Feng Lin. Abstract. Inspired by thewell-known motto of Henry David Thoreau [1], that government

Transcription

1 That Supervisor Is Best Which Supervises Least Feng Lin Department of Electrical and Computer Engineering Wayne State University, Detroit, MI Abstract Inspired by thewell-known motto of Henry David Thoreau [1], that government is best which governs least, we prove that in a general discrete event system, whose behavior is described by strings of discrete events, a supervisor (or controller) that supervises less (that is, gives more freedom to the system) is indeed better in the sense that the system will evolve faster (that is, events will occur earlier), if certain fairness condition is satised. This interesting result is a consequence of recent development of a supervisory control theory for discrete event systems [2,3,4]. Consider the following scenario: a manager sitting in his oce supervises the operations of a plant. He may conceive of these operations as a ow of discrete events. For him, events are generic abstractions of various occurrences, activities, decisions, tasks, etc.. Initiating a new program, marketing a new product, signing a contract, negotiating a deal, placing an order, renovating certain equipments, hiring new personal, etc., are typical example of events in his plant. His aim is to ensure the orderly ow ofevents leading to the realization of certain objectives. Formally, we denote by the set of events. A string of events describes a possible evolution of the plant. One of such strings could be, for example, initiate a new program install new equipments hire more workers 1

2 Each occurrence of events has a lifetime which, for generality, is assumed to be random and generated by some stochastic process (of which deterministic process is a special case). The event lifetime describes the time elapsed from the initiation of the event and the completion of the event. Because of our general assumption, dierent occurrences of the same event mayhave dierent lifetimes. We denote the set of all strings of eventsofby. Clearly, in general, not all strings in can physically occur in the plant. We call the set of feasible (or physically possible) strings of, in the absence of supervision, the uncontrolled behavior UB. The set UB may be thought of as the set of all free trajectories, in the classical sense, and thus serves as a model for the plant. Naturally, every prex (or initial substring) of a string in UB is itself in UB.For example, if the aforementioned string is in UB, then initiate a new program install new equipments is also a string in UB. In other words, UB is closed under prex operation. This will be assumed for all behaviors dened in this paper. The manager as a supervisor decides on the strings of events that are permissible (legal or desirable) and these that are not. Thus, he determines the target behavior TB of his plant. In general, TB is a proper subset of UB (TB UB). The manager pursues the objective of realizing this target behavior through a string of enablement/disablement. Each timehe authorizes a transaction, approves a plan, rejects a request, etc., he is enabling or disabling event, as the case may be. We assume, without lose of generality, that his role is merely supervisory. That is, he refrains from issuing orders or enforcing actions (which is implicit in the model itself). Rather, he responds to his environment (that is, his subordinates or the proposal put forward by them) through approval or disapproval. It is only in this \permissive" manner that he exercises control. In short, he leaves initiation of actions to the internal dynamics of his plant and those involved in the operational detail, responds to the events of his environment by his approval or disapproval, and leave enforcement entirely 2

3 to his subordinates. Clearly, hedoesnothave control over all events. For instance, unpredictable events that are often also unpreventable may occur. Machine may break down, market may be lost to competition, personnel may resign, workers may go on strike, contractors may renegeon their promises of timely delivery, etc. These events the manager cannot control (may not disable). Therefore we partition into controllable events c and uncontrollable events uc. Furthermore, it is reasonable to assume that the manager cannot observe the occurrence of all events (some information is either not reported to him or is not available at the time he needs to make a relevant decision). Thus, we also partition into observable events o and unobservable events uo. The manager's task is thus to enable or disable the controllable events based on the record of occurrences of the observable events in such away that only the strings belonging to the target behavior survive. To formalize the manager's function, we rst dene the projection P :! o as P = P (s) = where denotes the empty string 8 >< >: P (s) if 2 uo P (s) if 2 o : Thus, P erases these events that are not observable to the manager. Projecting UB on o results in P (UB):=fs 2 o :(9t 2 UB)Pt = sg: The set P (UB) describes the behavior of the plant observable by the manager. Based on the observed behavior, the manager would like to implement a policy which is dened as a feedback map : P (UB)!f0 1g 3

