REFERENCES. Minimal Communication for Essential Transitions in a Distributed Discrete-Event System V. DISCUSSION

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST Theorem 5: W.p.1, the IPA erivatives of the lost-sales time average with respect to the prouction rate parameter are given for all T > 0 an 2 2 as follows: + L (T;)=0 1 T m28(t;) [minfv + m ();Tg0V + m01()] (4.6) 0 L (T;)=0 1 T n29(t;) [minfv 0 n ();Tg0V 0 n01()]: (4.7) Furthermore, letting st enote stochastic orering, the sie IPA erivatives satisfy + L (T;) st L (T;); T > 0 0; 2 2: (4.8) Stochastic equality hols above, provie fi(t; ) =Sg f(t) < (t; )g hols for all t 2 [0;T] an fi(t; ) =0gf(t) >(t; )g hols for all t 2 [0;T]. Proof: We only prove (4.6) since the proof of (4.7) is analogous. By stanar arguments (see [2], [6], an [8]), the set 8(T;) is locally inepenent of in a right neighborhoo of, an U m() + is locally ifferentiable with respect to in a right neighborhoo of. Thus + L (T;)= 1 T + m28(t;) U minfv () By part 1) of Assumption 4, for each m 2 8(T;) + U minfv () ();T g (t; ) t: (4.9) ();T g (t; ) t = 0[(U m())0(u + m();)] U + m()0 U minfv () ();T g t: (4.10) To compute the term associate with (= + )U m(), + consier U () [(t) 0 (t; )] t for m 2 8(T;). A moicum V () U () of algebra shows that [(t) 0 (t; )] t is locally V () inepenent of. Taking the right erivative of this integral yiels 0[(V + m01 ()) 0 (V + m01 ();)](=+ )V + m01 () + [(U m()) + 0 (U m();)](= + + )U m() + 0 U () t = 0, V () because the first term vanishes, resulting in 0[(U m()) + 0 (U m();)](= + + )U m() + = 0 U () t. Finally, substituting the previous equation into (4.8) yiels (= + minfv ();T g V () ) U () minfv ();T g (t; ) t = 0 t = 0[minfV + V () m ();Tg0V + m01 ()]. Equation (4.6) now follows by substituting the previous equation into (4.10). Finally, inequality (4.8) follows irectly from (4.6) an (4.7) an the efinition of V m + () an Vn 0 () by stanar arguments. Theorem 6: Uner Assumptions 1 an 4, the IPA erivatives with respect to the prouction rate parameter, (= 6 )L I (T;) an (= 6 )L (T;), are unbiase for all T > 0 an 2 2. Proof: Part ) of Assumption 4 ensures that for all T > 0, [5, Assumption C4 of Lemma A2] hols for both L I (T;) an L (T;). Moreover, Theorems 4 an 5 imply that 0 (= 6 )L I(T;) T=2 an j(= 6 )L (T;)j 1. Since the one-sie erivatives exist for every an all IPA erivatives are boune, [5, Assumption C of Lemma A2] is in force, which completes the proof. V. DISCUSSION Reference [8] an the current note can provie a theoretical basis for new online control algorithms of prouction-inventory systems, incluing those where the unerlying stochastic processes (i.e., eman an prouction capacity processes) may be subject to nonstationary probability laws. One irection of future research is the extension of the current results to more general supply chains with multiple proucts, such as assemble-to-orer systems (e.g., such as those implemente by Dell Computer Corporation, Roun Rock, TX), where eman patterns fluctuate consierably over time. REFERENCES [1] C. G. Cassanras an S. Lafortune, Introuction to iscrete event systems. Boston, MA: Kluwer, [2] C. G. Cassanras, Y. Wari, B. Melame, G. Sun, an C. G. Panayiotou, Perturbation analysis for on-line control an optimization of stochastic flui moels, IEEE Trans. Autom. Control, vol. 47, no. 8, pp , Aug [] M. C. Fu an J. Q. Hu, Conitional Monte Carlo: Graient Estimation an Optimization Applications. Boston, MA: Kluwer, [4] I. Paschaliis, Y. Liu, C. G. Cassanras, an C. Panayiotou, Inventory control for supply chains with service level constraints: A synergy between large eviations an perturbation analysis, Ann. Oper. Res., vol. 126, pp , [5] R. Y. Rubinstein an A. Shapiro, Discrete Event Systems: Sensitivity Analysis an Stochastic Optimization by the Score Function Metho. New York: Wiley, 199. [6] Y. Wari, B. Melame, C. G. Cassanras, an C. G. Panayiotou, Online IPA graient estimators in stochastic continuous flui moels, J. Optim. Theory Appl., vol. 115, no. 2, pp , [7] Y. Zhao an B. Melame, Make-to-stock systems with backorers: IPA graients, presente at the Winter Simulation Conf. (WSC), Washington, D.C., Dec. 5 8, [8] Y. Zhao an B. Melame, IPA erivatives for make-to-stock prouction-inventory systems with backorers, Methoology Comput. Appl. Probability, vol. 8, no. 2, pp , Minimal Communication for Essential Transitions in a Distribute Discrete-Event System Feng Lin, Karen Ruie, an Stéphane Lafortune Abstract In a istribute iscrete-event system with ecentralize information, agents at the various sites (e.g., controllers or iagnosers) may be require to communicate in orer to correctly perform some prescribe tasks. Banwith, power, or security constraints motivate the esign of Manuscript receive December 1, 2006; revise March 26, Recommene by Associate Eitor S. Haar. This work was supporte in part by the Natural Sciences an Engineering Research Council of Canaa (NSERC) uner Grant 18887, the National Science Founation (NSF) uner Grants CCR , CCR , an INT , the National Aeronautics an Space Aministration (NASA) uner Grant NAG2-1279, the U.S. Office of Naval Research (ONR) uner Grant N , an the National Institute of Health (NIH) uner Grant 1 R21 EB A1. F. Lin is with the Department of Electrical an Computer Engineering, Wayne State University, Detroit, MI USA ( flin@ece.eng.wayne.eu). K. Ruie is with the Department of Electrical an Computer Engineering, Queen s University, Kingston, ON K7L N6 Canaa ( karen.ruie@queensu.ca). S. Lafortune is with the Department of Electrical Engineering an Computer Science, University of Michigan, Ann Arbor, MI USA ( stephane@eecs.umich.eu) /$ IEEE

