Keywords
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| Automata, Timed automata, discrete event systems, Petri nets, Timed Petri nets, KRC systems |
INTRODUCTION
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| The process synchronization and forbidden states avoidance are considered as the main problems in both normal and abnormal modes of automation systems [1]. These characteristics allow the system to be considered as Discrete Event System (DES) and allow the researchers to perform the analysis and control of such systems using Petri nets (PN) and automata. DES is a dynamic system with state evolution produced by the occurrence of physical events. For example, an event is opening or closing a gate in Kernel Railroad Crossing (KRC) systems [2-3]. DES can be found in domains such as manufacturing, robotic, traffic control, logistics, and communication systems, etc. |
| Important contributions in supervisory control of DES based on the Finite Automata (FA) and Petri Nets (PNs) are found in [4-8]. Petri net models are normally more compact than similar automata based models and are better suited for the representation of discrete event systems. They also have a good representational power [9]. On one hand, numerous approaches to the systematic construction of Petri net models have been proposed in [10]. On the other hand, because automata detail the process to be modelled, they are also employed to analysis and avoid forbidden states with ease. |
| Historically, the theory of DES for supervision was initiated by the research work of Ramadge and Wonham [8]. A process is considered to be a spontaneous generator of events. Its functioning is characterized by event sequences. A supervisor is a DES which can modify the functioning of the process in order to avoid forbidden states. Unfortunately, the theory of Ramadge and Wonham does not a maximally permissive in nature [11]. A maximal permissive supervisor is to ensure that the controlled behaviour of the DES must be maximally permissive within a given specifications, i.e. all events which do not contradict the given specifications are allowed to happen. Thus, the theory of Ramadge and Wonham cannot be used to control all DES. An extension of this theory to the control of DES is proposed to force the occurrence of certain events to follow a desired trajectory. This approach does not take into account explicitly the influence of time. However, an industrial process is subject to time constraints such as the beginning instant or the duration of tasks. This information can be used to compute more permissive control laws. Timing introduces a new dimension in DES modelling and control of considerable applied interest, but also of significant complexity. A first attempt to take into account the influence of time is presented in [12]. The authors extend the theory of Ramadge and Wonham by introducing forcible events and time constraints. The elapsing of time is modelled by the occurrence of a particular event. This approach provides good solutions for the control of time discrete event systems (TDES). However, the discrete nature of time generates the exponential explosion of the number of states. |
| particular event. This approach provides good solutions for the control of time discrete event systems (TDES). However, the discrete nature of time generates the exponential explosion of the number of states. |
| This paper deals with analysis and control of TDES whose behaviour is determined by the occurrence of events at moments specified by time intervals. First, it adopted the timed Petri net model (TPN) [14]. This tool inherits the modelling capacity specific to Petri nets based models. Moreover, it has the ability to represent time constraints specified by time intervals. In spite of their modelling capacity, only few properties can be determined using exclusively a TPN. The method currently used for analysing TPN properties is based on modelling its behaviour by a state class graph [14]. This tool provides an over-approximation of the TPN behaviour and it is appropriate for verification of safety properties. However, it is not adapted for control synthesis because it may represent forbidden behaviours that do not exist in the real system. |
| In [15], a method for building systematically the timed automaton (TA ) that models the exact behaviour of a TPN is introduced. This paper borrows this method to build a TA for KRC system. This method provides a clear graphical representation of all TPN possible forbidden behaviours and considers the forbidden states correspond to forbidden TPNs markings that are represented by forbidden TA locations. |
| This paper also focuses on the control synthesis of TDES modelled by TPNs that developed based on place invariants (P-invariant) [9]. The method that we propose is based on the TA with forbidden locations. For each forbidden location, the corresponding reachability path that acts on the transitions associated with controllable events is analysed. This paper also tries to define the exact time unit (t.u.) period of transitions that have undefined t.u [0, ∞]. |
| The KRC is a standard benchmark in real time systems verification and operates to ensuring safety and maximizing utility (the gate should be open as long as possible) [2]. Modelling and control of KRC systems using timed Petri nets and focusing on the control synthesis method, which consists in computing new firing conditions for the timed automaton transitions so that the forbidden locations are no longer reachable, is the main contribution of this paper. |
