CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systems - Stéphane Lafortune

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1 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune MODELING OF DISCRETE EVENT SYSTEMS Stéphane Lafortune The University of Michigan, USA Keywords: Autoaton, observer, parallel coposition, Petri net, process algebra, product, regular languages. Contents 1. Introduction 1.1. Foral Languages 2. Autoata 2.1. Basic Concepts 2.2. Languages Represented by Autoata Blocking: Deadlock and Livelock 2.3. Accessibility Properties 2.4. Nondeterinistic Autoata 3. Operations on Autoata 3.1. Product and Parallel Coposition 3.2. Exaple: Two Users of Two Coon Resources 3.3. Observer Autoata 4. Regular Languages and Finite-state Autoata 5. Petri Nets 5.1. Petri Net Languages 5.2. Matrix Algebra and Petri Net Dynaics 5.3. Coposition of Petri Nets 6. Process Algebras 7. Discussion on Tied Models Glossary Bibliography Biographical Sketch Suary The odeling foralis of autoata is considered and the anipulation of autoata for the construction of coplete syste odels fro individual coponent odels is discussed. These anipulations include the operations of product and parallel coposition. The notion of blocking in the context of language odels and autoaton odels is presented. Blocking captures the phenoena of deadlock and livelock that can occur in the behavior of discrete event systes. An algorith for the construction of observer autoata fro autoata with unobservable transitions is presented. The odeling foraliss of Petri nets and counicating sequential processes are also treated and contrasted to autoata in the context of a siple resource sharing exaple. 1. Introduction Discrete event systes are dynaic systes with discrete state spaces and event-driven Encyclopedia of Life Support Systes (EOLSS)

2 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune dynaics. The area of discrete event systes is ultidisciplinary and involves concepts and techniques fro coputer science theory, control theory, and operations research. A wide variety of odeling foraliss is being used to describe and study the behavior of discrete event systes. Two widely-used odeling foraliss in control engineering are autoata and Petri nets. A lot of progress has been ade in the last two decades in the developent of a control theory for discrete event systes odeled as autoata or Petri nets. For this reason, these are the foraliss discussed in this chapter; in addition, a brief discussion of counicating sequential processes is given in Section 6. Discrete event odels of dynaic systes are classified in ters of how they abstract tiing inforation and randoness in the syste behavior. Untied odels abstract away precise tiing issues by focusing only on the ordering of the events and not on the exact ties of their occurrence. Untied odels also abstract away statistical inforation about the probabilities of the events and consider all possible saple paths in the syste behavior. Untied odels are often referred to as logical odels. Tied odels enrich untied odels and explicitly include tiing inforation. This inforation ay be provided in a deterinistic anner or in a stochastic anner. The process of odel refineent fro untied autoata to stochastic tied autoata and their associated generalized sei-markov stochastic processes are discussed in Discrete Event Systes. This chapter explores in further depth discrete-event odeling by building on the discussion in Discrete Event Systes. The priary focus of this chapter is untied odels of discrete event systes. Deterinistic and stochastic tied odels are briefly discussed in Discrete Event Systes; stochastic tied odels are also considered in Saple Path Analysis of Discrete Event Dynaic Systes Foral Languages The concept of language is introduced in Discrete Event Systes for odeling the logical behavior of a discrete event syste. An untied language, or siply language, is a set of strings of events over an event set. Let E be the finite set of events (or alphabet ) associated with the discrete event syste under consideration. This set consists of all the events that can possibly be executed by the syste. A string (or trace) is a finite sequence of events fro E. The length of a string s, denoted by s, is a non-negative integer corresponding to the nuber of events coposing the string, counting ultiple occurrences of the sae event. The epty string, denoted by ε (not to be confused with the generic event e E), is the string containing no events, i.e., ε =. The concatenation of two strings s 1 and s 2 is the trace ss 1 2 (i.e., s 1 followed by s 2 ). Thus the epty string ε can be interpreted as the identity eleent for concatenation. Denote by E the set of all finite strings of eleents of E, including the epty string ε ; the * operation is called the Kleene closure. For exaple, if E = { abc,, }, then E = { ε, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, }. Encyclopedia of Life Support Systes (EOLSS)

