Behavior Of Discrete Event Systems Research Articles

Optimizing the max-min function with a constraint on a two-sided linear system

<abstract><p>The behavior of discrete-event systems, in which the individual components move from event to event rather than varying continuously through time, is often described by systems of linear equations in max-min algebra, in which classical addition and multiplication are replaced by $ \oplus $ and $ {\otimes} $, representing maximum and minimum, respectively. Max-min equations have found a broad area of applications in causal models, which emphasize relationships between input and output variables. Many practical situations can be described using max-min systems of linear equations. We shall deal with a two-sided max-min system of linear equations with unknown column vector $ x $ of the form $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $, where $ A $, $ B $ are given square matrices, $ c $, $ d $ are column vectors and operations $ \oplus $ and $ {\otimes} $ are extended to matrices and vectors in the same way as in the classical algebra. We give an equivalent condition for its solvability. For a given max-min objective function $ f $, we consider optimization problem of type $ f^\top{\otimes} x\rightarrow \max\text{ or } \min $ constraint to $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $. We solve the equation in the form $ f(x) = v $ on the set of solutions of the equation $ A{\otimes} x{\oplus} c = B{\otimes} x{\oplus} d $ and extend the problem to the case of an interval function $ {{\boldsymbol{f}}} $ and an interval value $ {{\boldsymbol{v}}} $. We define several types of the reachability of the interval value $ {{\boldsymbol{v}}} $ by the interval function $ {{\boldsymbol{f}}} $ and provide equivalent conditions for them.</p></abstract>

Crucial States Estimation in Radio Block Center Handover Using Petri Nets With Unobservable Transitions

The radio block center (RBC) is one of the most essential ground systems in a high-speed train control system both in Europe and in China. The RBC handover procedure is an important function of RBC, which affects the transport efficiency, reliability and safety of railways. Analysis of crucial states in the RBC handover procedure is helpful to determine whether there are potential risks in the procedure, and to locate the fault in time when a fault occurs. In this paper, we study a property, called C-detectability, of the RBC handover. This property has been defined in discrete event systems and requires that the crucial states can be determined uniquely by observing the system output. Taking the RBC handover procedure in the Chinese train control system level 3 (CTCS-3) as an example, we first model the RBC handover procedure using Petri nets, which are a graphical and mathematical modeling tool to formalize the behavior of discrete event systems. Then, based on the notion of basis reachability graph, an efficient approach is used to check C-detectability of the Petri net modeling the handover procedure. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The railway system is a safety-critical system. System safety is essential to the railway system. It is necessary to provide formal methods to model the system and analyze its properties. The motivation of this work is to present a general modeling framework of railway systems for the crucial states estimation. The crucial states estimation of the railway system has not been discussed yet in the literature despite its importance due to its relationship with the safety of the railway. In particular, estimating the crucial states in the railway system helps to determine whether there are potential risks in the system. In this paper, we investigate a new state estimation property (C-detectability), which is related to fault diagnosis, marking estimation and fault identification. Future research will consider the use of timed Petri nets to increase the modeling power and to allow the performance analysis of the railway system.

Transformational Nonblocking Verification*

Nonblocking verification can be applied to evaluate the behavior of discrete event systems. Performing nonblocking verification can be computationally costly. In this work, we consider discrete event systems that evolve over time. We study how to reuse results from a previous nonblocking verification, to more efficiently perform nonblocking verification when the system is adapted. We call this approach transformational nonblocking verification, and present an algorithm for the method. The efficiency of the method is evaluated by applying an academic and an industrial use case.

System Reliability using Simulation Models and Formal Methods

The general scope of the paper consists of improving the verification of simulation models through the integration of formal methods. We offer a formal verification approach of DEVS models based on Z notation. DEVS is a formalism that allows the description and analysis of the behavior of discrete event systems, i.e., systems whose state change depends on the occurrence of an event. A DEVS model is essentially validated by the simulation which permits of verifying whether it correctly describes the behavior of the system. However, a simulation does not cover all possible cases that means the model is validated only for the expected behaviors. For this reason, we have integrated the Z formal specification language in the DEVS formalism to detect errors before simulation which is still an important step for the validation. This integration consists of: (1) transforming a DEVS model into an equivalent Z specification and (2) verifying the consistency of the DEVS model on the resulting specification using the tools developed by the Z community. Such consistency is fulfilled by determinism and completeness. Thus, a DEVS model is subjected to an automatic formal verification before proceeding to its simulation.

