DISCRETE-event dynamic systems (DEDS) are dynamic

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1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH The Supervised Control of Discrete-Event Dynamic Systems François Charbonnier, Hassane Alla, and René David Abstract The supervisory control theory of discrete-event dynamic systems (DEDS), first introduced by Ramadge and Wonham, is based on automata concept. Given a process, the objective of this theory is to design a supervisor in such a way that the process coupled with the supervisor behaves according to various constraints. In this framework, the process is assumed to evolve spontaneously and the supervisor can only prevent some events from occurring, but cannot force them. In fact, most processes require the addition of an external control agent that forces some events to occur. This has led us to the supervised control concept where control and supervision are separated. This provides a hierarchical frame, thus enabling us to formalize and to systematize the transition from the synthesis to the implementation of the control. In this paper, Grafcet is intensively used for the supervision and the control design. Index Terms Automata, controllability, control systems, Grafcet, hierarchical systems, manufacturing. I. INTRODUCTION DISCRETE-event dynamic systems (DEDS) are dynamic systems that are basically asynchronous (not clock driven) and that evolve in accordance with the occurrence of events. The supervisory control of DEDS is a new research area which is receiving increasing recognition [11], [12], [15] [18], and that has led to the control of fairly complex systems [1], [2]. The supervisory control theory, first introduced by Ramadge and Wonham [15] [17], is based on automata and formal language models [9]. In these models, chronological time is not involved, i.e., we only focus on the ordering of events. A process is assumed to generate events spontaneously. Its behavior may be described by sequences of events and forms a language over the alphabet of events. Given a process, the objective of the theory [15] [17] is to design a supervisor in such a way that the supervised process, i.e., the process coupled with the supervisor, behaves according to various constraints. The techniques described in [12] and [16] allow us to synthesize a supervisor such that the behavior of the resulting supervised process: 1) does not contradict some behavioral specifications; 2) is maximally permissive within the specifications; and 3) is nonblocking. The theory has been extended to cover modular [18], decentralized [13], and hierarchical supervision [19]. In the Ramadge Wonham (RW) framework, the process is assumed to evolve spontaneously. Nevertheless, a process Manuscript received July 1, 1996; revised April 14, Recommended by Associate Editor, S. Kumagai. The authors are with the Laboratoire d Automatique de Grenoble, (CNRS- INPG-UJF), BP 46, St. Martin d Hères, France. Publisher Item Identifier S (99)01619-X. requires the addition of an external control agent that forces some events to occur. For implementation purposes, this has led to several extensions where the supervisor may force some of the events [1], [8]. Within these forcing event approaches, the supervision does not fit its original function and control and supervision concepts are mixed. This generates several limits for the synthesis of the control. In particular, one cannot use a modular approach that is consistent with the proof of controllability. This obliges us to build a unique supervisory control agent that observes all the events coming from the process, thus resulting in a loss of conciseness. Moreover, the resulting control automata may not be directly implemented into a programmable logic controller (PLC). In general, a major drawback is that the proof models are not the same as the implemented models. It thus follows that it cannot be guaranteed that the proofs also hold in the implemented models. These points will be discussed in Section III. This paper presents the supervised control concept [3], [4]. This new concept allows us to clearly separate control and supervision. Within this approach, the controller can force some events to occur in the process. For an external observer, i.e., from the supervisor viewpoint, it appears that the process coupled with the controller, i.e., the extended process, generates events spontaneously. The supervision is thus confined to prohibiting some of the events to be generated in the extended process. The hierarchical frame introduced by the supervised control clarifies the input and output of the system, thus enabling us to formalize and to systematize the transition from the synthesis to the implementation of the control. The main concepts we introduce are independent from the modeling tool that may be used. Grafcet [5], [6], [10] is a powerful description tool that allows modeling of large sequential machines and that is being used increasingly for PLC s. Moreover, Grafcet is a control tool that allows the enforcement of events and that provides a unique and unambiguous sequence of outputs, given a sequence of inputs. This makes this tool attractive for supervised control design and implementation purposes. In this paper, supervision and control are based on Grafcet. However, the proof of controllability is supported by automata models. Formally, the algorithms provided in this paper allow us to obtain automata models from grafcet models. This will ensure that the proofs also hold in the implemented models. This paper is organized as follows. After a recall on the supervisory control of discrete-event systems in Section II, the problems encountered in the forcing event approaches /99$ IEEE

2 176 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH 1999 Fig. 2. Example 1: A manufacturing system. Fig. 1. Supervisory control. Process P in isolation. Process P with a restricted behavior. (c) Supervisory control schema. [1], [8] are discussed in Section III. The supervised control concept is introduced in Section IV. Section V then refers to the Grafcet tool and the way it may be used for control purposes. The change over from Grafcet models to automata models is presented in Section VI. Finally, the synthesis of supervised control is discussed and illustrated in Section VII. II. RECALL ON SUPERVISORY CONTROL Discrete-event dynamic systems (DEDS) are systems that evolve in accordance with the occurrence of events. For qualitative analysis, chronological time is not involved. The behavior of a DEDS may be described as a set of sequences over the alphabet of events and defines a formal language [9]. A. Logical Model for DEDS A DEDS may be modeled by a state transition graph. In the RW approach, it is assumed one observes from outside the simultaneity of some events with some transitions. Wedo not know if the events are causes or consequences of the transitions. It is said that they are generated by the process. The transition graph of a process may be seen as a fourtuple deterministic automaton 1 also called an acceptor [9]: where: is the set of states ; is the alphabet of events; is a partial transition function; and is the initial state. The behavior of in isolation may be seen as illustrated in Fig. 1. For