2 1496 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 communication protocols among agents that minimize the number of require communications. This note presents an algorithm that can synthesize such a communication protocol given a set of essential transitions that constitute the initial require communications. The synthesize communication protocol is shown to be minimal. Inex Terms Discrete-event systems, istribute control, istribute iagnosis, minimal communication. I. INTRODUCTION The setting consiere in this note is that of networke systems, namely, istribute systems where the information is ecentralize but where the agents at the ifferent sites of the system may communicate over a network. We assume that the agents are working as a team in orer to achieve some prespecifie monitoring, iagnosis, or control task. However, successful completion of this task requires the agents to communicate in real time. Banwith, power, or security constraints motivate the esign of communication protocols among agents that minimize, in some technical sense, the number of require communications. This problem is approache in the framework of iscrete-event systems. Therefore, the agents coul be supervisors if the problem at han is one of supervisory control [], or they coul be iagnosers if the problem at han is one of fault etection an ientification [8]. Builing on our earlier work in [5] an [6], we present, in this note, an algorithm that can synthesize a communication protocol given a set of essential transitions that constitute the initial require communications. The synthesize communication protocol is shown to possess the properties of feasibility, implementability, an minimality, which are aopte from [5] an [6], where they were first efine. The set of essential transitions ifferentiates this work from the problem solve in [5] an [6], where communication among agents was require in orer to achieve state isambiguation in the moel of each agent. This work is motivate by the fact that a problem formulation in terms of essential transitions generalizes the earlier state isambiguation problem since the state isambiguation problem can be cast as an instance of the essential transition problem. The notion of valiity employe in [5] an [6] to eal with state isambiguation is replace in this work by the notion of legality, which is efine in a manner that captures the requirement on essential transitions. As in [5] an [6], we efine a notion of feasibility; we then provie a solution to the minimal communication problem for essential transitions, subject to the constraints on legality an feasibility. (All of the aforementione terminology is precisely efine in Section II.) To further motivate the communication problem consiere in this note, consier the well-stuie problem of ecentralize control of iscrete-event systems. As we know [7], one of the necessary an sufficient conitions for the existence of supervisors that exactly achieve a given safety specification is the property of coobservability. Coobservability crucially epens on the local knowlege, at each site, of the global system behavior, as emboie by the set of locally observable events. We can fin a minimal set of observable events for each supervisor so that coobservability is ensure (although this can be computationally ifficult, cf. [4]). 1 In this context, if a supervisor nees to know about the occurrence of an event that is not irectly observable by it, then every occurrence of that event will be communicate to the supervisor by another supervisor that oes observe the event. Unfortunately, this scheme may not result in a minimal set of communications Digital Object Ientifier /TAC We assume that communication of all observable event occurrences by all agents to all other agents is sufficient to ensure the existence of a solution, which is guarantee by the property of observability [2] with respect to the union of the observable event sets. because although an event is neee for coobservability, not all its occurrences (i.e., transitions labele by that event) necessarily nee to be communicate. The objective of this note is to aress the problem of minimal communication from the point of view of iniviual event occurrences. The intricacies that arise when agents communicate in a istribute system are iscusse in [6]. Roughly speaking, the ecision to communicate an event occurrence relies on which event sequences are inistinguishable to an agent, which, in turn, is a result of what has alreay been communicate to that agent. In other wors, the communication strategies of the agents are mutually epenent. This makes their synthesis conceptually an computationally challenging. Moreover, in the case of control problems, communication an control are themselves couple an cannot, in general, be separate. In orer to formulate a tractable problem, we consier a istribute system with two agents an assume as in [6] that the tasks that the agents nee to perform have been precompute uner the assumption that agents coul observe any event observable at any site. The resulting control or iagnosis task is compile in the form of a finite-state automaton. In contrast to the state isambiguation problem consiere in [6], it is require in this note that each agent observe the events labeling certain essential transitions in this automaton, immeiately upon their occurrence. The essential transitions, which are assume to be given, e facto efine an initial set of require event communications among the agents. The problem to solve is to enlarge this set of communications just enough so that a legal, feasible, implementable, an minimal communication strategy is obtaine. We