Arborescent Architecture for Decentralized Supervisory Control of Discrete Event Systems

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1 Arborescent Architecture for Decentralized Supervisory Control of Discrete Event Systems Ahmed Khoumsi and Hicham Chakib Dept. Electrical & Computer Engineering, University of Sherbrooke, Canada Abstract Decentralized control of discrete event systems consists in using local supervisors that observe locally the plant and compute local decisions to enable or disable events; those local decisions are transmitted to fusion modules that combine them to generate global enablement/disablement decisions that are actually applied to the plant. C&P and D&A controls are the two simplest systematic decentralized controls. Inference-based control is the most general systematic decentralized control, which generalizes significantly C&P and D&A controls. In this paper, we first propose a method that realizes a control objective C by an arborescent architecture (or tree). Each leaf of the tree is a decentralized control, and each node n is a disjunction or conjunction of the enabling/disabling decisions of the two children of n. We show that if the control objective C is realizable by inference-based control, then every leaf of the obtained tree is a C&P or D&A control. This means that by combining adequately C&P and D&A controls, we can realize every control objective that is realizable by inference-based control. Index Terms Discrete event systems (DES); decentralized supervisory control; C&P control; D&A control; inference-based control; arborescent architecture; multi-decision control. I. INTRODUCTION In decentralized (supervisory) control of discrete event systems (DES), several local supervisors cooperate to restrict the behavior of a plant so that it respects a specification. C&P (for Conjunctive and Permissive) and D&A (for Disjunctive and Anti-permissive) controls are the two simplest language-based decentralized controls [1], [2]. Inference-based control is the most general language-based decentralized control [3]. By language-based, we mean that the control methods depend on the languages used to model the plant and the specification. This is in

2 contrast with automata-based control methods, like multi-decision control [4] and intersectionbased control [5]. In this paper, we propose a method that constructs an arborescent architecture (or tree) which realizes a given control objective C. Each leaf of the tree is a decentralized control and each node n is a disjunction or conjunction of the control decisions of the two children of n. We consider the case where C is realizable by inference-based control and show that in the obtained tree, every leaf is a C&P or D&A control. Therefore, every control realizable by inference-based control is also realizable by disjunctions and conjunctions of C&P and D&A controls. Hence, we show that inference-based control (which is nonintuitive and difficult to apprehend) is equivalent to a combination of intuitive and simple controls. The rest of the paper is organized a follows. In Sect. II, we introduce decentralized control with an emphasis on C&P, D&A and inference-based controls. Sect. III presents the method to construct an arborescent architecture realizing a given control objective C. In Sect. IV, we study the tree construction when C is realizable by inference-based control. Section V concludes our study. For lack of space, the proofs of our results are omitted. A. Generalities II. DECENTRALIZED CONTROL A finite set of events is called alphabet, a finite sequence of events is called trace, and a set of traces is called language. ε denotes an empty trace. Henceforth, λ and σ denote a trace and an event, respectively. A prefix of λ is a trace µ s.t. λ = µα for some trace α. The set of prefixes of a language L is a language denoted L. If L and K are two languages, we denote by L \ K the language consisting of the traces of L from which are removed the traces of K. P i (X) and P 1 i (X) denote the usual projection and inverse projection of a language X. A regular language is modeled by a finite state automaton (FSA) A = (Q, Σ, δ, q 0, Q m ), where Q is a finite set of states, Σ is an alphabet, the transitions are specified by a partial function δ : Q Σ Q, q 0 Q is the initial state, and Q m Q is the set of marked states. The generated language L of A contains the traces leading from q 0 to the states in Q, while the marked language L m of A contains the traces of L leading to the states in Q m. We have L m L. An FSA is said trim if: 1) all its states are reachable from q 0, and 2) from any state in Q\ Q m we can reach a state in Q m. For a trim FSA, L m =L.

