1,393 research outputs found
Stability of quaternionic linear systems
The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework
Algebraic tools for the study of quaternionic behavioral systems
In this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations. As happens in the commutative case, it turns out that quaternionic polynomial matrices play an important role in this context. Therefore we pay special attention to such matrices and derive new results concerning their Smith form. Based on these results, we obtain a characterization of system theoretic properties such as controllability and stability of a quaternionic behavior
Dynamical properties of quaternionic behavioral systems
In this paper we study behavioral systems whose trajectories are given as solutions of
quaternionic difference equations. As happens in the commutative case, it turns out that
quaternionic polynomial matrices play an important role in this context. Therefore we focus
our attention on such matrices and derive new results concerning their Smith form. Based
on these results, we obtain characterizations of system theoretic properties of quaternionic
behaviors
Split Quaternions and Particles in (2+1)-Space
It is known that quaternions represent rotations in 3D Euclidean and
Minkowski spaces. However, product by a quaternion gives rotation in two
independent planes at once and to obtain single-plane rotations one has to
apply by half-angle quaternions twice from the left and on the right (with its
inverse). This 'double cover' property is potential problem in geometrical
application of split quaternions, since (2+2)-signature of their norms should
not be changed for each product. If split quaternions form proper algebraic
structure for microphysics, representation of boosts in (2+1)-space leads to
the interpretation of the scalar part of quaternions as wavelength of
particles. Invariance of space-time intervals and some quantum behavior, like
noncommutativity and fundamental spinor representation, probably also are
algebraic properties. In our approach the Dirac equation represents the
Cauchy-Riemann analyticity condition and the two fundamental physical
parameters (speed of light and Planck's constant) appear from the requirement
of positive definiteness of quaternionic norms.Comment: The version published in Eur. Phys. J.
Octonions and the Triple Articulation
Num artigo anterior descrevemos a nossa abordagem para
modelar as relações que um ecossistema de negócios pode sustentar
para as demandas multi-facetadas dos seus clientes. Esta abordagem
distinguia dois tipos de tempo, chronos e kairos, e era triplamente
articulada, descrevendo uma empresa como uma realização de
possíveis composições de 'capacidades tecnológicas', 'modelos sociais
de orquestração e sincronização' e as antecipações de satisfações ‘da
diferenciação das organizações dos clientes individuais'. Este artigo
descreve os trabalhos subsequentes que necessitaram abandonar os
Números Complexos como sua base matemática em favor dos
Quaterniones e finalmente adoptar os Octoniones que fornecem um
modelo da trialidade, necessário para abstrair as relações entre as
articulações. Identificam-se uma série de questões de pesquisa
derivadas da abordagem que fornece um meio para relacionar a
agilidade necessária para apoiar uma relação dinâmica entre a
situação de um cliente individual e a abordagem adoptada para
instituir a empresa como um todo.A previous paper described our approach to modeling the
relations that a business ecosystem can sustain to the multi-sided demands of its clients. This approach distinguished two kinds of time,
chronos and kairos, and was triply articulated, describing an enterprise
as a realization of possible compositions of ‘technological capabilities’,
‘social models of orchestration and synchronization’ and ‘the differing
organizations of individual clients' anticipations of satisfaction’. This
paper describes subsequent work that necessitated abandoning the
Complex Numbers as its mathematical basis in favour of the
Quaternions and finally adopting the Octonions which provide a
model of triality necessary for abstracting the relations between the
articulations. It identifies a number of research questions derived
from the approach which provides a means of relating the agility
needed to support a dynamic relation to an individual client’s
situation with the approach taken to instituting the enterprise as a
whole
An approach to CMG steering using feedback linearization
This paper presents an approach for controlling spacecraft equipped with control moment gyroscopes. A technique from feedback linearization theory is used to transform the original nonlinear problem to an equivalent linear form without approximating assumptions. In this form, the spacecraft dynamics appear linearly, and are decoupled from redundancy in the system of gyroscopes. A general approach to distributing control effort among the available actuators is described which includes provisions for redistribution of rotors, explicit bounds in gimbal rates, and guaranteed operation at or near singular configurations. A particular algorithm is developed for systems of double-gimbal devices, and demonstrated in two examples for which existing approaches fail to give adequate performance
Quaternion kinematics for the error-state KF
A complete compendium of Quaternion formulas, with applications to Kalman filtering for visual-inertial navigation.Preprin
Dirac Equation in Scale Relativity
The theory of scale relativity provides a new insight into the origin of
fundamental laws in physics. Its application to microphysics allows to recover
quantum mechanics as mechanics on a non-differentiable (fractal) space-time.
The Schr\"odinger and Klein-Gordon equations have already been demonstrated as
geodesic equations in this framework. We propose here a new development of the
intrinsic properties of this theory to obtain, using the mathematical tool of
Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in
standard physics, is merely postulated. The bi-quaternionic nature of the Dirac
spinor is obtained by adding to the differential (proper) time symmetry
breaking, which yields the complex form of the wave-function in the
Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries,
namely, the differential coordinate symmetry () and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.
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