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1 Controlled Petri Nets: A Tutorial Survey L. E. Holloway 1 and B. H. Krogh 2 1 Center for Manufacturing Systems and Dept. of Electrical Engineering, University of Kentucky, Lexington, KY USA, holloway@engr.uky.edu 2 Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA USA, krogh@ece.cmu.edu 1 Introduction This paper surveys recent research on the synthesis of controllers for discrete event systems (DESs) modeled by controlled Petri nets (CtlPNs). Petri nets have been used extensively in applications such as automated manufacturing [6], and there exists a large body of tools for qualitative and quantitative analysis of Petri nets [28]. For control, Petri nets oer a structured model of DES dynamics that can be exploited in developing ecient algorithms for controller synthesis. The following section presents the basic untimed CtlPN model, discusses special classes of CtlPNs, and gives an overview of the various control problems that have been formulated and solved in the literature. Section 3 describes state feedback controllers for CtlPNs and the general conditions that must be satised for a state feedback control policy to exist to prevent the CtlPN from reaching a given set of forbidden markings. In Sect. 4, control of the sequential behavior of a CtlPN is considered. We summarize methods for converting event-based sequential control problems into state feedback problems. We then present two general approaches to computing state feedback policies, namely, the linear integer programming approach of Li and Wonham (Sect. 5) and the path-based approach introduced by Holloway and Krogh (Sect. 6). The paper concludes with a discussion of several directions for further research in Sect Controlled Petri Nets Controlled Petri nets (CtlPNs) are a class of Petri nets with external enabling conditions called control places which allow an external controller to inuence the progression of tokens in the net. CtlPNs were rst introduced by Krogh [20] and Ichikawa and Hiraishi [19]. Formally, a CtlPN G = (P; T ; C; E; B) consists of a nite set P of state places, a nite set T of transitions, a nite set C of control places, a set E (P T ) [ (T P) of directed arcs connecting state places to transitions and vice versa, and a set B (C T ) of directed arcs connecting control places to transitions. The sets P; T ; C are mutually disjoint. For a transition t 2 T, we denote the set of input control places as (c) t :=

2 2 L. E. Holloway and B. H. Krogh fc j (c; t) 2 Bg, and for a control place c 2 C we denote the set of output transitions as c (t) := ftj(c; t) 2 Bg. Similarly, we denote the set of input (output) state places for transition t as (p) t := fp j (p; t) 2 Eg (t (p) := fp j (t; p) 2 Eg). The set of input (output) transitions for a state place p is dened as (t) p (p (t) ). A transition t is said to be a controlled transition if its set of control inputs (c) t is nonempty. The state of a CtlPN is given by its marking, which is the distribution of tokens in the state places. Formally, a marking is a function m : P?! IN, where m(p) is the number of tokens in place p. M denotes the set of all markings. A set of transitions T T is state enabled if for all p 2 P, m(p) j p (t) \ T j. A control for a CtlPN is a function u : C?! f0; 1g associating a binary value to each control place. The set of all such controls is denoted by U. A set of transitions T T is said to be control enabled if for all t 2 T, u(c) = 1 for all c 2 (c) t. A control u 2 U is said to be as permissive as control u 0 2 U if u(c) u 0 (c) for all c 2 C. Control u is said to be more permissive than control u 0 if u is as permissive as u 0 and u(c) > u 0 (c) for some c 2 C. The most permissive control is u one 1, and the least permissive control is u zero 0. Letting T e (m; u) 2 T denote the collection of sets of transitions that are both state enabled by m 2 M and control enabled by u 2 U, any set of transitions T 2 T e (m; u) can re, thereby changing the marking of the net to the marking m 0 dened by m 0 (p) = m(p)+ j (t) p \ T j? j p (t) \ T j : (1) In words, ring a set of transitions T 2 T e (m; u) causes one token to be removed from each p 2 (p) t, and one token to be added to each p 2 t (p), for each t 2 T. Given a marking m 2 M and control u 2 U, the set of immediately reachable markings, R 1 (m; u), is given by R 1 (m; u) = fmg [ fm 0 2 Mjm 0 is given by (1) for some T 2 T e (m; u)g: (2) The set of reachable markings under an arbitrary number of transition rings from a given marking m 2 M with a constant control u 2 U is denoted R 1 (m; u). An uncontrolled subnet (P; T; E) of G where P P, T T, and E = ((P T ) [ (T P )) \ E is called a marked graph structure if all state places p 2 P have at most one input transition and one output transition in the subnet. Marked graph structures can model synchronization of concurrent processes: tokens in places which share an output transition must progress synchronously. A subnet (P; T; E) is a state graph structure if all transitions t 2 T have at most one input state place and one output state place in the subnet. State graph structures with a single token are analogous to nite-state automata. Section 6 examines a family of control synthesis methods which exploit state graph and marked graph structures in CtlPNs to compute state feedback policies. It is sometimes useful to write the state transition equation (1) as a linear matrix-vector equation by introducing indices for the places and transitions,