3 A discrete event system (DES) is a system whose behavior is described by the possible traces it can execute. We consider DESs modeled by regular languages (hence also by FSA). B. Supervisory Control Consider a plant modeled by an FSA G = (Q, Σ, δ, q 0, Q m ), and let L and L m be its generated and marked languages, respectively. Consider a specification modeled by a trim FSA K = (R, Σ, ξ, r 0, R m ), and let K be its marked language. Since K is trim, its generated language is K. The objective of control is to restrict the behavior of the plant so that it conforms to the specification. Such objective is achieved by using a supervisor S that observes the evolution of the plant and decides to enable or disable events. Let then Sup(λ, σ) {1, 0} (1 for enable, 0 for disable) be the decision made by S on an event σ when the plant has executed a trace λ L. We define the following prefix-closed language L(S/G): - ε L(S/G); - [λ L(S/G) λσ L Sup(λ, σ)=1] λσ L(S/G). We also define the corresponding marked language L m (S/G) = L(S/G) L m. Intuitively, L(S/G) and L m (S/G) consist respectively of the traces of L and L m which are permitted by S. Consider the following languages E σ and D σ : Definition 1: For every event σ Σ, E σ = {λ K λσ K} and D σ = {λ K λσ L \ K}. Intuitively, E σ contains the traces of the specification after which σ is accepted by both the plant and the specification, while D σ contains the traces of the specification after which σ is accepted only by the plant. The alphabet Σ is partitioned into Σ c and Σ uc, the set of controllable and uncontrollable events, respectively. Here are two important notions related to S: Definition 2: S is said feasible if it enables uncontrollable events. Formally: λ L : σ Σ uc Sup(λ, σ) = 1. Definition 3: S is said nonblocking if L m (S/G)=L(S/G). The objective of control is to find a non-blocking feasible supervisor that actuates so that Eqs. (1,2) are satisfied: L(S/G) = K (1) L m (S/G) = K (2)

4 Here are two important notions related to K: Definition 4: K is said L-controllable if D σ = σ Σ uc. Definition 5: K is said L m -closed if K = K L m. We have the following proposition: Proposition 1: If K is L m -closed: Eqs. (1,2) are equivalent to: σ Σ λ E σ Sup(λ, σ) = 1 λ D σ Sup(λ, σ) = 0 (3) C. Decentralized Supervisory Control There exist many studies of decentralized supervisory control, a few examples are [1], [2], [3], [6], [7]. The conventional approach in decentralized control is that n local supervisors (S i ) 1 i n take local decisions which are combined to generate the global decision which is applied to the plant. Each S i has its set of observable events Σ o,i and its set of controllable events Σ c,i. Let Σ o = Σ o,1 Σ o,n, Σ c = Σ c,1 Σ c,n, and I σ {1,, n} be the indexing set of the local supervisors controlling σ Σ c. To every decentralized control architecture is associated a property of coobservability s.t.: a non-blocking feasible supervisor satisfying Eqs. (1,2) exists iff K is L m -closed, L-controllable and coobservable. We are specifically interested by [1], [2], [3], [7]. The authors of [1], [2] propose the two simplest relevant decentralized architectures, which are called C&P (for Conjunctive & Permissive) and D&A (for Disjunctive & Anti-permissive) controls, respectively. The authors of [7] propose conditional C&P and D&A controls that generalize the (unconditional) C&P and D&A controls. The authors of [3] propose inference-based control that generalizes significantly conditional and unconditional C&P and D&A controls. Sects. II-D, II-E and II-F present respectively C&P, D&A and inference-based controls. D. Conjunctive and Permissive (C&P) Control [1], [2] In C&P control, the local supervisors are permissive (i.e. each S i enables an event when it is unsure of the adequate decision), and their local decisions are combined conjunctively to generate the global decision. This can be realized as follows:

5 After the execution of λ L, S i has observed P i (λ) and computes, for every σ Σ c,i, its local decision Sup i (P i (λ), σ): Sup i (P i (λ), σ) = The global decision Sup(λ, σ) is conjunctive, i.e.: Sup(λ, σ) = 1, if P i (λ) P i (E σ ) 0, if P i (λ) P i (E σ ) (4) i I σ Sup i (P i (λ), σ) (5) Let then C&P supervisor denote the feasible supervisor defined by Eqs. (4,5). Coobservability associated to C&P control for σ Σ c is: Definition 6: (E σ, D σ ) is said C&P-coobservable if i I σ P 1 i P i (E σ ) D σ =. The following theorem is an adaptation of results from [2]: Theorem 1: The C&P supervisor is non-blocking and satisfies Eqs. (1,2) iff : (E σ, D σ ) is C&P-coobservable σ Σ c, and K is L-controllable and L m -closed. E. Disjunctive and Anti-permissive (D&A) Control [2] In D&A control, the local supervisors are anti-permissive (i.e. each S i disables an event when it is unsure of the adequate decision), and their local decisions are combined disjunctively to generate the global decision. This can be realized as follows: After the execution of λ L, S i has observed P i (λ) and computes, for every σ Σ c,i, its local decision Sup i (P i (λ), σ): Sup i (P i (λ), σ) = The global decision Sup(λ, σ) is disjunctive, i.e.: Sup(λ, σ) = 0, if P i (λ) P i (D σ ) 1, if P i (λ) P i (D σ ) (6) i I σ Sup i (P i (λ), σ) (7) Let then D&A supervisor denote the feasible supervisor defined by Eqs. (6,7). Coobservability associated to D&A control for σ Σ c is: Definition 7: (E σ, D σ ) is said D&A-coobservable if i I σ P 1 i P i (D σ ) E σ =. The following theorem is an adaptation of results from [2]: Theorem 2: The D&A supervisor is non-blocking and satisfies Eqs. (1,2) iff : (E σ, D σ ) is D&A-coobservable σ Σ c, and K is L-controllable and L m -closed.

