Matrix Differential Equation Research Articles

A generalized differential scheme for the effective conductivity of percolating microinhomogeneous materials with the Hall effect

In this paper, we propose a self-consistent scheme for the calculation of the components of the effective electrical conductivity tensor. The calculations were fulfilled for a microinhomogeneous material, the components of which have the Hall effect. The presence of the Hall effect leads to appearance of asymmetry of the components of the conductivity tensor and to dependence of these components on the magnitude of the magnetic field applied to the material. Our approach is based on the Generalized Differential Effective Medium (GDEM) method. This method generalizes the classical differential scheme (DEM) for the case of several inclusion types instead of one. In this case, the GDEM scheme leads to a system of matrix differential equations that were solved numerically. This solution was obtained for materials containing spherical or cylindrical inclusions (3D and 2D-problems). In the case of cylindrical inclusions, the results were obtained for inclusions with the symmetry axes orthogonal to the magnetic field. The application of the GDEM method allows us to consider the percolation effect for 2D and 3D-microheterogeneous materials. The results obtained are compared to the experimental data and the calculation results obtained by other self-consistent schemes.

Coupled of Semi Analytic Approach Associated with Laplace Transform First Step for Solving Matrix Differential Equations Quadratic Form when the Time-Delay in Noise Term

في هذا البحث تناولنا طريقة فعالة وجديدة وهي الدمج بين طريقتي الادوميان والهوموتوبي مع إستخدام طريقة الخطوات لتسهيل المسألة والتي تخص المعادلات التفاضلية الاعتيادية التباطئية لحل معادلة المصفوفات التباطئية التربيعية الغير خطية . كلتا الطريقتين على درجة عالية من التأثير والفعالية. جزء التكلمل الكلي لطريقة الهوموتوبي سيستخدم بدلا من جزء التكامل الخاص ب الادوميان . الميزة الرئيسية لهذه التقنية هي الحصول على نتائج اكثر دقة و لفترة و منطقة اوسع واطول ولمعرفة دقة هذه النتائج تحت تأثير التأخير. الجزء الخاص بالتأخير يختفي بعد استخدام طريقة الخطوات. تم حساب الخطأ المتبقي . لتقليل الوقت والعمليات الحسابية المعقدة تم أستخدام تحويلات لابلاس. أخيرا النتائج التي تم الحصول عليها بينت ان التقنية فعالة وسريعة التقارب للحل المظبوط ولفترة ومنطقة اوسع . يمكن استخدام هذه التقنية لحل مسائل غير خطية مختلفة. طريقة الادوميان هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية ؛ جزئية ؛ كسرية و تباظؤية . هذه الطريقة تطورت بواسطة جورج ادوميان. هي سريعة التقارب للحل المظبوط وتستخدم للخطية و غير الخطية و المتجانسة و غيلر المتجانسة. متعددة ادوميان تسمح للحل التقارب للحل المظبوط للمسألة قيد الدراسة دون الحاجة الى أي تحوير. طريقة الهوموتوبي هي تقنية شبه تحليلية لحل معادلات تفاضلية مختلفة أعتيادية؛ جزئية ؛ كسرية و تباطؤية وانواع مختلفة. هذه الطريقة تطورت عن طريق العالم ليو . هي سريعة التقارب للحل المظبوط وتستخدم للخطية وغير الخطية و المتجانسة وغير المتجانسة. جاء مفهوم الهوموتوبي من مفهوم التبلوجي في توليد متسلسلة متقاربة للحل المظبوط. هذه الطريقة تم ايجادها من قبل ليو خلال اطروحته. تحتوي الطريقة على متقير او معلمة خلاله تمكننا التقارب.

Modeling and control of dynamical biomechanical arm systems with elastic joints sensitive to the effect of an external load

In this study, a biomechanical model mimicking the human hand-arm system under heavy disk excitation is developed to define the stability threshold between the preload vibration and the stress of the human hand-arm system. The fully electromechanical system, consisting of a DC motor, a transmission system, and three discrete masses representing the upper arm, forearm, and hand lifting a heavy disk, is connected by shock absorbers and various springs. The challenge of mimicking the angular activity of the elbow joint in torsion and flexion was also included to assess its contribution to system stability and to study the absorption of mechanical energy in the human hand-arm system. Finally, the movements of the model were described by a differential matrix equation, and its analysis facilitates the understanding of the threshold of the chaotic behaviors observed in an oscillating arms system. Specifically, it explores how the motion of a DC motor can be regulated using a single controller parameter within the context of a multi-degree-of-freedom human hand-arm system. The study uses numerical integrations to analyze system behaviors, with an emphasis on the use of frequency spectra, Poincaré maps, and bifurcation diagrams for visualization and interpretation. The results indicate that the system exhibits chaotic behavior when subjected to certain conditions, particularly when the mass load exceeds 35 kg. Imposing motion on the mass block further intensifies the chaos, leading to manifestations such as strong oscillations and frequency-synchronization effects. These characteristics, exhibited by the idealized model, which imitates the human body through a mechanical model and computation of the motions of the body, play a vital role in various fields of ergonomics and sports biomechanics. Understanding the dynamic behaviors of such systems, especially in response to varying conditions, has significant implications for engineering applications.