3 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune A language is then forally defined as a subset of E. If st = s with ss,, t E, then s is called a prefix of s and t a suffix of s. Both ε and s are prefixes of s, by definition. The usual set operations, such as union, intersection, difference, and copleent (with respect to E ) are applicable to languages since languages are sets. In addition, consider the following operations ( := denotes equal to by definition ): Concatenation: Let L1, L2 E, then { ( ) ( ) ( )} LL := s E : s= ss s L s L Prefix-closure: Let L E, then { ( )} L:= s E : t E st L. Thus the prefix-closure L of L is the language consisting of all the prefixes of all the strings in L. In general, L L. L is said to be prefix-closed if L= L. 2. Autoata 2.1. Basic Concepts The autoaton odeling foralis is introduced in Discrete Event Systes. For the sake of copleteness, the definition of an autoaton is recalled. Autoaton: A deterinistic autoaton, denoted by G, is a six-tuple G = ( X, E, f,γ, x, X ) where: X is the set of states, which could be infinite; E is the finite set of events associated with the transitions in G ; f : X E X is the transition function: f ( xe, ) = y eans that there is a transition labeled by event e fro state x to state y (in general, f is a partial function on its doain); Γ: X 2 E is the feasible event function: Γ ( x) is the set of all events e for which f ( xe, ) is defined; x is the initial state; X is the set of arked states. (Given a set A, the notation 2 A eans the X power set of A, i.e., the set of all subsets of A.) Proper selection of which states to ark is a odeling issue that depends on the proble of interest. By designating certain states as arked, we ay for instance be recording that the syste, upon entering these states, has copleted soe operation or task (cf. exaple in Section 3.2). It is convenient to represent graphically autoata where X is finite and sall by eans of their state transition diagras. The state transition diagra of an autoaton is a directed graph where nodes represent states and labeled arcs between nodes are used Encyclopedia of Life Support Systes (EOLSS)

4 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune to represent the transition function f : if f ( xe, ) = y, then an arc labeled by e is drawn fro x to y. Special notation is used to identify the initial states and arked states. In Figs. 2-4 below, the initial state is identified by an arrow pointing into it and arked states are differentiated by eans of a double circle or box. For the sake of convenience, the transition function f of an autoaton is extended fro doain X E to doain X E in the following recursive anner: f( x, ε) := x f ( xse, ):= f( f( xs, ), e) for s E ande E Languages Represented by Autoata Autoata are used to represent and anipulate languages. The language generated by G = ( X, E, f,γ, x, X ) is LG ( ):= { s E : f( x, s)isdefined}. The language arked by G is L ( G) := { s L( G) : f( x, s) X }. The language LG ( ) represents all the directed paths that can be followed along the state transition diagra of G, starting at the initial state; the string corresponding to a path is the concatenation of the event labels of the transitions coposing the path. Therefore, a string s is in LG ( ) if and only if it corresponds to an adissible path in the state transition diagra, equivalently, if and only if f is defined at ( x, s). LG ( ) is prefixclosed by definition, since a path is only possible if all its prefixes are also possible. If f is a total function over its doain, then necessarily LG ( ) = E. The second language represented by G, L ( G ), is the subset of LG ( ) consisting only of the strings s for which f ( x, s) X, i.e., these strings correspond to paths that end at a arked state in the state transition diagra. Since not all states of X need be arked, the language arked by G, L ( G ), need not be prefix-closed in general. The language arked is also called the language recognized by the autoaton, and the given autoaton is often referred to as a recognizer of the given language. An autoaton G thus represents two languages: LG ( ) and L ( G ). In the standard definition of autoaton in autoata theory, the function f is required to be a total function and the notion of language generated is not eaningful since it is always equal to E. Allowing f to be partial is a consequence of the fact that a discrete event syste ay not be able to produce (or execute) all strings in E. Encyclopedia of Life Support Systes (EOLSS)