every current state evolves when generating an event in, where is the set of all the eligible events such that is defined. B. Supervised Control Concept and Controllability The process coupled with a supervisor and receiving, in input, a list of forbidden events will be denoted and will be referred to as a supervised process.if is in state then it behaves as shown in Fig. 1: may only evolve 1 One may also find a process modeled by five-tuple deterministic automaton (Q; 6; ;q 0 ;Qm) where Qm Q is a set of marking states that defines a set of particular strings, i.e., the strings of events that lead to a marking state from the initial state of the automaton. This allows checking for the nonblocking property [14], [15]. (c) by generating an event 2 of. This means that an event is eligible in iff is eligible in [i.e., ] and is not forbidden by supervisory control (i.e., ). Note that a supervised process can be defined in a dual way by specifying, in input, the list of authorized events [3]. We assume the following partition:, where and are the set of controllable and uncontrollable events, respectively [16]. As uncontrollable events may not be disabled through supervisory control, it is always required for that. Formally, a supervisor may be defined as a Moore machine [9] by a sixtuple where is the state set; is the finite input set; is the transition function; is the initial state; is the finite output set; and is the output function. The behavior of a process coupled with is illustrated in Fig. 1(c). At any logical time, provides with. The supervised process may then generate the th event, which is an event in. The occurrence of may drive into a new state. From this state, [with possibly ] is supplied to. And so on. More generally, one can define a specification as a Moore machine where ; ; ; keep the same meaning as for a supervisor; and where the output function is extended to. It follows that an uncontrollable event can belong to the list of forbidden events associated with a state. This means that (although one cannot forbid the occurrence of ) must not be generated by the process when the trace of events leads the specification to state. The language associated with the Moore machine may be defined in a recursive way as the less restrictive language on * (i.e., the set of all finite strings over including the empty string ) satisfying: 1) and 2) if then for if. Given a process, and a specification, the existence of a supervisor such that the behavior of coupled with respects, lies in the concept of controllability [15] [17]. C. Example Example 1: Let us consider a small manufacturing system (Fig. 2) composed of two identical machines and, working independently. As shown in Fig. 3, the start of working (i.e., event ) is assumed to be simultaneous with the pick-up of a part upstream. However, the transport downstream is explicitly modeled: from the busy state (corresponding to the work being over and the part being on ), the start of transport, i.e., event, leads to transport state. Then the end of transport (event ) leads the machine to the idle state where a new cycle may start. 2 6e(q)n8 defines the set of events that belong to 6e(q) and that do not belong to the set 8.

3 CHARBONNIER et al.: SUPERVISED CONTROL 177 Fig. 5. An I/O supervisor according to [1]. Fig. 3. Automata models of the manufacturing system. Model of M1. Model of M2. Fig. 4. Automata models of the behavioral specification. A Moore machine. The corresponding acceptor. Fig. 6. Automaton model of the I/O supervisor. Specification 1: The two machines are assumed to work in tandem and the behavior of the system must respect the presence of a buffer that is assumed to be placed between and. This means that: 1) a part will first visit then ;2) may not put a part in the buffer if the latter is full; and 3) may not start a cycle if it cannot take a part in the buffer, i.e., when the buffer is empty. The buffer is assumed to have a capacity of one and to be empty in its initial state. This behavioral specification can be modeled by the Moore machine in Fig. 4. Whenever the buffer is empty (state ), the list of forbidden events associated with means that the event is forbidden ( may not start a cycle). The label associated with the self loop means that any occurring event except does not change the state. The occurrence of ( finishes its transport) leads to state where the buffer is full. In this state, the occurrence of is forbidden: this prevents depositing a new part in the buffer that is full. We assume that the start of working and the start of transport are controllable events, i.e.,, then one can check that the language of specification embodied by the automaton of Fig. 4 is controllable. One can also define an acceptor of the language of specification. This may be obtained from the Moore machine by deleting, from every state, the transitions labeled by where and by deleting all the states that may not be reached from the initial state. In our example, the acceptor obtained from Fig. 4 is shown in Fig. 4. Let us recall that in state, event is forbidden. Then this event is deleted from the list associated with the self loop of state in Fig. 4, i.e.,. We obtain the list as shown in Fig. 4. Similarly, one obtains the set of events associated with the self loop of state. In the acceptor, the outputs are implicit in the state transition structure: an event is forbidden in a state iff there is not an output transition labeled by from.in practice, Fig. 4 and contain the same information on the language. Although the Moore machine provides a clearer understanding of inputs and outputs, the acceptor model is commonly used in the RW approach. III. FORCING EVENT APPROACHES In RW theory, one assumes that a process evolves spontaneously. Nevertheless, this is not the case in practice. For implementation purposes, one can find several extensions of the RW theory where the supervisor may force some of the events [1], [8]. Let us focus on the approach proposed in [1] (however, the subsequent comments 1) 4) will also hold for [8]). In Fig. 5, the input output (I/O) supervisor [1] controls the process by forcing its outputs. The outputs of the I/O supervisor are controllable events, whereas its inputs, i.e., the events generated by the process, are uncontrollable events. One can make the following remarks about the forcing event approaches. 1) Let us consider Example 1: using a forcing event approach, the events and become outputs of I/O supervisor. However, one can note that the supervisor of Fig. 4 cannot force the starts of working and transport of the machines according to their work cycles as described in Fig. 3. In fact, it is necessary to compute the synchronous product of the supervisor [Fig. 4] with the process models (Fig. 3): this ensures the completeness [1] of the process with regard to the I/O supervisor. Then we obtain the controller as shown in Fig. 6. 2) In the model of Fig. 6, control and supervision concepts are mixed: from a state, an output transition associated with means either 1) is forced when the I/O supervisor is in state if is controllable or 2) is authorized (not forbidden) from state if is uncontrollable. It follows that this model does not provide a clear understanding of inputs and outputs.

