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On Petri Net Models of Infinite State Supervisors Where L(G) = {Wlw~~*And6*(W,G,)~Q}

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On Petri Net Models of Infinite State Supervisors Where L(G) = {Wlw~~*And6*(W,G,)~Q} 274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 37, NO. 2, FEBRUARY 1992 On Petri Net Models of Infinite State Supervisors where L(G) = {wlw~~*and6*(w,g,)~Q}. R. S. Sreenivas and B. H. Krogh Given a language L G X* we use the symbol zto denote its prefix Abstract-We consider a class of supervisory control problems that closure. require infinite state supervisors and introduce Petri nets with inhibitor To control a DES we assume the set C is partitioned into two sets arcs (PN's) to model the supervisors. We compare this PN-based ap- C, and X,, where C, is the set of events that can be disabled. We proach to supervisory control to automata-based approaches. The pri- mary advantage of a PN-based supervisory controller is that a PN-based let r denote the set of control patterns, where controller provides a finite representation of an infinite state supervisor. I'={yIy:X+{O,l}andy(a)=l foreachuEC,}. For verification, implementation, and testing reasons, a finite PN-based representation of aninfinite state supervisor is preferred over an au- An event u E C is said to be enabled by y when y( u j = 1, and is tomata-basedsupervisor. We show that thismodeling advantage is disabled otherwise. accompanied by a decision disadvantage, in that in general the controlla- bility of a language that can be generated by the closed-loop system is A supervisor 0 is asystem that changes the controlpattern undecidable. dynamically.An automata-based supervisor 0 is apair (S,+), I. INTRODUCTION where S = (X,C, f , x,) is an automaton with a (possibly infinite) state set X, input alphabet C, partial transition function E: X X C Supervisory control of discrete-event systems (DES'S) was intro- -+ X, initial state x,, and 6:X -+ r. As defined by Ramadge and duced by Ramadge and Wonham [l], [2] and has since been studied Wonham [l], the supervisor state transitions are synchronous with extensively [3]-[7]. A DES is a dynamical system with a (possibly identically labeled events in the plant G. At each state of S a control infinite) set of stares and a finite set of events. For a given DES G, pattern is selected basedon the 6 function.We usethe symbol it is of interest to synthesize a supervisor DES 0 that prevents the 0 1 G to represent the closed-loop plant-supervisor system described occurrence of certain events of G to enforce some specifications on above, and the symbol L(O I G) torepresent the language generated the behavior of the controlled DES. The classes ofspecifications by the system 0 I G. thathave been consideredhitherto fall intotwo categories: state To havea well-defined closed-loopbehavior a completeness avoidance problems [2], where the objective is to avoid a collec- condition [l] must be imposed on thesupervisor 0, namely, V tion of states.and string avoidance problems [I], wherethe w E C* and u E X the following statement must be true: objective is to avoid a collection of event strings. In this note. we consider string avoidance problems. weL(Q1G) and W"UEL(G)and $(E*(xo, w))(u) We introduce a method of modeling infinite state supervisors by = 1 * w'uo~L(0IG) Petri nets (PN's) of finite size. We relate supervisory control in the where isthe stringconcatenation operator and (*: X X C* -+ X PN-based models tothose in thecurrent literature thatuse is an extension of (: X X C + X. automata-based models.We then introducea class of supervisory Following Ramadge and Wonham [l] a language K E C* is said controlproblems that require infinitestate supervisors.We show to be controllable with respect to G if that for these problems there is a PN-based supervisor thathas a finite representation. For verification,implementation, and testing PC,n L(G) c E. reasons, a finite PN-based representation of an infinite state supervi- There exists a complete supervisor 0 suchthat L(0I G) = K if sor is preferred over an automata-based supervisor. and only if the language K E C* is prefix-closed and controllable Thisnote is organizedas follows. In Section 11, wepresent an with respect to G [I]. The supervisor 0 thus prevents the genera- overview of automata-based supervisorycontrol. Section 111 states tion ofstrings in L(G) that are notin K, while making sure all tworesults onthe use of infinite state models for plantsand strings in K can be generated by the system 0 I G. We