3 Controlled Petri Nets: A Tutorial Survey 3 P := fp 1 ; :::; p n g and T := ft 1 ; :::; t m g, and, with a slight abuse of notation, redening the marking as a column vector, m 2 IN n, where m i := m(p i ). Given a sequence of sets of transitions ft 0 ; :::; T k g (2 T ) k, the ring vector v(ft 0 ; :::; T k g) 2 IN m has components v j (ft 0 ; :::; T k g) dened as the number of times transition t j 2 T occurs in the sequence of sets T 0 ; :::; T k. Given a marking m 2 M and control u 2 U, if T 2 T e (m; u), the state transition equation (1) can then be written as m 0 = m + Ev(T ) (3) where E 2 f?1; 0; 1g nm is the incidence matrix for the CtlPN graph [28]. Thus, Petri nets dynamics can be viewed from a linear algebraic perspective, as described in Sect. 5. This approach has been pursued in a control context by Giua, DiCesare, and Silva [11], and by Li and Wonham (which they call vector discrete event systems) [26, 27]. In much of the Petri net literature it is assumed that only a single transition can re at any instant. We will refer to this case as the no concurrency (NC) assumption, under which the state transition equation (1) applies only to singleton sets. Given a CtlPN G and an initial marking m 0 2 M, under the NC assumption L(G; m 0 ) T denotes the language of valid sequences of transition rings for G starting from m 0 and the control u one, where T is the set of all nite-length strings of elements of T. The ring of a transition corresponds to an event in the usual DES terminology. In general, Petri net languages are dened in terms of a separate set of event labels which can be assigned to some or all of the transitions. By identifying the transitions with the event labels, each transition is a distinct event, which corresponds to a so-called free-labeled Petri net in the theory of Petri net languages [30]. The notion of the language of a CtlPN is useful for specifying and analyzing problems of sequential control which are considered in Sect State Feedback and State Specications A state feedback policy for a CtlPN is a function U : M?! 2 U. The state feedback policy is deterministic if U(m) is a singleton for all markings m 2 M. We extend the notation for immediately reachable markings from a marking m 2 M for a feedback policy U by dening R 1 (m; U) = S u2u (m) R 1(m; u). Similarly, the set of markings reachable under an arbitrary number of rings from a marking m for a state feedback policy U is denoted R 1 (m; U). Extending the concept of relative permissiveness of controls to state feedback policies, we say state feedback policy U 1 is as permissive as state feedback policy U 2, denoted by U 1 U 2, if for each m 2 M, U 1 (m) U 2 (m). It follows that U 1 U 2 implies R 1 (m; U 1 ) R 1 (m; U 2 ) for any marking m 2 M. State feedback policies for CtlPNs have been investigated by a number of researchers [15, 21, 2, 26, 18, 1, 16, 35]. In most cases the fundamental problem is to design a state feedback policy that guarantees the system remains in a specied set of allowed states, or, equivalently, that the marking of the CtlPN is never in a specied set of forbidden markings. Given a CtlPN G with initial

4 4 L. E. Holloway and B. H. Krogh marking m 0, let M F denote the set of forbidden markings. T The objective is to nd a state feedback policy U F for which: (1) R 1 (m 0 ; U F ) M F = ;; and (2) U F is more permissive than any other state feedback policy satisfying (1). We call a state feedback policy satisfying these two conditions a maximally permissive state feedback policy for the given forbidden state specication M F. From the general theory of controlled DES developed by Ramadge and Wonham, a necessary and sucient condition for the existence of a maximally permissive state feedback policy is determined by an analysis of the CtlPN behavior under the control u zero. Specically, dene the set of admissible markings for a CtlPN G with respect to a set of forbidden markings M F as A(M F ) = fm 2 MjR 1 (m; u zero ) \ M F = ;g: (4) Necessary and sucient conditions for the existence of a state feedback control policy that keeps a CtlPN out of a given set of forbidden markings is then given by the following theorem. Theorem 1. [21] Given a CtlPN G with initial marking m 0 and a forbidden marking specication M F, a unique maximally permissive state feedback policy exists if and only if m 0 2 A(M F ). The unique maximally