Abstract
The objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is “linear” in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.
Highlights
In recent years both industry and the academic world have become more and more interested in techniques to model, analyze, and control complex discrete-event systems (DESs) such as flexible manufacturing systems, telecommunication networks, multiprocessor operating systems, railway networks, traffic control systems, logistic systems, intelligent transportation systems, computer networks, multi-level monitoring and control systems, and so on
In general DESs lead to a nonlinear description in conventional algebra, there exists a subclass of DESs for which this model becomes “linear” when it is formulated in the max-plus algebra (Baccelli et al 1992; Cuninghame-Green 1979; Heidergott et al 2006; Butkovic 2010), which has maximization and addition as its basic operations
We provide several worked examples for basic max-plus concepts, we include several references to recent literature, and we present some results not included in previous surveys
Summary
In recent years both industry and the academic world have become more and more interested in techniques to model, analyze, and control complex discrete-event systems (DESs) such as flexible manufacturing systems, telecommunication networks, multiprocessor operating systems, railway networks, traffic control systems, logistic systems, intelligent transportation systems, computer networks, multi-level monitoring and control systems, and so on. DESs in which only synchronization and no concurrency or choice occur can be modeled using the operations maximization (corresponding to synchronization: a new operation starts as soon as all preceding operations have been finished) and addition (corresponding to the duration of activities: the finishing time of an operation equals the starting time plus the duration). This leads to a description that is “linear” in the max-plus algebra. We provide several worked examples for basic max-plus concepts, we include several references to recent literature, and we present some results not included in previous surveys (such as, e.g., two-sided systems of linear maxplus equations, systems of max-plus-algebraic polynomial equations and inequalities, and model-based predictive control for max-plus linear systems)
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