callthe supervisors. Section IV introduces Petri nets with inhibitor arcs as problem of synthesising a supervisor to realize a restricted language finite-sizemodels of infinite statesupervisors. We showthat any K as the string avoidance problem. Turing acceptable language can be realized with such supervisors. III. INFINITE STATE PLANTSAND SUPERVISORS The concluding section indicates directions for future research. Inthis section. we consider two ways in whichinfinite size 11. Ah: OVERVIEWOF AUTOMATA-BASEDSUPERVISORY automata may be required for the string avoidance problem. First, CONTROL the plant G might be ofinfinite size,for example, when the languagegenerated by the plant L(G) is nonregular.Second, the Following Ramadgeand Wonham [7], wedefine a controlled desired closed-loop behavior K E L(G)may lead to an infinite state DES or a plant asa 4-tuple G = X, 6, qo), where Q is a (Q, supervisor,for example, when K is acontrollable, prefix-closed, (possibly infinite) state sef, 40 E Q is the inirial state, C is a finite and nonregular language. The following two propositions deal with alphabet usedto label transitionsbetween states (events), and 6: each of these cases in turn. The second case is illustrated with an Q x C --t Q is a partial function that describes the dynamics of the example following Proposition 2. system. Events are assumed to be instantaneous and asynchronous. Proposition I: If G = (Q, 6,6, q,,) is an infinite size plant and We extendthe function 6 to afunction 6*: Q X C* + Q inthe K is a prefix-closed language such that K E L(G) and K is usual way. Also, we define the size of an automaton to be card G C* (Q). controllable with respect to G, then there exists a finite size plant The language generated by G is denoted by the symbol L(G), G' = (Q. C, 6', 46) (i,e,, Q' is finite), such that Manuscript received January 12, 1990. This work was supported in part I) L(G) G L(G'j, and b} the National Science Foundation under Grant DMC-8451493. 2) 3 aK' E C* suchthat K' 1 K, and K' is prefix-closed and The authors are with the Laboratory for Automated Systems and Informa- tion Processing, Department of Electrical Engineering and Computer Engi- controllable with respect to G', and any complete supervisor 0' that neering. Carnegie Mellon University, Pittsburgh, PA 15213. satisfiesthe property L(0' 1 G') = K' also satisfies theproperty IEEE Log Number 9104397. L(0' 1 G) = K. 0018-9286/92$03,00 0 1992 IEEE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 37, NO. 2, FEBRUARY 1992 275 Proof.. The first part of the proposition follows trivially from a b the fact that for any language that is based on a finite alphabet there is alwaysa regular language that is a superset. For example, if L(G) E C*, then the language of the trivial automaton G' with only one state for which L(G') = X* contains L(G). For the second part, let L(G') = E* as described above. Now, Fig. 1. A plant automation. considerthe language K' = K'X;. Notethat K' is prefix-closed since K is prefix-closed.Furthermore, K'"C, fl L(G3 = K'OE, TABLE I = K'. This implies that K' is controllable with respect to G'. Since K' is prefix-closed and controllable with respectto G', there exists a complete supervisor 0' such that L(0' I G') = K'. Since L(G) E 40 a 90 L(G'), 0' 1 G is well-defined and L(0' 1 G) E L(0'I G3. We now 90 b 41 91 b q1 show that L(Q' 1 G) = K. Let weL(Q' 1 G) and suppose w#K. Since L(0' 1 G) C L(0' 1 G') = K"Xz, 3 w, EK and 3 w2 E Et such that w = wIaw2 be a finite state automaton as in the previous section. However, the and wlouu# K, where u, is the first element in the string w2. This supervisor is a PN. In the following paragraphs, we describe the contradictsthe assumption that K is controllable.Therefore, closed-loop behavior when a PN based supervisor is used in con- L(Q 1 G) C K. junction with a finite automata based plant. Now, suppose w E K and w $L(Q' 1 G). Clearly, w # E: be- Dejnition I: A Petri net with inhibitor arcs (PN) is an ordered cause this would imply that weL(0' l G). So, if w #LC@'l G), Stuple, N = (n,T, a, 4,m0), where II = { pl,p2; . ., pn) is a then 3 wl, wzE C*, and 3 uc E E,, such that set of n places, T = { t,, tZ;9 * , tm} is a collection of m transi- 1) w = wloucc'wz,and tions, @ E (XI X T)U (Tx n) is a set of arcs, 'Ir E (II X T)is 2) wlouc#L(O' I G). a set of inhibitor arcs, mO:II + N is the initial markingfunction These observations imply that wlooc~L(O'1 G'), since the supervi- (or the initial marking), and N is the set of nonnegative integers.
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