permissive state feedback policy in Th. 1 is nondeterministic because the ring rule (1) allows multiple transitions in the CtlPN to re simultaneously [20, 25]. In general, there will not be a unique deterministic maximally permissive policy because the set of controls U F (m) does not have necessarily a unique maximal element. On the other hand, under the NC assumption where the ring rule (1) is restricted to singleton sets, a unique deterministic maximally permissive state feedback policy exists [26]. When the initial marking for a CtlPN satises the condition m 0 2 A(M F ) in Th. 1, the maximally permissive state feedback policy can be described simply as the policy which does not allow any state transitions to markings outside A(M F ). Theorem 2. [21] Given a CtlPN G with a forbidden marking specication M F and an initial condition m 0 2 A(M F ), if m 2 R 1 (m 0 ; U F ), then U F (m) = fu 2 UjR 1 (m; u)? A(M F ) = ;g: One approach to compute the set of admissible controls for a given marking is to simply create the equivalent controlled automaton for the CtlPN, which is a matter of generating the reachability graph for the Petri net structure with the associated control information. Since the reachability graph can grow exponentially with respect to the size of the CtlPN model [22], alternative methods are desirable for computing feedback policies. Sections 5 and 6 present the two approaches that have been developed to use the structure of the CtlPN model directly, thereby avoiding the generation of the equivalent controlled automaton. We conclude this section with a brief summary of various extensions and generalizations of state feedback policies considered in the literature. Baosheng

5 Controlled Petri Nets: A Tutorial Survey 5 and Haoxun consider partial observability in the context of distributed state feedback control systems [13]. Control capability is distributed among several predened controllers, and each controller has only limited observations of the net marking. Li and Wonham consider partial state observability for centralized controllers [26], and prove that a maximally permissive state feedback policy exists if and only if the given predicate (set of admissible states) is both controllable and observable [26]. They also consider modular state feedback policies where the overall specication for the admissible states for the closed-loop system are given as the conjunction (intersection) of a collection of subspecications, similar to the forbidden marking specications considered in Sect Event Feedback and Sequential Specications Event feedback control for a CtlPN is based on observations of the transition rings rather than the marking. Under the NC assumption (which is assumed throughout this section), an event feedback policy is a function V : T?! U mapping each sequence of transitions onto a control. L(VjG; m 0 ) T denotes the language generated by the net G from initial marking m 0 under an event feedback policy V. The goal of event feedback control is to restrict the operation of the Petri net such that it produces a desired (prex-closed) language K L(G; m 0 ). From the general Ramadge and Wonham theory, there exists an event feedback policy V such that L(VjG; m 0 ) = K if and only if the language K is controllable, where K is dened to be controllable with respect to L(G; m 0 ) if K L(G; m 0 ) and KT u \ L(G; m 0 ) K [32]. 3 As with state feedback control policies, one approach to synthesizing event feedback controllers for a CtlPN G is to create the equivalent controlled automata model for G and apply the supervisor synthesis algorithms from the standard Ramadge and Wonham theory. As an alternative to automata-based supervisors, some researchers have considered using Petri nets as supervisors. This increases the class of controllable languages that can be realized by event feedback policies, but computational complexity becomes a problem for more general Petri net languages. Indeed, Sreenivas has shown that controllability is undecidable for the most general Petri net languages, but it is decidable if the specication language is accepted by a free-label Petri net [34]. Giua and DiCesare obtained necessary and sucient conditions for the existence of a Petri net supervisor when the controllable language specication can be accepted by a conservative Petri net [8, 9]. An alternative to constructing dynamic event feedback controller directly in the form of an automaton or Petri net is to augment the dynamics of the plant with an additional automaton or Petri net that \encodes" the desired sequential behavior of the system in the state of the augmented system. One can then apply state feedback to the augmented system to achieve the desired sequential behavior. This approach has been developed by Kumar and Holloway [23] using a Petri net to augment the plant dynamics, and by Li and Wonham [26] using 3 KTu denotes the set of all strings of the form! such that! 2 K and 2 Tu.