4 178 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH ) Let us consider a modular supervision. In the RW approach, a controllable event can be generated only if it is authorized by all the supervisors. However, in [1], it suffices that one of the supervisors forces for to be generated. For example, we consider a process and a sequence of events with and. Let and be two I/O supervisors for with:,, and. The event can be forced by after and the sequence can occur [although ]. Thus, although is controllable, the constraints given by are not necessarily respected. Hence, in a modular forcing event approach, the controllability of the different specifications does not necessarily imply that they will be respected in the closed-loop behavior. 4) Because of 1) and 3) a unique I/O supervisor must be built that observes all the events coming from the process. In practice, this leads to a state explosion that restricts the applicability of the approach to small systems. 5) Some states in Fig. 6 have several output transitions associated with a controllable event (e.g., state, represented by a bold circle in Fig. 6, has two output transitions labeled by and ). This means that the I/O supervisor must force the occurrence of or in state. However, in this model, both events may not occur simultaneously (in practice, one will have to force one of the two events, only). It follows that, in general, an I/O supervisor may not be directly implemented into a PLC. IV. SUPERVISED CONTROL CONCEPT Fig. 7 presents the supervised control concept [3], [4]. A process coupled with its logic controller is shown in Fig. 7. The process to be controlled is seen as a DEDS by the logic controller that is itself a DEDS. The inputs of the logic controller are the outputs of the process to be controlled and vice versa. Basically, the inputs and outputs of the logic controller, namely and, are Boolean variables or Boolean vectors. However, this is not a restriction; as it has been observed in [6] and [7], one can always encode the state of a DEDS by Boolean variables and furthermore any event can be associated with the rising edge or falling edge of a Boolean variable. It follows that one can have events for and. This will be discussed in Section V. For an external observer, the extended process, i.e., the process coupled with the logic controller, may be seen as a device that evolves spontaneously (i.e., with no external forcing). If one wants some behavioral specifications to be respected by the extended process, a supervisor may be coupled with the extended process as shown in Fig. 7. The inputs of the supervisor are the outputs of the extended process. The supervisor bases its observation on the alphabet (i.e., the events produced in the extended process). At any logical time, the supervisor provides the extended process with a list of forbidden events. This means that the extended process may only generate events in. (c) Fig. 7. Supervised control. The extended process. The supervisory schema. (c) The supervised control schema. The supervised control schema [3], [4] is shown in Fig. 7(c). In this hierarchical schema, control and supervision concepts are clearly separated. The process may be seen as a device that evolves by producing events in. The logic controller may also be seen as a DEDS that controls the process by producing events in. For an external observer, the extended process may be seen as a device that evolves by generating events in. Remark 1: In the schema of Fig. 7(c), the supervision fits its original function (the supervisory task is confined to prohibiting some of the events from occurring). It follows that the control task is independent from the supervisory task. Therefore, if one modifies the behavioral specifications that must be applied to the extended process, the control system does not need to be rebuilt. It is assumed that the events produced by the process are uncontrollable events. In fact these are information coming from sensors. This implies that. Moreover, we assume that, if needed, one can always disable the logic controller to produce events in, i.e., orders for actuators of the process. However, for some given behavioral specifications, all the events in do not need to be controllable events, then in general, can be defined as a subset of (and ). Note that in the Fig. 7(c), the outputs of the supervisor are some of the inputs of the logic controller, thus enabling the supervisor to disable, at any time, some of the controllable events to be produced by the logic controller. Remark 2: From a theoretical viewpoint, any constraint given by the supervisor could be added to the controller (i.e., the limit case which is the union of the controller and the supervisor is a controller), while the reciprocal property is not true (the union of the controller and the supervisor is not a supervisor). One considers that there is a single controller and a single supervisor. However, the controller may be made up of several

5 CHARBONNIER et al.: SUPERVISED CONTROL 179 parts and the supervisor may also be made up of several parts (an example of controller with several parts is given in Section VI). The design of a supervised control system is based on the following two steps: 1) the design of the controller (if it does not already exist) and 2) the design of the supervisor that restricts the behavior of the extended process. Note that the first step necessarily precedes the second one: the design of the supervisor depends on the extended process. This represents a major difference with respect to [1]. The supervised control approach will be illustrated on the Grafcet tool. V. RECALL ON GRAFCET Grafcet is a tool whose aim is the specification of logic controllers. It draws inspiration from Petri nets (a general mathematical tool allowing DEDS of various kinds to be described [14]). It is the basis of the preparation of sequential function charts (SFC), an international standard in 1987 [5], [10]. Fig. 8. Example 2: tank filling. A. Boolean Variables and Events Let us consider a Boolean variable that is assumed to be known in the range of time [0, ). Then the times when the events (rising edge) and (falling edge) occur are known. Reciprocally, if all the occurrences of and between times 0 and are known, and the value of is known at, then the value of is known for any time in the range [0, ). It follows that one can choose to model the inputs [ in Fig. 7] and outputs [ in Fig. 7] of a DEDS either by Boolean variables or by events or by both. Grafcet allows the use of both Boolean variables and events for inputs and for outputs. Let us note that although the time when a Boolean value is constant has a finite duration, the events and have no duration. Notation: We shall use a capital letter for a Boolean variable, e.g., and for its complement. An event will be represented by a lower case letter, for example and. For simplicity, a device (sensor or actuator) will be represented by the same symbol as its corresponding Boolean variable. B. Basic Notions Let us illustrate Grafcet with a simple example. More details can be found in [5]. Example 2 Tank Filling: The device in question is represented in Fig. 8. Both tanks are used in a similar way. Tank 1 is empty when the level is less than, i.e.,, and is full when the level is greater than, i.e.,.at the initial state both tanks are empty. As soon as button is pressed (event ), both tanks are filled by opening valves ( means that valve is open) and. When a tank is full, e.g., tank 1, filling stops (by closing valve ) and its content starts to be used (by opening valve ). When tank 1 is empty, valve is closed. Filling may only start up again when both tanks are empty, the next time that button will be pressed. Fig. 9. Two grafcets corresponding to the same specifications. Inputs are events and Boolean variables, outputs are Boolean variables. All the inputs and outputs are events. The grafcet 3 in Fig. 9 represents these specifications. The squares labeled 1 to 6 are steps, i.e., components of states. At initial time, the steps in the set are active (represented by a double square). Then transition (1) that follows these steps can be fired as soon as the event, associated with (1), occurs. After this firing, the steps that are active are 2 and 5. When step 2 is active, the output (represented by written in a rectangle associated with step 2) and transition (2) can be fired if. And so on. The concurrency is explicitly represented in this model. The steps 1, 2, and 3 correspond to the states of tank 1 (empty, during filling, and during emptying, respectively) and the steps 4, 5, and 6 correspond to the states of tank 2. One can observe that the times when the transitions are fired depend on the inputs of the controller [ in Fig. 7]; as a matter of fact the event, associated with transition (1), and the Boolean variables, and, associated with transitions (2), (4), (3), and (5), respectively, correspond to information coming from the process. The event (for example, ) or condition (for example, ) associated with the transitions are called receptivities. The actions, and associated with some steps of the grafcet correspond to the outputs of the controller [ in Fig. 7]. In this example, all the actions are level actions, i.e., Boolean variables. However, 3 We shall write Grafcet (with a capital G) when speaking of the tool in general, and grafcet (with a small g) when referring to a particular logic controller model.