| This contribution can be achieved as follows. |
| 1- Modeling and supervising the KRC system using timed Petri nets (TPN) |
| 2- Building the TA associated to the developed TPN |
| 3- Analysis the TA using phaver-software |
| 4- Control synthesis of the KRC system ensures safety and maximizes utility properties of the KRC |
| The paper is organized as follows. In Section II, the essential background used in this paper to TPN and timed automata is presented. In Section III, modelling and supervision of the KRC system is detailed. The main idea of building the TA associated to a TPN is described in section IV. Section V creates the TA associated with the developed TPN of the KRC system and proposes its control synthesis. Using phaver-software, simulation results are depicted in section VI. Some concluding remarks and future work are given in section 7. |
ESSENTIAL BACKGROUND TO TPN AND AUTOMATA
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| Petri nets are classified based on their structural and behavioural properties [1]. Basically, a Petri net is a particular kind of bipartite directed graphs characterized by three types of objects, places, transitions, and directed arcs. The latter connects places to transitions or transitions to places. This elementary Petri net may be employed to represent various aspects of the system to be modelled. In order to study the dynamic behaviour of a system structured with a Petri net in terms of its states and state changes, each place may potentially hold with positive number of tokens. A set of definitions related to Petri nets used in this paper can be briefly discussed below [16-19]. |
| Definition 1: A Petri net structure is a weighted bipartite graph; G =(P,T , F ,w), where P={ n P p , p ,..., p 1 2 } is a finite set of places, P={ n T t , t ,..., t 1 2 } is a finite set of transitions, F (PxT) TxP is a set of arcs from places to transitions and from transitions to places in the structure, and : I O 1,2,3, ,... is the weight function on the arcs. |
| The way in which tokens are assigned to a Petri net graph defines a marking vector, M, which can be formally defined i n M m p ,...m p ,..., m p 1 } where n, is the number of places in the Petri net. The state transition mechanism is provided by moving tokens through the Petri net and hence changing its state. |
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| Definition 3: Given a Petri net with the initial marking Mo; n=(G,M0) , the set of reachable states of this Petri net is called the reachability set and is signified by R=(N M0) . |
| Definition 4: A Petri net is said to be live if it guarantees no stop operations (deadlock-freeness in general), regardless what firing sequence is chosen. Deadlock-freeness may not necessarily be completely live. A deadlocked Petri net contains at least one empty siphon. A net with an unmarked siphon is not live. For more details the reader can be directed to read [1]. |
| In contrast to the behavioural properties defined above that depend on the initial marking vector of a Petri net, structure properties do not depend on the initial marking vector of the net. Based on the structural properties, Petri nets are classified also into many classes. These important classes are described below. |
| Definition 5: A state machine (finite state automata) is a Petri net in which all transitions have one input and one output place. |
| The timed automaton (TA) model is defined as a finite state machine extended with a set of continuous variables called clocks [14]. These variables are piece-wise continuous real-valued functions of time that precisely measure the time elapsed between events. |
| Definition 6: A timed automaton is 6-tuple TA= 0 L L X I T , where: |
| - L is the finite set of locations; |
| L0εL is the initial location; |
| - X is the finite set of clocks; |
| -Σ is a set of labels associated with the events; |
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| within a TA location is .xas will be shown in section 4. Each TA location is characterized by a condition on the value of clocks, called invariant of the location. A system may sojourn in a location as long as the value of clocks satisfies the invariant of the location. Firing a TA transition takes no time to complete. The occurrence of this event may cause the initialization of certain clocks. This discrete evolution of a clock x is modelled by the relation x = 0 . The set of clocks initialized by a transition firing is described by an assignment. Each transition is characterized by a condition on the value of clocks called guard or firing condition. This condition describes the possible dates for firing the corresponding transition. A TA transition may be fired if the value of clocks satisfies its guard. |
| Definition 7: Let X be the finite set of clocks. A clock valuation is a mapping v : X →R+ that assigns a non-negative real number to each clock. |
| A clock valuation defines the value of clocks at a given moment. V denotes the set of clock valuations. Let v εV be a clock valuation, t εR+a positive rational number and m n A , an assignment. We use the notation: |
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| Definition 8: The state of a TA is defined by the pair (L, v), where L ε L 0 is a TA location and v εV is a clock valuation. |
