Jorge Júlvez

On fluidization of discrete event models: observation and control of continuous Petri nets

As a preliminary overview, this work provides first a broad tutorial on the fluidization of discrete event dynamic models, an efficient technique for dealing with the classical state explosion problem. Even if named as continuous or fluid, the relaxed models obtained are frequently hybrid in a technical sense. Thus, there is plenty of room for using discrete, hybrid and continuous model techniques for logical verification, performance evaluation and control studies. Moreover, the possibilities for transferring concepts and techniques from one modeling paradigm to others are very significant, so there is much space for synergy. As a central modeling paradigm for parallel and synchronized discrete event systems, Petri nets (PNs) are then considered in much more detail. In this sense, this paper is somewhat complementary to David and Alla (2010). Our presentation of fluid views or approximations of PNs has sometimes a flavor of a survey, but also introduces some new ideas or techniques. Among the aspects that distinguish the adopted approach are: the focus on the relationships between discrete and continuous PN models, both for untimed, i.e., fully non-deterministic abstractions, and timed versions; the use of structure theory of (discrete) PNs, algebraic and graph based concepts and results; and the bridge to Automatic Control Theory. After discussing observability and controllability issues, the most technical part in this work, the paper concludes with some remarks and possible directions for future research.

A Continuous Petri Net Approach for Model Predictive Control of Traffic Systems

Traffic systems are often highly populated discrete event systems that exhibit several modes of behavior such as free flow traffic, traffic jams, stop-and-go waves, etc. An appropriate closed loop control of the congested system is crucial in order to avoid undesirable behavior. This paper proposes a macroscopic model based on continuous Petri nets as a tool for designing control laws that improve the behavior of traffic systems. The main reason to use a continuous model is to avoid the state explosion problem inherent to large discrete event systems. The obtained model captures the different operation modes of a traffic system and is highly compositional. In order to handle the variability of the traffic conditions, a model predictive control strategy is proposed and validated.

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.

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.

Performance analysis of concurrent systems with early evaluation

Early evaluation allows to execute operations when enough information at the inputs has been received to determine the value at the outputs. Systems that can tolerate variable-latency units, such as latency-insensitive or asynchronous systems, can enhance their performance by using early evaluation. The most relevant example of a unit with early evaluation is the multiplexor: the output can be determined as soon as the information of the selected channel arrives, without waiting for the other channels. This paper analyzes the potential impact of early evaluation in concurrent systems. An analytical model, based on a Petri net extension with early firing is proposed to estimate the performance. The reduction of the analytical model to a linear programming formulation for an efficient estimation of the upper bound for the system throughput is proposed. The results show the accuracy of the model and the benefits of early evaluation

MODELLING AND CONTROLLING TRAFFIC BEHAVIOUR WITH CONTINUOUS PETRI NETS

Abstract Traffic systems are discrete systems that can be heavily populated. One way of overcoming the state explosion problem inherent to heavily populated discrete systems is to relax the discrete model. Continuous Petri nets (PN) represent a relaxation of the original discrete Petri nets that leads to a compositional formalism to model traffic behaviour. This paper introduces some new features of continuous Petri nets that are useful to obtain realistic but compact models for traffic systems. Combining these continuous PN models with discrete PN models of traffic lights leads to a hybrid Petri net model that is appropriate for predicting traffic behaviour, and for designing traffic light controllers that minimize the total delay of the vehicles in the system.

Performance optimization of elastic systems using buffer resizing and buffer insertion

Buffer resizing and buffer insertion are two transformation techniques for the performance optimization of elastic systems. Different approaches for each technique have already been proposed in the literature. Both techniques increase the storage capacity and can potentially contribute to improve the throughput of the system. Each technique offers a different trade-off between area cost and latency. This paper presents a method that combines both techniques to achieve the maximum possible throughput while minimizing the cost of the implementation. The provided method is based on mixed integer linear programming. A set of experiments is designed to show the feasibility of the approach.

On Observability and Design of Observers inTimed Continuous Petri Net Systems

This paper is devoted both to the study of observability criteria and the design of observers in the continuous Petri net setting. The concept of structural observability, regarding the possibility of estimating the marking of places, i.e., the system state, for any speed of the transitions is introduced and studied for the subclass of join-free Petri nets (JF). For non-join-free Petri nets, conditions to compute suitable state estimates are established. The proposed observers are piecewise linear systems that assure the continuity of the estimate even when a switch occurs. The system simulation may allow us to estimate even the unobservable space of the net system during a given time period.

Relaxed continuous views of discrete event systems: considerations on Forrester diagrams and Petri nets

Petri nets (PNs) constitute a formal paradigm for discrete systems. Some discrete models can be relaxed into continuous models. Infinite server semantics continuous Petri nets (ISSCPNs) is one of the most relevant timed interpretation of continuous PNs. ISSCPNs can be seen as piecewise linear systems. Forrester diagrams (FD) are specific modelling tools inside system dynamics, a methodology for the analysis of complex continuous systems. This paper explores and compares the modelling power of both formalisms, ISSCPNs and FD. Previous comparative views focused on the formalisms and on positive and compartmental systems, constitute the basis of this work. The comparison is complemented taking into account the interpretation of linear ordinary differential equation systems (LODES), the information delays and some methodological considerations. ISSCPNs permit to model any LODES when known upper and lower bounds of the state variables exists. Therefore systems with cyclic behaviour or delays in the information can be modelled.

Event-driven optimal control of continuous Petri nets

Optimally controlling a hybrid system is a challenging problem for which mainly continuous-time and discrete-time methods have been suggested. In this paper, the problem of optimal control is addressed in the framework of continuous Petri nets, a kind of hybrid systems whose state evolution is piecewise linear. The proposed approach consists of transforming the continuous Petri net into an equivalent hybrid system whose evolution is described by means of discrete-event steps. In particular, each step coincides with the occurrence of an event in the continuous Petri net. Thus, the number of steps required to know the behavior of the Petri net is minimum, while the accuracy is completely preserved. It is shown how to design a mixed integer linear programming problem in order to compute the optimal control solution of different performance criteria.