6 180 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH 1999 one can also have impulse actions corresponding to events. In that case, the impulse action associated with a step is executed exactly when step passes from the state inactive to the state active if it is an unconditional action (the case where it is conditional will be illustrated in Section VII). C. Obtaining a Grafcet Such That All the Inputs and Outputs Are Events As an example, let us show in Fig. 9 that all the receptivities and all the actions may be events. 1) Event Instead of Conditions for Receptivities: When Step 2 becomes active [at firing of (1)], tank 1 is empty because Step 1 is reached when this tank is empty. It follows that at this time. Now the time when the receptivity, associated with transition (2), will take the value one is exactly the time when event will occur. Hence, the receptivity (condition) may be replaced by (event). This reasoning is the same for the receptivities associated with (3), (4), and (5). 2) Impulse Actions (Events) Instead of Level Actions (Boolean Variables): The level action associated with Step 2 is such as exactly when Step 2 is active. This means that occurs when Step 2 becomes active and that occurs when Step 2 becomes inactive, i.e., when Step 3 becomes active. Then the level action in Fig. 9 is replaced by two impulse actions in Fig. 9: is associated with Step 2 and associated with Step 3. A similar transformation can be performed for the level actions, and. Remark 3: In the example of Fig. 9, a grafcet has been directly obtained where all inputs and outputs are events. However, this is not always the case and the above transformation does not provide a general algorithm. We only emphasize the fact that, if the initial states of the Boolean variables in input of the logic controller are known, then a grafcet can always be obtained where all inputs and outputs are events. In general, it may be necessary to go back to the specifications in order to design such a grafcet. VI. MOVING FROM THE GRAFCET TO THE AUTOMATON MODEL Grafcet is a powerful description tool that allows the modeling of large sequential machines. This tool will thus be advantageously used for supervised control design. However, the proof of controllability will be based on automata models. This section provides algorithms for obtaining automata models from the grafcet of control and from the grafcet of supervision (that embody the specifications of control and the specifications of supervision, respectively). The supervised control design using Grafcet will be illustrated on our manufacturing system example of Fig. 2. A. Specifications of Control Let us consider the following specifications of control for the manufacturing system (Fig. 2). Specifications of Control: We impose each machine to work as follows: in its initial state, is idle. must start working immediately (this is assumed to occur simultaneously Fig. 10. Specifications of control for the manufacturing system. Grafcet for M1. Grafcet for M2. with the pick-up of a part upstream). When the end of working occurs then the transfer of the part downstream must start. A new cycle may begin as soon as the transfer is finished. Within these specifications, the behavior of the machines is similar to those in Fig. 3. These specifications of control can be modeled by the grafcet in Fig. 10 where, and have the same meaning as in Fig. 3. Let us consider the grafcet of control for in Fig. 10: the steps 1 ( being working) and 2 (the transfer being processed downstream ) correspond to the states of. At initial state, step 1 is activated and the impulse action is executed. As soon as event occurs, is executed. Then the occurrence of leads the grafcet of Fig. 10 in its initial state. One can observe in Fig. 10 that all the inputs and all the outputs of the logic controller are events. According to Fig. 7(c), the set of events produced by the actual process and by the logic controller are and, respectively. From the grafcet of Fig. 10, an automaton model of the logic controller can be obtained. This is illustrated in Fig. 11. When the system is started up, the steps 1 and 11 in Fig. 10 become active (i.e., state ) and the impulse actions and are immediately executed. Let us denote the event that is the simultaneous occurrence of and. So, an entering arc labeled by is associated with the initial state in Fig. 11. (Let us recall that the label of an arc between two states is not necessarily the cause of the transition: it corresponds to the event which is simultaneous with the transition. Similarly, the arc labeled means that the event occurs at initialization). Hypothesis 1: Two external events (i.e., events of ) cannot occur simultaneously. Hypothesis 1 means that we consider continuous-time discrete-event processes: in our example, and cannot be simultaneous from state. Let us assume that occurs first, then as soon as occurs, step 2 of Fig. 10 is activated (i.e., state ) and the impulse action is immediately executed. For an external observer, events and occur simultaneously. This is represented by the event ( denotes the event that is the simultaneous occurrence of and ) associated with the transition in Fig. 11, and so on. We obtain the automaton in Fig. 11 where the meaning of the events is represented. Remark 4: If a controller is composed of several grafcets, one could consider obtaining an automaton model of the controller from the automata models of the different grafcets.