| A system may sojourn in a TA location n L as long as the locations invariant n L is satisfied by the value of clocks. Consequently, a location may not correspond to a state of the TA, but to a space of states. A space of clock values Q within a location n L is called region and it is denoted by the couple (Q, n L ). |
| The main target of the control synthesis is to guarantee that the forbidden states or the forbidden locations, are not reachable during the evolution of the TA. Therefore, we are interested in computing: (1) the set of states reachable form a given region, called the successor of the region; and (2) the set of states from which a given region may be reached, called the predecessor of the region. |
| The evolution of a TA is characterized both by a continuous and a discrete dynamic. The continuous dynamic is represented by the evolution of clocks with the elapsing of time while resting in the same location. The discrete dynamic consists in transitions firing. Therefore, a region may have two kinds of successors (predecessors): continuous and discrete ones [15]. |
| The method for computing the successors and the predecessors of a region are described in [20]. This paper uses these methods in our control synthesis approach. Although, these procedures are implemented in a software dedicated to verification of timed and hybrid systems, this paper employs phaver-software due to its power in analysing and verifying the TA model. For more details, the reader can be directed to read [15]. |
| The method for computing the successors and the predecessors of a region are described in [20]. This paper uses these methods in our control synthesis approach. Although, these procedures are implemented in a software dedicated to verification of timed and hybrid systems, this paper employs phaver-software due to its power in analysing and verifying the TA model. For more details, the reader can be directed to read [15]. |
MODELLING AND SUPERVISION KRC SYSTEMS USING TPN
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| This section develops the TPN model of the KRC and its supervisor that is synthesized based on P-invariant method [9], [21]. |
| III.1. The KRC system description and its Petri nets |
| The kernel railroad crossing (KRC) is a standard benchmark in real time systems [1-2] and considers a railway crossing composed of sensors, gates and a supervisor. When a train is sensed to approach the crossing, a signal is sent to the supervisor that sends a command to the specified gate that is closed to prevent cars crossing that survives our ourselves. The KCR problem is a simplified version, where there is only one track and hence only one train at a time may enter the crossing zone. Having more than one track and more than one train may enter crossing zone, leads to a complicated situation that is out of our paper scope. The KRC system is depicted in Fig. 1. For simplicity let us merge the two depicted zones as one zone (region). The Petri net of the system is depicted in Fig. 2 and its places/transitions are described in Table 1. In Fig. 3, the train needs one time unit (t.u.) to enter the R-Y zoon and five (t.u.) to leave it for departure phase. The gate needs no time to start closing and requires two (t.u.) to be completely closed. It needs extra two (t.u.) to be completely opened after firing the transition T5. The problem arises at the beginning of opening the gate that has unknown time units "[0, ∞]". This is due to the reason that no expectation for beginning the departure phase of the train; its departure depends on the passenger riding. The analysis should be performed to get the exact period time unit required to activate the transition T5 every departure phase. This means that this transition is continuously evaluated as detailed in the subsequent sections. The system comprises two tasks, train task and gate task. We consider that the controllable events are the beginning of each task. However, the accomplishing of the tasks is uncontrollable. Therefore, our goal is to control the beginning of the tasks in order to obtain safe arrival and departure of the train. Also, the synthesized control scheme should avoid the forbidden state (P2, P4). This means that the train in the R-Y zone and the gate still opened. |
| III.2. Analysis the PN of the KRC system |
| Using Petri net tool software ver. 2.1, the system incidence matrix is: |
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| Note that this Petri net tool is supported by MATLAB ver. 6 or higher. In this simulation, there are 11 reachable states starting from the initial marking vector M0 to the final vector M11. The firing sequence shows that the marking vector, M1 = [0, 1, 0, 1, 0, 0, 0] is a forbidden state. It is clear that the marking vector M1 includes the forbidden state (P2, P4). Based on P-invariant , a supervisory control scheme for the KRC system to avoid this state is synthesized in subsection 3.3. |
| III.3. Supervisory Control Synthesis for the KRC system |
| This subsubsection develops the supervisory control scheme that aims to restrict the reachable markings of a plant, p m such that: |
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| The slack variable c M in this case contains new places that hold the extra tokens required to meet the equality. The slack variable that enforces the equality defined in (2) is a part of a separate net called the supervisor (controller). If the initial marking does not violate the given set of constraints defined in (1), these constraints can be enforced by the supervisor with the incidence matrix [4]. |
