Control of Discrete-Event Systems pp 451-469

Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 433)

# Discrete-Event Systems in a Dioid Framework: Control Theory

• Laurent Hardouin
• Olivier Boutin
• Bertrand Cottenceau
• Thomas Brunsch
• Jörg Raisch
Chapter

## Abstract

In this chapter, we will recall contributions to dioid theory dealing with control that were achieved during the last two decades. Just like in classical control engineering, control is to be understood as having an action on the inputs so as to adapt to given specifications. For instance, one could aim at finding an optimal control in order to track an a priori known output trajectory. Since inversion, which would be necessary for such a computation, does not exist in general in a dioid framework, we will also present notions of residuation theory, which introduces pseudo-inverses that are suitable to our needs. Provided a model of a system and a specified output for it, it can be shown that there exists a greatest input that leads to an output which is lower than or equal to the specified one. In practice, this greatest solution implies that all the events occur as late as possible while ensuring that the output events occur before the ones given by the specified output. In a production management context, this comes down to delaying as much as possible the input of raw parts in the manufacturing system, while ensuring a predefined throughput; hence the internal stock is reduced as much as possible. The control strategy is then optimal according to the just-in-time criterion. This chapter will provide the results allowing to synthesize this optimal control. But this kind of open-loop strategy does not take the real-time response of the system into account. So we will also extend control strategies to closed-loop ones, which allow to react to possible disturbances. For each control strategy an illustrative example dealing with a High-Throughput Screening system, which has served as a case study within the DISC project, will be given.

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## Authors and Affiliations

• Laurent Hardouin
• 1
• Olivier Boutin
• 2
• Bertrand Cottenceau
• 1
• Thomas Brunsch
• 1
• 3
• Jörg Raisch
• 3
• 4
1. 1.LUNAM, University of Angers, LISA, ISTIAAngersFrance
2. 2.Calle Santiago 2 – 4∘CCadizSpain
3. 3.Fachgebiet RegelungssystemeTechnische Universität BerlinBerlinGermany
4. 4.Fachgruppe System- und RegelungstheorieMax-Planck-Institut für Dynamik komplexer technischer SystemeMagdeburgGermany