4 satisfying (s )=1 (s ) 2f0 1g 2 uc s2 P (UB) 2 c s 2 P (UB): In other words, after a string of events t has occurred, the manager sees Pt, next possible events, generically denoted by, are subject to the supervision (Pt ). If (Pt )=0, then is disabled (prohibited from occurring), while if (Pt ) = 1, then is enabled (initiated). Therefore, the supervised behavior under policy is given by CB := fs 2 UB :(8t s) (Pt )=1g where t s means that t is a prex of s. We would like to consider whether, given a target behavior TB, there exists a policy such that CB = TB In other words, whether the target behavior TB is \realizable". To obtain a condition that ensures the existence of such a policy, we need to introduce the notions of controllability and observability of behaviors. Denition 1 (2) A target behavior TB is controllable (with respect to UB)if (8s 2 TB)(8 2 uc )s 2 UB ) s 2 TB Intuitively, a target behavior TB is controllable if uncontrollable events are tolerable in the following sense: An uncontrollable event 2 uc will not lead to undesirable (or illegal) behavior (s 2 TB)ifitisphysically possible (s 2 UB). The idea is that we do not wish the occurrence of an event, designated as uncontrollable, after a legal string s, renders the resulting string illegal. 4

5 Denition 2 (3) A target behavior TB is observable (with respect to UB)if where (8s s 0 2 )(Ps = Ps 0 ) consistent(s s 0 )) consistent(s s 0 ), (8 2 )(s 2 TB ^ s 0 2 TB ^ s 0 2 UB ) s 0 2 TB) In words, for an arbitrary pair of strings s and s 0, the predicate consistent(s s 0 ) holds if and only if an arbitrary event has the same legal/illegal status after both s and s 0. Observability requires that if s and s 0 looks the same (Ps = Ps 0 ) then they must be consistent (consistent(s s 0 )). The idea is that we do not want to put the manager in a position that his action after two dierent strings that nonetheless have been presented to him as the same string renders one string legal and the other illegal. With these two denitions, we can state the following result. Theorem 1 (3) Given a nonempty target behavior TB, there exists a policy such that CB = TB if and only if TB is both controllable and observable. Thus we have shown that realizable target behaviors of the manager are precisely those target behaviors that are controllable and observable. It is however unlikely that the target behavior as initially specied will be controllable and observable to begin with. Because specication of the target behavior is usually independent of the controllability and observability consideration. This being the case, the manager can only hope to realize a part of the target behavior. We would like to show that in the process of selecting a part of the target behavior, he should try to make the resulting behavior (called modied target behavior) as large as possible. In other words, he should give the plant maximum freedom and thus supervise as little as possible. By doing so, the plant willevolve fastest in the following sense. The pace of the plant, described by the occurrence times of events, is determined by the policy and the event lifetimes. Given a realization! of the stochastic process 5

6 governing the event lifetimes, the occurrence times of events under can be determined as follows: Each event is assigned a clock whose time is set according to!. Initially, all the clocks corresponding to the enabled events under are running concurrently untile the event with the shortest lifetime (as specied by!) occurs when its clock runs out. After the occurrence of, a dierent set of enabled events is determined by. If an event is newly enabled, its clock will now start running. If an event is newly disabled, its clock will be put on hold. Otherwise, the clock will run continuously until the next occurrence of an event. This procedure will then repeat itself. Under this general scheme, we can show that the policy generating the largest CB (that is, supervising least) is best in the sense that events will occur fastest, if a fairness condition is satised. To dene fairness, we denote, for a string s, the number of occurrences of an event by jj(s). Denition 3 (5) A supervised behavior CB is fair if (8s s 0 2 CB )s s 0 ) s s 0 where s s 0, (8 2 )jj(s) jj(s 0 ) s s 0, (8 2 )(s 2 CB ^jj(s)=jj(s 0 ) ) s 0 2 CB ) In words, a supervised behavior CB is fair if, given two strings with the number of occurrences of each event in the second string no less that the number of occurrences of the same event in the rst, then an event that is enabled by after the rst string is also enabled after the second, provided that it has not yet occurred. In a sense, events have a fair chance of occurring. With this denition, we can state our main result. Theorem 2 (5) Consider two policies 1 and 2 with supervised behaviors CB 1 and CB 2 respectively. Assuming no simultaneous occurrences of events in the plant. If (1) CB 1 CB 2, and (2) CB 2 is fair, then s 1 (t!) s 2 (t!) for all t and!, where 6

