3,498 research outputs found
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.Comment: 22 pages, 3 figures (6 eps files
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
The tropical analogue of polar cones
We study the max-plus or tropical analogue of the notion of polar: the polar
of a cone represents the set of linear inequalities satisfied by its elements.
We establish an analogue of the bipolar theorem, which characterizes all the
inequalities satisfied by the elements of a tropical convex cone. We derive
this characterization from a new separation theorem. We also establish variants
of these results concerning systems of linear equalities.Comment: 21 pages, 3 figures, example added, figures improved, notation
change
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 discussed. An
application to the study of transportation networks which evolve according to a
timetable is considered.Comment: 24 pages, 1 Postscript figure, proof of Lemma 1 and some references
adde
The tropical double description method
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be solved
in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an application
to static analysis) showing that the method outperforms experimentally the
previous ones by orders of magnitude. The present tools also lead to worst case
bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on
Theoretical Aspects of Computer Science, 2010, Nancy, Franc
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, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande
Green's J-order and the rank of tropical matrices
We study Green's J-order and J-equivalence for the semigroup of all n-by-n
matrices over the tropical semiring. We give an exact characterisation of the
J-order, in terms of morphisms between tropical convex sets. We establish
connections between the J-order, isometries of tropical convex sets, and
various notions of rank for tropical matrices. We also study the relationship
between the relations J and D; Izhakian and Margolis have observed that for the semigroup of all 3-by-3 matrices over the tropical semiring with
, but in contrast, we show that for all full matrix semigroups
over the finitary tropical semiring.Comment: 21 pages, exposition improve
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