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| The initial marking of the proposed supervisor is computed by: |
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| where p0 m is the n*1 initial plant marking vector of non-negative integers. The supervisor is a Petri net with incidence matrix c D made up of the process net's transitions and a separate set of supervisor places. The supervised net is also called the controlled system or the closed loop system and is defined as: |
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| Uncontrollable transitions cannot be prevented from firing by the supervisor. They can cause problems for the PN based supervisors due to limitations in their modeling power. For an invariant-based Petri net supervisor to be realizable on a plant with uncontrollable transitions, the constraints enforced by the supervisor must be admissible. Inadmissible constraints cannot be directly enforced on a plant due to the uncontrollability of certain plant transitions. Procedures are given for identifying all admissible linear constraints for a plant with uncontrollable transitions, and methods for transforming inadmissible constraints into admissible. |
| Let Duc be a matrix representing the columns in the plant incidence matrix that corresponds to uncontrollable transitions. The matrix LDuc must contain non-positive elements when constructing the supervisor. Thus for the constraints to be admissible, it must satisfy: |
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| If this is not the case, matrix L and vector B must be transformed so that (6) is to be satisfied, while the supervisor designed to enforce the new set of constraints that will also maintain the original set of constraints. This can be achieved by performing row operations on Duc and LDuc. Matrices R1 and R2 are used to transform the inadmissible constraints Lmp B ≤ into the admissible constraints [4]: |
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| where |
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| The developed supervised time Petri net of the KRC system is depicted in Fig. 4. There are 10 reachable states starting from the initial marking vector M0 to the final vector M10 and the supervisor eliminates the forbidden state vector M = [0,1,0,1,0,0,0]. |
| Eliminating the forbidden state is the first objective of this paper, however, determining the time units required for the transition T5 is also its second objective. So, the timed Petri net shown in Fig. 4 is converted to a timed automaton for analysing purposes. This analysis should eliminate the forbidden state (location) and determine the time unit of the transition T5. This matter is detailed in section 5. |
TPN AND TIMED AUTOMATA
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| This section describes the relation between TPN and TA. It also designs a TA for the KRC system and its control synthesis. |
| 1. TPN and TA relationship |
| In a TPN, the occurrence of an event is subject to time constraints similar to that of TA [22]. Thus, a transition may be fired if it is enabled by a marking vector and if some time constraints are satisfied [14], [23]. The marking vector remains unchanged between two consecutive transition firing. Thus, each marking vector of the TPN can be modelled by a TA location, while the firing of a TPN transition can be modelled by a TA transition. The time information memorized by a time Petri net can also be represented by elements of a timed automaton as follows. First, the time elapsed since a transition enabling moment is measured by a clock. Second, the time interval of a TPN transition is modelled by the guard of the corresponding TA transition. Third, the initialization of the clocks associated to the newly enabled transitions is modelled by the assignment of the corresponding TA transition. Thus, the behaviour of a TPN be can be modelled by a TA. |
| Structuring the TA associated with a TPN consists in building the set of reachable states. This set is composed of locations corresponding to the discrete marking vectors. In order to simplify the presentation of the proposed method, this paper considers that none of the TPN transitions may become enabled more than once simultaneously. Generally, the solution is to dynamically allocate with each transition a number of clocks equal to the number of simultaneous enabling |
| 2. Elements of a TA |
| The elements of the TA which models the behaviour of a TPN are described as follows: |
| 2.1. Locations |
| Each TPN marking vector i M is modelled by a unique TA location i L . |
| 2.2. Clocks |
| A clock j x is associated with each TPN transition j T . It serves to count the time elapsed since the last enabling moment of the transition j T and determines if the firing condition of this transition is satisfied or not. |
| 2.3. Transitions |
| Firing a TPN transition is modelled by a TA transition. The guard of the TA models the firing interval of the TPN transition while its assignment initializes the clocks associated with the newly enabled TPN transitions. |
| Constructing the TA associated with a TPN |
| The initial TPN marking vector is modelled by the TA initial location. The TA is structured by analysing the evolution of the TPN from its initial marking vector. Fig. 5 depicts the principle of the building method. |
| Computing a location invariant |