7 CHARBONNIER et al.: SUPERVISED CONTROL 181 (c) Fig. 11. Automaton of the extended process, obtained from Fig. 11. Global model. Model for M1. (c) Automaton model for M1, similar to those of Fig. 3. If the alphabet of events of the different grafcets are pair-wise disjoint, this can be achieved by composing their corresponding automata (this will be discussed in Section VII-B). In our example, we should compose the automata models obtained from Fig. 10 and. The automaton model for, obtained from Fig. 10, is shown in Fig. 11. In this model, the states and correspond to the Steps 1 and 2 of the grafcet [Fig. 10]. Let us recall that as soon as Step 1 of Fig. 10 becomes active (i.e., the machine becomes idle), is executed (i.e., the work cycle starts). Thus, stays a zero time in the idle state. Comparing with the automaton model of in Fig. 3, one can note that the states idle and working are merged in the grafcet of Fig. 10. This is illustrated in Fig. 11(c). The merging of these two states corresponds to state in Fig. 11 (similarly, the merging of the states busy and transport corresponds to state because, when the machine is isolated, a finished part is immediately transported). The simultaneity of some inputs with some outputs is only due to Grafcet. So, regardless of the concurrency, Grafcet results in models [e.g., Fig. 10 and its corresponding automaton model of Fig. 11] that are more concise than those that would be given in the RW theory. Remark 5: The nonblocking issue can be addressed by specifying a set of marked steps in the grafcet model. In the corresponding automaton model, a state could be marked if and only if each step that is active in is marked. Note that this is consistent with the definition of a marked state using the synchronous product in the RW approach [16]. In our example, assume Steps 1 and 11 in Fig. 10 are marked, then state in Fig. 11 should be a marked state. Note that the grafcet in Fig. 10 evolves when an event of (i.e., input of the logic controller) occurs. Assume that the event of occurs: the transitions associated with and for which the steps upstream are active, are fired. This is a basic rule of Grafcet (a Grafcet interpretation algorithm is provided in [5]). The automaton model of Fig. 11 can be directly obtained from the grafcet model of the controller of Fig. 10 by applying the following algorithm. Algorithm 1: From a grafcet to its corresponding automaton model. Step 1) Let be the initial state (i.e., the set of initial steps) of the grafcet. We initialize the automaton model with the state. The input arc of is labeled by the event that models the simultaneous occurrence of the impulse actions associated with the initial steps of the grafcet. We note as the list of states to be treated, and we set. Step 2) Consider a state of (i.e., the set of active steps in the grafcet). Define all the states that may be reached from. For every state : 1) if does not exist in the automaton model then add to the automaton model and add to and 2) add an arc associated with the event that corresponds to the changeover from to. Step 3) Remove from. If is not empty, return to Step 2. Note that no hypothesis is made on the structure of the grafcet (this one can explicitly model concurrency). However, we only focus on grafcets that have events as inputs and outputs. In that case, assume that an event of causes the changeover from state to state, then the transition has to be labeled by the event where is an impulse action associated with a step that is just activated from to (Step 2). Hence, an event of the automaton model is the simultaneity of one event of and a set of events of (except the event associated with the initial state that does not contain any events of ). In general, an action has a result that has to be observed by the logic controller. In our example, the occurrence of [associated with transition (1) in Fig. 10] has to be observed by the logic controller as a consequence of, in order to start the transfer (event ). In the grafcet of Fig. 10, the events and are, respectively, the consequences of the actions and associated with the preceding steps. Many control automata have the same feature. Then the following hypothesis will be considered for further results. Hypothesis 2: The event associated with a receptivity is the consequence of the execution of the actions that are associated with the steps upstream from the transition. Note that may be produced by the process if and only if the level action has been executed, i.e., when Step 1 of Fig. 10 is active. In fact, it results from Hypothesis 2 that an event can occur in the actual process if and only if the logic controller can take it into account (i.e., there is a transition of the controller that can be fired when this event occurs). Therefore, the logic controller contains sufficient information to describe the behavior of the extended process. It follows that the model of the logic controller of Fig. 11 is equivalent to the model of the extended process, i.e., and coupled with the logic controller of Fig. 10. Remark 6: Hypothesis 2 allows us to obtain the model of the extended process from knowledge of the control system, only. In practice, it seems that Hypothesis 2 is not restrictive.

8 182 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH 1999 Fig. 12. Specifications of supervision. However, if Hypothesis 2 does not hold, it is necessary to go back to the specifications to obtain a model of the extended process. B. Specifications of Supervision Grafcet can also be used for the specifications of supervision. In our example, we consider the specifications as described in specification 1 (Section II-C). These can be modeled by the grafcet in Fig. 12. This grafcet provides two Boolean outputs: and, i.e., forbidding of events and, respectively. In its initial state,, the level action one as long as Step 21 is active). This means that from state that corresponds to the buffer being empty, the start of working of (i.e., event ) is forbidden by the supervision. Whenever event occurs then state is reached and the buffer is full. In this state, means that the supervision forbids the start of transfer on. Remark 7: According to Fig. 7(c), the inputs of the grafcet of supervision correspond to the outputs of the extended process, i.e.,. In general, the outputs of the grafcet are Boolean variables, where meaning that the event is forbidden when. Let us consider the case where (this necessarily implies that ). In this case, although the supervision cannot forbid the occurrence of means that the occurrence of may not occur. Note that this is consistent with the definition of a specification (Section II-B). From the grafcet of Fig. 12, one can obtain the Moore machine for the specifications of supervision. This is shown in Fig. 4 where the states and correspond to the states and, respectively, of the grafcet in Fig. 12 (in general, we associate a state of the automaton with every state of the grafcet). From state, the occurrence of leads the grafcet of Fig. 12 in state. This is modeled in Fig. 4 by the transition labeled by. However, if an event