Σ Σ i, and permanently enable all events in Σ Σ ci. Then it is easy to see that L( S i /G) = L(G) T i [L(S i /G i )], i I. From [, Theorem ], we have the following result for global specification under the general decentralized control of Figure. Theorem Let G be a plant, Σ be the global event set, Σ i Σ, i I, be the local event sets. Let T i be the natural projection from Σ to Σ i, Σ ci Σ i be the local controllable event set, M i be the local observation mask. Given a nonempty prefix-closed sublanguage K of L(G), there exists a group of local supervisors {S i, i I} such that L(( i S i )/G) = K if and only if K = L(G) ( i T i [infp CO i (T i (K))]), where S i is the extension of S i to the global system G, and infp CO i (T i (K)) is the infimal prefix closed, (L(G i ), Σ ui )-controllable and (L(G i ), M i )- observable superlanguage of T i (K) (Σ ui := Σ Σ ci ; and L(G i ) := T i [L(G)] is the ith local behavior of the system). Consider the following target specification problem: Given a group of nonempty prefix-closed languages {K i L(G i ), i I}, find local supervisors {S i, i I} such that T j [L(( S i i )/G)] = K j, j I, i.e., T j [L(G) ( i T i [L(S i /G i )])] = K j. For the existence of the local supervisors, a necessary and sufficient condition is obtained in the following theorem. Theorem Let G be a plant, Σ be the global event set, Σ i Σ, i I, be the local event sets. Let T i be the natural projection from Σ to Σ i, Σ ci Σ i be the local controllable event set, M i be the local observation mask. Given a group of nonempty prefixclosed languages {K i L(G i ), i I}, there exists a group of local supervisors {S i, i I} such that T j [L(( S i i )/G)] = K j for all j I if and only if j I, K j = T j [L(G) ( i T i (infp CO i (K i )))], where S i is the extension of S i to the global system G, and infp CO i (T i (K)) is the infimal prefix closed, (L(G i ), Σ ui )-controllable and (L(G i ), M i )- observable superlanguage of T i (K) (Σ ui := Σ Σ ci ; and L(G i ) := T i [L(G)] is the ith local behavior of the system). Proof: The sufficiency is obvious, since we can choose the local supervisor S i by letting L(S i /G i ) = infp CO i (K i ). Then the result follows from the definition of L(( i S i )/G) and the hypothesis. To prove the necessity, if the local supervisors exist, we let K = L(( S i i )/G), i.e., K = L(G) ( i T i [L(S i /G i )]). Then from hypothesis we have T i (K) = K i for all i I. Since K is a global specification, and it can be achieved by the local supervisors {S i, i I}, from Theorem we know K = L(G) ( i T i [infp CO i (T i (K))]). Further from T i (K) = K i we have: j I, K j = T j (K) = T j [L(G) ( i T i (infp CO i (K i )))]. This completes the proof.
Remark Theorem 3 provides a modular approach for the design of supervisors of large systems. For the large systems having many subsystems, and each subsystem having its own local specification, we first check whether these local specifications satisfy the condition in Theorem 3. If the local specifications do satisfy the condition, then we can design the local supervisors S i by letting L(S i /G i ) = infp CO i (K i ). Having solved the target specification problem, we next consider the following range specification problem: Given two groups of languages {A i L(G i ), i I} and {E i L(G i ), i I} with A i E i for all i I, find local supervisors {S i, i I} such that j I, A j T j [L(( S i i )/G)] E j. This problem should be viewed as a generalization of the target specification problem studied above, since when A i = E i = K i for all i, it reduces to the target specification problem. The following theorem provides a necessary condition and a separate sufficient condition for the range specification problem. Theorem 3 Let G be a plant, Σ be the global event set, Σ i Σ, i I, be the local event sets. Let T i be the natural projection from Σ to Σ i, Σ ci Σ i be the local controllable event set, M i be the local observation mask. Given two groups of languages {A i L(G i ), i I} and {E i L(G i ), i I} with A i E i for all i I, there exists a group of local supervisors {S i, i I} such that j I, A j T j [L(( i S i )/G)] E j if j I, A j T j [L(G) ( i T i (infp CO i (A i )))] E j, where infp CO i T i (A) is the infimal prefix-closed, (L(G i ), Σ ui )-controllable, and (L(G i ), M i )-observable superlanguage of T i (A). Also the condition j I, T j [L(G) ( i T i (infp CO i (A i )))] E j, is necessary for the existence of the local supervisors. Proof: The sufficiency of the first condition is obvious, since we can choose the local supervisor S i such that L(S i /G i ) = infp CO i (A i ) for each ith local system G i. For the necessity of the second condition, if the local supervisors exist, then we let K = L(( S i i )/G), i.e., K = L(G) ( i T i [L(S i /G i )]). Then we have A i T i (K) E i. Since L(S i /G i ) is prefix-closed, (L(G i ), Σ ui )-controllable, and (L(G i ), M i )-observable, which together with the fact that A i T i (K) T i [ j T j (L(S j /G j ))] L(S i /G i ) gives infp CO i (A i ) L(S i /G i ) for all i. So the following holds for all j I, T j [L(G) ( i T i (infp CO i (A i )))] T j [L(G) ( i T i (L(S i /G i )))] = T j (K) E j. This completes the proof. So far we have not assumed any structural properties of the system considered. Now we suppose that the system G is composed of concurrent subsystems, G i, with event sets Σ i, i.e., G := i G i, and Σ = i Σ i. From [9, Proposition 3.], we have L(G) = i T i [L(G i )]. (Note that in the earlier analysis of Section 3 we also had the local systems {G i } with L(G i ) = T i L(G), but we only have the property that L(G) i T i [L(G i )].) We also note that for the concurrent system the property L(G i ) T i L(G) holds instead of the definition L(G i ) := T i L(G) used for non-concurrent systems. It can be seen that a concurrent system is a special case of the systems we considered up to now. So we can derive the results for the concurrent system as before, by applying L(G) = i T i [L(G i )]. We give the similar results to those of Theorem and 3, omitting the proofs. Theorem 4 Let G be a concurrent system with G = i G i, i I, Σ be the global event set, which is the union of local event sets, i.e., Σ = i Σ i, i I. Let T i be the natural projection from Σ to Σ i, Σ ci Σ i be the local controllable event set, M i be the local observation mask. Then for a given group of nonempty prefix-closed languages {K i L(G i ), i I}, there exists a group of local supervisors {S i, i I} such that T j [L(( i S i )/G)] = K j for all j I if and only if j I, K j = T j [ i T i (infp CO i (K i ))], where infp CO i T i (K) is the infimal prefix closed, (L(G i ), Σ ui )-controllable, and (L(G i ), M i )-observable superlanguage of T i (K), and Σ ui := Σ i Σ ci is the local uncontrollable event set. Theorem 5 Let G be a concurrent system with G = i G i, i I, Σ be the global event set, which is the union of local event sets, i.e., Σ = i Σ i, i I. Let T i be the natural projection from Σ to Σ i, Σ ci be the local controllable event set, M i be the local observation mask.
Given two groups of languages {A i L(G i ), i I} and {E i L(G i ), i I} with A i E i for all i I, there exist local supervisors {S i, i I} such that j I, A j T j [L(( i S i )/G)] E j if j I, A j T j [ i T i (infp CO i (A i ))] E j. where G, G, and G 3 represent C, T, and W respectively, and are shown in Figure 3. Here we consider only closed languages, therefore all states are considered to be marked states. Also the condition j I, T j [ i T i (infp CO i (A i ))] E j, (a) (b) τ (c) is necessary for the existence of the local supervisors. Remark As stated in Remark, we only need the local information L(G i ) when designing the local supervisor S i, since S i achieves the language infp CO i (K i ). However to verify the condition in Theorem, we still need the global information L(G). Only local information suffices for the concurrent systems as is obvious from the condition of Theorem 4. So for the concurrent systems, we need not compute L(G). This provides tremendous computational savings for large concurrent systems. 4 Illustrative example In this section, we apply our results to a simple manufacturing system. The system, shown in Figure, consists of a transporter T, a part conveyor C, and a workstation W, which includes a buffer with a capacity of. angle sensor rack Transporter Gearbox motor TRANSPORTER pinion Unloading position Workstation FIXTURE Part Conveyor Conveyor Gearbox motor Processed part loading position Figure : A manufacturing system Workpieces are taken from the input port of the system and transfered to the buffer of W by the conveyor C; then they are processed on workstation W ; after the processing, they are taken away from W to the output port by the transporter T. We assume that there are an infinite number of workpieces at the input port of the system, and a workpiece is taken away from the output port immediately. The system is modeled as G = G G G 3, Figure 3: Subsystems G, G, and G 3 The event labels in Figure 3 represent the following actions: : a workpiece is taken from the input port by C; : a workpiece is taken from W by T ; : a workpiece is sent to the buffer of W by C; : a workpiece is sent to the output port by T τ : a workpiece is processed on W. The event τ is the only uncontrollable event. At first, we consider the case that all events are observable. The global event set is Σ = {,,,, τ }, and the local event sets are Σ = {, }, Σ = {, }, and Σ 3 = {, τ, }. We have Σ c = Σ, Σ c = Σ, Σ 3c = {, }, T i : Σ Σ i, M i = Id, for i =,, 3. Figure 4: Automaton for A 3 τ τ τ Figure 5: Automaton for E 3 The local range specifications are given as {A, A, A 3, E, E, E 3 }, where E = A = L(G ), E = A = L(G ). A 3 is generated by the automaton in Figure 4, requiring that the system can run continuously. E 3 is generated by the automaton in Figure 5, requiring that the overflow and the underflow of W be avoided.