Communicating Processes with Data for Supervisory Coordination

We employ supervisory controllers to safely coordinate high-level discrete(-event) behavior of distributed components of complex systems. Supervisory controllers observe discrete-event system behavior, make a decision on allowed activities, and communicate the control signals to the involved parties. Models of the supervisory controllers can be automatically synthesized based on formal models of the system components and a formalization of the safe coordination (control) requirements. Based on the obtained models, code generation can be used to implement the supervisory controllers in software, on a PLC, or an embedded (micro)processor. In this article, we develop a process theory with data that supports a model-based systems engineering framework for supervisory coordination. We employ communication to distinguish between the different flows of information, i.e., observation and supervision, whereas we employ data to specify the coordination requirements more compactly, and to increase the expressivity of the framework. To illustrate the framework, we remodel an industrial case study involving coordination of maintenance procedures of a printing process of a high-tech Oce printer.

Mathematical programming formulations for approximate simulation of multistage production systems

Mathematical programming representation has been recently used to describe the behavior of discrete event systems as well as their formal properties. This new way of representing discrete event systems paves the way to the creation of simpler mathematical programming models that reduce the complexity of the system analysis. The paper proposes an approximate representation for a class of production systems characterized by several stages, limited buffer capacities and stochastic production times. The approximation exploits the concept of a time buffer, modeled as a constraint that put into a temporal relationship the completion times of two customers in a sample path. The main advantage of the proposed formulation is that it preserves its linearity even when used for optimization and, for such a reason, it can be adopted in simulation–optimization problems to reduce the initial solution space. The approximate formulation is applied to relevant problems such as buffer capacity allocation in manufacturing systems and control parameters setting in pull systems.

A Constraint Programming Approach for Generating Firing Sequences in Timed Petri Nets With Token Identification

Timed Petri Nets are a good modeling framework to express the behavior of discrete event systems, such as transport or manufacturing systems. They allow to represent easily the distribution of tasks within a complex system, with the capacity to handle time constraints on the duration of these tasks.However, when synchronization is needed between several layers of the system, such models suffer from a lack of mechanisms to identify the components that are handled. Thus, it is difficult to obtain firing sequences avoiding confusion of tokens when activation or coordination patterns are used.Such issues are well addressed with High Level Petri Nets, like Colored PNs where data structures are attached to tokens, allowing to represent a complex behaviour while keeping a compact PN model. However, such models introduce a modeling material that makes it difficult to use direct incremental solving approaches like mathematical or constraint programming, i.e. approaches that do not need to unfold the net and build its whole reachability graph.In this paper, we propose to extend the expressivity of Timed Petri Nets by associating integer identifiers with tokens. Such approach can avoid the confusion of parts in a plant, or trains in a railway network, and allow to model much more possible coordination situations. A constraint programming model is built to follow this new firing semantics, and benchmarks are given to assess its efficiency.

Mathematical programming representation for approximate model of flow lines

AbstractMathematical programming representation has been recently used to describe the behavior of discrete event systems as well as their formal properties. This paper proposes an approximate mathematical programming model for the simulation of flow lines with finite buffer capacities. The approximation exploits the concept of time buffer, modeled as a constraint that put into a temporal relationship the completion times of two jobs in a sample path. The system performance estimated by the approximate formulation are between two extreme values calculated from the solution of a linear programming model. When the linear model is coupled with a buffer optimization model, the extreme values represent the upper and lower bounds containing the optimal solution. The main advantage of the proposed formulation is that it preserves its linearity even when used for buffer optimization.