other than occurs from state, the grafcet of Fig. 12 remains in the same state. Then the list is associated with the self loop of state in Fig. 4. Similarly, the transition is labeled by and the list is associated with the self loop of state. Furthermore, we associate the lists and with the states and of the automaton: for example, associated with Step 21 of Fig. 12 corresponds to in Fig. 4. Formally, the automaton of Fig. 4 can be obtained from the grafcet of Fig. 12 by applying the following algorithm. Algorithm 2: From a grafcet of supervision to its corresponding Moore machine. Step 1) Define the automaton model of the grafcet, applying Algorithm 1. Step 2) Consider all the states of the automaton. Add to every state, a self loop associated with all the events such that there is not an output transition labeled by from. Step 3) Add to every state the list of forbidden events, where if the level action is associated with a step of the grafcet that is active in state. The first step allows us to obtain the reachable states of the grafcet. Let us note that none of the outputs of the grafcet of supervision can be an impulse action. Therefore, a transition of the automaton obtained in Step 1 by Algorithm 1 is labeled by an event of (in our example the events and are associated with the transitions and ). Step 2 means that we add to the self loop of every state all the events which have no influence in input of the supervisor. Finally, an output list of forbidden events is associated with every state (this is performed in Step 3). VII. SUPERVISED CONTROL In our example, the specifications of supervision can be enforced by coupling the grafcet of Fig. 12 with the extended process (Fig. 10). Let us recall that this grafcet of supervision has to forbid the occurrence of the potentially controllable events and. Hence, these events have to be controllable and we set: and. Remark 8: Although and are potentially controllable events, we never need to forbid them by the supervision. Then we let them in the set of uncontrolled events. (However, one could consider all the events of as being controllable events.) If the routing has to be inverted, i.e., a part must visit then, then and would be controllable events. One has to modify the controller in order to be able to forbid some of its outputs (in fact, one cannot forbid the occurrence of or in Fig. 10). Within the Grafcet tool, one may condition the actions corresponding to the controllable events (this will be discussed in Section VII-A). We will refer to such a system as the extended process under supervision. The proof of controllability will be supported by automata models (Section VII-C). Hence, one must first obtain an automaton model of the extended process under supervision (Section VII-B). A. The Extended Process Under Supervision In our example, the grafcet model of the extended process under supervision, obtained from Fig. 10, is shown in Fig. 13. Let us consider the specifications of control for [Fig. 13]. The impulse action, associated with step 2 [Fig. 10], has been replaced by the conditional impulse action 4 [ if ] in Fig. 13 (where is the Boolean variable which is the complement of ). Two cases are possible and illustrated in Fig. 14. Let us denote as the Boolean value that is equal to one exactly when Step 2 of Fig. 13 is active. In Fig. 14, when the value of changes from zero to one, i.e., Step 2 becomes active, the 4 For clarity, a conditional impulse action, e.g., t1 if F 0 t1, is written in brackets.

9 CHARBONNIER et al.: SUPERVISED CONTROL 183 Fig. 13. for M2. Extended process under supervision. Grafcet for M1. Grafcet Fig. 14. Execution of a conditional impulse action. Fig. 16. Models of the extended process under supervision, obtained from Fig. 15. Model for M1. Model for M2. Fig. 15. Two grafcets that are equivalent to those of Fig. 13. value of is equal to zero (i.e., ), and the impulse action is executed. The case where when Step 2 becomes active is illustrated in Fig. 14: will be executed exactly the next time takes the value zero. It follows that, if the condition is true, a conditional impulse action associated with a step is executed as soon as step becomes active, but if the condition is not true whenever step becomes active, the action is executed on the first rising edge of the condition. Similarly for, one replaces the impulse action associated with Step 11 in Fig. 10 by the conditional impulse action [ if ] in Fig. 13. Formally, a grafcet model of the extended process under supervision can simply be obtained from the extended process by replacing every impulse action, with, by the conditional impulse action: [ if ]. B. Obtaining the Automaton of the Extended Process Under Supervision One can obtain an automaton model of the extended process under supervision by transforming its grafcet model. In our example, the grafcet in Fig. 13 can be transformed according to the grafcet in Fig. 15. This actually corresponds to splitting the two states represented as busy and transport in Fig. 11(c), since the machine is no longer isolated. This is based on the following rule. Rule of Construction 1: For every step associated with a conditional impulse action (e.g., [ if ] associated with Step 2 in Fig. 13: 1) we add a transition [i.e., in our case in Fig. 15] downstream from step, associated with the condition of the action (i.e., ); 2) we add a Step (i.e., step ) downstream from the transition ; and 3) we remove the conditional impulse action from Step (i.e., [ if ] at Step 2) and we add its corresponding unconditioned impulse action at Step (i.e., at Step ). The above rule stands only for grafcets that have at most one conditional impulse action associated with a step. However, this rule can easily be generalized [4]. Assuming Hypothesis 2 holds we then have the following property [4]. Property 1: Let be a grafcet obtained from a grafcet by rule of construction 1. Then has the same input/output behavior as. It follows that the grafcet of Fig. 15 is equivalent from the control and supervision viewpoint to the grafcet in Fig. 13. The automata obtained from the grafcets of Fig. 15 and are shown in Fig. 16 and, respectively. Let us consider [Fig. 16]: at initial time, Step 1 of Fig. 15 is activated and is immediately executed. This is modeled in Fig. 16 by the entering arc labeled by the event associated with state. Whenever occurs, Step 2 of Fig. 15 is activated. At this time, if is equal to zero (i.e., ), Step 2 is immediately deactivated, step is activated, and is executed (the state where Step 2 is active is said to be unstable). For an external observer, the events and occur simultaneously. This is modeled in Fig. 16 by the transition labeled by the event ( represents the simultaneous occurrence of and whenever ). From a theoretical viewpoint, this is consistent since the product of an event by a Boolean variable is an event [5], [6]. On the contrary, if whenever event occurs, the transition may not be fired (the state where Step 2 is active and is said to be stable). The event that leads from state to state in Fig. 16, models the occurrence of whenever, and so on.

