On the Enforcement of a Class of Nonlinear Constraints on Petri Nets

International audienceThis paper focuses on the enforcement of nonlinear constraints in Petri nets. First, a supervisory structure is proposed for a nonlinear constraint. The proposed structure consists of added places and transitions. It controls the transitions in the net to be controlled only but does not change its states since there is no arc between the added transitions and the places in the original net. Second, an integer linear programming model is proposed to transform a nonlinear constraint to a minimal number of conjunc-tive linear constraints that have the same control performance as the nonlinear one. By using a place invariant based method, the obtained linear constraints can be easily enforced by a set of control places. The control places consist to a supervisor that can enforce the given nonlinear constraint. On condition that the admissible markings space of a nonlinear constraint is non-convex, another integer linear programming model is developed to obtain a minimal number of constraints whose disjunctions are equivalent to the nonlinear constraint. Finally, a number of examples are provided to demonstrate the proposed approach

Controller synthesis with very simplified linear constraints in PN model

This paper addresses the problem of forbidden states for safe Petri net modeling discrete event systems. We present an efficient method to construct a controller. A set of linear constraints allow forbidding the reachability of specific states. The number of these so-called forbidden states and consequently the number of constraints are large and lead to a large number of control places. A systematic method for constructing very simplified controller is offered. By using a method based on Petri nets partial invariants, maximal permissive controllers are determined.Comment: Dependable Control of discrete Systems, Bari : Italie (2009

Structural methods to improve the symbolic analysis of Petri nets

Symbolic techniques based on BDDs (Binary Decision Diagrams) have emerged as an efficient strategy for the analysis of Petri nets. The existing techniques for the symbolic encoding of each marking use a fixed set of variables per place, leading to encoding schemes with very low density. This drawback has been previously mitigated by using Zero-Suppressed BDDs, that provide a typical reduction of BDD sizes by a factor of two. Structural Petri net theory provides P-invariants that help to derive more efficient encoding schemes for the BDD representations of markings. P-invariants also provide a mechanism to identify conservative upper bounds for the reachable markings. The unreachable markings determined by the upper bound can be used to alleviate both the calculation of the exact reachability set and the scrutiny of properties. Such approach allows to drastically decrease the number of variables for marking encoding and reduce memory and CPU requirements significantly.Peer ReviewedPostprint (author's final draft

Control of Safe Ordinary Petri Nets Using Unfolding

International audienceIn this paper we deal with the problem of controlling a safe place/transition net so as to avoid a set of forbidden markings "F" . We say that a given set of markings has property REACH if it is closed under the reachability operator. We assume that all transitions of the net are controllable and that the set of forbidden markings "F" has the property REACH. The technique of unfolding is used to design a maximally permissive supervisor to solve this control problem. The supervisor takes the form of a set of control places to be added to the unfolding of the original net. The approach is also extended to the problem of preventing a larger set "F" of impending forbidden marking. This is a superset of the forbidden markings that also includes all those markings from which—unless the supervisor blocks the plant—a marking in "F" is inevitably reached in a finite number of steps. Finally, we consider the particular case in which the control objective is that of designing a maximally permissive supervisor for deadlock avoidance and we show that in this particular case our procedure can be efficiently implemented by means of linear algebraic techniques