| The location invariant (L ) is stemmed from the time intervals of the transitions i T and j T , and is enabled by the marking vector n M . |
To clarify this point, if the firing interval of i T is [ai, bi], then i T cannot be continuously enabled more than bi t.u. Accordingly, the system may sojourn in the location n L until the clock i x reaches the value bi. For the same reason, the  |
| Calculating the space of clocks in a location |
| When a location n L is reached by firing a transition m n T , , there are a set of possible values for the clocks. These values define the space of clock values denoted by m n E , . The space of clocks within a location n L , denoted m,n Suc E is the continuous successor of the space of clocks at the access of Ln. Property 1: The space of clocks in a location n L represents the set of all possible evolution trajectories while the system sojourns in n L [15]. |
CREATING A TA FOR THE TPN
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| This section develops the TA associated with the developed TPN of the KRC system. It also creates a timed automaton transition and defines the space of clocks at the entrance of a location. |
| V.1. A timed automaton transition and the space of clocks |
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| V.2. The timed automaton of the KRC system |
| This subsection constructs the TA associated with the developed TPN. Fig. 6 models the exact behaviour of the TPN developed in this section 3 and depicted in Fig. 4. Our goal is to avoid the forbidden state (location 6; L6) and to define the proper guard 3, 6 g to define the right period of the transition T5 depicted in Fig. 4. |
| A clock i x is associated with each TPN transition i T . The initial TA location 0 L corresponds to the state where the train arrived at the first point of the R-Y zoon and the gate of the KRC system is completely opened. Let us suppose that the train in the R-Y zoon, P2, and the gate starts its opening task by firing the transition T5, the system reaches the forbidden location 6 L . These forbidden locations must be avoided. Avoiding these states solves two problems addressed by this paper, the forbidden states and infinity problem that is defined as " [0, ] 5 T ". It also achieves the safety and utility properties of the system. Therefore, the clocks of the transitions that link 3 L and 6 L should be determined and consequently the time units of the transition T5 can be determined. These problems will be solved in subsection 5.3. by synthesizing a controller for the KRC system. The TA built has the following properties: |
| Property 2: The TDES forbidden behaviors are modeled by forbidden locations. |
| Property 3: The guard gm n , of each transition Tm n , is described by an expression g m,n [ai≤xi≤bi] where xi is a clock and ai, bi are rational numbers. |
| Property 4: The TA constructed by this method is non blocking. |
| V.3. The control synthesis of the KRC system |
| In time discrete event systems (TDES), events are classed according to the possibility to control their occurrence dates into three kinds, controllable, uncontrollable and forcible [12], [15]. In this paper, the first two types are more adapted to TDES control. An event is said to be controllable if it is possible to fix its occurrence instant within a given time interval. In contrast to the controllable events, an event is considered to be uncontrollable if it is not possible to act on its arriving moment (sensors). The time interval associated with an uncontrollable event may be considered as an uncertainty on its arriving moment. The aim of the control synthesis is to compute new time constraints on the occurrence of controllable events so that the functioning of the KRC system respects the specifications. Synthesizing the controller for the KRC system is detailed below. |
| The TDES control synthesis method proposed is based on the timed automaton model respecting the properties presented in section 4. Each timed automaton transition models the occurrence of an event and its guard models the time constraint imposed on the occurrence of the associated event. The main idea of synthesizing the proposed controller in this paper is to modifying the time constraint associated with a controllable event. By this modification this event occurrence at certain time instants can be forced such that the functioning of the system follows the desired specifications. Also, the time units needed for the unknown time units transitions e.g. T5 and its clock x5 in our application can be determined. |
| In fact, a transition can be controllable or uncontrollable according to the nature of the associated event. The guards of the controllable transitions can be modified by the control synthesis while the guards of uncontrollable transitions follow Property 3 during the control synthesis. In the KRC system, a forbidden state is the consequence of firing the transition T5. In fact, reaching a forbidden behaviour by executing a controllable event corresponds to the situation where the operator has determined a fault. Excluding this situation, this paper considers that a forbidden location is always reached by firing a timed transition with unknown time units. |
