Supervisory Control Of Petri Nets Research Articles

Supervisory Control of Petri Nets in the Presence of Replacement Attacks

This article addresses the robust control problem of discrete event systems assuming that replacement attacks may occur, thus making it appear that an event that has occurred looks like another event. In particular, we assume that this is done by tampering with the sensor-readings in the sensor communication channel. Specifically, we use Petri nets as the reference formalism to model the plant and assume a control specification in terms of a generalized mutual exclusion constraint. We propose three different methods to derive a control policy that is robust to the possible replacement attacks. The first two methods lead to an optimal (i.e., maximally permissive) policy but are computationally inefficient when applied to large-size systems. On the contrary, the third method computes a policy more efficiently and reveals more easily implementable in practice. However, this is done at the expense of optimality.

Extended Place-Invariant Control in Automated Manufacturing Systems Using Petri Nets

In supervisory control of Petri nets (PNs), the place-invariant ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariant) control principle is the most typical and principal method to deal with the siphon control problem. Although it has a relatively narrow application, this principle is widely acknowledged due to its simplicity and efficiency. In this article, we first propose the extended <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariant control principle in order to extend the application of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariants and provide a general methodology for the control of siphons. Second, three types of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariants, from the special to the general, are developed to implicitly or explicitly invariant control the siphons. In the most general case, the virtual <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariants are constructed in the PNs. Third, the extended principle is further applied to the supervisor simplification. In the paradigm of the extended principle, it presents the redundancy from a structural perspective in contrast to several typical methods, and shows the significant importance of structural analysis in PNs, especially the important role of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariants. As a consequence, the extended <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> -invariant control principle can be considered as the fundamental principle of siphon control as well as its supervisor simplification.

Static and Dynamic Partitions of Inequalities: A Unified Methodology for Supervisor Simplification

In the context of automated manufacturing systems, supervisor simplification is drawing increasing attention. As a special class of state specifications, generalized mutual exclusion constraints (GMECs) are a typical kind of supervisory control of Petri nets. In this note, we first propose static and dynamic partitions to reduce the size of supervisor by partitioning inequalities, i.e., GMECs, into redundant and necessary ones. The former considers the specifications themselves and exhibits higher efficiency; while the latter takes into account both specifications and reachable markings and achieves more economical supervisors. In an incremental manner, we develop Type I supervisor simplification method which has a compromised efficiency and complexity. In an integrated manner, we propose Type II supervisor simplification method that further improves efficiency and eases computational complexity. All techniques proposed in this note can be realized in an algebraic manner. Examples are presented for the sake of illustration as well as demonstration.

Compact Supervisory Control of Discrete Event Systems by Petri Nets With Data Inhibitor Arcs

This work proposes a novel structure in Petri nets, namely data inhibitor arcs, and their application to the optimal supervisory control of Petri nets. A data inhibitor arc is an arc from a place to a transition labeled with a set of integers. A transition is disabled by a data inhibitor arc if the number of tokens in the place is in the set of integers labeled on it. Its formal definitions and properties are given. Then, we propose a method to design an optimal Petri net supervisor with data inhibitor arcs to prevent a system from reaching illegal markings with respect to control specifications. Two techniques are developed to reduce the supervisor structure by compressing the number of control places. Finally, a number of examples are used to illustrate the proposed approaches and experimental results show that they can obtain optimal Petri net supervisors for the net models that cannot be optimally controlled by pure net supervisors. A significant result is that the proposed approach can always lead to an optimal supervisor with only one control place for bounded Petri nets on the premise that such a supervisor exists.

On the Notion of Uncontrollable Marking in Supervisory Control of Petri Nets

We show that the notion of uncontrollable marking commonly used in the literature on supervisory control theory of Petri nets is not sound, by means of a counter-example. We also show how the definition can be corrected and provide an adaptation of a decidability proof for the problem of checking controllability for specifications expressed as deterministic Petri net languages.

New Petri Net Structure and Its Application to Optimal Supervisory Control: Interval Inhibitor Arcs

This paper presents a new Petri net structure, namely, an interval inhibitor arc, and its application to the optimal supervisory control of Petri nets. An interval inhibitor arc is an arc from a place to a transition labeled with an integer interval. The transition is disabled by the place if the number of tokens in the place is between the labeled interval. The formal definition and the firing rules of Petri nets with interval inhibitor arcs are developed. Then, an optimal Petri net supervisor based on the interval inhibitor arcs is designed to prevent a system from reaching illegal markings. Two techniques are developed to simplify the supervisory structure by compressing the number of control places. The proposed approaches are general since they can be applied to any bounded Petri net models. A marking reduction approach is also introduced if they are applied to Petri net models of flexible manufacturing systems. Finally, a number of examples are provided to demonstrate the proposed approaches and the experimental results show that they can obtain optimal Petri net supervisors for some net models that cannot be optimally controlled by pure net supervisors. Furthermore, the obtained supervisor is structurally simple.

