Synchronization For Timed Event Graph In Max-Plus Algebra

Timed event graph is one kind of timed discrete-event systems model which can beexpressed by linear equations in max-plus algebra. The synchronization of the timedevent graph has important theoretical and practical significance in many applications,such as the periodic time of legged locomotion and the periodic time of manufacturingsystems.In the thesis, we study the synchronization of the timed event graph. In the au-tonomous systems, we control the system by adding constraints between the stronglysubsystems. And the properties and theorems of the synchronized systems are given. Inthe non-autonomous systems, we get the synchronization by state feedback control oroutput feedback control when each event is caused by input. And the minimum periodictime is given. The application of the synchronization of timed event graph in railway andmanufacturing system is given in the last.The thesis is divided into six parts.In the part of introduction, we introduce the related background and status of thesynchronization of the timed event graph.In the first chapter, we give the basic knowledge, such as the definition and propertiesof max-plus algebra, the correspondence relation between timed event graphs and thestate equations, and the theorems about the matrix and its precedence graph.In the second chapter, we study the synchronization of autonomous systems in threeconditions. We give the definition of the synchronization of timed event graph and neces-sary and sufficient conditions for synchronization of system when ?? = 1, ?? = 1. In thesynchronized systems, the occur of events is cyclical and the periodic time is the maxi-mum weights of all elementary circuits of the precedence graph of ??. We also pointed outthat the periodic time is the eigenvalue of matrix ??, the state vectors is correspondingeigenvectors. Finally, the coupling time is given. When ?? = 1, ?? ≥ 2, we find outa synchronized timed event graph and make the system synchronized by constructing asimilar one. Similar synchronization is a special case of the synchronization. A modelof similar synchronized system is given, and it is proved that the periodic time is theeigenvalue of , the state vectors is corresponding eigenvectors.In the third chapter, we study the synchronization of non-autonomous systems. Weobtain the synchronization of non-autonomous systems by constructing state feedbackor output feedback. And we get the causal feedback matrix ?? by restricting the states.We also proved that the periodic time is the eigenvalue of ????, the state vectors iscorresponding eigenvectors if the system is synchronized. Let all of the elements of ?? are?? to make the interval time of inputs to the same. In this way, we get the synchronizationof non-autonomous systems.In Chapter 4, the application of the synchronization of autonomous systems in rail-way and the application of the synchronization of non-autonomous systems in manu-facturing system are introduced. We also get the timetable of railway stations and theperiodic time of manufacturing system.Finally, we summarize the main conclusions of this thesis, and raise the issue whichneeds further research, such as the synchronization of complex timed event graph.