Duality and interval analysis over idempotent semirings

Brunsch, Thomas; Hardouin, Laurent; Maia, Carlos Andrey; Raisch, Jörg

doi:10.1016/j.laa.2012.06.025

Cited by 18 publications

(19 citation statements)

References 23 publications

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“…Interval arithmetic is presented in [33] and extended to Max Plus Algebra in [34], [35], [6], [36]. An interval can be defined by a pair of entries of the type (a, ) as follows:…”

Section: A Interval Analysismentioning

confidence: 99%

An Algorithm to Compute the Inverse Image of a Point With Respect to a Nondeterministic Max-Plus Linear System

Cândido

Hardouin

Lhommeau

et al. 2021

IEEE Trans. Automat. Contr.

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Max Plus Linear (MPL) systems are often described by a transition function, which models the state evolution of the system, and a measurement function, which binds the measures with the system states. Methods for computing the inverse image of a point w.r.t. the measurement function are particularly interesting in applications where it is desirable to obtain informations about the system states based on the output observations. The inverse image of a set w.r.t. a nondeterministic MPL system, called uncertain MPL (uMPL) system, can be computed by using the Difference-Bound Matrices (DBM) approach. In this work we aim to use an interval analysis to propose a method to compute the inverse image of a point w.r.t. an uMPL system. The algorithm proposed has a lower worst-case complexity compared with the DBM approach as previously proposed in the literature.

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“…Interval arithmetic is presented in [33] and extended to Max Plus Algebra in [34], [35], [6], [36]. An interval can be defined by a pair of entries of the type (a, ) as follows:…”

Section: A Interval Analysismentioning

confidence: 99%

An Algorithm to Compute the Inverse Image of a Point With Respect to a Nondeterministic Max-Plus Linear System

Cândido

Hardouin

Lhommeau

et al. 2021

IEEE Trans. Automat. Contr.

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“…Les premiers travaux sur l'analyse par intervalles dans l'algèbre max-plus sont issus de (Litvinov, Sobolevskii, 2001). Depuis, de nombreux auteurs se sont intéressés à l'utilisation des intervalles dans l'algèbre max-plus (Lhommeau et al, 2004), (Cechlárová, 2005), (Brunsch et al, 2012).…”

Section: Généralités Sur Les Fonctions Min-max-plusunclassified

Sur la stabilisation de systèmes min-max-plus incertains

Lhommeau¹,

Hardouin²,

Santos-Mendes³

2013

Journal Européen des Systèmes Automatisés

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Une grande variété de problèmes issus des réseaux informatiques, des systèmes de production, etc... peuvent être modélisés par des systèmes min-max-plus. Ces systèmes sont une extension non-linéaire des systèmes max-plus. Dans ce papier, on s'intéresse aux systèmes minmax-plus où les paramètres sont mal connus mais supposés bornés. On montre que pour ces systèmes il est possible de définir un correcteur qui permet de stabiliser le système incertain. ABSTRACT. A variety of problems in computer networks, manufacturing systems, etc... can be modeled as min-max-plus systems. They are non-linear extension of the well known linear max-plus systems. In this paper, we focus on the min-max-plus systems where the parameters are unknown but assumed bounded. We show that for these uncertain systems it is possible to define a state feedback that stabilizes the uncertain system.

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“…Gaubert [8] examines the greatest solution of the K ⊗ X L equation with K ∈ S n×p and X ∈ S p×1 . Furthermore, Hardouin [4] also discusses the greatest solution of the K ⊗ X L equation, but K ∈ S n×p and X ∈ S p×m .…”

Section: Introductionmentioning

confidence: 99%

“…As you are familiar, while idempotent has the multiplicative operation of ⊗ distributive to the sum of ⊕. Subsequently, Hardouin [4] introduced the duality of the ⊗ multiplication operation, which is a dual product that has the distributive distributive property of ∧. In addition to dual products on idempotent semiring, dual products are also given on the top matrix of idempotent semiring notated K X, defined by operation (K X) ij = p=1,...,n (k ip x pj ) with ∧ is the greatest lower bound.…”

Section: Introductionmentioning

confidence: 99%

“…Ouerght and Hardouin [15] use the greatest solution of this equation to solve optimal control problems for P-temporal event graphs, that is, looking for the greatest control that contented the output and the constraints imposed on the adjusted model problem (the model matching problem). Furthermore, Hardouin together with Brunsch, Maia, Raisch [4] explained the characteristics of the equation K ⊗ X X L X related to the set of solutions. The author completes the proof of the properties of the equation K ⊗ X X L X.…”

Section: Introductionmentioning

confidence: 99%

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THE GREATEST SOLUTION IN THE INEQUALITY OF K X X LX WITH K 2 ISnn;L 2 ISnn;X 2 ISnm ARE A COMPLETE IDEMPOTENT SEMIRINGS OF INTERVAL

Susilowati

2021

JMT

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The greatest solution of an inequality KX X LX to solve the optimalcontrol problem for P-Temporal Event Graphs, which is to nd the optimal control thatmeets the constraints on the output and constraints imposed on the adjusted model prob-lem (the model matching problem). We give the greatest solution K X X L Xand X H with K; L;X;H matrices whose are entries in a complete idempotent semir-ings. Furthermore, the authors examine the existence of a sucient condition of theprojector in the set of solutions of inequality K X X L X with K; L;X matrixwhose entries are in the complete idempotent semiring. Projectors can be very necessaryto synthesize controllers in manufacturing systems that are constrained by constraintsand some industrial applications. The researcher then examines the requirements forthe presence of the greatest solution was called projector in the set of solutions of theinequality K X X L X with K; L;X matrices whose are entries in an completeidempotent semiring of interval. Researchers describe in more detail the proof of theproperties used to resolve the inequality K X X L X. Before that, we givethe greatest solution of the inequality KX X LX and X G with K; L;X;Gmatrices whose are entries in an complete idempotent semiring of interval

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Duality and interval analysis over idempotent semirings

Cited by 18 publications

References 23 publications

An Algorithm to Compute the Inverse Image of a Point With Respect to a Nondeterministic Max-Plus Linear System

An Algorithm to Compute the Inverse Image of a Point With Respect to a Nondeterministic Max-Plus Linear System

Sur la stabilisation de systèmes min-max-plus incertains

THE GREATEST SOLUTION IN THE INEQUALITY OF K X X LX WITH K 2 ISnn;L 2 ISnn;X 2 ISnm ARE A COMPLETE IDEMPOTENT SEMIRINGS OF INTERVAL

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