Quaternionic Systems Research Articles

Stabilization and synchronization control of quaternion-valued fuzzy memristive neural networks: Nonlinear scalarization approach

In this paper, we present new conditions ensuring existence, uniqueness, and stabilization of the equilibrium point (EP) for a large class of quaternion-valued memristive neural networks. The results are applicable to both complex-valued memristive neural networks and quaternion-valued system with uncertain terms. Subsequently, by employing suitable controller, the synchronization control conclusion is also considered. It is worth mentioning that, the nonlinear scalarization approach is employed to ensure the closed convex hull consisted by the quaternion is meaningful, which preserves the integrity of the quaternion system and makes the conclusion more reasonable. Finally, this paper provides two examples, so that the superiority of the main results can be illustrated.

Representing complex vague soft set by quaternion numbers

Based on the quaternion system, we give a new representation of the complex vague soft set, and related logical operations. This new representation contains more information than before. Three quaternion distance measures are proposed and a decision model is established. The disease diagnosis of breast cancer is applied to the model to reflect the superiority of the model. By comparing the diagnostic errors under the different distance measures, the most suitable distance measure for this dataset is selected.

A Block Conjugate Gradient Method for Quaternion Linear

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

Stabilization control of quaternion-valued fractional-order discrete-time memristive neural networks

In this paper, the stabilization control of the memristive neural networks with discrete-time terms and fractional derivative is proposed. By employing the Lyapunov functional method and fully considering the integrity of the quaternion system, some sufficient criteria are derived based on nonlinear scalarization approach and several inequality techniques. The stabilization criteria are established in the form of algebraic inequality, which can be directly resolved by a simple calculation. Finally, two numerical examples are proposed to demonstrate the validity of theoretical results.

Fully actuated systems in terms of quaternions for spacecraft attitude control

Attitude control is an essential operation for spacecraft to perform space missions, and control strategies play a decisive role. In this paper, a second-order quaternion system of rigid spacecraft attitude dynamics is derived based on the fully actuated system approach. The quaternion system has two constraints that lead it to be a fully actuated system. Two closed-loop control strategies are proposed to achieve attitude tracking and stabilization for the fully actuated system. The stabilization strategy can control the scalar quaternion to reach the desired value and be used to deal with the unwinding phenomenon. Numerical simulations demonstrate the second-order quaternion system’s validity and the control strategies’ effectiveness.

A generalization of complex, dual and hyperbolic quaternions: Hybrid quaternions

Hybrid numbers are a new non-commutative number system which is a generalization of the complex (i2 = ?1), dual (?2 = 0), and hyperbolic numbers (h2 = 1). In this article, firstly we define a new quaternion system called hybrid quaternions by taking the coefficients of real quaternions as hybrid numbers. This new quaternion system is a combination of complex quaternions (biquaternions), hyperbolic (perplex) quaternions, and dual quaternions, and it can be viewed as a generalization of these quaternion systems. Then, we present the basic properties of hybrid quaternions including fundamental operations, conjugates, inner product, vector product, and norm. Finally, we give a schematic representation of numbers and quaternions.

МОДЕЛЬ РУХУ БЕЗПІЛОТНИХ ЛІТАЛЬНИХ АПАРАТІВ НА ОСНОВІ АЛГЕБРИ ДУАЛЬНИХ КВАТЕРНІОНІВ

The widespread use of unmanned aerial vehicles during warfare has intensified the problem of their management, especially when they are used in large groups. One of the main tasks is to ensure coordinated movement of the group's aircraft in space. Optimizing the movement of each device of the group in three-dimensional space is expedient to use mathematical models. The movement of any unmanned aerial vehicle can be presented as a combination of translational and rotational movements, and its speed as a combination of translational and rotational velocities. Previously, these movements were modeled separately using a system of differential equations or quaternions. In this article, a mathematical model of rotational and translational movements of an aircraft based on the algebra of dual quaternions is developed. Dual quaternions consisting of eight scalars are a compact representation of rigid transformations in space. Therefore, their properties determine the advantage in the course of motion simulation, as they reduce the amount of calculations. Thus, with the help of one dual quaternion, it is possible to provide both translational and rotational motions at once, and the operation of non-commutative multiplication of dual quaternions is used to simulate the movement. The model assumes that the real part of the dual quaternion determines the orientation of the UAV in space, and the dual part determines its position in three-dimensional space. In order to connect aircraft coordinate systems with the model, expressions for the transition from aircraft orientation angles (roll, yaw, and pitch) to dual quaternion parameters and vice versa are obtained. The functionality of the proposed model was confirmed using the developed software for modeling the coordinated movement of aircraft. The software is adapted for graphical display of a large number of aircraft in web browsers with WebGl support. Keywords: motion modeling; rotational and translational movement; unmanned aerial vehicles; quaternions; dual quaternions; algebra of quaternions.

