Control of Temporal Constraints Based on Dioid Algebra for Timed Event Graphs

Abstract

We consider a class of controlled timed event graphs subject to strict temporal constraints. Such a graph is deterministic, in the sense that its behavior only depends on the initial marking and on the control that is applied. As it is well-known, this behavior can be modelled by a system of difference equations that are linear in the min-plus algebra (/spl Ropf/ /spl cup/{+/spl infin/}, min, plus). The temporal constraint is represented by an inequation, that is also linear in the min-plus algebra. Then, a method for the synthesis of a control law ensuring the respect of the constraint is described. Two sufficient conditions are given, in terms of initial tokens and delays along the graph. We give explicit formulas characterizing a control law, which, if the conditions are satisfied, ensures the validity of the temporal constraints. This control law is also defined as a linear system over the min-plus algebra. It is a causal state feedback, involving delays. The method is illustrated on a production system.