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

Abstract

The class of timed event graphs (TEGs) has widely been studied thanks to an approach known as the theory of max-plus linear systems. In particular, the modeling of TEGs via formal power series in a dioid called Min <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ax</sup> [[ γ,δ]] has led to input-output representations on which some model matching control problems have been solved. Our work attempts to extend the class of systems for which a similar control synthesis is possible. To this end, a subclass of timed Petri nets that we call weight-balanced timed event graphs (WBTEGs) will be first defined. They can model synchronization and delays (WBTEGs contain TEGs) and can also describe dynamic phenomena such as batching and event duplications (unbatching). Their behavior is described by rational compositions (sum, product and Kleene star) of four elementary operators γ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> , μm, and βb on a dioid of formal power series denoted ε* [[ δ]]. The main feature is that the transfer series of WBTEGs have a property of ultimate periodicity (such as rational series in M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ax</sup> [[ γ,δ]]). Finally, the existing results on control synthesis for max-plus linear systems find a natural application in this framework.