Mittag–Leffler Synchronization of Caputo-Delayed Quaternion BAM Neural Networks via Adaptive and Linear Feedback Control Designs

The Mittag–Leffler synchronization (MLS) issue for Caputo-delayed quaternion bidirectional associative memory neural networks (BAM-NNs) is studied in this paper. Firstly, a novel lemma is proved by the Laplace transform and inverse transform. Then, without decomposing a quaternion system into subsystems, the adaptive controller and the linear controller are designed to realize MLS. According to the proposed lemma, constructing two different Lyapunov functionals and applying the fractional Razumikhin theorem and inequality techniques, the sufficient criteria of MLS on fractional delayed quaternion BAM-NNs are derived. Finally, two numerical examples are given to illustrate the validity and practicability.

A family of quaternion-valued pipelined second-order Volterra adaptive filters for nonlinear system identification

This paper primarily proposes a family of quaternion Volterra filters based on the feedforward pipelined structure (QPSOVAFs) for nonlinear quaternion system identification to reduce the computational complexity. Then, the strictly nonlinear QPSOVAF (SNL-QPSOVAF), semi-widely nonlinear QPSOVAF(SWNL-QPSOVAF) and widely nonlinear QPSOVAF (WNL-QPSOVAF) are proposed. This architecture consists of several quaternion-valued second-order Volterra (SOV) modules. The structure’s nonlinear subsection executes a nonlinear mapping from the input space to an intermediate space using the feedforward SOV; the linear combiner subsection performs a linear mapping from the intermediate space to the output space. Moreover, the theoretical analysis expresses the effectiveness of the proposed QPSOVAFs in a specific condition. Finally, simulation results further prove that the proposed QPSOVAFs have good performance in identifying the quaternion-valued nonlinear system.

Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class

We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al. In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem. The model is invariant under the 1:1:2 weighted SO(2) symmetry and spin inversion. There is one formal control parameter in addition to four dynamical variables of the slow subsystem. We demonstrate that the most elementary qualitative phenomenon associated with the rearrangement of the energy super-bands of our model consists of the rearrangement of one energy level between two energy superbands which takes place when the formal control parameter takes the special isolated value associated with the conical degeneracy of the semi-quantum eigenvalues. This qualitative phenomenon is of topological origin, and is characterized by the second Chern class of the associated semi-quantum system. The correspondence between the number of redistributed energy levels and the second Chern number is confirmed through a series of examples.

