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Some special cases in the stability analysis of multi-dimensional time-delay systems using the matrix Lambert W function

2015
*
Automatica
*

This paper revisits a recently developed methodology based

doi:10.1016/j.automatica.2015.01.016
fatcat:rap64fykjjebhiuvbl2gtimkra
*on**the*matrix*Lambert**W**function*for*the*stability analysis of linear time invariant, time delay systems. ... By studying a particular, yet common, second order system, we show that in general there is no*one*to*one*correspondence between*the*branches of*the*matrix*Lambert**W**function*and*the*characteristic roots ...*The*authors thank*the*comments received from*the*anonymous reviewers, which helped to improve*the*presentation of*the*paper, specially in*the*second example. ...##
###
A Robust Approximation to a Lambert-Type Function
[article]

2015
*
arXiv
*
pre-print

*The*

*function*$y = g(x) = \mathrm{log}\big(

*W*(e^x)\big)$, where $

*W*()$ denotes

*the*

*Lambert*

*W*

*function*, is

*the*solution to

*the*equation $y + e^y = x$. ...

*The*algorithm does not assume that

*the*chosen computer language implements

*the*

*Lambert*

*W*

*function*. ... When

*w*and z are restricted to having

*real*values,

*the*graph of

*the*

*Lambert*

*W*

*function*is as shown in figure 1. For further background regarding

*the*

*Lambert*

*W*

*function*, see [1, 2] . ...

##
###
Finding of roots of the matrix transcendental characteristic equation using Lambert W function

2011
*
Lietuvos matematikos rinkinys
*

*The*examples of

*the*application of

*the*method are presented. ...

*The*method of finding roots of

*the*matrix transcendental characteristic equation, corresponding to linear matrix differential equation with delayed argument, is analyzed. ... z n . (8)

*One*of

*the*advantages of

*the*

*Lambert*

*W*

*function*method is that

*one*can compute analytically values of

*the*

*Lambert*

*W*

*function*in all its branches using commands already installed in various software ...

##
###
Some remarks on the time of flight and range of a projectile in a linear resisting medium

2011
*
Journal of Engineering Science and Technology Review
*

In view of

doi:10.25103/jestr.041.04
fatcat:fnkm2tgp7va2rorf7p6si4smyi
*the*recent work by Karkantzakos [Journal of Engineering Science and Technology Review 2 (2009) 76-81], a number of remarks highlighting*the*connection between*the**Lambert**W**function*and*the*... In particular, we show how each of these quantities can be expressed in closed form in terms of*the**Lambert**W**function*and indicate how*the*analysis of*the*motion becomes greatly simplified by its introduction ... Appendix*The**Lambert**W**function*, denoted by*W*(z), is defined to be*the*inverse of*the**function*f(z) = ze z satisfying (11) Referred to as*the*defining equation for*the**Lambert**W**function*, Eq. (11) has ...##
###
Some physical applications of generalized Lambert functions

2016
*
European journal of physics
*

In this paper we review

doi:10.1088/0143-0807/37/6/065802
fatcat:k4qqzmg2sjgm7gvcrge2tbdi2y
*the*physical applications of*the*generalized*Lambert**function*recently defined by*the*first author. ... We also point out that*the*inverse Langevin*function*is nothing else but a specially parametrized generalized*Lambert**function*. ... Depending*on**the*parameter r,*the*r-*Lambert**function*has*one*, two or three*real*branches and so*the*above equations can have*one*, two or three solutions (we restrict our investigation to*the**real*line) ...##
###
Another Incarnation of the Lambert W Function
[article]

2017
*
arXiv
*
pre-print

*The*

*Lambert*

*W*

*function*was introduced by Euler in 1779, but was not well-known until it was implemented in Maple, and

*the*seminal paper of Corless, Gonnet, Hare, Jeffrey and Khuth was published in 1996 ... In this note we describe a simple problem, which can be straightforwardly solved in terms of

*the*

*W*

*function*. ...

*The*author is thankful to Professor German Kalugin for useful remarks, which help to improve

*the*manuscript. ...

##
###
Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2

2002
*
IEEE Transactions on Signal Processing
*

We show that this transformation is expressable in terms of a special

doi:10.1109/tsp.2002.801912
fatcat:lsadwmiw65gr3puopq5hstnckm
*function*known under*the*name of*the**Lambert**W**function*. ... We review*the*main methods for numerical evaluation of*the*relevant branch of*the*(multivalued)*Lambert**W**function*with controlled accuracy and complement them with an original rational*function*approximation ... ACKNOWLEDGMENT*The*authors would like to thank R. Corless for valuable informations*on**the**Lambert**W**function*. ...##
###
REALIZATION OF A LAMBERT W-FUNCTION FOR A CHAOTIC CIRCUIT