| The control synthesis method proposed in this paper comprises two steps, analysing one by one each forbidden location, and computing new guards for the controllable transitions so that the forbidden location is not reachable. It is executed in two phases, forward treatments and backward treatment. The principle of this method using the part of a timed automaton depicted in Fig. 7. In this figure, the forbidden location q L can be reached from n L by firing the transition, n q T , . The main objective of the control synthesis method is to avoid the forbidden locations in general e.g. q L in this situation. Therefore, the system must leave the location n L by firing another output transition leading to the not forbidden location p L . So, this firing condition should be forced before the guard of the transition n q T , becomes enabled by the value of clocks. Another important pint is that if the transition n q T , to the forbidden location is uncontrollable, its guard cannot be modified and we must compute a new initial condition of the continuous dynamic of clocks in the location. |
| As mentioned above, the control synthesis is executed in two phases, forward treatments and backward treatment. The former concerns only the location n L . This subsection can be summarizes the scheme as follows. |
- The forward treatment phase
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| If a transition n p T , is controllable, compute a new guard d n p g , for the transition n p T , in order to force the firing of this transition before the guard of the transition n q T , becomes satisfied by the clocks and update the location invariant ( ) n L according to the new guard computed for the transition n p T , . |
| 1- Compute the space of desired clock values, d n D in the location n L . This step restrains the space of clocks values in the location n L so that the guard n q g , of n q T , is never verified during the sojourn of the system in this location. |
| 2- Compute the space d n E of the desired clock values, d n D at the entrance of the location n L . This new space symbol, d n E , is due to the fact that the space of desired clock values, d n D within a location is characterized by the continuous dynamic of clocks and the initial condition for this dynamic. The former is fixed. The only way to modify the space of clocks values in location n L is to act on the initial condition, i.e. the space n E of clock values at the entrance of n L . The space of desired clock values has three situations: |
| (a) If n d n E E then the space clock values within the location n L does not allow the KRC system to reach the forbidden location q L . Thus, the control synthesis problem for q L is solved. |
| (b) If d n E then the evolution from n L to the forbidden location q L cannot be avoided and the former becomes a forbidden location. |
| (c) If the conditions (a) and (b) cannot be satisfied i.e. n d n E E and d n E then go to the backward treatment phase detailed below. |
The backward treatment phase
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| The backward treatment consists in going back in the automaton structure and compute new guards for controllable transitions such that all possible clock values at the entrance of the location n L belong to d n E . The main idea of the backward treatment is shown in Fig. 8. It can be summarized as follows. |
| This phase goes back through a set of locations as shown in Fig. 8. Suppose that this backward phase reaches the location j L and computes the desired space clock values d n E . |
| 1- If a transition i j T , is controllable, compute a new guard d i j g , and update the location invariant n L according to the new guard computed for the transition i j T , . |
| 2- Compute the space of desired clock values, d i D in the location i L . |
| 3- Compute the space d i E of the desired clock values, d i D at the entrance of the location i L . The space of desired clock values has three situations: |
| (a) If i d i E E then the space clock values within the location i L does not allow the KRC system to reach the forbidden location q L . Thus, the control synthesis problem for q L is solved. |
| (b) If d i E then the location i L becomes a forbidden location. |
| (c) If the conditions (a) and (b) cannot be satisfied i.e. i d i E E and d i E and if i L not equals the initial location, 0 L then go one transition backward in the automaton structure and return to the step #1, otherwise the control synthesis cannot avoid the forbidden state q L . |
SIMULATION RESULTS AND DISCUSSIONS
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| This section simulates the proposed control synthesis on the KCR system using phaver software. The scheme changes the guards of the forbidden states if its transition is controllable or modifies the initial conditions otherwise. Using phaver software, the space of clocks values before starting the control synthesis procedure is shown in Fig. 9. This figure shows that the location 6 L is forbidden state and must be avoidable. To avoid this forbidden state, two guards, 3, 6 g and 2,5 g must be changed if the transition 5 T is controllable, otherwise the backward treatment phase is applied to reinitialize the KRC system. Let us consider the transition 5 T is controllable; the forward treatment phase detailed in section 5 is employed. Fortunately, changing the guard 3, 6 g is enough in our application due to its similarity with 2,5 g as depicted in Fig. 6. Changing this guard, leads to adapting the firing condition of the transition 3,4 T to be fired before enabling the forbidden location 6 L . |