Effect of dissimilatory iron reduction and

Environmental context Dissimilatory iron reduction and Carex-produced dissolved organic matter (DOM) have an important influence on Cr(VI) reduction by the Fe(III)-reducing bacterium Enterobacter. The role of Carex DOM and ferrihydrite in the biotransformation of Cr(VI) by the bacterial isolate was investigated. The findings should help underpin the remediation and detoxification of chromium in anaerobic environments, and provide promising insights into the quaternary system of bacterium/Fe(III)/Cr(VI)/DOM. Rationale FeIII oxides and organic matter are important factors influencing CrVI degradation in wetland soils. However, it remains unclear how they interact in anaerobic systems. Methodology In this study, a strain of iron-reducing bacterium was isolated from Poyang Lake Wetland and identified as Enterobacter sp. PY16 (PY16) by PCR-16S-rDNA sequence analysis. Moreover, microbial reduction of FeIII/CrVI by PY16 and their mutual transformation in the quaternionic system of PY16/ferrihydrite/CrVI/dissolved organic matter (DOM, extracted from Carex cinerascens) were investigated. Results The results showed that PY16 could directly participate in the reduction of ferrihydrite and CrVI. The rate of CrVI reduction decreased with the increase of initial CrVI concentration, while it was enhanced by 2.78–42.99% in the presence of ferrihydrite. Moreover, 15 mg L−1 CrVI was almost eliminated after 72 h and the content of FeII increased by 78.21 mg L−1 in the presence of DOM. Discussion The fastest CrVI reduction rate occurred when ferrihydrite and DOM coexisted in the system, mainly because the promoting effect of DOM on ferrihydrite synergistically promoted CrVI reduction. DOM and FeII produced during the ferrihydrite reduction process served as electron shuttles that promoted CrVI reduction by a biochemical redox pathway. However, the electron transfer and donation capacity of DOMox/DOMred and FeIII/FeII in the reaction process still need to be further studied. Implications for future research The results underscored the importance of FeIII oxides and DOM on microbial CrVI reduction, thus providing a valuable technique to remove and detoxify chromium in wetland soils.

Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

Image colour edge detection using hypercomplex convolution

Quaternions are considered for colour image edge detection. Most work on quaternions is based on a linear quaternion system (LQS) which applies multi-directional kernels (horizontal, vertical, and diagonal) using hypercomplex convolution, each kernel producing an edge map for a specific direction, and the final result is a combination of these maps. This paper introduces a new colour image edge detection filter based on LQS convolution. The process starts by applying quaternion convolution with the proposed filter, and then generating the final edge map by computing the magnitude of the result. The proposed filter is able to highlight both colour and greyscale edges in multiple directions using a single LQS convolution pass. The validity of the proposed filter is demonstrated, and its performance is supported experimentally through a set of comparisons with state-of-the-art methods.

Intelligent OFDM telecommunication system. Part 1. Model of complex and quaternion systems

In this paper, we aim to investigate the superiority and practicability of many-parameter transforms (MPTs) from the physical layer security (PHY-LS) perspective. We propose novel Intelligent OFDM-telecommunication systems based on complex and quaternion MPTs. The new systems use inverse MPT (IMPT) for modulation at the transmitter and MPT for demodulation at the receiver. The purpose of employing the MPT is to improve: 1) the PHY-LS of wireless transmissions against to the wide-band anti-jamming and anti-eavesdropping communication; 2) the bit error rate (BER) performance with respect to the conventional OFDM-TCS; 3) the peak to average power ratio (PAPR). Each MPT depends on finite set of independent parameters (angles). When parameters are changed, many-parametric transform is also changed taking form of a set known (and unknown) orthogonal (or unitary) transforms. For this reason, the concrete values of parameters are specific “key” for entry into OFDM-TCS. Vector of parameters belong to multi-dimension torus space. Scanning of this space for find out the “key” (the concrete values of parameters) is hard problem. MPT has the form of the product of the Jacobi rotation matrixes and it describes a fast algorithm for MPT. The main advantage of using MPT in OFDM TCS is that it is a very flexible anti-eavesdropping and anti-jamming Intelligent OFDM TCS. To the best of our knowledge, this is the first work that utilizes the MPT theory to facilitate the PHY-LS through parameterization of unitary transforms.

Widely linear estimation for multisensor quaternion systems with mixed uncertainties in the observations

The optimal widely linear state estimation problem for quaternion systems with multiple sensors and mixed uncertainties in the observations is solved in a unified framework. For that, we devise a unified model to describe the mixed uncertainties of sensor delays, packet dropouts and uncertain observations by using three Bernoulli distributed quaternion random processes. The proposed model is valid for linear discrete-time quaternion stochastic systems measured by multiple sensors and it allows us to provide filtering, prediction and smoothing algorithms for estimating the quaternion state through a widely linear processing. Simulation results are employed to show the superior performance of such algorithms in comparison to standard widely linear methods when mixed uncertainties are present in the observations.