10 184 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH 1999 Fig. 18. Automaton model of the specifications of supervision, obtained from Fig. 4. Fig. 19. Synchronous product of Fig. 17 and Fig. 18. Fig. 17. Global model of the extended process under supervision, obtained from Fig. 16. An automaton model. The corresponding acceptor. Similarly, one obtains the automaton for in Fig. 16 from the grafcet of Fig. 15. The models of Fig. 16 are constructed on the set of events where each event in is the simultaneous occurrence of some events of (possibly conditioned by some Boolean variables, with ). The events and keep the same meaning as those in Fig. 11. However, the events and that contain the controllable events and, respectively, in Fig. 11, are replaced by and by in Fig. 16. Remark 9: The automata of Fig. 16 can be obtained from the grafcets of Fig. 15 by applying algorithm 1 (this consists in searching for all the stable states that may be reached in the grafcet, regarding the values that can take the Boolean variables and ) [4]. One can obtain a global model of the extended process under supervision. This can be done simply by computing the synchronous product [16] of the automata of Fig. 16 and. We obtain the automaton of Fig. 17 where each state corresponds to a state of Fig. 16 (i.e., state of ) and to a state of Fig. 16 (i.e., state of ). Moreover, we label the entering arc of the initial state (i.e., ) by the event that corresponds to the simultaneous occurrence of the events associated with the entering arc of the initial states in Fig. 16 and in. Thus, is associated with the entering arc of in Fig. 17. This means that the occurrence of occurs at initial time. Therefore, the occurrence of precedes the occurrence of any other events and one can easily obtain an acceptor model of the extended process under supervision. In our example, the acceptor of Fig. 17 is obtained from the automaton of Fig. 17 by the following transformation: 1) add a new state that becomes the initial state in Fig. 17; 2) add an arc from state to the state that corresponds to the initial state in Fig. 17, i.e., ; and 3) label this arc by the event associated with the entering arrow of the initial state in Fig. 17, i.e., event. Note that the events and (Fig. 16) contain the controllable event and are conditioned with the Boolean variable. This implies that it is possible to preempt at any time the occurrence of and. In fact, if is equal to 1 (i.e., is forbidden), and cannot occur. Corollary, an event of that does not contain any controllable event of is uncontrollable. Then the following property holds. Property 2: Any event in that contain a controllable event of (i.e., that is conditioned by any Boolean variable ) is controllable. In our example, and are the sets of controllable and uncontrollable events over, respectively. C. Proof of Controllability and Implementation of Supervised Control In order to prove the controllability of the specifications of supervision of Fig. 4, one must obtain its corresponding

11 CHARBONNIER et al.: SUPERVISED CONTROL 185 Fig. 20. Supervised control schema of the manufacturing system example. automaton model constructed on the set of events. This is shown in Fig. 18: the transition labeled by in Fig. 4 is labeled, in Fig. 18, by every event in that contains, i.e.,. For every other event than the automaton of Fig. 18 remains in state. So, the list of events is first associated with the self loop of state. The controllable event is forbidden in state, then all the events of that are conditioned by, i.e., and, are forbidden in this state. The events and are then deleted from the list previously associated with the self loop of state, i.e.,. Finally, we obtain the list as shown in Fig. 18. Similarly, one obtains the set of events associated with the self loop of state. Formally, the acceptor of Fig. 18 can be obtained from Fig. 4 by applying the following rule. Rule of Construction 2: Obtaining of an acceptor of the specification of supervision (defined on the set of events ) from a Moore machine (defined on the set of events ). Step 1) Associate a state of with every state of. Step 2) Associate a transition of (where and are distinct states) with every transition of. Label the transition in by all the events in that contain an event of associated with of. Step 3) Associate with the self loop of every state in, every event of that is not associated with an output transition from state in. Step 4) Remove from the list of events associated with the self loop of every state in, every event of that is conditioned by if belongs to the list of forbidden associated with state in. Step 5) Remove from the list of events associated with the self loop of every state in, every event of that contains the uncontrollable event (i.e., ) such that belongs to the list of forbidden events associated with state in. Steps 4 and 5 mean that if an event is forbidden from a state in, every event of that contain may not occur from in. One can obtain the model of the desired behavior of the manufacturing system. This is illustrated by the automaton in Fig. 19, obtained from the synchronous product of the automata in Figs. 17 and 18. Let us note that the 15-state automaton in Fig. 19 corresponds to the 28-state automaton in Fig. 6 obtained from the RW theory, in which some of the states have been merged (this comes from the simultaneity of some events). From the models of Figs. 17 and 19, one now can prove the controllability of the specifications of supervision (Fig. 12) with respect to the extended process. This can be done using the techniques described in [12] and [17]. This ensures that the closed-loop system respects the behavioral constraints imposed by the specifications of supervision. As illustrated in Fig. 20, the supervised control of our manufacturing system can simply be achieved by jointly implementing the grafcet of control in Fig. 13 and the grafcet of supervision in Fig. 12. Remark 10: 1) Note that given a set of marked states in the automaton model of the extended process, it is also possible to check for the nonblocking property. This property can be proved using the techniques in [14] and [15], thus guaranteeing that for any nonmarked state that can be reached by the extended process coupled with the supervisor, there is a nonforbidden sequence of events that leads the process in any marked state. In our example, assume Steps 1 and 11 in Fig. 13 are marked

