Analysis of non-linear discrete event dynamic systems in (min, +) algebra

Under the name discrete event dynamic systems are grouped some systems whose dynamic behaviour cannot be described by differential equations. This class of systems includes many industrial systems, for which we study the flow entities (material, resources). This paper deals with the analysis of discrete event systems which can be modelled by timed event graphs with multipliers (TEGM). These models do not admit a linear representation in (min, +) algebra. This non-linearity is due to the presence of the weights on arcs. To mitigate this problem of non-linearity and to apply some basic results used to analysis the performances of linear systems in dioid algebra, we propose a linearisation method of mathematical model reflecting the behaviour of a TEGM in order to obtain a (min, +) linear model

Just-in-time control of time-varying discrete event dynamic systems in (max,+) algebra

We deal with timed event graphs whose holding times associated with places are variable. Defining a first-in-first-out functioning rule, we show that such graphs can be linearly described in (max,+) algebra. Moreover, this linear representation allows extending the just-in-time control synthesis existing for timed event graphs with constant holding times. An example is proposed in order to illustrate how the approach can be applied as a just-in-time strategy for production lines

On Steady State of Continuous Min-Plus Systems

We study the steady state of a class of continuous min-plus linear systems by using the notion of system type. The aim is to control systems in order that outputs asymptotically track certain polynomial reference inputs in relation with the just-in-time criterion. As in conventional system theory, the use of system type property gives very simple expression for the resulting controllers. Disturbances acting on the system output are also considered

Model reference control for timed event graphs in dioids

This paper deals with feedback controller synthesis for timed event graphs in dioids. We discuss here the existence and the computation of a controller which leads to a closed-loop system whose behavior is as close as possible to the one of a given reference model and which delays as much as possible the input of tokens inside the (controlled) system. The synthesis presented here is mainly based on residuation theory results and some Kleene star properties

Synthesis of greatest linear feedback for timed-event graphs in dioid

This paper deals with the synthesis of greatest linear causal feedback for discrete-event systems whose behavior is described in dioid. Such a feedback delays as far as possible the input of the system while keeping the same transfer relation between the input and the output. When a feedback exists in the system, the authors show how to compute a greater one without decreasing the system\u27s performance

Modeling and Control of Weight-Balanced Timed Event Graphs in Dioids

The class of Timed Event Graphs (TEGs) has widely been studied for the last 30 years thanks to an algebraic approach known as the theory of Max-Plus linear systems. In particular, the modeling of TEGs via formal power series has led to input-output descriptions for which some model matching control problems have been solved. In the context of manufacturing applications, the controllers obtained by these approaches have the effect of regulating material flows in order to decrease internal congestions and intermediate stocks. The objective of this work is to extend the class of systems for which a similar control synthesis is possible. To this end, we define first a subclass of timed Petri nets that we call Balanced Timed and Weighted Event Graphs (B-TWEGs). B-TWEGs can model synchronisation and delays (B-TWEGs contains TEGs) and can also describe some dynamic phenomena such as batching and event duplications. Their behavior is described by some rational compositions of four elementary operators γ n , δ t , μm and βb on a dioid of formal power series. Then, we show that the series associated to B-TWEGs have a three dimensional graphical representation with a property of ultimate periodicity. This modeling allows us to show that B-TWEGs can be handled thanks to finite and canonical forms. Therefore, the existing results on control synthesis, in particular the model matching control problem, have a natural application in that framework

Le produit synchrone des automates (max,+)

Une extension des automates (max,+) est étudiée dans le but de modéliser le parrallélisme (occurrence simultanée d\u27évements). Pour cela, on introduit une composition synchrone des automates (max, +) vus comme des automates temporisés. Ceci nous amène à introduire des automates (max, +) avec multi-événements qui correspondent à une classe des automates temporisés avec plusieurs horloges. Nous obtenons la formule pour le comportement de produit synchrone d\u27automates (max,+) et montrons que dans le cas général il n\u27est pas possible de définir le produit synchrone des comportement (séries formelles) sans prendre en compte leurs représentations par automates (max,+)