Topic

# Max-plus algebra

About: Max-plus algebra is a research topic. Over the lifetime, 150 publications have been published within this topic receiving 2269 citations.

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TL;DR: The conventional model predictive control framework is extended and adapted to max-plus-linear systems, i.e., discrete-event systems that can be described by models that are ''linear'' in the (max,+) algebra.

285 citations

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TL;DR: The specialization to max-plus algebra of Howard’s policy improvement scheme is described, which yields an algorithm to compute the solutions of spectral problems in the max- plus semiring.

200 citations

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TL;DR: The Hamilton--Jacobi--Bellman (HJB) equation associated with the {robust/\hinfty} filter (as well as the Mortensen filter) is considered.

Abstract: The Hamilton--Jacobi--Bellman (HJB) equation associated with the {robust/\hinfty} filter (as well as the Mortensen filter) is considered. These filters employ a model where the disturbances have finite power. The HJB equation for the filter information state is a first-order equation with a term that is quadratic in the gradient. Yet the solution operator is linear in the max-plus algebra. This property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off-line. The precomputed solutions are stored as vectors of coefficients of the basis functions. These coefficients are then used directly in the real-time computations.

122 citations

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TL;DR: In this article, the authors introduced a new structure of commutative semiring, generalizing the tropical semiring and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum.

Abstract: This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.

113 citations

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TL;DR: A max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems and derives a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\ Delta x(\ delta)^-1$, depending on the choice of the approximation.

Abstract: We introduce a max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup-norm can be bounded from the difference between the value function and its projections on max-plus and min-plus semimodules when the max-plus analogue of the stiffness matrix is exactly known. In general, the stiffness matrix must be approximated: this requires approximating the operation of the Lax-Oleinik semigroup on finite elements. We consider two approximations relying on the Hamiltonian. We derive a convergence result, in arbitrary dimension, showing that for a class of problems, the error estimate is of order $\delta+\Delta x(\delta)^{-1}$ or $\sqrt{\delta}+\Delta x(\delta)^{-1}$, depending on the choice of the approximation, where $\delta$ and $\Delta x$ are, respectively, the time and space discretization steps. We compare our method with another max-plus based discretization method previously introduced by Fleming and McEneaney. We give numerical examples in dimensions 1 and 2.

111 citations