Feedback control logic synthesis for non safe Petri nets
This paper addresses the problem of forbidden states of non safe Petri Net (PN) modelling discrete events systems. To prevent the forbidden states, it is possible to use conditions or predicates associated with transitions. Generally, there are many …
Authors: Abbas Dideban, Hassane Alla (GIPSA-lab)
Real -life discrete-event systems (DES) are becoming more and more complex and highly automated which makes it tricky the realization of an efficient and realistic control system. Given a discrete-event model of the plant and the specification of the desired behaviour, the objective is to synthesize appropriate supervisor that will act in closed-loop with the plant according to the desired behaviour. Finite-state machines and formal languages are the modelling framework considered in the approach of Ramadge and Wonham (1989). The main limitation in such an approach is the lack of structure in controlled automata.
Petri nets have been proposed as an alternative modelling formalism for DES control. There have been many attempts to solve the control problem for DES with PN modelling. Li and Wonham (1994) have presented an algorithm, which calculates the optimal solution for nets whose uncontrollable subnets are loop-free. The theory of regions (Ghaffari et al. 2003a), allows the design of a maximally permissive PN controller. However, the number of control places is equal to the number of forbidden states and sometimes this leads to complex solutions. Holloway and Krogh have presented a method for controller calculating in real time for a safe and cyclic marked graph (Holloway and Krogh,1990). An effective method for controller synthesis was presented in (Dideban and Alla, 2006), however this method is applicable only on safe PNs. Moreover the final model may be complex.
In this paper, a method is presented to solve the problem of forbidden states for controlled Petri Nets. We develop the method presented in (Dideban and Alla, 2006) for non safe PNs. Moreover, in comparison with (Ghaffari et al. 2003b) and (Holloway et al. 1996),, the final condition will be very simple. The disadvantage of this approach is the calculation of the reachability graph that is fortunately performed offline. In this paper we use the "over-state" concept that was presented in (Dideban and Alla, 2008). This paper is organized as follows: In the second section, the fundamental definitions will be presented. The motivations for this approach will be presented in Section 3 and in Section 4, a method for calculating the condition of forbidding transitions will be presented. Then, in Section 5, the method for simplification of the conditions in safe PNs is called. In Section 6, a compact algorithm will formalize this method and solving the problem of forbidden states will be illustrated via an example. In Section 7, this method will be extended for non safe PN. Finally, the conclusion is given in the last section.
A PN is presented by a 4-uple N = {P, T, W, C} where: 1) P is the set of places, 2) T is the set of transitions, 3) W: (P×T) ∪ (P×T), is the incidence matrix, and 4) C is the firing conditions associated with each controllable transition.
The reachability graph consists of nodes, which correspond to the accessible markings M i , and arcs to the firing of the transitions. In the reachability graph, there are two types of states: the authorized state M A and the forbidden state M F . Among the forbidden states, a particular and important subset is constituted by the border forbidden states, which are denoted by the set M B . These states are such that all the input transitions are controllable.
In this paper, we use the word state instead of marking.
Definition 1: The set {0,1} N represents all the Boolean vectors of dimension N.
The set of the marked places of a marking M is given by a Support function that is defined in the following.
Definition 2: The function Support(X) of a vector X ∈ {0,1} N is: Support(X) = The set of marked places in vector X.
The support of vector M 0 T = [1, 0, 1, 0, 0, 1, 0] is:
Definition 3: Let M 1 and M 2 be 2 states of the system, and P = {P 1 , P 2 , …, P N } the PN set of places, M 2 is an over-state of
This relation is represented as shown bellow:
Definition 4: Informally, the forbidden states are:
-The states reachable in the process but not authorized by the specification.
-Deadlock states.
In the PN modelling, when a controllable event is associated with a transition, a controller can be calculated for this transition. Then we use the controllable transition instead of the controllable event. Firing of some controllable transitions can lead to forbidden states. This set is named set of critical transitions. The rest of the controllable transition is named sound transitions.
