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 ($dx^{\mu} \leftrightarrow - dx^{\mu}$) and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.