procee in Section II with the efinitions of these four properties an the problem statement. The proceure we evelop in Section III to solve the minimal communication problem for essential transitions can be summarize as follows. We examine all possible pairs of communication sets that contain the essential transitions; we check each caniate pair to see if the communications yiele by the pair are feasible. If so, we remove any superset of that pair because a superset woul most certainly not be minimal if some subset of it is. We then provie a metho for yieling the communication protocol for any pair of communication sets that is feasible an will lea to a minimal solution. Our solution to the minimal communication problem for essential transitions prouces potentially many (incomparable) pairs of minimal solutions an, in that sense, might seem not as pleasing as a strategy that prouces only one pair. On the other han, this allows for specific applications to take avantage of specific minimal pairs that might be more well suite for that application than some arbitrary solitary minimal solution. Secon, although there is an exhaustive search in our solution which on the face of it can appear to be computationally emaning in fact, each check for feasibility can be one in polynomial time. II. DEFINITIONS AND PROBLEM STATEMENT To formally efine the minimal communication problem stuie in this note, we use the following notations, which are introuce in [5] an [6]. We associate with an agent i (i = 1; 2) a(finite-state) automaton R i R i =(6;X i ; i ;x i;0) where 6 is an alphabet of event labels, X i is a set of states, x i;0 2 X i is the initial state, an i : X i 2 6! X i is the transition function. We assume that i is efine over its entire omain. (This is in contrast to the efinition of automata usually use in the iscrete-event systems literature.) For the case where X i is finite, R i can be represente by a

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST Fig. 1. Legal language E an system G. irecte graph whose noes are states an whose eges are transitions efine by i. For the sake of convenience, we introuce the notation Transition(R i)=f(x i;; i(x i;)) jx i 2 X i; 2 6g for the relation associate with the function i. We will assume that each agent only observes irectly the occurrence of some subset of events in 6. For an agent i, which observes only the events in 6 i;o, the natural projection P i : 6! 6 i;o is interprete as the agent s view of the strings in 6. The function P i is efine in the usual manner, namely, for s 2 6 an 2 6, P i (s) =P i (s)p i (), where P i () = if 2 6 i;o ; otherwise, P i () = (empty string symbol). The objective of agent i, i =1; 2, is unspecifie (an not necessary to the problem at han) but it is require that the agent observe the events labeling certain transitions, immeiately upon their occurrence. We call these transitions essential transitions an enote by ETR i the essential transition set of agent i, i =1; 2, where ETR i Transition(R i ).We assume the following: 1) either ETR 1 or ETR 2 or both are nonempty an 2) event labels of transitions in ETR 1 are irectly observe by agent 2 but not by agent 1. In other wors Similarly (x 1 ;; 1 (x 1 ;)) 2 ETR 1 ) 2 6 2;o n 6 1;o : (x 2;; 2(x 2;)) 2 ETR 2 ) 2 6 1;o n 6 2;o: These assumptions imply that communication is neee in orer to achieve the objective. If each agent communicates all its observations of occurrences of its observable events to the other agent, then both agents will know all event occurrences for transitions in ETR i, an hence, achieve their objectives. For the later use of the proof of our results, we ientify the set of all transitions irectly observe by or essential to an agent TR i = ETR i [f(x i ;; i (x i ;))2Transition(R i )j 2 6 i;o g: Example: Let us consier a ecentralize supervisory control problem with two supervisors. The event set is 6 = f; ; g. Supervisor 1 cannot observe an supervisor 2 cannot observe an event can be controlle by both supervisors. That is, 6 1;o = f; g, 6 2;o = f; g, an 6 1;c = 6 2;c = fg. The supervisors must control the system G to achieve the legal language E L(G); both are shown in Fig. 1, where the otte transition is possible in G but is not in the legal language E. Essentially, is legal after either or occurs but illegal after n, n 1. Since supervisor 1 oes not see, it will not be able to istinguish, which is legal, from, say,, which is illegal [i.e., P 1() = P 1()], an since supervisor 2 oes not see, it will not be able to istinguish, which is legal, from [i.e., P 2 () = P 2 ()]; so, without communication, the ecentralize supervisors cannot guarantee that only legal sequences an all legal sequences are generate. In the parlance of [7], the legal language E is not coobservable with respect to G. In this example, we can see that to guarantee that the supervisors can, jointly, achieve the global goal, it is necessary an sufficient for either supervisor 1 to be able to istinguish between n an n, n 1 or for supervisor 2 to be able to istinguish between an. Consequently, we can see that it is essential either for supervisor 1 to know whether the from state 2 to state 5 has occurre or for supervisor 2 to know whether the from state 1 to state 2 has occurre. For the sake of simplicity, we take R 1 an R 2 to be isomorphic to G; refer to Fig. 2. Then, on the basis of the preceing iscussion, we choose the transition (2,, 5) to be the only essential transition; namely, we choose ETR 1 = f(2;;5)g ETR 2 = ;: The essential transitions (in this case, only one) are inicate by bol eges in Fig. 2. We will return to this example in Section III. We consier the