Multi‐directional colour edge detector using LQS convolution

A new linear colour image filter based on linear quaternion systems (LQSs) is introduced. It detects horizontal, vertical, left- and right-diagonal edges with a single LQS convolution mask. The proposed filter is a canonic minimal filter of four LQS filters, each with different angles of rotation combined parallel wise. Different angles of rotation are a key features of the new filter such that horizontal, vertical, left, and right-diagonal LQS filter masks rotate pixels through angles , , , and , respectively. Although, the four LQS masks are combined parallel to make a single LQS mask but derived using four quaternion convolutions, one for each direction of edges, the LQS filter produces a result without the combination of results from four separate edge detectors. This methodology could be generalised to design more elaborate LQS filters to perform other geometric operations on colour image pixels. The proposed filter translates smoothly changing colours to different shades of grey and produces coloured edges in multiple directions, where there is a sudden change of colour in the original image. Another key idea of the proposed filter is that it is linear because it operates in homogeneous coordinates.

Unscented and square‐root unscented Kalman filters for quaternionic systems

SummaryProper construction of an unscented Kalman filter (UKF) for unit quaternionic systems is not straightforward due to the incompatibility between the algebraic properties of the unit quaternions and the common real vector space operations (additions and scalar multiplications) needed in the steps of a filter algorithm. This work studies, in detail, all UKFs and square‐root UKFs for quaternionic systems proposed in the literature. First, we classify the algorithms according to the preservation of the unity norm of the quaternion variables. Second, we propose two new algorithms: the quaternionic additive unscented Kalman filter (QuAdUKF) and a square‐root variant of it. The QuAdUKF encompasses all known UKFs for quaternionic systems of the literature preserving, in all steps, the norm of the unit quaternion variables. Besides, it can also yield new UKFs with this norm preservation property. The QuAdUKF's square‐root variant has better properties in comparison with all the square‐root UKFs for quaternionic systems of the literature. Numerical experiments for a spacecraft attitude estimation problem illustrate the theoretical results.

Some new linear representations of matrix quaternions with some applications

In this paper, we construct several new attractive and interested linear representations of matrix quaternions by using Kronecker structures in order to obtain the general partitioned linear representation form of matrix quaternions. In addition, we present the general solutions of three important partitioned quaternions systems by using our new representations and Kronecker structure. These systems are: the partitioned linear quaternion equations, general linear matrix quaternion system and coupled Sylvester matrix quaternion system.

Fast adaptive pose tracking control for satellites via dual quaternion upon non-certainty equivalence principle

This paper presents the implementation of the high performance adaptive control algorithm for 6-DOF systems described by dual quaternions. The fast adaptation is achieved from the introduced stable adaptation dynamics in the closed-loop system which cannot be expected in the conventional adaptive controller designed upon the certainty equivalence framework. The development of the proposed adaptive control algorithm relies on the two newly introduced algebraic operations. The one is a dual filter for a dual relative error signal. The dual filter essentially has a linear low-pass filter structure and enables us to recast the system dynamics into a linear parameter-affine system dynamics in dual quaternions combined with dual control inputs. The other is an isomorphism between a dual quaternion and a vector for the error measurement. Using this isomorphism, the stability proof of the closed-loop dual quaternion system based on the proposed Lyapunov-like function is rewritten in the form of familiar matrix/vector algebra. In order to demonstrate the performance improvement, numerical simulation results are given, which compare the proposed adaptive control method with the conventional one from the certainty-equivalence principle.

Quaternion-Based Algorithm for Direct Kinematic Model of a Kawasaki FS10E Articulated Arm Robot

This article follows a detailed description of development and validation for the direct kinematic model of six degrees of freedom articulated arm robot - Kawasaki FS10E model. The development of the kinematic model is based on widely used Denavit-Hartenberg notation, but, after the initial parameter identification, the mathematical algorithm itself follows an approach that uses the quaternion number system, taking advantage of their efficiency in describing spatial rotation - providing a convenient mathematical notation for expressing rotations and orientations of objects in three-dimensional space. The proposed algorithm concludes with two quaternion-based relations that express both the position of robot tool center point (TCP) position and end-effector orientation with respect to robot base coordinate system using Denavit-Hartenberg parameters and joint values as input data. Furthermore, the developed direct kinematic model was validated using the programming and offline simulation software Kawasaki PC Roset.