Identification of the unobservable behaviour of industrial automation systems by Petri nets

This paper addresses the problem of identifying the model of the unobservable behaviour of discrete event systems in the industrial automation sector. Assuming that the fault-free system structure and dynamics are known, the paper proposes an algorithm that monitors the system on-line, storing the occurred observable event sequence and the corresponding reached states. At each event observation, the algorithm checks whether some unobservable events have occurred on the basis of the knowledge of the Petri net (PN) modelling the nominal system behaviour and the knowledge of the current PN marking. By defining and solving some integer linear programming problems, the algorithm decides whether it is necessary to introduce some unobservable (silent) transitions in the PN model and provides a PN structure that is consistent with the observed event string. A case study describing an industrial automation system shows the efficiency and the applicability of the proposed algorithm.

最悪ケースを考慮した最適スーパバイザの強化学習

The supervisory control initiated by Ramadge and Wonham is a framework for logical control of discrete event systems. In the original supervisory control, the costs for occurrence and disabling of events have not been considered. Then, the optimal supervisory control based on quatitative measures has also been studied. This paper proposes a synthesis method of the optimal supervisor based on the worst-case behavior of discrete event systems. We introduce the new value functions for the assigned control patterns. The new value functions are not based on the expected total rewards, but based on the most undesirable event occurrence in the assigned control pattern. In the proposed method, the supervisor learns how to assign the control pattern based on reinforcement learning so as to maximize the value functions. We show the efficiency of the proposed method by computer simulation.

Identification of DES Unobservable Behaviour by Petri Nets

The paper addresses the problem of the on-line identification of Petri Nets (PN) modelling the unobservable behaviour of Discrete Event Systems (DES). Starting from a previous specification of an identifier that monitors the DES events and the corresponding available place markings, the paper proves some results that make the on-line identification of reasonable complexity. Assuming that the PN system modelling the observable events is known, at each event occurrence an identification algorithm recursively updates and provides as an output the complete PN system describing both the observable and unobservable DES behaviour. An example shows an application of the proposed technique.

Diagnostic décentralisé des systèmes à événements discrets

We consider the detection and identification of unobservable events in the behavior of discrete event systems. We study decentralized diagnosis architectures where several sites observe the system behavior, each site having its own set of observable events. These sites do not communicate among each other but they send their local diagnostic decisions to a coordinating site that implements a simple memoryless Boolean fusion rule. Several new results are presented in the context of such architectures, with special focus on the use of conditional decisions by the local sites.

Control of stochastic discrete event systems modeled by probabilistic languages

In Garg et al. (1999) and Garg (1992) the formalism of probabilistic languages for modeling the stochastic qualitative behavior of discrete event systems (DESs) was introduced. In this paper, we study their supervisory control where the control is exercised by dynamically disabling certain controllable events thereby nulling the occurrence probabilities of disabled events, and increasing the occurrence probabilities of enabled events proportionately. This is a special case of "probabilistic supervision" introduced in Lawford and Wonham (1993). The control objective is to design a supervisor such that the controlled system never executes any illegal traces (their occurrence probability is zero), and legal traces occur with minimum prespecified occurrence probabilities. In other words, the probabilistic language of the controlled system lies within a prespecified range, where the upper bound is a "nonprobabilistic language" representing a legality constraint. We provide a condition for the existence of a supervisor. We also present an algorithm to test this existence condition when the probabilistic languages are regular (so that they admit probabilistic automata representation with finitely many states). Next, we give a technique to compute a maximally permissive supervisor online.

A constructive fixed point theorem for min-max functions

Min-max functions, F :R n R n , arise in modelling the dynamic behaviour of discrete event systems. They form a dense subset of those functions which are homogeneous, F i (x 1 + h, .. . , x n + h) = Fi(x 1 , … , x n ) + h, monotonic , x y F(x) F(y), and nonexpansive in the l infinity norm-so-called topical functions-which have appeared recently in the work of several authors. Our main result characterizes those min-max functions which have a (generalized) fixed point, where Fi(x) = x i + h for some h R. We deduce several earlier fixed point results. The proof is inspired by Howard's policy improvement scheme in optimal control and yields an algorithm for finding a fixed point, which appears efficient in an important special case. An extended introduction sets the context for this paper in recent work on the dynamics of topical functions.