6 F. Inference-Based Control [3] After the execution of λ L, every S i has observed P i (λ) and computes, for every σ Σ c,i, a pair Sup i (P i (λ), σ) = (c i (P i (λ), σ), n i (P i (λ), σ)), where c i (P i (λ), σ) {1, 0, ϕ} is a local decision, and n i (P i (λ), σ) is a nonnegative integer representing an ambiguity level. ϕ denotes an unsure decision. The global decision Sup(λ, σ) is obtained by selecting the local decision with the smallest ambiguity level. The computation of c i (P i (λ), σ) and n i (P i (λ), σ) is quite complex and nonintuitive and will not be presented. However, we present the following iterative languages E σ [k] and D σ [k] which are fundamental in inference-based control and in our study: Basis: E σ [0] = E σ and D σ [0] = D σ, Inductive step: for k 0 E σ [k + 1] = E σ [k] [ D σ [k + 1] = D σ [k] [ Pi 1 i I σ Pi 1 i I σ P i (D σ [k])] P i (E σ [k])] Proposition 2: Some properties of E σ [k] and D σ [k]: 1) E σ [k+1] E σ [k] and D σ [k+1] D σ [k] k 0. 2) If E σ [k] = for some k 0, then E σ [i] = D σ [i] = i > k. The property holds if we switch E σ and D σ. 3) If E σ [k + 2] = E σ [k] for some k 0, then E σ [i] = E σ [k] and D σ [i + 1] = D σ [k + 1] i k. The property holds if we switch E σ and D σ. 4) If E σ [k + 1] = E σ [k] and D σ [k + 1] = D σ [k] for some k 0, then E σ [i] = E σ [k] and D σ [i]=d σ [k] i k. The above properties can be easily deduced from [3]. Inference-based control is denoted Inf N -control if N is the maximum ambiguity level which is computed. In [3], it is proved that Inf 0 -control is equivalent to a combination of C&P and D&A controls, and Inf 1 -control is equivalent to a combination of conditional C&P and conditional D&A controls. Coobservability associated to Inf N -control for σ Σ c is: Definition 8: (E σ, D σ ) is said Inf N -coobservable for some nonnegative integer N, if E σ [N+ 1] = or D σ [N +1] =. The following theorem is an adaptation of results from [3]:

7 Theorem 3: The feasible supervisor computed by Inf N -control is non-blocking and satisfies Eqs. (1,2) iff : (E σ, D σ ) is Inf N -coobservable σ Σ c, and K is L-controllable and L m -closed. G. Reformulation of the Control Objective Note that Theorems 1, 2 and 3, are not expressed as usual: instead of writing there exists a supervisor..., we consider a specific supervisor previously defined. The two formulations are equivalent in any architecture, but the advantage of the new formulation is that it is more concrete since it is related to a known supervisor. L-controllability and L m -closure of K are independent of the control architecture, and hence do not influence our study of the following sections which is purely related to control architectures. Therefore, without loss of generality, we will use the following assumption, which allows to consider uniquely coobservability as condition of existence of supervisor. Assumption 1: K is L-controllable and L m -closed. From Assump. 1 and Prop. 1, the control objective to satisfy Eqs. (1,2) is reformulated as the objective to satisfy Eq. (3), σ Σ. The advantage of this reformulation is that Eq. (3) is more concrete than Eqs. (1,2), since Eq. (3) expresses explicitly a constraint on the global decision of the supervisor. III. SYNTHESIS OF AN ARBESCENT ARCHITECTURE For an event σ, let a pair (E, D) s.t. E E σ and D D σ. Controlling (E, D) means applying control decisions to σ that satisfy Eq. (3) w.r.t (E, D). Controlling a plant can hence be defined as controlling (E σ, D σ ) for every σ Σ. As we will explain it later in this section, our arborescent methodology is based on decomposing E σ and D σ, s.t. the control of (E σ, D σ ) is transformed into a composition (by disjunctions and conjunctions) of controls of pairs, where each pair (E, D) is s.t. E E σ and D D σ. For the purpose of our methodology, we generalize the definitions of D&A and C&P controls and their corresponding equations (4,5) and (6,7) for any pair (E, D), instead of only (E σ, D σ ). We also generalize the control objective to satisfy Eq. (3) for any pair (E, D), instead of only (E σ, D σ ). We also define the iterative languages E[k] and D[k] like E σ [k] and D σ [k] have been defined in Sect. II-F. In Sects. III-B and III-C, we show how to decompose the control of (E, D) into a conjunction or disjunction of two controls inspired from the multi-decision control [4]. The cases where E = or