Decomposition of sequential and concurrent models

Le macchine a stati finiti (FSM), sistemi di transizioni (TS) e le reti di Petri (PN) sono importanti modelli formali per la progettazione di sistemi. Un problema fodamentale è la conversione da un modello all'altro. Questa tesi esplora il mondo delle reti di Petri e della decomposizione di sistemi di transizioni. Per quanto riguarda la decomposizione dei sistemi di transizioni, la teoria delle regioni rappresenta la colonna portante dell'intero processo di decomposizione, mirato soprattutto a decomposizioni che utilizzano due sottoclassi delle reti di Petri: macchine a stati e reti di Petri a scelta libera. Nella tesi si dimostra che una proprietà chiamata ``chiusura rispetto all'eccitazione" (excitation-closure) è sufficiente per produrre un insieme di reti di Petri la cui sincronizzazione è bisimile al sistema di transizioni (o rete di Petri di partenza, se la decomposizione parte da una rete di Petri), dimostrando costruttivamente l'esistenza di una bisimulazione. Inoltre, è stato implementato un software che esegue la decomposizione dei sistemi di transizioni, per rafforzare i risultati teorici con dati sperimentali sistematici. Nella seconda parte della dissertazione si analizza un nuovo modello chiamato MSFSM, che rappresenta un insieme di FSM sincronizzate da due primitive specifiche (Wait State - Stato d'Attesa e Transition Barrier - Barriera di Transizione). Tale modello trova un utilizzo significativo nella sintesi di circuiti sincroni a partire da reti di Petri a scelta libera. In particolare vengono identificati degli errori nell'approccio originale, fornendo delle correzioni.Finite State Machines (FSMs), transition systems (TSs) and Petri nets (PNs) are important models of computation ubiquitous in formal methods for modeling systems. Important problems involve the transition from one model to another. This thesis explores Petri nets, transition systems and Finite State Machines decomposition and optimization. The first part addresses decomposition of transition systems and Petri nets, based on the theory of regions, representing them by means of restricted PNs, e.g., State Machines (SMs) and Free-choice Petri nets (FCPNs). We show that the property called ``excitation-closure" is sufficient to produce a set of synchronized Petri nets bisimilar to the original transition system or to the initial Petri net (if the decomposition starts from a PN), proving by construction the existence of a bisimulation. Furthermore, we implemented a software performing the decomposition of transition systems, and reported extensive experiments. The second part of the dissertation discusses Multiple Synchronized Finite State Machines (MSFSMs) specifying a set of FSMs synchronized by specific primitives: Wait State and Transition Barrier. It introduces a method for converting Petri nets into synchronous circuits using MSFSM, identifies errors in the initial approach, and provides corrections

Optimal trajectory generation for Petri nets

Recently, the increasing complexity of IT systems requires the early verification and validation of the system design in order to avoid the costly redesign. Furthermore, the efficiency of system operation can be improved by solving system optimization problems (like resource allocation and scheduling problems). Such combined optimization and validation, verification problems can be typically expressed as reachability problems with quantitative or qualitative measurements. The current paper proposes a solution to compute the optimal trajectories for Petri net-based reachability problems with cost parameters. This is an improved variant of the basic integrated verification and optimization method introduced in [11] combining the efficiency of Process Network Synthesis optimization algorithms with the modeling power of Petri nets

Petri net modeling and analysis of an FMS cell

Petri nets have evolved into a powerful tool for the modeling, analysis and design of asynchronous, concurrent systems. This thesis presents the modeling and analysis of a flexible manufacturing system (FMS) cell using Petri nets. In order to improve the productivity of such systems, the building of mathematical models is a crucial step. In this thesis, the theory and application of Petri nets are presented with emphasis on their application to the modeling and analysis of practical automated manufacturing systems. The theory of Petri nets includes their basic notation and properties. In order to illustrate how a Petri net with desirable properties can be modeled, this thesis describes the detailed modeling process for an FMS cell. During the process, top-down refinement, system decomposition, and modular composition ideas are used to achieve the hierarchy and preservation of important system properties. These properties include liveness, boundedness, and reversibility. This thesis also presents two illustrations showing the method adopted to model any manufacturing systems using ordinary Petri nets. The first example deals with a typical resource sharing problem and the second the modeling of Fanuc Machining Center at New Jersey Institute of Technology. Furthermore, this thesis presents the analysis of a timed Petri net for cycle time, system throughput and equipment utilization. The timed (deterministic) Petri net is first converted into an equivalent timed marked graph. Then the standard procedure to find the cycle time for marked graphs is applied. Secondly, stochastic Petri net is analyzed using SPNP software package for obtaining the system throughput and equipment utilization. This thesis is of significance in the sense that it provides industrial engineers and academic researchers with a comprehensive real-life example of applying Petri net theory to modeling and analysis of FMS cells. This will help them develop their own applications

Optimal Supervisory Control Synthesis

The place invariant method is well known as an elegant way to construct a Petri net controller. It is possible to use the constraint for preventing forbidden states. But in general case, the number forbidden states can be very large giving a great number of control places. In this paper is presented a systematic method to reduce the size and the number of constraints. This method is applicable for safe and conservative Petri nets giving a maximally permissive controller.Comment: Journ\'ee sur l'Instrumentation Industrielle J2I, ORAN : Alg\'erie (2009