A Max-Plus Algebra of Signals for the Supervisory Control of Real-Time Discrete Event Systems

This paper defines a max-plus algebra of signals for the evaluation of timing behavior of discrete event systems modeled by timed event graphs. The scope of this work is limited to infinite. periodic sequences for whhich finite representations called signals can be Computed. The implementation of this Inax-plus algebra allows a user to Compute supremal controllers for real-time. discrete event systems

Optimization of the Transient and Steady-State Behavior of Discrete Event Systems

We present a general framework for applying simulation to optimize the behavior of discrete event systems. Our approach involves modeling the discrete event system under study as a general state sp...

Optimization of the Transient and Steady-State Behavior of Discrete Event Systems

We present a general framework for applying simulation to optimize the behavior of discrete event systems. Our approach involves modeling the discrete event system under study as a general state space Markov chain whose distribution depends on the decision parameters. We then show how simulation and the likelihood ratio method can be used to evaluate the performance measure of interest and its gradient, and we present conditions that guarantee that the Robbins-Monro stochastic approximation algorithm will converge almost surely to the optimal values of the decision parameters. Both transient and steady-state performance measures are considered. For steady-state performance measures, we consider both the case when the Markov chain of interest is regenerative in the standard sense, as well as the case when this Markov chain is Harris recurrent, and thereby regenerative in a wider sense.

Supervision of Infinite Behavior of Discrete-Event Systems

Some basic results of supervisory control theory are extended to the setting of $\omega$-languages, formal languages consisting of infinite strings. The extension permits the investigation of both liveness and safety issues in the control of discrete-event systems. A new controllability property appropriate to the infinitary setting ($\omega$-controllability) is defined; this language property captures in a natural way the limitations of available control actions. It is shown that every specification language contains a unique maximal $\omega$-controllable sublanguage, representing the least upper bound of the set of achievable closed-loop sublanguages. This supremal $\omega$-controllable sublanguage allows a simple formulation of necessary and sufficient conditions for the solvability of an infinitary supervisory control problem. The problems of effectively deciding solvability of the control problem and of effectively synthesizing appropriate supervisors are solved for the case where the plant is represented by a deterministic Buchi automaton and the specification of legal behavior by a deterministic Rabin automaton.

Efficient simulation of hierarchical stochastic activity network models

Stochastic extensions to Petri nets have gained widespread acceptance as a method for describing the dynamic behavior of discrete-event systems. Both simulation and analytic methods have been proposed to solve such models. This paper describes a set of efficient procedures for simulating models that are represented as stochastic activity networks (SANs, a variant of stochastic Petri nets) and composed SAN-based reward models (SBRMs). Composed SBRMs are a hierarchical representation for SANs, in which individual SAN models can be replicated and joined together with other models, in an iterative fashion. The procedures exploit the hierarchical structure and symmetries introduced by the replicate operation in a composed SBRM to reduce the cost of future event list management. The procedures have been implemented as part of a larger performance-dependability modeling package known asUltraSAN, and have been applied to real, large-scale applications.

Stability of Discrete Event System Behavior

Abstract This paper studies the stability and stabilizability of Discrete Event Dynamical Systems (DEDS’s) in the framework of Ramadge and Wonham. We define stability and stabilizability in terms of the behavior of the DEDS’s, i. e. the language generated by the state machines (SM’s) used to model the DEDS. The notions of stability and stabilizability of DEDS’s are presented in a more general setting than in earlier works where they were defined in terms of legal and illegal states rather than strings. The notion of revarsal of languages is used to obtain algorithms for determining the stability and stabilizability of a given system. We also present an algorithm of linear complexity for computing the regions of attraction which is used for determining the stability and stabilizability of a given system defined in terms of legal states. This algorithm is then used to obtain more efficient tests of sufficiency of stability and stabilizability of systems defined in terms of languages.