8 D = are trivial. Indeed, controlling (E, ) is achieved by always enabling σ, controlling (, D) is achieved by always disabling σ, and controlling (, ) is achieved by taking any decision. Hence, we assume: Assumption 2: E = and D =. A. Series (r i ) i 1 and (s i ) i 1 We define two series (r i ) i 1 and (s i ) i 1 and their respective indexes ν and µ. They will be useful for presenting our methodology. r 1 = (E σ,d σ ), r 2k+2 = (E σ [2k + 1], D σ [2k]) and r 2k+3 = (E σ [2k + 1], D σ [2k + 2]) for k 0. Let r ν be the first element of (r i ) i 1 that has an empty component, if any. s 1 = (E σ \ E σ [1], D σ ), s 2k+2 = (E σ [2k + 1], D σ [2k] \ D σ [2k + 2]) and s 2k+3 = (E σ [2k + 1] \ E σ [2k + 3], D σ [2k + 2]) for k 0. Let s µ be the first element of (s i ) i 1 that has an empty component, if any. Note that each r i<ν and s i<µ is a pair (E, D) that can be controlled, with the objective to satisfy Eq. (3) w.r.t (E, D). B. Disjunctive Control Based on a Decomposition of E Consider a decomposition of E into E 1 and E 2, i.e. E = E 1 E 2. The disjunctive control of (E 1, D) and (E 2, D) is obtained by combining disjunctively the decisions of the controls of (E 1, D) and (E 2, D). From [4] we have: Proposition 3: The disjunctive control of (E 1, D) and (E 2, D) realizes the control of (E 1 E 2, D). By applying Prop. 3 to r 2k+1 for k 0, we obtain the following proposition: Proposition 4: The disjunctive control of s 1 and r 2 realizes the control of r 1. The disjunctive control of s 2k+3 and r 2k+4 realizes the control of r 2k+3, for k 0. We have the following proposition, which implies that every s 2k+1 can be controlled by a D&A architecture: Proposition 5: Every s 2k+1 for k 0 is D&A-coobservable. Prop. 4 and Prop. 5 are illustrated in Fig. 1.

9 control decision control decision Control of (E, D) (a1) (a2) (E \ E[1], D) Control of (E[1], D) control decision Control of (E[2k+1], D[2k+2]) (b1) (b2) control decision Control of (E[2k+1]\E[2k+3], D[2k+2]) (E[2k+3], D[2k+2]) Fig. 1. (a1) realized by (a2); (b1) realized by (b2). C. Conjunctive Control Based on a Decomposition of D Consider a decomposition of D into D 1 and D 2, i.e. D = D 1 D 2. The conjunctive control of (E, D 1 ) and (E, D 2 ) is obtained by combining conjunctively the decisions of the controls of (E, D 1 ) and (E, D 2 ). From [4]: Proposition 6: The conjunctive control of of (E, D 1 ) and (E, D 2 ) realizes the control of (E, D 1 D 2 ). By applying Prop. 6 to r 2k+2 for k 0, we obtain the following proposition: Proposition 7: The conjunctive control of s 2k+2 and r 2k+3 realizes the control of r 2k+2, for k 0. We have the following proposition, which implies that every s 2k+2 can be controlled by a C&P architecture: Proposition 8: Every s 2k+2 for k 0 is C&P-coobservable. Prop. 7 and Prop. 8 are illustrated in Fig. 2. control decision Control of (E[2k+1], D[2k]) (a1) (a2) (E[2k+1], D[2k]\D[2k+2]) control decision Control of (E[2k+1], D[2k+2]) Fig. 2. (a1) realized by (a2).

