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

Rudy Cepeda-Gomez, Wim Michiels
2015 Automatica  
This paper revisits a recently developed methodology based 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.  ... 
doi:10.1016/j.automatica.2015.01.016 fatcat:rap64fykjjebhiuvbl2gtimkra

A Robust Approximation to a Lambert-Type Function [article]

Ken Roberts
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] .  ... 
arXiv:1504.01964v1 fatcat:ldcsuxho2fab3dpgwb2wwncvia

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

Irma Ivanovienė, Jonas Rimas
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  ... 
doi:10.15388/lmr.2011.mt02 fatcat:wck64lv2infndnsdi4qowzhb7e

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

S. M. Stewart
2011 Journal of Engineering Science and Technology Review  
In view of 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  ... 
doi:10.25103/jestr.041.04 fatcat:fnkm2tgp7va2rorf7p6si4smyi

Some physical applications of generalized Lambert functions

István Mező, Grant Keady
2016 European journal of physics  
In this paper we review 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)  ... 
doi:10.1088/0143-0807/37/6/065802 fatcat:k4qqzmg2sjgm7gvcrge2tbdi2y

Another Incarnation of the Lambert W Function [article]

Alexander Kheyfits
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.  ... 
arXiv:1703.06833v1 fatcat:ddc4vvdqwzg4nfh5t257eqrqzu

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

F. Chapeau-Blondeau, A. Monir
2002 IEEE Transactions on Signal Processing  
We show that this transformation is expressable in terms of a special 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.  ... 
doi:10.1109/tsp.2002.801912 fatcat:lsadwmiw65gr3puopq5hstnckm


2013 Journal of Circuits, Systems and Computers  
Both parts form a new circuit for a 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.  ... 
doi:10.1142/s0218126613500758 fatcat:cvhvia5wgrawpgro6f2d44pnbu

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.  ... 
doi:10.15546/aeei-2014-0002 fatcat:6jm6tqtyuzb4jg4t2z3drsrn6q

Lambert W Function In Hydraulic Problems

Dejan Brkić
2012 Zenodo  
But unfortunately, 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  ... 
doi:10.5281/zenodo.890013 fatcat:4j3i6apjdvg5jpbit73wmrmdtu

W solutions of the CW equation for flow friction

Dejan Brkić
2011 Applied Mathematics Letters  
Alternate mathematical equivalents to 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.  ... 
doi:10.1016/j.aml.2011.03.014 fatcat:jxzismoulfeodb2ermggcpybg4

On the generalization of the Lambert $W$ function

István Mező, Árpád Baricz
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)  ... 
doi:10.1090/tran/6911 fatcat:z6fxguaznfe45lj4tcwtoksgjq

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

Huang-Nan Huang, Chew Chun Yong
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.  ... 
arXiv:1803.09953v2 fatcat:impmugjbbrgozmzo7bjxmmzywi

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

Sun Yi, Sangseok Yu, Jung H. Kim
2011 Proceedings of the 2011 American Control Conference  
For stability analysis, 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.  ... 
doi:10.1109/acc.2011.5991085 fatcat:qolodfyzwzbr3nyk6mhdjxhtha

Transcendental Numbers and the Lambert-Tsallis Function [article]

J. L. E. da Silva, R. V. Ramos
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  ... 
arXiv:2004.07101v1 fatcat:l772ohljyfcnnbo4ectqvmeb2a
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