Optimal Control for (max,+)-linear Systems in the Presence of Disturbances

This paper deals with control of (max,+)-linear systems when a disturbance acts on system state. In a first part we synthesize the greatest control which allows to match the disturbance action. Then, we look for an output feedback which makes the disturbance matching. Formally, this problem is very close to the disturbance decoupling problem for continuous linear systems

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

Kanban policy improvement thanks to a (max,+)-algebra analysis

This article deals with a possible improvement of a Kanban policy due to a (max,+)-algebra analysis. We show that for a given Kanban system, it is possible to change the original feedback loop by a (max,+)-linear controller which keeps the same quality of service but reduces the work in process. This new control policy contains a (max,+)-linear dynamic behaviour for the recycling of Kanban cards

Modeling of Time-Varying (max,+) Systems by means of Weighted Timed Event Graphs

International audienceThe (max,+) theory allows to describe the behavior of Timed Event Graphs (TEG) with constant holding times. For time-varying systems, a class of First-In First-Out TEGs with periodic holding times has already been studied in the literature. We show here that such time-varying TEGs can be modeled by equivalent Weighted TEGs for which an input-output model exists. In summary, we can describe FIFO TEGs with periodic holding times by means of ultimately periodic formal series in a dioid denoted E * per δ