10 D. Tree Construction We construct an arborescent architecture to control (E σ, D σ ); the construction method is based on Props. 4, 5, 7 and 8, which are illustrated in Figs. 1 and 2. In a first step the objective of controlling r 1 = (E σ, D σ ) is split disjunctively (as shown in Fig. 1(a1,a2)) into the control of s 1 (left child) and the control of r 2 (right child). The latter control of r 2 is itself split conjunctively (as shown in Fig. 2 for k = 0) into the control of s 2 (left child) and the control of r 3 (right child). The latter control of r 3 is itself split disjunctively (as shown in Fig. 1(b1,b2) for k = 0) into the control of s 3 (left child) and the control of r 4 (right child). And so on, conjunctive and disjunctive decompositions are applied alternately to the successive r i until we obtain a r i whose left child (s i ) or right child (r i+1 ) has an empty component. Such r i is a leaf. Here are necessary notations to define formally the construction of the tree. Notations 1: Let x and y be a pair (E, D) to be controlled: x denotes a leaf representing the control of x; x DA and x CP denote leafs representing the D&A and C&P controls of x, respectively; x DA y is a node with the left child x DA and the right child y ; x CP y is a node with the left child x CP and the right child y. The tree (which we denote T ) is constructed as follows: Initialization: T = r 1 = (E σ, D σ ); i = 1. While s i and r i+1 are without empty component: - if i is odd: replace r i by s i DA r i+1 ; - if i is even: replace r i by s i CP r i+1 ; - i = i + 1. The obtained tree is represented in Fig. 3. Note that in this tree, E σ [i] and D σ [i] have odd and even indexes, respectively. The tree realizes the control of (E σ, D σ ), i.e. the decisions (enable, disable) to control (E σ, D σ ) are provided by the operator of the root of the tree. Each node n of the tree has two children: the left child of n is a leaf which provides the decisions of a C&P or D&A control, and the right child of n is a subtree which provides the decisions of a control. Note that (r i ) i 1 are the pairs written at the upper right sides of the nodes of Fig. 3. The successive r i are controlled by the successive nested subtrees. Note also that (s i ) i 1 are the pairs

11 written in the left leafs of Fig. 3. The successive s i are controlled by the successive left leafs. The fact that each left child (which is a leaf) of a node is a C&P or D&A control is due to Props. 5 and 8. (E \ E[1], D) (E[1], D\D[2]) (E[1]\E[3], D[2]) (E[3], D[2]\D[4]) Control of (E, D) Control of (E[1], D) Control of (E[1], D[2]) Control of (E[3], D[2]) Control of (E[3], D[4])... (E[2k 1]\E[2k+1], D[2k]) (E[2k+1], D[2k]\D[2k+2]) (E[2k+1]\E[2k+3], D[2k+2]) Control of (E[2k 1], D[2k]) Control of (E[2k+1], D[2k]) Control of (E[2k+1], D[2k+2]) Control of (E[2k+3], D[2k+2])... Fig. 3. Tree obtained if we start with a disjunction. The tree of Fig. 3 has been obtained by starting with a disjunction and a decomposition of E σ. If we start with a conjunction and a decomposition of D σ, we obtain a symmetrical tree, in the sense that each of the two trees is obtained from the other tree by making the following switches: E σ [x] D σ [x],, D&A C&P. Consequently, the results obtained from the two trees are symmetrical in the same sense. Therefore, we will consider only the tree of Fig. 3. However, we will explain later that in some examples, only one of the two trees is constructible. We see that in the construction of the tree, the while-loop stops with emptiness of a component of s i or r i+1. Emptiness of a component of r i+1 is studied in Sect. IV, while emptiness of a component of s i is left for a future study. We will use the following definitions of convergence: Definition 9: (E σ [i]) i 0, (D σ [j]) j 0 converge after (u, v) steps, if u and v are the smallest indexes s.t. ( i u, E σ [i] = E σ [u]) and ( j v, D σ [j] = D σ [v]). We specify that the convergence is to when E σ [u] = D σ [v] =. (It is easy to show that 1 u v 1.)

