Partial synchronicity and the (max,+) semiring

Rational semimodules over the max-plus semiring and geometric approach of discrete event systems

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule S^n over a semiring S is rational if it has a generating family that is a rational subset of S^n, S^n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules are stable under the natural algebraic operations (union, product, direct and inverse image, …

Nonlinear left and right eigenvectors for max-preserving maps

It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean $n$-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate stable dynamical systems. For these monotone maps, the closure is used to define suitable notions of left and right eigenvectors that are characterized by inequalities. Some explicit examples are given and applications in the construction of Lyapunov functions are described.

Duality between invariant spaces for max-plus linear discrete event systems

We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system.

Max-plus (A,B)-invariant spaces and control of timed discrete event systems

The concept of (A,B)-invariant subspace (or controlled invariant) of a linear dynamical system is extended to linear systems over the max-plus semiring. Although this extension presents several difficulties, which are similar to those encountered in the same kind of extension to linear dynamical systems over rings, it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings. Sufficient conditions are given for computing the maximal (A,B)-invariant subspace contained in a given space and the existence of linear state feedbacks is d…

关于一个离散的max-plus运输问题

On a discrete max-plus transportation problem

本文提出了解决热带最优运输问题的显式算法，指出在特定条件下最优热带计划可能不对应于完美匹配，并证明了解的唯一性罕见。

无线传感器网络中鲁棒最大共识的分析与设计

Analysis and Design of Robust Max Consensus for Wireless Sensor Networks

本文提出了一种新的分布式算法，用于在存在加性通信噪声的情况下估计网络中节点初始状态值的最大值。传统方法在每个节点本地估计最大值，但加性信道噪声导致估计值在每次迭代中漂移，从而使得节点远离真实最大值。本文利用max-plus代数研究这一遍历过程，并通过分析max-plus Lyapunov指数来研究噪声引起的状态值增长速率。最终提出了一种对网络中加性噪声具有鲁棒性的两阶段算法，并利用集中不等式分析其方差。