It is easy to check that Using the fact we can verify that infp CO 3 (A 3 ) = A 3 E 3. infp CO i (A i ) = A i = E i, i =,, j, T j [ i T i (infp CO i (A i ))] = A j. So from Theorem 5 we know that there exist local supervisors {S i, i =,, 3} such that j, A j T j [ i T i (L(S i /G i ))] E j. In fact, {S i, i =,, 3} can be synthesized as follows: S and S enable all events at all states; S 3 = (R 3, ψ 3 ), R 3 is the automaton in Figure 6, ψ 3 disables at states {x 0, x } and at state x 4. τ τ x x x 0 Figure 6: Automaton for R 3 x x 3 4 Now suppose that, owing to a failure in a sensor, event becomes unobservable in G 3, i.e., M 3 ( ) = ɛ. Then we can calculate infp CO 3 (A 3 ), which is the language generated by the automaton in Figure 7. τ τ Figure 7: Automaton for infp CO 3 (A 3 ) From Figure 5 and Figure 7, it can be seen that infp CO 3 (A 3 ) E 3, since ( τ ) 3 infp CO 3 (A 3 ) and ( τ ) 3 E 3. Also we can verify that T 3 [ i T i (infp CO i (A i ))] = infp CO 3 (A 3 ). So T 3 [ i T i (infp CO i (A i ))] E 3. So the necessary condition in Theorem 5 does not hold, which implies that no local supervisors exist for this case. specifications for ensuring the satisfaction of a group of local specifications. The main result is the derivation of a necessary and sufficient condition for the existence of decentralized supervisors. This result was also specialized to the setting of decentralized control of concurrent systems. The ideas were illustrated by application to a simple manufacturing system. References [] R. Cieslak, C. Desclaux, A. Fawaz, and P. Varaiya. Supervisory control of discrete event processes with partial observation. IEEE Transactions on Automatic Control, 33(3):49 60, 988. [] S. Jiang and R. Kumar. Decentralized Control of Discrete Event Systems with Specializations to Local Control and Concurrent Systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 30(5), Oct. 000. [3] R. Kumar and V. K. Garg. Modeling and Control of Logical Discrete Event Systems. Kluwer Academic Publishers, Boston, MA, 995. [4] F. Lin and W. M. Wonham. On observability of discrete-event systems. Information Sci., 44(3):73-98, 988. [5] F. Lin and W. M. Wonham. Decentralized supervisory control of discrete-event systems. Information Sci., 44(3):99-4, 988. [6] F. Lin and W. M. Wonham. Decentralized control and coordination of discrete-event systems with partial observation. IEEE Transactions on Automatic Control, 35():330-337, 990. [7] P. J. Ramadge and W. M. Wonham. Supervisory control of a class of discrete event processes. SIAM Journal of Control and Optimization, 5():06 30, 987. [8] K. Rudie and W. M. Wonham. Think globally, act locally: decentralized supervisory control. IEEE Transactions on Automatic Control, 37():69-708, 99. [9] Y. Willner and M. Heymann. Supervisory control of concurrent discrete-event systems. Int. J. Control, 54(5):43-69, 99. 5 Conclusions In this paper we studied the decentralized supervisory control problem of discrete event systems with local