6 6 L. E. Holloway and B. H. Krogh an automaton called a memory to augment the plant dynamics. In both cases it is shown that the maximally permissive state feedback control for avoiding an appropriately dened predicate or set of forbidden states on the augmented state space results in the supremal controllable sublanguage K " for a given sequential specication K. Since event feedback control problems for sequential specications can be converted into state feedback control problems, we turn our attention in the following two sections to methods for computing state feedback policies for CtlPNs. 5 Linear Integer Programming Approach Linear algebraic methods and linear integer programming have been standard tools for solving many problems in the Petri net literature based on the matrixvector representation of the CtlPN state transition equation (3) [28]. In the context of control, Giua, DiCesare, and Silva use linear integer programming to evaluate properties of DESs controlled by a class of Petri net controllers [10, 11]. In this section we describe the linear integer programming method proposed by Li and Wonham [24, 27] for computing state feedback policies. Under the NC assumption and the assumption each control place is connected to a single controlled transition, Li and Wonham consider the synthesis of maximally permissive feedback policies when the allowable states are specied by a linear predicate P of the form P = fm 2 Mja T m bg (5) where a is an n-vector and b is a scalar. The control objective is to guarantee the linear constraints on the marking are satised for all markings reachable under the control. The set of admissible markings corresponding to a predicate P of the form (5) is denoted as [P ] by Li and Wonham, and the unique maximally permissive control input for a given m 2 [P ] is denoted by u [P ](m). From the general theory in Sect. 3, computing u [P ](m) is a matter of determining if markings of the form ^m = m + Ev(c (t) ) are in [P ]. Li and Wonham show that this problem can be reduced to solving a linear integer program provided the CtlPN satises a particular structural condition, namely, the uncontrolled portion of the CtlPN must be loop free. To describe this result, we use the language notation for strings of admissible transitions to express [P ] as: [P ] = fm 2 Mja T m + a T Ev (m)) bg; (6) where v (!) is the solution to the following optimization problem max a T Ev(!): (7)!2L(u zerojg;m) The reduction of the optimization problem (7) to a linear integer program is based on the following general result for Petri nets (which we state for CtlPNs).