problem of minimizing communications between two agents, uner the requirement that event labels of essential transitions must be communicate upon their occurrence. Our problem is to evelop an algorithm to fin the minimal set of communications neee, in the sense that if any one event occurrence is not communicate from an agent to the other, the agents will not be able to observe all the event labels of the transitions in ETR i. As in [6], we assume that both observation an communication are instantaneous. The formulation in terms of essential transitions is more flexible than the state isambiguation one consiere in [6]. To the best of our knowlege, it is not possible to write a general proceure that woul transform an essential transitions problem to a state isambiguation one (as formulate in [6]). On the other han, by marking all transitions in R i that are not self-loops an are not observe by agent i as essential an using the approach in this note, the goal of state isambiguation will effectively be achieve. Informally, the characteristics of a esirable solution are as follows: 1) an agent must be able to observe the event labels of all the transitions in TR i upon their occurrence, the essence of what we will term a legal solution (efine precisely in the following); 2) an agent must make consistent communication ecisions in the sense that if the agent cannot istinguish between two sequences of events, then it must make the same communication ecision after the occurrence of either of those sequences, a requirement which we will call feasibility; () for our results to be synthesize using finite-state automata, we will restrict our attention to solutions that possess a property which makes them implementable. The concepts of feasibility an implementability are as introuce in [6] an their efinitions are recalle in the following for the sake of completeness. However, legality is a new concept that can be thought of as the counterpart of the concept of valiity in [6] (geare there to the problem of state isambiguation) for the problem of essential transitions aresse in this note. Formally, we efine the observations of each agent an the communications between two agents as follows. As mentione earlier, each agent can observe the occurrences of events in 6 i;o;i =1; 2, via the natural projection P i :6! 6 i;o. Each agent can also communicate

4 1498 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 Fig. 2. R, R, TR, an TR. some occurrences of events to the other agent via a communication mapping com ij :6 o! 2 6 where com ij escribes communications from agent i to agent j an where 6 o = 6 1;o [ 6 2;o. For a string s of (observable) events, com ij (s) is the set of events that will be communicate from agent i to agent j. That is, if event may occur after sequence s an if is an element of com ij (s), then upon its occurrence after s, event will be communicate. We refer the reaer to [6] for further iscussion on the efinition of com ij, in particular the motivation for its omain 6o. We assume that if an event nees to be communicate, then it will be communicate immeiately upon its observation, without elay or loss. Given (P 1 ;P 2 ) an (com 12 ; com 21 ), we procee as in [6] an use mutual inuction to efine the information mapping ( 1; 2):(6! 6o; 6! 6o) 1 () = (8s 2 6 )(8 2 6) 1 (s) = n6 1(s); if 2 61;o [ com21 (2(s)) : 1 (s); otherwise In other wors, after the occurrence of s, the next event is known to agent 1 if an only if it is either irectly observe by agent 1 or communicate by agent 2 to agent 1. The mapping 2 is efine similarly. Clearly, from this efinition, i is a mapping from 6 to 6 o. This efinition shows how to erive ( 1 ; 2 ) from (com 12 ; com 21 ) for the given (P 1;P 2). We enote this operation by ( 1; 2)=C(com 12; com 21): It is important to note that not any arbitrary (com 12; com 21) pair will be feasible base on the information available to the agents. To guarantee feasibility, it is require that any two sequences of events that are inistinguishable to an agent must be followe by the same communication. Namely, com 12 an com 21 must be compatible with the 1 an 2 that are built from them. Formally, (com 12 ; com 21 ) is feasible with respect to (P 1;P 2) if (8s; s ) 1(s) = 1(s 0 )=) com 12 (P (s)) = com 12(P (s 0 )) ^ 2 (s) = 2 (s 0 )=) com 21 (P (s)) = com 21 (P (s 0 )) (1) where ( 1 ; 2 )=C(com 12 ; com 21 ) an P :6! 6o is the natural projection onto 6o. The projection onto the set of events observe by agent 1 or agent 2 is use because the mappings com 12 an com 21 take sequences of events in 6 o as inputs. To check feasibility, we first calculate ( 1; 2)=C(com 12; com 21) an then check if (1) hols. We are now reay to present a formal efinition of the notion of legality introuce intuitively earlier, which replaces the notion of valiity employe in [6]. Formally, legality is expresse as (8s 2 6 )(8 2 6) ( 1 (x 1;0 ;s);; 1 (x 1;0 ;s))2 TR 1 ) 1 (s) = 1 (s) ^ ( 2(x 2;0;s);; 2(x 2;0;s))2 TR 2 ) 2(s) = 2(s): When a communication map (com 12; com 21) gives rise to 1 an 2 that satisfy (2), we say that (com 12 ; com 21 ) is legal with respect to (TR 1 ;TR 2 ). Following the strategy aopte in [6], we restrict attention to a boune state space over which to synthesize communication maps, namely, the set of states of R 1 2 R 2, where 2 is the stanar prouct operation for automata (see, e.g., [1]). (Note that R 1 an R 2 have the same event set 6.) In this regar, let us efine R := R 1 2 R 2 =(6;X;;x 0 )=(6;X 1 2 X 2 ;;(x 1;0 ;x 2;0 )) : The structural requirement impose on the communication maps is the notion of implementability with respect to (R 1;R 2) that was introuce in [6]. (See [6] for further iscussion of this notion.) To guarantee that (com 12; com 21) is implementable with respect to (R 1;R 2), it is require that any two strings leaing to the same state in R (an, hence, in both R 1 an R 2 ) must be followe by the same communications. Formally, (com 12; com 21) is sai to be implementable with respect to (R 1 ;R 2 ) if (8s; s )(x 0;s)=(x 0;s 0 )=) com 12(P (s)) = com 12(P (s 0 )) ^com 21 (P (s)) = com 21 (P (s 0 )): The remaining property to be efine is the notion of minimality of communication maps. We say that a communication map (2)