5 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune Blocking: Deadlock and Livelock The definitions of G, LG ( ), and L ( G ) iply that in general L ( G) L ( G) L( G), since X ay be a proper subset of X. It is worth exaining the second set inclusion in ore detail. An autoaton G could reach a state x where Γ ( x) = but x X. This is called a deadlock because no further event can be executed. Given the interpretation of arking, this eans that the syste blocks because it enters a deadlock state without having terinated the task at hand. If deadlock happens, then necessarily L ( G ) will be a proper subset of LG ( ), since any string in LG ( ) that ends at state x cannot be a prefix of a string in L ( G ). Another issue to consider is when there is a set of unarked states in G that fors a strongly connected coponent (i.e., these states are reachable fro one another), but with no transition going out of the set. If the syste enters this set of states, then livelock results. While the syste is live in the sense that it can always execute an event, it can never coplete the task started since no state in the set is arked and the syste cannot leave this set of states. If livelock is possible, then again L ( G ) will be a proper subset of LG ( ). Any string in LG ( ) that reaches such an absorbing set of unarked states cannot be a prefix of a string in L ( G ), since it is assued that there is no way out of this set. Again, the syste is blocked in the livelock. The iportance of deadlock and livelock in discrete event systes leads to the following definition. Blocking: Autoaton G is said to be blocking if L ( G) L( G), where the set inclusion is proper, and nonblocking when L ( G) = L( G). Thus, if an autoaton is blocking, this eans that deadlock and/or livelock can happen. The notion of arked states and the definitions of language generated, language arked, and blocking, provide an approach for considering deadlock and livelock that is useful in a wide variety of applications Accessibility Properties Fro the definitions of LG ( ) and L ( G ), all the states of G that are not accessible or reachable fro x by soe string in LG ( ) can be deleted without affecting the languages generated and arked by G. When deleting a state, all the transitions that are attached to that state are also deleted. This operation is denoted by Ac( G ), where Ac stands for taking the accessible part. Encyclopedia of Life Support Systes (EOLSS)

6 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune A state x of G is said to be cessible to X, or siply cessible, if there is a path in the state transition diagra of G fro x to a arked state. The operation of deleting all the states of G that are not cessible is denoted by CoAc( G ), where CoAc stands for taking the cessible part. Taking the cessible part of an autoaton eans building CoAc( G) := ( X, E, f, x, X ) where,, { ( ( ) )} X = x X : s E f x, s X x f x if x X = undefined otherwise = f, X E X where the notation f X E X eans the restriction of function f to doain X E. The CoAc operation ay shrink LG ( ), since states that are accessible fro x ay be deleted; however, the CoAc operation does not affect L ( G ), since any deleted state is not on any path fro x to X. If G = CoAc( G), then G is said to be cessible; in this case, L( G) = L ( G). Coaccessibility is closely related to the concept of blocking. Blocking necessarily eans that L ( G ) is a proper subset of LG ( ) in which case there are accessible states that are not cessible. Note that if the CoAc operation results in X = (this would happen if X = for instance), then the epty autoaton is obtained. The ter epty autoaton refers to an autoaton whose state space is epty; an epty autoaton necessarily generates and arks the epty set. An autoaton that is both accessible and cessible is said to be tri. The Tri operation is defined as follows: Tri( G) := CoAc[ Ac( G)] = Ac[ CoAc( G)] where the coutativity of Ac and CoAc is easily verified Nondeterinistic Autoata In the definition of autoaton, state transitions describe how an event e causes a transition fro soe state x to a unique new state y. Suppose, however, that an event e at state x ay cause transitions to ore than one new states. This is a useful anner of odeling randoness in the syste behavior. In this case, f ( xe, ) should no longer represent a specific new state x X, but rather a set of possible new states. In addition, let the label ε be allowed in the state transition diagra of an autoaton, i.e., allow transitions between distinct states to have the epty string as label. These transitions ay represent events that cause a change in the internal state of the syste but are not observable by an outside observer iagine that there is no sensor that records this state transition. Thus the outside observer cannot attach an event label to such a Encyclopedia of Life Support Systes (EOLSS)