7 Controlled Petri Nets: A Tutorial Survey 7 Theorem 3. [19] Given a loop-free CtlPN G and a marking m 2 M,! 2 L(G; m) if and only if! 2 T and m + Ev(!) 0: (8) The signicance of this theorem is that any ring vector satisfying (8) corresponds to a valid ring sequence. Giua and DiCesare have called Petri nets with this property easy Petri nets [7]. To apply this result to the optimization problem (7), observe that the language L(u zero jg; m) is the same as the language for the Petri net obtained when all controlled transitions are removed from G, which Li and Wonham denote by G u. Thus, if G u is loop free, (7) becomes a linear integer program. The maximally permissive feedback control can be computed using this linear integer program by observing that u [P ](m)(c) = 1 when c (t) is state enabled if and only if a T ^m + a T Ev ( ^m) b; where ^m = m + Ev(c (t) ). Li and Wonham develop this basic approach in several ways, including the generalization to multiple linear predicates (modular synthesis) and developing a closed-form expression for the maximally permissive control under further structural assumptions. They also consider sequential specications in the form of linear predicates on the ring vector which they call linear dynamic specications. For sequential specications, they convert the problem to a state feedback problem using a memory as described in the previous section. 6 Path-Based Algorithms Path control algorithms decompose the forbidden state control problem into the regulation of tokens in individual paths in the CtlPN. For some classes of CtlPNs it has been shown that this exploitation of the net structure can lead to significant gains in computational eciency for on-line control synthesis [22]. Path control has been used to address a variety of control specications, including state control [15, 21], sequence control [18], and distributed control [13]. Path control assumes the set of forbidden states is expressed in the form of forbidden conditions, which are specications of sets of forbidden markings based on linear inequalities on the marking vectors [21, 2]. A forbidden condition is represented by the triple (F; v; k), where F P is a subset of places, v : F?! IN is a weighting function over the places in F, and k is the threshold of the forbidden condition. The forbidden condition species a set of forbidden markings as M (F;v;k) := fm 2 M j X p2f m(p)v(p) > kg (9) Given a collection F of set conditions, the set M F of forbidden markings for F is dened as the union of M (F;v;k) for all (F; v; k) 2 F.

8 8 L. E. Holloway and B. H. Krogh In general, arbitrary forbidden marking sets cannot be represented by forbidden conditions; however, for the special case of cyclic controlled marked graphs, any subset of live and safe markings can be dened by a set F of forbidden set conditions [14]. Previous work has considered specic cases of the generalized forbidden condition specication. Holloway and Krogh consider live and safe cyclic controlled marked graphs where for each (F; v; k), v(p) = 1 for all p 2 F, and k = j F j?1 in [15] and 1 k < j F j in [21]. Boel et al. consider a class of controlled state graphs without restrictions on v or k [2]. Holloway and Guan consider a general class of Petri nets with forbidden conditions where v(p) = 1 for all p 2 F and k = j F j?1 [17]. Path control methods characterize the reachability of forbidden conditions in terms of the markings of directed paths in the CtlPN. We consider only paths which begin with a transition and end with a state place. The following notation is used in this section. For a given path = (t 0 p 0 : : : t n p n ), the starting transition t 0 is denoted t. We dene set operations on paths to be over the set of places and transitions in the path. Thus, \ P is the set of places in the path, and \ T is the set of transitions in the path. We extend the marking notation such that m() is the sum of m(p) over all p 2. c (p) denotes the set of all precedence paths for a place p, where = (t 0 p 0 : : : t n p n ) 2 c (p) implies that: (1) path ends at place p; (2) t is a controlled transition or has (p) t \ 6= ;; (3) all other transitions t 2 ( \ T )? ft g are uncontrolled transitions; and (4) for any 0 i; j n then p i 6= p j and if t i = t j then either i = 0 or j = 0. The last condition (4) restricts a path to not include a cycle. The key question in path control is: For a given marking m, does there exist an uncontrollably reachable marking, m 0 2 R 1 (m; u zero ), for which m 0 (p) k? This problem is referred to as the uncontrollable k-coverability problem. The role of precedence paths in determining the uncontrollable k-coverability problem is most easily illustrated by considering the case where k = 1. For a marking m and a precedence path, dene the predicate 1 if m(p) m () := 1 for some p 2 0 else (10) Theorem 4. [15] Given a CtlPN G and place p such that c (p) = f 1 ; 2 ; : : : n g is a marked graph structure, for a marking m there exists an uncontrollably reachable marking m 0 2 R 1 (m; u zero ) with m 0 (p) 1 if and only if m ( 1 ) ^ m ( 2 ) ^ m ( n ) = 1 : (11) Theorem 5. (follows from both [2] and [17]) Given a CtlPN G and place p such that c (p) = f 1 ; 2 ; : : : n g is a state graph structure, for a marking m there exists an uncontrollably reachable marking m 0 2 R 1 (m; u zero ) with m 0 (p) 1 if and only if m ( 1 ) _ m ( 2 ) _ m ( n ) = 1 : (12)