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST (com 12; com 21) communicates strictly less than (com 12; com 21), enote by (com 12 ; com 21 ) < (com 12 ; com 21 ),if (8s 2 6 o) com 12(s) com 12(s) ^ com 21(s) com 21(s) ^ (9t 2 6o) com 12(t) com 12(t) _ com 21(t) com 21(t): A communication pair (com 12; com 21) is minimal if there is no (com 12; com 21) that communicates strictly less than (com 12; com 21). Now that all relevant notions have been efine, the problem treate in the subsequent section can be formally state. Problem Statement: Given R 1, R 2, P 1, P 2, TR 1, an TR 2, fin a minimal pair of communication maps (com 12 ; com 21 ) that is legal with respect to TR 1, TR 2, subject to the following constraints: 1) (com 12; com 21) is feasible with respect to (w.r.t.) (P 1;P 2); 2) (com 12 ; com 21 ) is implementable w.r.t. (R 1 ;R 2 ). Constraint 1) cannot be relaxe since feasibility is necessary to make the problem well pose, as explaine earlier in this section. Constraint 2), on the other han, restricts the range of solutions for the esire communication maps. III. ALGORITHM In this section, we present an algorithm that calculates a minimal legal communication pair (com12; com 21) that is feasible an implementable. We first introuce some notations necessary for the evelopment of the algorithm. Let the set of transitions unobservable to agent i be Transition uo (R i )=f(x i ;; i (x i ;)) j x i 2 X i ;26 j;o n6 i;o g Transition(R i ): Clearly ETR i Transition uo(r i) For a set of transitions of R i, U i Transition(R i), we exten it to the set of transitions in R as follows: EXT(U i)=f((x 1;x 2);;( 1(x 1;); 2(x 2;))) 2 Transition(R)j (x i ;; i (x i ;)) 2 U i g : For example, EXT(ETR i ) is the set of transitions in R whose event labels must be communicate to agent i. To specify com ij base on R, we introuce a set of transitions in R whose event labels will be communicate from agent i to agent j as V ij = f(x; ; (x; )) ;...g : Obviously, 2 6 i;o n 6 j;o for such transitions. The set V ij is calle the communication set. How to obtain com ij from V ij will be given in the main algorithm below after further notation is introuce. Since agent j sees only the events in 6 j;o (via irect observation) an the events in the transitions in V ij (via communication from agent i), from the point of view of agent j, all the other transitions in R are unobservable an shoul be replace by the empty string. We call the resulting transition function j. The resulting automaton R j(v ij )= 6;X; j ;x 0 is a noneterministic finite-state automaton (NFA). We transform R j (V ij ) into a reachable eterministic finite-state automaton (DFA) in the usual way [1] an enote the result by ~ R j (V ij ) ~R j(v ij) =DA R j(v ij) := (6; ~ X j; ~ j ; ~x j;0): We rop the argument to ~ R j when it is unerstoo from the context. Since some transitions are replace by in the process of constructing R j, not all events are efine in every state of ~ R j.toefine ~ j as a total function, we a self-loops for all transitions that are not efine an enote the resulting automaton by SL( ~ R j )= 6; ~ X j ; j ; ~x j;0 : The propose algorithm uses the table technique evelope in [9] to ientify confusable states ; that is, for an NFA Rj(V ij ), we can construct a table or a relation T (Rj(V ij)) X2X (see [9] for etails) such that Proposition 1 is true. Proposition 1: Table T (Rj(V ij )) has the following property: For all x k ;x l 2 X (x k ;x l ) 2 T R j(v ij), (9 ~x 2 ~ X j)x k 2 ~x ^ x l 2 ~x: Furthermore, the computational complexity of constructing T (R j(v ij )) is O(j6kXj 2 ). Proof: See [9]. Proposition 1 says that there is a polynomial-time algorithm for fining all pairs of states (x k ;x l ) in R such that x k an x l cannot be istinguishe by agent j. Base on this proposition, we propose the following algorithm, which is ivie into two parts. The first part is a function calle Check(V 12 ;V 21 ). For a pair of communication sets (V 12;V 21), the function Check(V 12;V 21) checks the feasibility of the communication using the table technique. If the pair of communication sets is feasible, then Check(V 12 ;V 21 ) = 1; otherwise, Check(V 12;V 21) =0. Function Check(V 12;V 21) Input: R =(6;X;Transition(R);x 0 ), V 12, V 21 Output: Check =1(yes) or 0 (no) 1) Construct T 1 := T (R 1(V 21)); T 2 := T (R 2(V 12 )); 2) Check := 1; ) For all (x i;;(x i;)) 2 V 12 an (x i;x j) 2 T 1 o If (x j;;(x j ;)) 62 V 12, then Check := 0; Return. (If states x i an x j are inistinguishable to agent 1, then if has to be sent by agent 1 to agent 2 at state x i,it must also be sent by agent 1 to agent 2 at state x j.) 4) For all (x i ;;(x i ;)) 2 V 21 an (x i ;x j ) 2 T 2 o If (x j;;(x j ;)) 62 V 21, then Check := 0; Return. (If states x i an x j are inistinguishable to agent 2, then if has to be sent by agent 2 to agent 1 at state x i,it must also be sent by agent 2 to agent 1 x j.) The secon part is the MAIN program, which checks all possible pairs of communication sets (V 12 ;V 21 ) within a certain range an fins the set of all minimal pairs that are feasible an legal.