7 s i (t!) i =1 2is the string of events occurred before t in the plant supervised by i with event lifetimes specied by!. On other words, under the fairness condition, if 2 supervises less (disables less and hence gives more freedom to the plant), then more events will occur in the plant. This is true measured at any time t and for any given event lifetimes!. The assumption of no simultaneous occurrences of events is natural because, since time is dense, if twoevents do occur exactly at the same instance, as in the case of synchronization, we can rename them as one event. This important result, rigorously proved now [5], was intuitively understood at least since 1845 [1]. There are some interesting ways to interpret this result, and we will leave them to the reader's imagination. Let us now turn to a more practical question of how to synthesize an optimal policy. Recall that a policy can only be synthesized for a controllable and observable behavior. So, if a given TB is not both controllable and observable, we would like to nd the largest subset of TB that is both controllable and observable, that is, to nd the supremal element in the set of all controllable and observable subset of TB. The policy corresponding to this supremal element is optimal. Unfortunately, such a supremal element may not exist [3]. This leads us to seek a suboptimal solution, for which weintroduce a strong version of observability, called normality. We will show that the supremal element of the set of all controllable and normal subset of an arbitrary TB does exist. Denition 4 (3) A target behavior TB is normal (with respect to UB)if (8s 2 UB)s 2 TB, Ps 2 P (TB) Intuitively, a target behavior is normal if one can check whether a string s 2 UB is legal by checking whether its projection Ps belongs to P (T B). In other words, information on occurrences of unobservable events is not needed in deciding wthether s 2 UB. Therefore, one expects normality to be stronger that observability. Infactwehave 7

8 Theorem 3 (3) If TB is normal, then TB is observable. On the other hand, if every controllable event is observable, then normality is equivalent to observability assuming controllability as shown in the following theorem. Theorem 4 (6) Under the condition c o,iftb is controllable and observable, then TB is controllable and normal. Therefore, instead of nding controllable and observable behaviors, we can nd controllable and normal behaviors. The set of normal behaviors of UB is algebraically better behaved than that of observable behaviors, in the sense that the set of normal behaviors is closed under arbitrary unions as illustrated by the following theorem. Theorem 5 (3) For a given nonempty TB, the set of behaviors CN(TB):=fK TB : K is controllable and normalg is nonempty and closed under arbitrary unions. Therefore, the supremal element ofcn(tb), supcn(t B), exists. Hence, if TB is not both controllable and observable, we can modify TB by taking the modied target behavior as supcn(t B). We can use the following formulas to computer supcn(tb). Theorem 6 The supremal element of CN(TB)is given by the following formula. supcn(tb)=ub \ P ;1 (P (NTB) ; ((P (UB) ; P (NTB))= uc ) o ) where NTB = TB ; P ;1 P (UB ; TB) (P (UB) ; P (NTB))= uc := fs 2 :(9u 2 uc)su 2 P (UB) ; P (NTB)g 8

9 After computing supcn(t B), the corresponding policy can be easily obtained as follows: (s )= 8 >< >: 1 if sp () 2 P (supcn(tb)) 0 otherwise Thus, we have developed a systematic way to design an optimal supervisor, which supervises least. 1 References [1] H. D. Thoreau, On the Duty of Civil Disobedience. [2] R. J. Ramadge and W. M. Wonham, Supervisory control of a class of discrete event processes. SIAM J. Control and Optimization, 25(1), pp [3] F. Lin and W. M. Wonham, On observability of discrete event systems. Information Sciences, 44(3), pp [4] P. J. Ramadge and W. M. Wonham, The control of Discrete Event Systems. Proceedings of IEEE, 77(1), pp [5] F. Lin, Analysis of temporal performance of supervised discrete event systems. Automatica, 30(3), pp [6] F. Lin and W. M. Wonham, Supervisory control of timed discrete-event systems under partial observation. IEEE Transactions on Automatic Control, 40(3), pp [7] R. D. Brandt, V. Garg, R. Kumar, F. Lin, S. I. Marcus, and W. M. Wonham, Formulas for calculating supremal controllable and normal sublanguages. Systems & Control Letters, 15(1), pp