| The philosophy of the control synthesis proposed in this paper is that determining the clock space of the unknown transition time unit e.g. T5 shown in Fig. 4 and consequently avoiding the forbidden locations. Suppose that a transition 3, 6 T is controllable, compute a new guard g d 3, 6 to satisfy the property 3 in order to force the firing of the transition 3, 6 T after the guard of the transition 3, 4 T becomes satisfied by the clocks. Starting with the controlled system behaviour shown in Fig. 10, the space of clocks values in 6 L for the new guard 3,6 g is 12 16 2 5 2 X ≤ + X + X ≤ . It is to be noticed that there still remains some clocks values can be modified to lead the optimal clocks. |
| Running the control synthesis with new conditions, leads to the space of clocks values in 6 L for the new guard 3,6 g is 10 2 5 2 X≤ X≤ X + as depicted in Fig. 11. In This figure, at X2=30, the guard 3,6 g becomes [30 40] 3, 6 5 g ≤ X≤ . Consequently, the guard 7,8 g can be estimated based on its t.u. needed; 2 t.u. Accordingly, it becomes [40≤ X 5≤ 42] 7,8 g This guard should be continuously evaluated each time arrival of a train. |
| Updating the location invariant ( ) 6 L according to the new guard computed for the transition 4, 6 T , is as follows. The maximum and minimum limits of the location invariant ( ) 6 L can be updated using (10) and (11) respectively. |
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| Using phaver-software, leads to the result that the desired space is &6 6 Ed E . Thus, the control synthesis problem for 6 L is solved with its new guard and there is no need to the backward phase. Furthermore, by applying the control synthesis method for this forbidden location, the TA for the control of the KRC system is obtained and shown in Fig. 12. Compared with Fig. 6, the forbidden location L6 is eliminated. Also, the number of locations is decreased that helps in searching purposes over the TA of the system |
| This paper discussed TDES modelling and control using Petri nets and automata. P-invariant based supervisor was designed to control the Petri net model of the KRC system. Although this method controls the system, it fails to define the time units needed for the undefined transition, "T5=[0, ïÂÂ¥]". A timed automaton model that represents the supervised Petri net model of the system is developed to overcome this problem. Phaversoftware was employed for analysing the developed TA model and defining the guards of each transition. The proposed scheme ensures safety and maximizes the utility property. |
| The issues initiated in this paper are discussed and solved using a simplified Petri net model of the KRC system. The proposed supervisor and TA scheme of the KRC system is valid for more complicated cases as depicted in Fig. 13. In this case, the authors supposed that the process is reversible. This assumption increases the forbidden locations by one and constructs a more complex TA model. Based on the methods proposed in this paper, the TA associated with the TPN built in Fig. 13 is depicted in Fig. 14. To avoid the repetition and due to the lack of space, the researchers who are interest can perform the same analysis made in this paper to the TA developed and depicted in Fig. 13 using the control synthesis proposed in this paper. Therefore, using phaver-software, the space of clocks values of locations should be defined to avoid the forbidden state 11 L and 12 L respectively. To avoid these forbidden states, all guards must be also computed and updated if the transitions 5 T and 7 T is controllable, otherwise the backward treatment phase is applied to reinitialize the KRC system. |
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| Using phaver software, the control synthesis problem for the locations 11 L and 12 L is solved with new guards. Other locations analyses are computed and updated in the same manner. The final analysis of the reversible supervised time Petri net of the KRC system eliminates the forbidden states that maximize the utility property of the system. Fig. 16 shows a clear view of the supervised and unblocked TA model. In this figure, all forbidden states are eliminated and a well structured TA associative with the TPN of the KRC system is obtained. |
CONCLUSIONS
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| This paper modelled and supervised the kernel railroad crossing (KRC) system using timed Petri nets. The developed Petri net model is analysed using its timed automata so that the functioning of the system respects a given specification. This paper also proposed the control synthesis method, which consists in computing new firing conditions for the timed automaton transitions so that the forbidden states are no longer reachable. The proposed control synthesis comprises two phases, forward and backward. Simulation results using phaver-software shows the effectiveness of the proposed controller for TDES for ensuring the safety conditions of the system and for maximizing the utility property. Although the KCR is employed for testing the proposed scheme, it is valid for more complex systems. Extending the proposed scheme to more complex problems is our future work. Furthermore, the malfunction of sensors, actuators, and erroneous actions of human operators can have some disastrous consequences in high risk systems [24-27]. These faults can lead to undesirable actions. Our future work also is to design and implement fault detection and isolation schemes for the KRC systems. |
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