The border forbidden states are reached from the admissible states by the occurrence of controllable events. Preventing the occurrence of the controllable events can forbid entering to a forbidden state. By constructing the reachability graph, we can divide the admissible states for each controllable transition (t i ) into 3 groups:
-The states from which the firing of t i is possible and allowed;
-The states from which the firing of t i is possible but is not allowed.
-The states from which the firing of t i is not possible;
The first group is named sound states and corresponds to the states from which by firing transition t i , the admissible states can be reached.
The second group of these states corresponds to the states leading to a border forbidden state by firing t i . This group is named critical states.
The third group are the states for which the firing of this transition is not possible. The first and third groups are non critical states. The first and second group can be defined as below;
Definition 5: Let M B be the set of border forbidden states and M A the set of admissible states. The sets of t i critical states
M , and t i sound states S t i M , are defined as follows:
Where Σ c is the set of controllable transitions Definition 6: Let
M be the set of critical states for the critical transition t i . Control Ut i : (
The control relation is modelled in Figure 1.
This is similar with the approach presented in (Holloway et al. 1996). The difference between both approaches is the method of calculation of the control U ti . As it will be shown the advantage of our approach is to provide a method to determine simple forbidding conditions. To achieve this goal, we need to build the reachability graph as an intermediate step. Our approach is applicable to ordinary PNs. Firstly we present it on safe PN and then for non safe PN. It is supposed that all of the events are independent.
We, first present our ideas via a simple example. Consider the classical system composed of two machines M 1 and M 2 and one buffer S 1 . The specification constraint is the capacity of the buffer (Figure 2). Firstly we suppose that the capacity of S 1 is one part, it will be changed later in order to have a non safe model.
We suppose here that only the starts of the tasks (event c 1 , c 2 , i.e. transitions t 1, t 3 ) are controllable and the ends of task (event f 1 , f 2, i.e. transitions t 2, t 4 ) are uncontrollable. The desired functioning in closed loop for this system is given in Figure 3.
The goal is to find a control such that the border forbidden states are never reached. For this, we must construct the rechability graph of the closed loop model. The rechability graph for this example is given in Figure 4.
In the behaviour of this PN, some transitions associated with uncontrollable events lead to forbidden states. For example, the firing of t 2 is possible when place P 2 is marked and event f 1 occurs even place P 3 is empty. These states are called the forbidden states and correspond to the set of border forbidden states for this example:
We can compute the forbidden states by the method that is presented in (Kumar and Holloway 1996).
This can be accomplished by adding conditions to transition t 1 resulting to the transition to be blocked when the system is in states M 3 and M 4 . This condition can be computed for each state M j taking into account the presence of marks in the places: Ut 1 ( C t M 1 ) = ((m(P 1 )Λm(P 4 )Λm(P 5 )) ∨ (m(P 1 )Λm(P 4 ) Λm(P 6 )))`
Remark2: Variable m j (P i ) represents the number of marks in place P i in state M j and then for a safe PN is a Boolean variable. Moreover the condition Ut 1 ( )is also Boolean.
Logical expression Ut 1 ( )means that transition t 1 would not be fireable when the system is in states M 3 or M 4 . The situation in M 3 is presented in Figure 5.
In this state the firing of t 1 is not possible. Now, the generalization of this idea is given in the following section and a method will be presented for the simplification of these conditions thanks to the concepts of over-state which corresponds to a significant contribution.
From the sets M σ C and M σ S for each controllable event or transition, there are two ways to construct the controller: the calculation of the conditions of forbidding or the calculation of the conditions for enabling each controllable event. In this paper, the first method is called but in general case it is better to examine both methods and to select the simpler solution. Now we explain how these states can be forbidden (Dideban and Alla, 2006).
Property 1: Let M 1 be a critical state for transition t i . By using the control Ut i (M 1 ), we can forbid the firing of t i in state M 1 .
By this method, it is possible to forbid the firing from state M 1 , but other states can forbid the firing of t i by the same control. For example if there is a sound state M 2 such that M 2
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