Analysis and control of max-plus linear discrete-event systems: An introduction

Schutter, Bart De; Boom, T.J.J. van den; Xu, Jie; Farahani, Samira S.

doi:10.1007/s10626-019-00294-w

Cited by 28 publications

(20 citation statements)

References 99 publications

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“…Therefore, in this section, we introduce some concepts and definitions related to this algebra. For more details on academic background, consult a historical review of Max-Plus tropical algebra [2,17,18]. "Tropical algebra" is a generic name for a family of idempotent algebraic structures, which can be a semiring or a semifield depending on the application.…”

Section: Tropical Algebramentioning

confidence: 99%

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Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

2020

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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.

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Section: Tropical Algebramentioning

confidence: 99%

“…Let n and m be the number of rows and columns of R, respectively. With this definition, we can rewrite (17) as max…”

Section: Tropical Fractional Linear Programmingmentioning

confidence: 99%

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

2020

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“…Generally, discrete event systems in conventional algebra often lead to a nonlinear description. However, the model is capable of reducing to "linear" for certain subclass of discrete event systems when formulated in the max-plus algebra with max (⨁) and plus (⨂) as its basic operations [3,5,7]. Precisely, discrete event systems whereby no concurrency, which means only the synchronization is been modeled via the operations plus (⨂) (conforming to the period of events: the starting time plus the duration equal the finishing time of an operation), and max (⨁) (conforming to synchronization: a new operation begins immediately the current operations finish) [8,9].…”

Section: Introductionmentioning

confidence: 99%

On Application of Max-Plus Algebra to Synchronized Discrete Event System

Aminu¹,

Olowo²,

Sulaiman³

et al. 2021

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“…The role of max-plus algebra in the field of discrete event systems and wide variety of topics in this algebra were presented in [2]. How the concept of max-plus algebra can be used in modeling discrete event systems was described in [3]. Theory regarding fuzzy sets and their applications was published in [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, since each cycle of length 2 is connected to a loop, by Lemma 4 all such cycles are in a common non-trivial strongly connected component. Using Theorem 5 each matrix A ∈ A M is robust, thus A M is universally robust (3). Let the non-trivial strongly connected component of G(A, 1) be generated by more than two nodes.…”

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confidence: 99%

Robustness of Interval Monge Matrices in Fuzzy Algebra

2020

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Max–min algebra (called also fuzzy algebra) is an extremal algebra with operations maximum and minimum. In this paper, we study the robustness of Monge matrices with inexact data over max–min algebra. A matrix with inexact data (also called interval matrix) is a set of matrices given by a lower bound matrix and an upper bound matrix. An interval Monge matrix is the set of all Monge matrices from an interval matrix with Monge lower and upper bound matrices. There are two possibilities to define the robustness of an interval matrix. First, the possible robustness, if there is at least one robust matrix. Second, universal robustness, if all matrices are robust in the considered set of matrices. We found necessary and sufficient conditions for universal robustness in cases when the lower bound matrix is trivial. Moreover, we proved necessary conditions for possible robustness and equivalent conditions for universal robustness in cases where the lower bound matrix is non-trivial.

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Analysis and control of max-plus linear discrete-event systems: An introduction

Cited by 28 publications

References 99 publications

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

On Application of Max-Plus Algebra to Synchronized Discrete Event System

Robustness of Interval Monge Matrices in Fuzzy Algebra

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