Laura Recalde

On Fluidification of Petri Nets: From Discrete to Hybrid and Continuous Models

Petri Nets (PNs) is a well-known modelling paradigm for discrete event systems (DES). As in other paradigms, hybrid and continuous PN formalisms have appeared in the literature, some of them being used in different engineering application domains. Hybridization may be obtained for example ‘‘by direct addition’’ of capabilities to model continuous subsystems. The approach adopted in this work is different: hybrid and continuous models appear because natural variables of a PN–DES model are transformed into non-negative reals. This relaxation may be quite reasonable when very populated or high traffic systems are considered. It is a classical relaxation applied to fight against the state explosion problem appearing when dealing with the analysis and synthesis of models. In tune with this, the paper presents ‘‘a biased’’ view of works in the hybrid PN arena. Partly an overview, this work revisits hybrid and continuous PNs, all of them being in essence hybrid models. Limitations to (partial) fluidification, and analysis and synthesis problems in this evolving field are considered. Several optimization problems (at design and at control) are also introduced here. # 2004 Published by Elsevier Ltd.

A deadlock avoidance approach for non-sequential resource allocation systems

The paper concentrates on the deadlock-avoidance problem for a class of resource allocation systems modeling manufacturing systems. In these systems, a set of production orders have to be executed in a concurrent way. To be executed, each step of each production order needs a set of reusable system resources. The competition for the use of these resources can lead to deadlock problems. Many solutions, from different perspectives, can be found in the literature for deadlock-related problems when the production orders have a sequential nature [sequential resource allocation systems (S-RAS)]. However, in the case in which the involved processes have a nonsequential nature [nonsequential resource allocation systems (NS-RAS)], the problem becomes more complex. In this paper, we propose a deadlock avoidance algorithm for this last class of systems. We also show the usefulness of the proposed solution by means of its application to a real system.

Steady-State Control Reference and Token Conservation Laws in Continuous Petri Net Systems

This paper addresses several questions related to the control of timed continuous Petri nets under infinite server semantics. First, some results regarding equilibrium states and control actions are given. In particular, it is shown that the considered systems are piecewise linear, and for every linear subsystem the possible steady states are characterized. Second, optimal steady-state control is studied, a problem that surprisingly can be computed in polynomial time, when all transitions are controllable and the objective function is linear. Third, an interpretation of some controllability aspects in the framework of linear dynamic systems is presented. An interesting finding is that noncontrollable poles are zero valued.

Autonomous Continuous P/T Systems

Discrete event dynamic systems may have extremely large state spaces. For their analysis, it is usual to relax the description by removing the integrality constraints. Applying this idea, continuous P/T systems are defined by allowing fractional firings of transitions, and thus the existence of non-discrete markings [4,5,1]. In this paper we compare the behaviors of discrete and continuous systems, and observe that they are not necessarily similar. The problems that appear lead to the definition of two extensions of reachability. Many properties shall be extended differently depending on which reachability definition is being considered. Here, we concentrate on liveness and deadlock-freeness, proposing extensions and relating them to their discrete counterparts.

Steady-state performance evaluation of continuous mono-T-semiflow Petri nets

The number of states in discrete event systems can increase exponentially with respect to the size of the system. A way to face this state explosion problem consists of relaxing the system model, for example by converting it to a continuous one. In the scope of Petri nets, the firing of a transition in a continuous Petri net system is done in a real amount. Hence, the marking (state) of the net system becomes a vector of non-negative real numbers. The main contribution of the paper lies in the computation of throughput bounds for continuous Petri net systems with a single T-semiflow. For that purpose, a branch and bound algorithm is designed. Moreover, it can be relaxed and converted into a linear programming problem. Some conditions, under which the system always reaches the computed bounds, are extracted. The results related to the computation of the bounds can be directly applied to a larger class of nets called mono T-semiflow reducible.

Optimal Model Predictive Control of Timed Continuous Petri Nets

This paper addresses the optimal control problem of timed continuous Petri nets under infinite servers semantics. In particular, our goal is to find a control input optimizing a certain cost function that permits the evolution from an initial marking (state) to a desired steady-state. The solution we propose is based on a particular discrete-time representation of the controlled continuous Petri net system, as a certain linear constrained system. An upper bound on the sample period is given in order to preserve important information of the timed continuous net, in particular the positiveness of the markings. The reachability space of the sampled system in relation to autonomous continuous Petri nets is also studied. Based on the resulting linear constrained model, the optimal control problem is studied through model predictive control (MPC). Implicit and explicit procedures are presented together with a comparison between the two schemes. Stability of the closed-loop system is also studied.

On reachability in autonomous continuous Petri net systems

Fluidification is a common relaxation technique used to deal in a more friendly way with large discrete event dynamic systems. In Petri nets, fluidification leads to continuous Petri nets systems in which the firing amounts are not restricted to be integers. For these systems reachability can be interpreted in several ways. The concepts of reachability and lim-reachability were considered in [7]. They stand for those markings that can be reached with a finite and an infinite firing sequence respectively. This paper introduces a third concept, the δ-reachability. A marking is δ-reachable if the system can get arbitrarily close to it with a finite firing sequence. A full characterization, mainly based on the state equation, is provided for all three concepts for general nets. Under the condition that every transition is fireable at least once, it holds that the state equation does not have spurious solutions if δ-reachability is considered. Furthermore, the differences among the three concepts are in the border points of the spaces they define. For mutual lim-reachability and δ-reachability among markings, i.e., reversibility, a necessary and sufficient condition is provided in terms of liveness.

Modeling and analysis of sequential processes that cooperate through buffers

Deterministically synchronized sequential processes (DSSP) are a subclass of Petri nets, well suited for the methodical construction of models of concurrent systems where several agents cooperate through asynchronous message passing, In this paper, DSSP are presented, illustrating their usability in manufacturing systems. They are compared to other more restrictive subclasses of Petri nets used for similar purposes. In spite of being more expressive, DSSP still enjoy many strong analytical results, some of which are derived and illustrated in the paper.

CONTINUOUS PETRI NETS: EXPRESSIVE POWER AND DECIDABILITY ISSUES

State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of the corresponding discrete model. The expected gains are twofold: improvements in complexity and in decidability. In the case of autonomous nets we prove that liveness or deadlock-freeness remain decidable and can be checked more efficiently than in Petri nets. Then we introduce time in the model which now behaves as a dynamical system driven by differential equations and we study it w.r.t. expressiveness and decidability issues. On the one hand, we prove that this model is equivalent to timed differential Petri nets which are a slight extension of systems driven by linear differential equations (LDE). On the other hand, (contrary to the systems driven by LDEs) we show that continuous timed Petri nets are able to simulate Turing machines and thus that basic properties become undecidable.

On performance monotonicity and basic servers semantics of continuous Petri nets

Continuous Petri nets were introduced as an approximation to deal with the state explosion problem which can appear in discrete event models. When time is introduced, the flow through a fluidified transition can be defined in many ways. The most used in literature are infinite and finite servers semantics. Both can be seen as derived from stochastic Petri nets. The practical problems addressed in this contribution are: (1) a sufficient condition for the performance monotonicity, and (2) a study of the transition semantics, always related to continuous Petri nets. We prove that under some conditions, the subclass of mono-T-semiflow is monotone and also for the same class of nets we prove a property for which infinite servers semantics offers a better approximation than finite servers semantics for the discrete model