Model decomposition of timed event graphs under periodic partial synchronization: application to output reference control

Timed Event Graphs (TEGs) are a graphical model for decision free and time-invariant Discrete Event Systems (DESs). To express systems with time-variant behaviors, a new form of synchronization, called partial synchronization (PS), has been introduced for TEGs. Unlike exact synchronization, where two transitions t1,t2 can only fire if both transitions are simultaneously enabled, PS of transition t1 by transition t2 means that t1 can fire only when transition t2 fires, but t1 does not influence the firing of t2. This, for example can describe the synchronization between a local train and a long distance train. Of course it is reasonable to synchronize the departure of a local train by the arrival of long distance train in order to guarantee a smooth connection for passengers. In contrast, the long distance train should not be delayed due to the late arrival of a local train. Under the assumption that PS is periodic, we can show that the dynamic behavior of a TEG under PS can be decomposed into a time-variant and a time-invariant part. It is shown that the time-variant part is invertible and that the time-invariant part can be modeled by a matrix with entries in the dioid ${\mathcal{M}}_{in}^{ax}\left [\!\left [\gamma ,\delta \right ]\!\right ]$ M i n a x γ , δ , i.e. the time-invariant part can be interpreted as a standard TEG. Therefore, the tools introduced for standard TEGs can be used to analyze and to control the overall system. In particular, in this paper output reference control for TEGs under PS is addressed. This control strategy determines the optimal input for a predefined reference output. In this case optimality is in the sense of the ”just-in-time” criterion, i.e., the input events are chosen as late as possible under the constraint that the output events do not occur later than required by the reference output.