6 1500 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 Fig.. Output of the algorithm: R an R. MAIN-ET Input: R 1, R 2, ETR 1, an ETR 2 Output: MINSET 1) R := R 1 2 R 2 =(6;X 1 2 X 2 ;;(x 1;0;x 2;0)); 2) A 12 := EXT(ETR 2 ); A 21 := EXT(ETR 1); B 12 := EXT(Transition uo(r 2)); B 21 := EXT(Transition uo (R 1 )); ) MINSET: ;; 4) For all (V 12;V 21) such that A 12 V 12 B 12 an A 21 V 21 B 21,o (Check all possible pairs of communication sets.) If :(9(V12;V 0 21) 0 2 MINSET)V12 0 V 12 ^ V21 0 V 21, then (A caniate pair is not minimal if it contains one that is alreay minimal.) If Check(V 12;V 21) =1, then REMOVE := f(v12;v MINSET : V 12 V12^V 0 21 V21g; 0 MINSET := MINSET [f(v 12;V 21)gnREMOVE; (If a caniate pair is minimal, then any pair strictly larger than it is not minimal so it is remove from MINSET.) 5) Pick (V12V 21) 2 MINSET 6) R ~ 1 := DA(R1(V 21)) = (6; X ~ 1 ; ~ 1 ; ~x 1;0); ~R 2 := DA(R2(V 12)) = (6; X ~ 2; ~ 2; ~x 2;0); 7) R1 := SL( R ~ 1)=(6; X ~ 1;1 ; ~x 1;0); R 2 := SL( ~ R 2 )=(6; ~ X 2 ; 2 ; ~x 2;0); 8) Define ( 12 ; 21 ) accoring to 2 12(~x) () (9x 2 ~x)(x; ; (x; )) 2 V21; 2 21 (~x) () (9x 2 ~x)(x; ; (x; )) 2 V21; Define (com 12; com 21) accoring to 9) En. com 12(s) := 12 ( 1 (~x 1;0;s)); com 21(s) := 21 ( 2 (~x 2;0;s)); Example (Revisite): Let us revisit the example introuce in the preceing section. Using the algorithm MAIN-ET, we can calculate R 1 an R 2 for the R 1 an R 2 epicte in Fig. 2, with ETR 1 = f(2;;5)g an ETR 2 = ;. The final results are shown in Fig.. (Intermeiate results are not shown ue to space constraints.) The transitions in boxes nee to be communicate. In this example, the label of the transition from state f1; 2g to state f; 5g in R 2 is the only event communicate. This communication will be seen by S 1 as either the transition from f1g to f; 4g or the transition from f2g to f5g in R 1, epening on the state of R 1 when the communication is receive. (Note that the other transitions in R 1 are self-loops, since they are not observe or communicate.) Returning to the ecentralize control problem that motivate this example, we can see that supervisor S 1 can be realize by R 1 (minus self-loops) with the only isablement action being to isable at state f5g. The occurrence of at state 1 of G (hence, at state 1 of R 1 ) i not nee to be communicate for supervisor 1 to exercise control. That is, it was not eeme essential because before supervisor 1 sees an, it knows that it can just allow to occur because only has the potential to lea to illegality after has occurre. Therefore, supervisor 1 oes not nee to know whether occurs from the initial state of the system. However, since supervisor 2 is the one communicating an since supervisor 2 must communicate the that occurs at state 2 of G, namely, the that occurs after, it must also communicate the that occurs at state 1 of G since supervisor 2 oes not see the that occurs from state 1. The foregoing is reflecte in the fact that the boxe transition in R 2 is from state f1; 2g. In other wors, while the transition (1,,)inR 1 was not essential, the occurrence of that must nevertheless be communicate to satisfy the feasibility requirement in this example.

