Invertible incline matrices and Cramer's rule over inclines

Han, Song-Chol; Li, Hongxing

doi:10.1016/j.laa.2004.03.025

Cited by 20 publications

(6 citation statements)

References 13 publications

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“…We are rather motivated by the basic problems of the tropical linear algebra, which we are going to consider here in the context of max-Lukasiewicz semiring. We also remark that max-Lukasiewicz semiring can be seen as a special case of incline algebras of Cao, Kim and Roush [9], see also, e.g., Han-Li [28] and Tan [40]. One of the main problems considered in that algebra is to study the periodicity of matrix powers over a larger class of semirings, using lattice theory, lattice ordered groups and residuations.…”

Section: Introductionmentioning

confidence: 99%

Tropical linear algebra with the Łukasiewicz T-norm

Gavalec

Němcová

Sergeev

2015

Fuzzy Sets and Systems

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The max-Lukasiewicz semiring is defined as the unit interval [0, 1] equipped with the arithmetics "a + b" = max(a, b) and "ab" = max(0, a + b − 1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.

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Section: Introductionmentioning

confidence: 99%

Tropical linear algebra with the Łukasiewicz T-norm

Gavalec

Němcová

Sergeev

2015

Fuzzy Sets and Systems

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“…The invertibility of matrices on general algebraic structures were also discussed in [7], [9], [19], [21], and [24].…”

Section: Introductionmentioning

confidence: 99%

A Heuristic Method to Compute the Approximate Postinverses of a Fuzzy Matrix

2014

IEEE Trans. Fuzzy Syst.

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1347 pure-feedback system (55) with unmeasured states and unknown deadzone input. VI. CONCLUSIONIn this paper, an adaptive fuzzy backstepping control approach has been proposed for a class of pure-feedback nonlinear systems with unmeasured states and unknown dead zone. The fuzzy logic systems are used to approximate the unknown nonlinear functions and a state filter observer is designed to estimate the unmeasured states. Based on the information of the dead-zone slopes as well as treating the unknown inputs coefficients as a system uncertainty, the designed state filter observer and backstepping design technique, a novel adaptive fuzzy backstepping output feedback control scheme has been developed and the stability of the closed-loop system, and the convergences of the observer and tracking errors have been proved. The proposed control method has extended the existing results on strict-feedback control to the counterparts on pure-feedback control. REFERENCES[1] D. X. Gao, Z. Q. Sun, and J. H. Liu, "Dynamic inversion control for a class of pure-feedback systems," Asian

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“…Zhao [31] formulated necessary and sufficient conditions for the existence of the inverses over Brouwerian lattices. Han and Li [17] gave some necessary and sufficient conditions for an incline matrix to be invertible and presented Cramer's ruler over inclines. Tan [30] considered the invertible matrices over general commutative zerosumfree semirings (a commutative zerosumfree semiring is called an antiring in [30]), obtained some necessary and sufficient conditions for a matrix over a commutative antiring to be invertible and showed Cramer's ruler, therefore generalized the corresponding results in the literature for Boolean matrices, fuzzy matrices, lattice matrices and incline matrices.…”

Section: Introductionmentioning

confidence: 99%