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 t 1 , t 2 can only fire if both transitions are simultaneously enabled, PS of transition t 1 by transition t 2 means that t 1 can fire only when transition t 2 fires, but t 1 does not influence the firing of t 2 . 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 M i n a x γ , δ ${\mathcal{M}}_{in}^{ax}\left [\!\left [\gamma ,\delta \right ]\!\right ]$ , 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.

中文翻译：

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周期性部分同步下定时事件图的模型分解：应用于输出参考控制

定时事件图 (TEG) 是一种用于无决策和时不变离散事件系统 (DES) 的图形模型。为了表达具有时变行为的系统，已为 TEG 引入了一种称为部分同步 (PS) 的新同步形式。与精确同步不同，其中两个转换 t 1 ，t 2 只能在两个转换同时启用时触发，转换 t 1 的 PS 由转换 t 2 意味着 t 1 只能在转换 t 2 触发时触发，但 t 1 不影响t 2 的点火。例如，这可以描述慢车和长途列车之间的同步。当然，为了保证乘客的顺利接驳，长途列车到站同步慢车发车也是合理的。相比之下，长途列车不应因慢车晚点而延误。在 PS 是周期性的假设下，我们可以证明 PS 下 TEG 的动态行为可以分解为时变部分和时不变部分。结果表明，时变部分是可逆的，并且时变部分可以通过一个矩阵建模，该矩阵的元素为二元组 M inax γ , δ ${\mathcal{M}}_{in}^{ax} \left [\!\left [\gamma ,\delta \right ]\!\right ]$ ，即时不变部分可以解释为标准 TEG。因此，为标准 TEG 引入的工具可用于分析和控制整个系统。特别是，在本文中，PS 下 TEG 的输出参考控制得到了解决。该控制策略确定预定义参考输出的最佳输入。