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST IV. CORRECTNESS OF THE ALGORITHM Using a contrapositive proof, if 1(s) 6= 1(s), then we have We first prove that (com 12; com 21) is feasible an implementable. Theorem 1: The communication (com12; com 21) prouce by MAIN-ET is feasible with respect to (P 1 ;P 2 ) an implementable with respect to (R 1;R 2). Proof: Let (com12; com 21) be obtaine from (V12;V 21) 2 MINSET. Since (V12;V 21) 2 MINSET, by MAIN-ET, Check(V12;V 21) =1.ByFunction, Check(V12;V 21) =1implies ( ;o n 6 2;o)(8~x 2 X ~ 1) ((8x 2 ~x)((x; ; (x; )) 2 V12)) _ ((8x 2 ~x)((x; ; (x; )) 62 V12)) () ( ;o n 6 1;o )(8~x 2 X ~ 2 ) ((8x 2 ~x)((x; ; (x; )) 2 V21)) _ ((8x 2 ~x)((x; ; (x; )) 62 V21)) : (4) A communication strategy satisfying () an (4) is calle consistent in [6]. Then, by [6, Th. 1], (com 12; com 21) is feasible with respect to (P 1 ;P 2 ) an implementable with respect to (R 1 ;R 2 ). It is worth noting that in view of () an (4) in the proof of Theorem 1, the functions ( 12; 21) efine in step 8) of MAIN-ET also satisfy the following (stronger) conition: 2 12(~x) () (8x 2 ~x)(x; ; (x; )) 2 V (~x) () (8x 2 ~x)(x; ; (x; )) 2 V 21: In other wors, if nees to be communicate at ~x, then is require to be communicate at all x 2 ~x. Next, we prove that (com 12; com 21) is legal. Theorem 2: The communication (com 12; com 21) prouce by MAIN-ET is legal with respect to (TR 1 ;TR 2 ). That is, for ( 1; 2):=C(com 12; com 21) (8s 2 6 )(8 2 6) ( 1(x 1;0;s);; 1(x 1;0;s)) 2 TR 1 ) 1(s)= 1(s) ^( 2 (x 2;0 ;s);; 2 (x 2;0 ;s)) 2 TR 2 ) 2(s)= 2(s): Proof: Let us only prove that for all s 2 6 an for all 2 6 ( 1 (x 1;0 ;s);; 1 (x 1;0 ;s)) 2 TR 1 ) 1(s) = 1(s) because the ual for agent 2 is similar. By the efinition of 1 an the algorithm 1(s) =1(s), 2 6 1;o [ com 21 (2(s)) (by the efinition of 1), 2 6 1;o _ 2 21 ( 2 (~x 2;0;2(s))) (by Step 8 of MAIN-ET), 2 6 1;o _ (9x 2 2 (~x 2;0;2(s))) (x; ; (x; )) 2 V21) (by Step 8 of MAIN-ET): ;o ^ (8x 2 2 (~x 2;0;2(s))) (x; ; (x; )) 62 V21) ) ;o ^ (8x 2 2 (~x 2;0 ;2(s))) (x; ; (x; )) 62 EXT(ETR 1 ) (by Steps 2 an 4 of MAIN-ET) ) ;o ^ (8(x 1 ;x 2 ) 2 2 (~x 2;0 ;2(s))) (x 1 ;; 1 (x 1 ;)) 62 ETR 1 (by the efinition of EXT) ) ;o ^ (8(x 1 ;x 2 ) 2 2 (~x 2;0 ;2(s))) ( ;o n 6 1;o _ (x 1 ;; 1 (x 1 ;)) 62 TR 1 ) (since ETR 1 = TR 1 \ (6 2;o n 6 1;o )) ) (8(x 1 ;x 2 ) 2 2 (~x 2;0 ;2(s))) ;o ^ ( ;o n 6 1;o _ (x 1;; 1(x 1;)) 62 TR 1) ) (8(x 1;x 2) 2 2 (~x 2;0;2(s))) ;o [ 6 2;o _ (x 1;; 1(x 1;)) 62 TR 1) : Clearly, ( 1 (x 1;0 ;s); 2 (x 2;0 ;s)) 2 2 (~x 2;0 ;2(s)). Denote (x 1;x 2) = ( 1(x 1;0;s); 2(x 2;0;s)). Then (x 1;x 2) must satisfy the previous conition; that is, either ;o [ 6 2;o or (x 1 ;; 1 (x 1 ;)) 62 TR 1 ). Case 1) If ;o [ 6 2;o, then ( 1(x 1;0;s);; 1(x 1;0;s)) 62 TR 1 because all transitions in TR 1 are labele by events in 6 1;o [ 6 2;o. Case 2) If (x 1;; 1(x 1;)) 62 TR 1), then ( 1 (x 1;0 ;s);; 1 (x 1;0 ;s)) = (x 1 ;; 1 (x 1 ;)) 62 TR 1 : This completes the contrapositive proof. Finally, we show that, in aition to being legal, feasible, an implementable, (com12; com 21) is also minimal among all other communication pairs that are legal, feasible, an implementable. Theorem : The (com12; com 21) prouce by the algorithm is a minimal pair of communication maps among all pairs of communication maps that are legal with respect to (TR 1;TR 2), feasible with respect to (P 1;P 2), an implementable with respect to (R 1;R 2). Proof: We procee by contraiction. Suppose that there is a (com 12; com 21) that is legal with respect to (TR 1;TR 2), feasible with respect to (P 1 ;P 2 ), an implementable with respect to (R 1 ;R 2 ) an (com 12 ; com 21 ) < (com12; com 21). Since (com 12; com 21) is legal with respect to (TR 1;TR 2), feasible with respect to (P 1 ;P 2 ), an implementable with respect to (R 1 ;R 2 ), there exists a pair (V 12 ; V 21 ) such that A 12 V 12 B 12, A 21 V 21 B 21, Check(V 12; V 21) =1, an (com 12; com 21) can be prouce from (V 12 ; V 21 ) by step 8 of MAIN-ET. Also, let (V12;V 21) be the pair proucing (com12; com 21) (com 12; com 21) < (com12; com 21) ) (8s 2 6o) com 12(s)com 12(s) ^ com 21(s)com 21(s) ^ (9t 2 6o) com 12 (t)com 12(t) _ com 21 (t)com21(t) ) V 12 V 12 ^ V 21 V 21 ^ V 12 V 12 _ V 21 V 21 : Step 4 of MAIN-ET then prouces a contraiction to the fact that (V12;V 21) 2 MINSET.