Supervisory control of Petri nets using polyhedral regions

We propose to use polyhedral computations to check the reachability and co-reachability of Petri nets, and design supervisors enforcing polyhedral invariance. To this aim, we first state a semi-algebraic behavioral model for Petri nets, derived from the marking equation. From it, we derive explicit characterizations of the sets of k-reachable marking vectors, in terms of polyhedral regions of a real vector space. This permits to calculate the region of reachable marking vectors, when the considered Petri net is bounded, as the projection of an integral polyhedron. The polyhedron is computed using the Γ-algorithm, in polynomial time. We propose an algorithm to this computation. We adapt the method to further compute the polytope of reachable, co-reachable and safe marking vectors, meeting a given set of polyhedral constraints, and finally apply it to design a supervising controller. We illustrate the method on the supervisor design of a simple AGV system.

On suboptimal supervisory control of Petri nets in the presence of uncontrollable transitions via monitor places

Recently, a study has been presented for suboptimal supervisory control of Petri nets via monitor (control) places in the presence of uncontrollable transitions. To enforce a generalized mutual exclusion constraint (GMEC) suboptimally on a plant Petri net with uncontrollable transitions, a set of monitor places has been provided to choose from. There are two main results of this study, the first of which is that “there is not an optimal solution to the problem of computing a monitor-based controller to enforce a GMEC on a plant net in the presence of uncontrollable transitions”. The second result deals with the computation of monitor places for the suboptimal supervisory control in such Petri nets. In this paper, it is shown that there may be an optimal solution to the problem of computing a monitor-based controller to enforce a GMEC on a plant net in the presence of uncontrollable transitions. To do this, an example Petri net is considered and then two different optimal (maximally permissive) solutions are provided for this Petri net.

Some New Results on Supervisory Control of Petri Nets with Decentralized Monitor Places

This paper presents two new results on the problem of determining a set of decentralized controllers for place/transition nets to enforce a global specification on the net behavior. Both the global specification and the decentralized specifications are given in terms of Generalized Mutual Exclusion Constraints (GMECs). First, an algorithm to select a decentralized specification that finds a compromise between fairness among variables and the maximal cardinality is proposed assuming that the support of each decentralized GMEC is a singleton. Then, a maximal solution in terms of permissiveness and fairness among places is proposed removing the previous assumption.

Supervision Based on Place Invariants: A Survey

The supervision based on place invariants (SBPI) is an efficient technique for the supervisory control of Petri nets. This paper reveals the significance of the SBPI based on a literature survey, applications, and an analysis of problems and supervisory settings that can be addressed using SBPI. Special attention is given to the various settings within which the problem can be formulated. Such settings can be distinguished based on the concurrency type, the type of controllability and observability, and the centralized or decentralized type of supervision. As we show, it is possible to approach the most general settings in a purely structural way, without resorting to reachability analysis. We begin by describing the SBPI problem and the literature methods that address this problem or are related to it. Then, we proceed to show classes of problems that can be reduced to the SBPI problem. In the SBPI, the specification is described as a system of inequalities that the Petri net marking must satisfy at any time. However, as we show, problems involving more general specifications can be approached in the SBPI setting, sometimes under additional assumptions, by performing net and/or specification transformations. Four of the specifications we will consider are logic constraints, language specifications, disjunctions of linear constraints, and liveness. We conclude with a presentation of possible applications of the SBPI approach to programming with semaphores, fault tolerance, and synchronic-distance based designs.

Suboptimal supervisory control of Petri nets in presence of uncontrollable transitions via monitor places

This paper deals with the problem of enforcing generalized mutual exclusion constraints (GMEC) on place/transition nets with uncontrollable transitions. An efficient control synthesis technique, which has been proposed in the literature, enforces GMEC constraints by introducing monitor places to create suitable place invariants. The method has been shown to be maximally permissive and to give a unique control structure in the case that the set of legal markings is controllable. This paper investigates on and formally shows that the class of controllers obtained by this technique may not have a supremal element for uncontrollable specifications. Moreover, it is shown that the family of monitor places enforcing an uncontrollable specification can be parameterized with respect to the solution of a linear system of equation. An algorithm to obtain such parameterization is presented here.

Structuring Acyclic Petri Nets for Reachability Analysis and Control

The incidence matrices—from places to transitions and vice versa—of an acyclic Petri net can obtain a block-triangular structure by reordering their rows and columns. This allows the efficient solution of some reachability problems for acyclic Petri nets. This result is further used in supervisory control of Petri nets; supervisors for Petri nets with uncontrollable transitions are constructed by extending the method of Yamalidou et al. (1996) to Petri nets where transitions can be executed simultaneously. A large class of Petri nets with uncontrollable transitions is given for which the maximally permissive supervisor can be realized by a Petri net. The original specification is algorithmically transformed—by using the results for acyclic Petri nets—into a new specification to take the presence of uncontrollable transitions into account. The supervisor is obtained by simple matrix multiplications and no linear integer programs need to be solved. Furthermore, a class of Petri nets is given for which the supervisor can be realized by extending the enabling rule with OR-logic.

Supervisory control of Petri nets using routing functions: starvation avoidance issues

In this paper, we present a new point of view on supervisory control of Petri nets by using routing functions instead of the traditional control places. We first show the relation between the two notions. In the second part of the paper, we illustrate the use of routing functions by showing how to compute a routing function in order to avoid starvation in general Petri nets. This control uses a continuous version of the net and a description of the evolution of the net under the form of linear algebraic equations. As for the computational part, we use algebraic polynomial geometry in the continuous case and Diophantine equations for the discrete version of the Petri net under study.