Cristian Mahulea

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.

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.

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.

Fault Diagnosis of Discrete-Event Systems Using Continuous Petri Nets

When discrete-event systems are used to model systems with a large number of possible (reachable) states, many problems such as simulation, optimization, and control, may become computationally prohibitive because they require some enumeration of such states. A common way to effectively address this issue is fluidization. The goal of this paper is that of studying the effect of fluidization on fault diagnosis. In particular, we focus on the purely logic Petri net (PN) model that results in the untimed continuous PN model after fluidization. In accordance to most of the literature on discrete-event systems, we define three diagnosis states, namely N, U , and F, corresponding respectively to no fault, uncertain, and fault state. We prove that, given an observation, the resulting diagnosis state can be computed solving linear programming problems rather than integer programming problems as in the discrete case. The main advantage of fluidization is that it enables to deal with much more general PN structures. In particular, the unobservable subnet needs not be acyclic as in the discrete case. Moreover, the compact representation of the set of consistent markings using convex polytopes can be seen in some cases as an improvement in terms of computational complexity.

An Automated Framework for Formal Verification of Timed Continuous Petri Nets

In this paper, we develop an automated framework for formal verification of timed continuous Petri nets (ContPNs). Specifically, we consider two problems: (1) given an initial set of markings, construct a set of unreachable markings and (2) given a Linear Temporal Logic (LTL) formula over a set of linear predicates in the marking space, construct a set of initial states such that all trajectories originating there satisfy the LTL specification. The starting point for our approach is the observation that a ContPN system can be expressed as a Piecewise Affine (PWA) system with a polyhedral partition. We propose an iterative method for analysis of PWA systems from specifications given as LTL formulas over linear predicates. The computation mainly consists of polyhedral operations and searches on graphs, and the developed framework was implemented as a freely downloadable software tool. We present several illustrative numerical examples.

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

A Petri net based approach for multi-robot path planning

This paper presents a procedure for creating a probabilistic finite-state model for mobile robots and for finding a sequence of controllers ensuring the highest probability for reaching some desired regions. The approach starts by using results for controlling affine systems in simpliceal partitions, and then it creates a finite-state representation with history-based probabilities on transitions. This representation is embedded into a Petri Net model with probabilistic costs on transitions, and a highest probability path to reach a set of target regions is found. An online supervising procedure updates the paths whenever a robot deviates from the intended trajectory. The proposed probabilistic framework may prove suitable for controlling mobile robots based on more complex specifications.

Petri net approach for deadlock prevention in robot planning

This paper provides a strategy for supervising the motion of some mobile robots that evolve in the same environment. Some regions of the environment are assumed to have a limited capacity in terms of the number of robots that can simultaneously occupy them, and a set of possible trajectories is available for each robot. The solution comprises the construction of a specific Petri net model for the available trajectories, and the use of resource-allocation techniques based on restricted-capacity regions and on deadlock-free execution.

Minimum-time control for structurally persistent continuous Petri Nets

This paper addresses the minimum-time control problem of structurally persistent timed continuous Petri Net systems (ContPN). In particular, an ON-OFF controller is proposed to drive the system from a given initial marking (state) to the desired (final) marking in minimum-time. The controller is developed first for the discrete-time system ensuring that all transitions are fired as fast as possible in each sampling period until the required total firing counts are reached. After that, they are stopped suddenly. By taking the limit of the sampling period, the controller for continuous-time systems is obtained. Simplicity and the fact that it ensures minimum-time are the main advantages of the controller.

A control method for timed distributed continuous Petri nets

A timed distributed continuous Petri net system (DcontPN) is composed of several subsystems which communicate through channels modeled by places. In this work, a reachability control problem for DcontPNs composed of two subsystems is considered. An algorithm is developed to calculate the control inputs for each subsystem. The application of the obtained control inputs drives the subsystems from the initial states to the target states in a finite amount of time. The algorithm allows the subsystems to reach their respective target markings at different time instants and keep them as long as required.