8 1502 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 8, AUGUST 2007 V. CONCLUSION The results in this note show that the concepts introuce in [6] for solving the minimal communication problem to achieve state isambiguation can be aapte to solve the more general problem of essential transitions. This aaptation require the introuction of the property of legality, which captures the requirements pertaining to essential transitions. Several issues remain open for future research. The etermination of well-pose sets of essential transitions in specific ecentralize supervisory control or fault iagnosis problems is of particular interest. This was one in an intuitive manner in the ecentralize control example in the note. Systematic proceures for generating these sets when coobservability is violate are currently being investigate. Also, the problem of synthesizing minimal communication maps in multiagent problems (three or more agents) is entirely open an most likely quite challenging. ACKNOWLEDGMENT The authors woul like to thank the reviewers for their careful reaing of their manuscript an their insightful comments that helpe improve the presentation of our results. They woul also like to thank W. Wang for bringing to the authors attention a gap in the proof of an earlier version of their algorithm. REFERENCES [1] J. E. Hopcroft an J. D. Ullman, Introuction to Automata Theory, Languages, an Computation. Reaing, MA: Aison-Wesley, [2] F. Lin an W. M. Wonham, On observability of iscrete-event systems, Inf. Sci., vol. 44, pp , [] P. J. G. Ramage an W. M. Wonham, The control of iscrete event systems, Proc. IEEE, vol. 77, no. 1, pp , Jan [4] K. Rohloff, T.-S. Yoo, an S. Lafortune, Deciing co-observability is PSPACE-complete, IEEE Trans. Autom. Control, vol. 48, no. 11, pp , Nov [5] K. Ruie, S. Lafortune, an F. Lin, Minimal communication in a istribute iscrete-event control system, in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp [6] K. Ruie, S. Lafortune, an F. Lin, Minimal communication in a istribute iscrete-event system, IEEE Trans. Autom. Control, vol. 48, no. 6, pp , Jun [7] K. Ruie an W. M. Wonham, Think globally, act locally: Decentralize supervisory control, IEEE Trans. Autom. Control, vol. 7, no. 11, pp , Nov [8] M. Sampath, R. Sengupta, S. Lafortune, K. Sinnamohieen, an Teneketzis, Diagnosability of iscrete event systems, IEEE Trans. Autom. Control, vol. 40, no. 9, pp , Sep [9] W. Wang, S. Lafortune, an F. Lin, An algorithm for calculating inistinguishable states an clusters in finite-state automata with partially observable transitions, Syst. Control Lett., 2007, to be publishe. PID Stabilization of MIMO Plants A. N. Güneş an A. B. Özgüler Abstract Close-loop stabilization using proportional integral erivative (PID) controllers is investigate for linear multiple-input multiple-output (MIMO) plants. General necessary conitions for existence of PID-controllers are erive. Several plant classes that amit PID-controllers are explicitly escribe. Plants with only one or two unstable zeros at or close to the origin (alternatively, at or close to infinity) as well as plants with only one or two unstable poles which are at or close to origin are among these classes. Systematic PID-controller synthesis proceures are evelope for these classes of plants. Inex Terms Integral action, proportional integral erivative (PID) controllers. I. INTRODUCTION Proportional integral erivative (PID) controllers are wiely use in many control applications an preferre for their simplicity. Due to their integral action, PID-controllers achieve asymptotic tracking of step-input references. The topic of PID-control is treate extensively in every classical control text, e.g., [11]. In spite of the importance an wiesprea use of these low-orer controllers, most PID esign approaches lack systematic proceures an rigorous close-loop stability proofs. Rigorous synthesis methos are explore recently in, e.g., [8] [10] an [1]. The simplicity of PID-controllers, which is esirable ue to easy implementation an from a tuning point-of-view, also presents a major restriction: PID-controllers can control only certain classes of plants. The problem of existence of stabilizing PID-controllers, which is practically very relevant (see []), is unfortunately not easy to solve. To gain insight into the problem s ifficulty, note that the existence of a stabilizing PID-controller for a plant G(s) is equivalent to that of a constant stabilizing output feeback for a transforme plant. Alternatively, the problem can be pose as etermining existence conitions of a stable an fixe-orer controller for G(s)((s +1)=s), which is also a ifficult problem [1], [14]. The restriction on the controller orer is a further major ifficulty. Strong stabilizability of the plant is a necessary conition for existence of PID-controllers, but it is not sufficient); e.g., G(s) =1=(s0p) cannot be stabilize using a PID-controller for any p 0, although the extene plant G(s)((s+1)=s) is stabilizable using a stable controller (whose inverse is also stable). The goal of this note is to fin sufficient conitions on PID stabilizability, an hence, to ientify plant classes that amit PID-controllers. Furthermore, explicit construction of the PID parameters for such plant classes is explore, leaing to systematic controller synthesis proceures for linear, time-invariant (LTI), multiple-input multiple-output (MIMO) plants of arbitrarily high orer using the stanar unity-feeback system shown in Fig. 1. The results obtaine here explore conitions for PID stabilizability of general MIMO unstable plants without /$ IEEE Manuscript receive April 19, 2006; revise January 8, 2007 an April 17, Recommene by Associate Eitor A. Hansson. This work was supporte in part by the TÜBİTAK-BAYG an the TÜBİTAK-EEEAG uner Grant EEEAG 105E065. A. N. Güneş is with the Electrical an Computer Engineering Department, University of California, Davis, CA USA ( angunes@ucavis. eu). A. B. Özgüler is with the Electrical an Electronics Engineering Department, Bilkent University, Ankara 06800, Turkey ( ozguler@ee.bilkent.eu.tr). Digital Object Ientifier /TAC