12 186 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 2, MARCH 1999 (i.e., state in Fig. 17 is a marked state), then the closed loop behavior of the manufacturing system can be proved to be nonblocking. 2) In the model in Fig. 19, the specifications of control and the specifications of supervision are mixed. Regardless of the simultaneity of events, the enforcement of the control for the manufacturing system using the techniques as described in [1] would oblige us to implement the model of Fig. 19. This would result in a loss of conciseness. Moreover, if the capacity of the buffer is to increase, e.g., let be the size of the buffer, then the automaton of the specifications of supervision would comprise states. Also, the automaton model of the supervisor to implement would be (at most) in, where is the size of the extended process. However, using the supervised control approach, the specifications of control would remain the same and, the grafcet of supervision to be implemented would only comprise steps. 3) In this paper, the proof of controllability requires the changeover to automata models. One of our objectives is to be able to make proofs and to synthetize supervisors directly from the grafcet models in order to avoid the state explosion we are faced within automata models. This would enlarge the scope of application of the approach. VIII. CONCLUSION The small number of supervisory control examples found in the literature highlights the difficulty of implementing a supervisory control system. In fact, the extension of the basic theory in which a supervisor can force some events [1], [8] often leads to models that are too complex to be implemented. Moreover, the synthesis techniques give back supervisory automata that may not be directly implemented in a PLC. It follows that systematization of the supervisory control implementation needs to be supported by formal tools. This has led us to define the supervised control concept. As illustrated in the manufacturing system example, separation of control and the supervision enables us to increase the conciseness of our control problem. The use of Grafcet makes the implementation of a supervised control system easier. Moreover, the Grafcet and the provided algorithms allow us to ensure that the proof of controllability also holds in the implemented models. In our example, note that from the behavioral specifications, one can obtain the controller in Fig. 13 (given the controller in Fig. 10) and the grafcet of supervision in Fig. 12. Figs. 11 and and the corresponding comments have been presented to show the consistency of the approach with the RW theory and with the Grafcet concepts and to prove controllability. The supervised control concept provides a good framework for further research. ACKNOWLEDGMENT The authors would like to thank Technology Ontario and Region Rhône-Alpes for their support of the Ontario/Rhône- Alpes Collaborative Project on Automated Manufacturing, as well as B. A. Brandin and W. M. Wonham for helpful discussions. REFERENCES [1] S. Balemi, G. J. Hoffman, P. Gyugyi, H. Wong-Toi, and G. F. Franklin, Supervisory control of a rapid thermal multiprocessor, IEEE Trans. Automat. Contr., vol. 38, July [2] B. A. Brandin and F. Charbonnier, The supervisory control of the automated manufacturing system of the AIP, in th Int. Conf. Integrated Manufacturing Automat. Technol., Oct. 1994, pp [3] F. Charbonnier, H. Alla, and R. David, The supervised control of discrete-event dynamic systems: A new approach, in 34th Conf. Decision Contr., New Orleans, Dec [4] F. Charbonnier, Commande supervisée des systèmes à evénements discrets, doctoral dissertation, Institut National Polytechnique de Grenoble, Grenoble, France, Jan [5] R. David and H. Alla, Petri Nets and Grafcet: Tools for Modeling Discrete-Event Systems. Englewood Cliffs, NJ: Prentice-Hall, [6] R. David, Grafcet: A powerful tool for specification of logic controllers, IEEE Trans. Contr. Syst. Technol., vol. 3, pp , Sept [7], Petri nets and Grafcet for specification of logic controllers, in IFAC Congr., Sydney, Australia, July 1993, vol. 3, pp [8] C. H. Golaszewski and P. J. Ramadge, Control of discrete-event processes with forced events, in 28th Conf. Decision Contr., Los Angeles, Dec. 1987, pp [9] J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation. Reading, MA: Addison-Wesley, [10] IEC, International Electrotechnic Commission, Preparation of function chart for control systems, publication 848, [11] B. H. Krogh and L. E. Holloway, Synthesis of feedback control logic for discrete manufacturing systems, Automatica, pp , July [12] R. Kumar, Supervisory synthesis techniques for discrete-event dynamical systems, Ph.D. dissertation, Univ. Texas, Austin, Aug [13] F. Lin and W. M. Wonham, Decentralized control and coordination of discrete-event systems with partial observation, IEEE Trans. Automat. Contr., vol. 35, pp , Dec [14] C. A. Petri, Kommunikation mit Automaten, Schriften des Rheinisch, Westfalischen Institutes für Intrumentelle Mathematik and der Universität Bonn, 1962, translation by C. F. Greene, Applied Data Research Inc., New York, Suppl.1 to Tech. Rep. RADC-TR , [15] J. G. Ramadge and W. M. Wonham, Supervisory control of a class of discrete-event processes, SIAM J., Contr. Optimization, vol. 25, pp , Jan [16], The control of discrete-event systems, Proc. IEEE, vol. 77, no. 1, pp , Jan [17] W. M. Wonham and J. G. Ramadge, On the supremal controllable sublanguage of a given language, Siam J. Contr. Optimization, vol. 25, pp , May [18], Modular supervisory control of discrete-event systems, Math. Contr., Signals, Syst., vol. 1, no. 1, pp , Jan [19] H. Zhong and W. M. Wonham, On consistency of hierarchical supervision of discrete-event systems, IEEE Trans. Automat. Contr., vol. 35, pp , François Charbonnier was born in Caen, France, in He received the engineering degree in 1993 from the Institut des Sciences et Techniques de Grenoble, and the Doctorat degree of automatic control from the Institut National Polytechnique de Grenoble in His research interests include the control of discrete-event systems and its application to manufacturing systems.

13 CHARBONNIER et al.: SUPERVISED CONTROL 187 Hassane Alla was born in Ain-Sefra, Algeria, in He received the diploma of automatic control engineering from the Ecole Nationale Supérieure d Ingénieurs Electriciens de Grenoble, France, and the Doctorat d Etat ès-sciences degree, in 1987, from the Institut National Polytechnique de Grenoble. He is currently a Professor at the University of Grenoble and Deputy Director of the Laboratoire d Automatique de Grenoble (in Institut National Polytechnique de Grenoble). His research interests include the analysis and control of discrete-event systems. The main used tools are Petri nets (ordinary, continuous, and hybrid Petri nets) and their application to the manufacturing systems. René David was born in Saint-Nazaire, France, in He received the engineering degree in 1962 from the Ecole Nationale Supérieure d Arts et Métiers. In 1963, an accident stopped his work and left him paraplegic. In 1965, he received the automatic control engineering degree from the University of Grenoble, and in 1969, he received the Doctorat d Etat és-sciences degree from the same university. He is now Directeur de Recherche at the Centre National de la Recherche Scientifique, working at the Laboratoire d Automatique de Grenoble (in Institut National Polytechnique de Grenoble). He was Deputy Director of this laboratory from 1987 to His works cover design of logic controllers, testing of digital circuits, modeling and performance evaluation of manufacturing systems. He is one of the creators of Grafcet and the creator (together with H. Alla) of continuous Petri nets. He is the author of Random Testing of Digital Circuits: Theory and Application (New York: Marcel Dekker, 1998). He is also coauthor of Du Grafcet aux Réseaux de Petri (Paris, France: Hermès, 1989), and Petri Nets and Grafcet: Tools for Modeling Discrete-Event Systems (Englewood Cliffs, NJ: Prentice-Hall, 1992). This book was also published in Chinese (Beijing: Machinery Press, 1996).

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