9 Controlled Petri Nets: A Tutorial Survey 9 The above theorems show that the marked graph structure and state graph structure lead to the complimentary characterizations of the uncontrolled 1- coverability of a place. These conditions are generalized by Holloway and Guan to cases where the uncontrolled region of the net leading to a given place may have a mixture of both marked graph structures and state graph structures [17]. We note that conditions for uncontrolled k-coverability do not have the same convenient Boolean characterization as the uncontrolled 1-coverability problem has. However, for state graph structures, Boel et al. show that uncontrolled k- coverability can be characterized through the sum of tokens among precedence paths leading to a place [2]. Given the characterization of the uncontrollably reachable markings in the above theorems, we now describe a method for avoiding forbidden states by controlling the markings of individual paths. Our discussion is primarily based on the method presented in [15] and [21] for controlled marked graphs. The reader is referred to [2] and [17] for control laws for other net structures. For the remainder of this section, we assume controlled marked graphs with binary markings, i.e. safe markings [28], where all cycles within the net contain at least one marked place under any initial marking. To be able to achieve the decomposition of control upon which path control methods depend, we also require the following assumption on the interaction of paths for places in a forbidden condition: Given a set F of forbidden conditions, for any (F; v; k) 2 F and any p; p 0 2 F with p 6= p 0, if 2 c (p) then p 0 62 and p 0 62 (p) t. This is referred to as the Specication Assumption (SA). We consider a set F of forbidden conditions of the form (F; v; k), where v(p) = 1 for all p 2 F and 1 k < j F j. For a forbidden condition (F; v; k) 2 F and marking m, dene m (p) from (11) for each p 2 F as the conjunction of m () for all 2 c (p). Furthermore, dene L F (m) := fp 2 F j m (p) = 1g : (13) From Th. 4 and the SA, it can be shown that L F (m) is the set of places in F which can become marked uncontrollably from the marking m [15]. From the denition of the admissible marking set A(M F ) in Sect. 3, it then can be shown that for any given marking m, m 2 A(M F ) if and only if L F (m) k [21]. To prevent a forbidden marking from being reachable, a control must be enforced to ensure that j L F (m 0 ) j k for all (F; v; k) 2 F for all immediately reachable markings under the control. To determine a control using path control techniques, we rst examine the markings of individual paths to determine which places in a condition F could potentially join the set L F (m 0 ) after the next transition set ring. Dene the predicate m () = 1 i t is state enabled under m, and dene 1 if m (p) = 0 and m (p) := m () _ m () = 1 for all 2 c (p) (14) 0 else B F (m) := fp 2 F j m (p) = 1g: (15) It can be shown that B F (m) is the set of places in F which could become uncontrollably marked following the next transition set ring unless some control

10 10 L. E. Holloway and B. H. Krogh is enforced. From the denition of m (p) above and from Th. 4, we note that keeping at least one path 2 c (p) unmarked is sucient to ensure that p is an element of B F (m 0 ) (and thus not an element of L F (m 0 )) for any immediately reachable marking m 0. This can be accomplished by disabling the controlled transition t for some path 2 c (p) which is unmarked but for which t is state enabled. From this observation we have that D F (m; u) := fp 2 B F (m) j u(c) = 0 for some c 2 (c) t for some 2 c (p) with m () = 1 and m () = 0g is the set of places in B F which will be prevented from joining L F (m 0 ) for a next reachable marking m 0 under the control u. The number j B F (m) j? j D F (m; u) j thus represents the maximum number of places in F that could join L F (m 0 ) after the next transition set ring. In order to ensure that j L F (m 0 ) j k for all immediately reachable markings m 0 2 R(m; u), the net control u then must satisfy the following equation. j L F (m) j + j B F (m) j? j D F (m; u) j k : (16) From Th. 2, the following result can be obtained: Theorem 6. [21] Given a CtlPN G with a forbidden marking specication F satisfying SA, let U F be the policy such that for each m 2 A(M F ), U F (m) is the set of controls such that (16) is satised for each (F; v; k) 2 F. The control policy U F is the maximally permissive state feedback policy. 