Abstract
Tropical Algebra is used to model the dynamics of Timed Event Graphs (TEG), a particular class of Timed Discrete-Event System (TDES) in which we are interested only in synchronization and delay phenomena. Whenever this TEG has control inputs, we can use them to control the synchronization of the system to achieve some objective. Thus, this paper formulates a framework based on tropical algebra and lexicographic optimization to synchronize a TEG when dealing with many synchronization objectives that are ranked in previous priority order. We call this kind of problem the Tropical Lexicographic Synchronization Optimization (TLSO). This work develops a solution to this problem, based on Tropical Fractional Linear Programming (TFLP) and lexicographic optimization concepts. In this way, the basics of tropical algebra are determined, including essential terms to this paper, such as left and right residuations, and the following stages of the solution to the TLSO problem are explained. Therefore, this work presents a general framework based on structured algebraic models with application to TEG synchronization. By synchronization, we mean balancing and organizing events chronologically in order to achieve the desired goal. So, we are dealing with concepts closely related to symmetry ones. An illustrative numerical example is presented, which demonstrates the implementation of the proposed algorithms. The acquired results confirm the efficiency of the proposed methodology. Codes used for implementing the algorithms are listed in the appendix section of the article.
Highlights
Timed Event Graphs (TEGs) are used to model a wide range of Timed Discrete-Event Systems (TDES), which is a class of dynamic systems in which the change of state is governed by events [1]
We propose the TLSO (Tropical Lexicographic Synchronization Optimization) framework
This framework is based on TFLP (Tropical Fractional Linear Programming), which allows the formulation and solution of problems in which it is desirable to synchronize transitions of
Summary
Timed Event Graphs (TEGs) are used to model a wide range of Timed Discrete-Event Systems (TDES), which is a class of dynamic systems in which the change of state is governed by events [1]. The authors propose a framework to synchronize the firing times of the transition of a TEG, considering a priority list of the pair of transitions that must be synchronized This framework models this problem as a Lexicographic variation of a TFLP [10,11]. As far as the authors’ knowledge goes, there are few works regarding linear fractional programming applications, and the authors do not find researches that deal with the insertion of lexicographic optimization in this context. This paper refers to Egmund and Olsder [13], who introduce the basics of Max-Plus algebra to deal with synchronization between phases in the traffic light system, with the purpose to generate the green waves. The software is very convenient to work with tropical algebra since it has many of the related functions native implemented. (Version 4.4) code for the algorithm
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