7 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune transition but it recognizes that the transition ay occur by using the ε label. These two changes lead to the notion of a nondeterinistic autoaton. Nondeterinistic autoaton: A nondeterinistic autoaton, denoted by six-tuple G nd, is a G = ( X, E { ε}, f,γ, x, X ) nd nd where these objects have the sae interpretation as in the definition of deterinistic autoaton, with the two differences that: 1. f nd is a function f ( { }) 2 X nd : X E ε, i.e., fnd ( xe, ) X whenever it is defined. 2. The initial state ay itself be a set of states, i.e., x X. 3. Operations on Autoata Discrete event odels of coplex dynaic systes are rarely built in a onolithic anner. Instead, a odular approach is used where odels of individual coponents are built first, followed by the coposition of these odels in order to obtain the odel of the overall syste. The synchronization, or coupling, between coponents can be captured by the use of coon events between syste coponents. Naely, if coponents A and B share event c, then event c should only occur if both A and B execute it. The process of coposing individual autoata (that odel interacting syste coponents) in a anner that captures the synchronization constraints iposed by their coon events is foralized by the product and parallel coposition operations that are studied in this section Bibliography TO ACCESS ALL THE 24 PAGES OF THIS CHAPTER, Click here Alur R. and Dill, D.L. (1994). A Theory of Tied Autoata. Theoretical Coputer Science. 126, pp [This paper presents a tied discrete-event odeling foralis that has been studied extensively in the coputer science literature.] Arnold A.(1994). Finite transition systes. International Series in Coputer Science. Prentice-Hall. [A ore advanced book on untied odeling of discrete event systes.] Cassandras C.G. and Lafortune S. (1999). Introduction to Discrete Event Systes. Kluwer Acadeic Publishers. [Chapters 2 and 4 of this textbook on discrete event systes present a detailed coverage of autoata and Petri net odels.] Encyclopedia of Life Support Systes (EOLSS)

8 CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XV - Modeling of Discrete Event Systes - Stéphane Lafortune David R. and Alla H. (1992). Petri Nets and Grafcet: Tools for Modelling Discrete Event Systes. Prentice-Hall. [A nice textbook on Petri net odels and their connection with Grafcet, a prograing ethod for prograable logic controllers (PLCs), which are widely-used in industrial autoation.] Harel D. and Politi M. (1998). Modeling Reactive Systes with Statecharts: The Stateate Approach. Wiley. [This book describes a class of hierarchical autoaton odels that has becoe popular in coercial software packages for odeling discrete-event dynaics.] Hoare C.A.R. (1985). Counicating Sequential Processes. International Series in Coputer Science. Englewood Cliffs, NJ, Prentice-Hall. [A textbook treatent of the odeling foralis of counicating sequential processes.] Hopcroft J.E. and Ullan J.D. (1979). Introduction to Autoata Theory, Languages, and Coputation. Reading, MA, Addison-Wesley. [A classic textbook on autoata and language theory.] Kurshan R.P. (1994). Coputer-Aided Verification of Coordinating Processes: The Autoata-Theoretic Approach. Princeton University Press. [A book on a class of autoaton odels in the context of foral verification.] Manna Z. and Pnueli A. (1992). The Teporal Logic of Reactive and Concurrent Systes: Specification. Springer-Verlag. [This book describes the use of teporal logic in the specification and verification of tied discrete event systes.] Murata T. (1989). Petri Nets: Properties, Analysis, and Applications. PIEEE, Vol. 77, 4, Apr, pp [An excellent tutorial/survey paper on Petri net odels.] Biographical Sketch Stéphane Lafortune received the B. Eng degree fro École Polytechnique de Montréal in 198, the M. Eng. degree fro McGill University in 1982, and the Ph.D. degree fro the University of California at Berkeley in 1986, all in electrical engineering. Since Septeber 1986, he has been with the University of Michigan, Ann Arbor, where he is a Professor of Electrical Engineering and Coputer Science. Dr. Lafortune is a Fellow of the IEEE (1999). He received the Presidential Young Investigator Award fro the National Science Foundation in 199 and the George S. Axelby Outstanding Paper Award fro the Control Systes Society of the IEEE in 1994 (for a paper co-authored with S. L. Chung and F. Lin) and in 21 (for a paper co-authored with G. Barrett). Dr. Lafortune was a eber of the Editorial Board of the Journal of Discrete Event Dynaic Systes: Theory and Applications in the period He was Associate Editor at Large ( ) and Associate Editor ( ) of the IEEE Transactions on Autoatic Control. His research interests are in discrete event systes, intelligent transportation systes, and counication networks. He co-authored, with C. Cassandras, the textbook Introduction to Discrete Event Systes (Kluwer Acadeic Publishers, 1999). Recent publications are available at the Web site Encyclopedia of Life Support Systes (EOLSS)