7 Directions for Future Research This paper surveys research on feedback control policies for discrete event systems using Petri net models. The primary objective in this research is to develop modeling, analysis and synthesis procedures that take advantage of the structural properties of Petri nets to reduce computational complexity. Toward this end there are several open directions for further research. It would be of interest to determine how more classes of Petri net structures such as free-choice nets can be exploited for feedback control. The computational complexity of Petri net methods versus unstructured automata-based approaches also needs futher investigation. One comparison is presented in [22] where it is shown that the complexity of computing maximally permissive state feedback policies for controlled marked graphs (CtlMGs) is polynomial in the number of transitions and the number of set conditions in the forbidden state specications, whereas the complexity grows exponentially for the equivalent automata-based models. Although there are ecient methods for solving similar problems independent concurrent automata [31], it has been shown that forbidden state problems for synchronized concurrent systems (of which CtlMGs are a subset) are in general computationally intractable [12]. Thus the boundary between tractable and intractable problems needs to be explored more deeply.

11 Controlled Petri Nets: A Tutorial Survey 11 Recently a number of researchers have been interested in developing methods for extending methods for synthesizing feedback control policies for untimed (logical) models of DESs to models and specications which include explicit representations of real time [4, 29, 3]. Petri nets oer an attractive framework for developing these extensions for controlled DESs because there are established methods for introducing and analyzing timing in uncontrolled Petri net models [28]. Two extensions of the methods discussed in this survey to controlled time Petri nets (CtlTPNs) are the work of Sathaye on synthesis of dynamic supervisors for sequential specications for CtlPNs [33], and the work of Brave and Krogh on extensions of the path-based approach for forbidden marking specications for the special case of controlled time marked graphs(ctltmgs) [5]. Given the complexity of timed DESs, we believe research in this direction should be guided by an understanding of problems arising in specic applications. Acknowledgments: L. E. Holloway has been supported in part by NSF grant ECS , NASA grant NGT-40049, Rockwell International, and the Center for Robotics and Manufacturing Systems at the University of Kentucky. B. H. Krogh has been supported by Rockwell International and the Alexander von Humboldt Foundation. References 1. Z. A. Banaszak and B. H. Krogh. Deadlock avoidance in exible manufacturing systems with concurrently competing process ows. IEEE Transactions on Robotics and Automation, 6(6), December R.K. Boel, L. Ben-Naom, and V. Van Breusegem. On forbidden state problems for a class of controlled Petri nets. IEEE Transactions on Automatic Control, to appear. 3. B.A. Brandin and W.M. Wonham. Supervisory control of timed discrete-event systems. In Proc. 27th IEEE Conf. on Decision and Control, pages 3357{3362, Austin, TX, Dec Y. Brave and M. Heymann. Formulation and control of real-time discrete event processes. In Proc. 27th IEEE Conf. on Decision and Control, pages 1131{1132, Austin, TX, Dec Y. Brave and B.H. Krogh. Maximally permissive policies for controlled time marked graphs. In Proceedings of 12th IFAC World Congress, Sydney, July F. Dicesare, G. Harhalakis, J. M. Proth, M. Silva, and F. B. Vernadat. Practice of Petri Nets in Manufacturing. Chapman and Hall, London, A. Giua and F. DiCesare. Easy synchronized Petri nets as discrete event models. In Proc. 29th IEEE Conf. on Decision and Control, pages 2839{2844, Honolulu, Dec A. Giua and F. DiCesare. Supervisory design using Petri nets. In Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK, December A. Giua and F. DiCesare. Blocking and controllability of Petri nets in supervisory control. Transactions on Automatic Control, 39(2), Feb to appear. 10. A. Giua and F. DiCesare. Petri net structural analysis for supervisory control. IEEE Transactions on Robotics and Automation, April to appear.

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