scholarly journals Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 818 ◽  
Author(s):  
Alejandro Arceo ◽  
Luis E. Garza ◽  
Gerardo Romero
Keyword(s):  
Orthogonal Polynomials ◽  
Robust Stability ◽  
Equations Of Motion ◽  
Hurwitz Polynomials ◽  
Algebraic Properties

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

scholarly journals THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

1994 ◽  
Vol 09 (30) ◽  
pp. 2783-2801 ◽  
Author(s):  
H. ARATYN ◽  
L. A. FERREIRA ◽  
J. F. GOMES ◽  
A. H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Equations Of Motion ◽  
Curvature Form ◽  
Two Dimensional ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Algebraic Properties ◽  
Toda Model ◽  
Infinite Sets ◽  
Poisson Commute

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

On Orthogonal Polynomials of Sobolev Type: Algebraic Properties and Zeros

1992 ◽  
Vol 23 (3) ◽  
pp. 737-757 ◽  
Author(s):  
M. Alfaro ◽  
F. Marcellán ◽  
M. L. Rezola ◽  
A. Ronveaux
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Type ◽  
Algebraic Properties

scholarly journals Equations of motion for zeros of orthogonal polynomials related to the Toda lattices

2011 ◽  
Vol 17 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Mourad E.H. Ismail ◽  
Wen-Xiu Ma
Keyword(s):  
Orthogonal Polynomials ◽  
Equations Of Motion ◽  
Zeros Of Orthogonal Polynomials ◽  
Toda Lattices

On the Robust Stability of Segmented Driveshafts with Active Magnetic Bearing Control

2005 ◽  
Vol 11 (3) ◽  
pp. 317-329 ◽  
Author(s):  
Hans A. Desmidt ◽  
K. W. Wang ◽  
Edward C. Smith ◽  
Andrew J. Provenza
Keyword(s):  
Robust Stability ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Stability Index ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Internal Damping ◽  
Shaft Speed ◽  
Load Torque ◽  
Loop Stability

Many researchers and engineers have employed active control techniques, such as active magnetic bearings (AMBs), to suppress imbalance vibration in various subcritical and supercritical speed rotors dynamic applications. One issue that has not yet been addressed in previous AMB driveline control studies is the effect of non-constant velocity (NCV) flexible couplings, such as U-joint or disk-type couplings, present in many segmented drivelines. The NCV effects introduce periodic parametric and forcing terms into the equations of motion that are functions of shaft speed, driveline misalignment, and load-torque, resulting in a linear periodically time-varying system. Previous research has found that both internal damping and NCV terms greatly impact stability; thus, they must be accounted for in the control law design in order to ensure closed-loop stability of any AMB-NCV-driveline system. In this paper, numerical Floquet theory is used to explore the closed-loop stability of a flexible segmented NCV-driveline supported by AMBs with a proportional-derivative (PD) type controller. To ensure robust stability with respect to internal damping and NCV effects, the robust P and D gains and AMB locations are selected based on maximizing a stability index over a range of shaft speeds, driveline misalignments, and load-torques. It is found that maximum robustness occurs within a finite range of P and D gains for several different AMB locations. Finally, the range of robustly stabilizing P gains versus the shaft speed is examined for several misalignment and load-torque bounds.

scholarly journals Orthogonal polynomials associated with an inverse spectral transform. The cubic case

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2477-2497
Author(s):  
Mabrouk Sghaier ◽  
Lamaa Khaled
Keyword(s):  
Orthogonal Polynomials ◽  
Monic Polynomial ◽  
Polynomial Sequence ◽  
Spectral Transform ◽  
Algebraic Properties ◽  
Inverse Spectral

The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn}n ? o defined by Qn(X) = Pn(X) + SnPn-1(X) + tnPn-2(X) + rnPn-3(X), n ? 1, where rn ? 0, n ? 3, and {Pn}n?0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters rn, sn, and tn can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.

On sequences of Hurwitz polynomials related to orthogonal polynomials

2018 ◽  
Vol 67 (11) ◽  
pp. 2191-2208 ◽  
Author(s):  
Noé Martínez ◽  
Luis E. Garza ◽  
Baltazar Aguirre-Hernández
Keyword(s):  
Orthogonal Polynomials ◽  
Hurwitz Polynomials

scholarly journals On the Galois group of Generalised Laguerre polynomials II

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Shanta Laishram ◽  
Saranya G. Nair ◽  
T. N. Shorey
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Galois Group ◽  
Laguerre Polynomials ◽  
Mathematical Physics ◽  
Negative Integer ◽  
Algebraic Properties ◽  
International Audience

International audience For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by$$L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}.$$These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ when $u$ is a negative integer.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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