2013
*
Journal of Circuits, Systems and Computers
*

Both parts form a new circuit for a

doi:10.1142/s0218126613500758
fatcat:cvhvia5wgrawpgro6f2d44pnbu
*Lambert**W*-*function*by enabling a term of an emitter current to be simultaneously appeared*on*both sides of an exponential equation. ...*The*new circuit is capable of both a*Lambert**W*-*function*and a current-tunable chaotic oscillator. ... Acknowledgments*The*authors are graceful to Mr. Theerawat Kiatdarakun for his useful suggestion. ...##
###
Application of Lambert W Function in Oscillation Theory

2014
*
Acta Electrotechnica et Informatica
*

*The*basic concepts and results keen to

*the*oscillatory nature of solutions, as well as stability analysis are re-investigated using

*the*properties of

*Lambert*

*W*

*function*and compared with existing

*ones*. ... A brief survey of

*the*

*function*origin and possible applications is presented together with a short behavior analysis. ...

*The*

*real*part of

*the*

*Lambert*

*function*is unbounded, although

*the*same property can be observed. ...

##
###
Lambert W Function In Hydraulic Problems

2012
*
Zenodo
*

But unfortunately,

doi:10.5281/zenodo.890013
fatcat:4j3i6apjdvg5jpbit73wmrmdtu
*the**Lambert**W**function*itself cannot be solved easily without approximation. ...*Lambert**W**function*is implicitly elementary but is not, itself, an elementary*function*. ...*On**the**Lambert**W*-*function**Lambert**W*-*function*can be noted as (3.1):*W*(x) · e*W*(x) = x (3.1) For*real*values of*the*argument, x,*the**W*-*function*has two branches, W0 (*the*principal branch) and*W*-1 (*the*...##
###
W solutions of the CW equation for flow friction

2011
*
Applied Mathematics Letters
*

Alternate mathematical equivalents to

doi:10.1016/j.aml.2011.03.014
fatcat:jxzismoulfeodb2ermggcpybg4
*the*original expression of*the*CW equation, but now in*the*explicit form developed using*the**Lambert**W*-*function*, are shown (with related solutions). ...*The**W*-*function*is also transcendental, but it is used more general compared with*the*CW*function*. ...*W*-*function**Lambert**W*and CW are transcendental*functions*.*The*(*real*-valued)*Lambert**W*-*function*is a solution of*the*nonlinear equation*W*· e*W*= x. ...##
###
On the generalization of the Lambert $W$ function

2017
*
Transactions of the American Mathematical Society
*

*The*

*Lambert*

*W*

*function*, giving

*the*solutions of a simple transcendental equation, has become a famous

*function*and arises in many applications in combinatorics, physics, or population dyamics just to mention ... In this paper we construct and study in great detail a generalization of

*the*

*Lambert*

*W*which involves some special polynomials and even combinatorial aspects. ... Depending

*on*

*the*parameter r,

*the*r-

*Lambert*

*function*has

*one*, two or three

*real*branches and so

*the*above equations can have

*one*, two or three solutions (we restrict our investigation to

*the*

*real*line) ...

##
###
Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
[article]

2018
*
arXiv
*
pre-print

*The*existence condition of

*the*desired eigenvalue is established when

*the*current and delay states are present in

*the*feedback loop. Design of

*the*feedback controller is then followed. ... Furthermore, eigenvalue assignment for

*the*input-delay system is also obtained as well. Numerical examples illustrate

*the*procedure of assigning

*the*desired eigenvalue. ... Acknowledgement This work was partially supported under

*the*grant No. MOST 106-2115-M-029-004. ...

##
###
Analysis of neural networks with time-delays using the Lambert W function

2011
*
Proceedings of the 2011 American Control Conference
*

For stability analysis,

doi:10.1109/acc.2011.5991085
fatcat:qolodfyzwzbr3nyk6mhdjxhtha
*the*dominant characteristic roots are obtained by using an approach based*on**the**Lambert**W**function*. ... Conditions for oscillation and robust stability are also given in term of*the**Lambert**W**function*. ... As seen in Fig. 4 , if*the**real*argument of*the**Lambert**W**function*, H, is equal to or greater than −1/e,*the*value of*W*0 (H) is*real*and*the*network does not show oscillation. ...##
###
Transcendental Numbers and the Lambert-Tsallis Function
[article]

2020
*
arXiv
*
pre-print

*The*Tsallis q-exponential, e_q (z), and

*the*

*Lambert*-Tsallis W_q (z)

*function*, where q is a

*real*parameter, are, respectively, generalizations of

*the*exponential and

*Lambert*

*functions*. ... To decide upon

*the*arithmetic nature of some numbers may be a non-trivial problem. Some cases are well know, for example exp(1) and

*W*(1), where

*W*is

*the*

*Lambert*

*function*, are transcendental numbers. ... Acknowledgements This study was financed in part by

*the*Coordenação de Aperfeiçoamento de Pessoal ...

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