12 E. Some Justifications Let us give justifications to two questions that may arise for readers: Why do we use E σ [j]\e σ [j + 2] and D σ [j]\d σ [j + 2] in the series (s i ) i 1, instead of E σ [j]\e σ [j + 1] and D σ [j]\d σ [j + 1]? This choice is due to Prop. 2(3), which guarantees that E σ [j]\e σ [j + 2] and D σ [j]\ D σ [j +2] before the convergence of (E σ [j]) j 0 and (D σ [j]) j 0, respectively. This result will be useful in Sect. IV. Why do we distinguish even and odd indices? This is due to the fact that the tree is based on the series (r i ) i 1 and (s i ) i 1, which have distinct forms with even and odd indices. IV. ARBESCENT ARCHITECTURE OF C&P D&A CONTROLS TO REALIZE AN INFERENCE-BASED CONTROL We make the following assumption, which implies (from Theorem 3) that inference-based control is applicable to (E σ, D σ ): Assumption 3: N 0 s.t. E σ [N +1]= or D σ [N +1]=. The tree of Fig. 3 is constructible iff E σ E σ [1], because E σ is initially decomposed into E σ \ E σ [1] and E σ [1]. Similarly, its symmetrical tree (introduced above) is constructible iff D σ D σ [1]. The situation where E σ = E σ [1] and D σ = D σ [1] is impossible, because it implies E σ [k] = E σ and D σ [k]=d σ k 0 (from Prop. 2(4)), which contradicts Assump. 3 (assuming Assump. 2). Therefore, in every situation respecting Assump. 3, at least one of the two trees is constructible. Due to the symmetry between the two trees, it is sufficient to present only the study of the tree of Fig. 3. Therefore, we assume E σ E σ [1]. We have the following lemma: Lemma 1: Assump. 3 is equivalent to the existence of ν 2 s.t. r ν is the first element of (r i ) i 1 with an empty component. In the remaining part of this Sect. IV, Assump. 3 is implicit when ν is mentioned. We have the following lemma: Lemma 2: s 1,, s ν 1 do not contain an empty component. From Lemma 2, in the construction of the tree, i = ν 1 is the first index where the while-loop has its condition unsatisfied. Therefore, the last iteration is for i = ν 2, and hence s ν 2 and r ν 1 are the last constructed elements.

13 The cases where ν is even or odd are treated in Sects. IV-A and IV-B, respectively. A. ν = 2k + 2 for k 0 Proposition 9: If ν = 2k + 2 for k 0, then r ν 1 is D&A-coobservable. Proposition 10: If ν = 2, D&A control is applicable to r 1 = (E σ, D σ ), hence the tree that realizes the control of (E σ, D σ ) consists of just one leaf. If ν = 2k + 4 for k 0, the tree of Fig. 4 realizes the control of (E σ, D σ ). (E \ E[1], D) (E[1], D\D[2]) (E[1]\E[3], D[2]) (E[3], D[2]\D[4]) Control of (E, D) Control of (E[1], D) Control of (E[1], D[2]) Control of (E[3], D[2]) Control of (E[3], D[4])... (E[2k 1]\E[2k+1], D[2k]) (E[2k+1], D[2k]\D[2k+2]) Control of (E[2k 1], D[2k]) Control of (E[2k+1], D[2k]) (E[2k+1], D[2k+2]) Fig. 4. Tree of Fig. 3 when ν = 2k + 4 for k 0. B. ν = 2k + 3 for k 0 If we proceed in the same way as in Sect. IV-A, we find the following results: Proposition 11: If ν = 2k + 3 for k 0, then r ν 1 is C&P-coobservable. Proposition 12: If ν = 3, the tree of Fig. 5 realizes the control of (E σ, D σ ). If ν = 2k + 5 for k 0, the tree of Fig. 6 realizes the control of (E σ, D σ ). C. General Result As already explained, the tree of Fig. 3 is constructible iff E σ E σ [1], and a symmetrical tree is constructible iff D σ D σ [1]. We have obtained Props. 10 and 12 from the tree of Fig. 3, and

14 (E \ E[1], D) Control of (E, D) (E[1], D) Fig. 5. Tree of Fig. 3 when ν = 3. (E \ E[1], D) (E[1], D\D[2]) (E[1]\E[3], D[2]) (E[3], D[2]\D[4]) Control of (E, D) Control of (E[1], D) Control of (E[1], D[2]) Control of (E[3], D[2]) Control of (E[3], D[4])... (E[2k+1]\E[2k+3], D[2k+2]) Control of (E[2k+1], D[2k+2]) (E[2k+3], D[2k+2]) Fig. 6. Tree of Fig. 3 when ν = 2k + 5 for k 0. we can obtain similar results from the symmetrical tree. Assump. 3 implies that E σ E σ [1] or D σ D σ [1], hence at least one of the two trees is constructible. Therefore: Theorem 4: Every inference-based control of (E σ, D σ ) is equivalent to an arborescent architecture consisting of disjunctions and conjunctions of C&P and D&A controls. D. Example We illustrate our results of Sect. IV with the example of [3], so that the interested reader can use this example to compare our approach with the inference-based method of [3]. The plant is modeled by the automaton G of Fig. 7, where the initial state is numbered 0 and the marked states are in bold. The specification is modeled by the automaton K which is obtained from G by removing the shaded (grey) states and the (dotted) transitions that lead to them. We have n = 2, I σ = {1, 2}, Σ o,1 = {a, a, d}, Σ o,2 = {b, b, d}, and Σ c,1 = Σ c,2 = {σ}. We compute: E σ = {ε, ab, ba, da, db}, D σ = {a, b, d, dab, dba },

15 σ 0 d b a a σ σ σ b a σ σ b σ σ a σ b σ Fig. 7. G is the whole automaton; K is obtained from G by removing the shaded (grey) states and the (dotted) transitions that lead to them. E σ [1] = {ε, da, db}, D σ [1] = {a, b, d}, E σ [2] = {ε}, D σ [2] = {d}, E σ [3] = D σ [3] =. ν = 4 because r 4 = (E σ [3], D σ [2]) = (, {d}) is the first element of (r i ) i 1 with an empty component. We are in the case ν = 2k + 4 of Prop. 10 with k = 0. We obtain therefore the tree of Fig. 4 for k = 0, which is represented in Fig. 8. (E \ E[1], D) (E[1], D\D[2]) Control of (E, D) Control of (E[1], D) (E[1], D[2]) Fig. 8. Arborescent architecture obtained for the example of Fig. 7. V. CONCLUSION We propose an arborescent architecture for decentralized control of discrete event systems. We show that when inference-based control is applicable to realize a given control objective C, then C is also realizable by an arborescent architecture combining uniquely C&P and D&A controls. This result shows that inference-based control is fundamentally based on C&P and D&A controls. We are now investigating the case where inference-based control is unapplicable to realize C. In this case, we obtain a tree whose leafs are C&P and D&A controls, with the exception of the lower right leaf. The latter corresponds to a control objective C where inference-based control is unapplicable. Therefore, the problem of finding a control architecture for C is reduced into the problem of finding a control architecture for C. The latter can be interpreted as the portion of

16 C which is responsible of the nonapplicability of inference-based control. This fact is relevant for multi-decision control [4] that has to find how to decompose a control objective which is unrealizable by inference-based control, into several control objectives that are realizable by inference-based control. Instead of finding a decomposition of C, now multi-decision control has to find a decomposition of C. REFERENCES [1] K. Rudie and W. Wonham, Think globally, act locally: Decentralized supervisory control, IEEE Transactions on Automatic Control, vol. 37, no. 11, pp , November [2] T.-S. Yoo and S. Lafortune, A general architecture for decentralized supervisory control of discrete-event systems, Discrete Event Dyna. Syst.: Theory Applicat., vol. 12, pp , [3] R. Kumar and S. Takai, Inference-based ambiguity management in decentralized decision-making: Decentralized control of discrete event systems, IEEE Transactions on Automatic Control, vol. 52, no. 10, pp , [4] H. Chakib and A. Khoumsi, Multi-decision supervisory control: Parallel decentralized architectures cooperating for controlling discrete event systems, IEEE Transactions on Automatic Control, vol. 56, no. 11, pp , November [5] X. Yin and S. Lafortune, Decentralized supervisory control with intersection-based architecture, IEEE Transactions on Automatic Control, vol. 61, no. 11, pp , November [6] S. L. Ricker and K. Rudie, Knowledge is a terrible thing to waste: using inference in discrete-event control problems, in American Control Conference (ACC), Denver, CO, USA, [7] T.-S. Yoo and S. Lafortune, Decentralized supervisory control with conditional decisions: Supervisor existence, IEEE Transactions on Automatic Control, vol. 49, no. 11, pp , November 2004.