Approximate solution of a zero-sum linear-quadratic differential game by the auxiliary parameter method: analytical/numerical study

ABSTRACT A zero-sum finite horizon linear-quadratic differential game is considered. First, the terminal-value problem for the game-theoretic Riccati matrix differential equation, associated with the considered game by the solvability conditions, is analysed. This problem may not have the solution in the entire time-interval of the game's duration. Using the method of auxiliary parameter, a sufficient condition (in the terms of the game's data) for the existence of the solution to the aforementioned terminal-value problem in the entire time-interval of the game's duration is derived. An approximate analytical solution to this problem also is obtained as a partial sum of some infinite series of functions arising in the method of auxiliary parameter. Based on this approximate solution, suboptimal state-feedback controls of the players are formally designed and justified. It is shown that the pair of these controls constitutes an approximate-saddle point in the considered game. The theoretical results of the article are illustrated by their application to approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.

Using the dispersion model of a chiral metamaterial to calculate the characteristics of homogeneous and heterogeneous reflective and waveguide structures

Currently, metamaterials are actively used in the creation of microwave and optic devices for various purposes. This is due to their non-traditional properties, compared to dielectric, conductive and other materials. One type of metamaterial is chiral media, which contain conductive microelements of mirror asymmetric shape. Such metamaterials have dispersion and, as a consequence, it is necessary to take it into account when developing various microwave structures. The work demonstrates taking into account the dispersion of the dielectric constant and the chirality parameter for calculating waveguide and reflecting structures with homogeneous and inhomogeneous chiral layers. The work carried out an electrodynamic analysis of the reflective properties of an inhomogeneous chiral layer and a rectangular waveguide, with a homogeneous layer of chiral metamaterial located in the transverse plane. The work has developed a method for calculating the reflection and transmission coefficients of an optical wave from the inhomogeneous chiral structure under consideration, which is based on the use of the differential sweep method. When constructing a mathematical model, we took into account the cross-polarization of the optical wave field, which consists in the appearance of orthogonal components during the interaction of the field with a chiral metamaterial. The solution to the problem was reduced to a matrix differential equation for the unknown reflection coefficients of the main and cross-polarized components of the optical field. The work examined chiral metamaterials with linear and parabolic inhomogeneity profiles. The results of the work can be applied in the design of new microwave and optics devices with layers of metamaterial, which can expand their functionality.

Cubature Kalman fusion filtering under stochastic communication protocol: Tackling dynamical bias, scheduling probability and channel noise

The paper addresses the problem of cubature Kalman fusion filtering (CKFF) for nonlinear systems subject to dynamical bias and stochastic communication protocol (SCP). To account for the engineering practice of networked environment, the channels from sensors to filters, affected by Gaussian channel noise, are considered in the system model incorporating dynamical bias. With the purpose of reducing communication load and enhancing resource utilization efficiency, the SCP is adopted for information transmissions, where the scheduling probability is allowed to deviate within bounds from its designated value owing to possible implementation errors. The objective of this paper is to design a CKFF algorithm in spite of the presence of dynamical bias, channel noise and SCP with uncertain scheduling probability. Specifically, the local filter is constructed such that an upper bound on the filtering error covariance (FEC) is calculated through the resolution of matrix difference equations, and the filter gain is subsequently designed to minimize this upper bound. Then, the local filters are fused by utilizing the inverse covariance intersection fusion rule. Moreover, the boundedness of the FEC’s upper bound is also investigated using the matrix theory. The feasibility and effectiveness of the proposed CKFF approach are demonstrated through two simulation experiments.

Non-fragile cubature Kalman filtering under innovation-based weighted try-once-discard protocol with channel noise

In this paper, the problem of non-fragile cubature Kalman filtering is addressed for nonlinear systems with stochastic uncertainties under innovation-based weighted try-once-discard protocol (WTODP) affected by channel noise. To reduce transmission burden and improve communication efficiency, the innovation-based WTODP, which is constructed by a set of quadratic functions regarding innovation, is employed in the network channel between the sensor and the filter to regulate the transmission of data signals. Meanwhile, Gaussian-distributed channel noise is taken into account during the signal propagation process to provide a better representation of the network media. An upper bound of the filtering error covariance (FEC) is derived by solving certain matrix difference equations, and the minimization of the obtained upper bound is achieved through the design of an appropriate filter gain. Furthermore, the uniform boundedness of the FEC's upper bound is investigated using the matrix theory to evaluate the performance of the filtering scheme. The effectiveness and superiority of the proposed non-fragile cubature Kalman filtering algorithm are demonstrated through two simulation experiments with comparisons.

Local exponential synchronization rate of commutative Kuramoto oscillators on spheres

For proportional Kuramoto oscillators on spheres, the local exponential synchronization rate for complete phase synchronization was obtained in existing literature. But for the general nonidentical Kuramoto oscillators on spheres, the problem of local exponential synchronization rate has not been solved. In this paper, for commutative Kuramoto oscillators on spheres, the matrix differential equations involving the synchronization errors are constructed, and the local exponential synchronization rate is obtained by calculating the spectrum of the linearized error system limited to an invariant subspace.

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Quaternion Differential Matrix Equations with Singular Coefficient Matrices

Jost Solution of the Matrix Difference Equations

Jost Solution of the Matrix Difference Equations

On the generalized Apostol-Kolodner differential equation of the second-order

This paper concerns another approach for solving the matrix differential equations of the second order $X''(t)=AX'(t)+BX(t)$, where $A$, $B$ are square matrices of order ${d\times d}$. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between $A$ and $B$, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.

A study of Horn matrix functions and Horn confluent matrix functions

Abstract In this paper, we give the matrix version of Horn’s hypergeometric function and their confluent cases. We also discuss the regions of convergence, system of matrix differential equations of bilateral type, differential formulae and infinite summation formulae satisfied by these hypergeometric matrix functions. With the study of these 23 matrix functions, matrix generalization of Horn’s list of 34 hypergeometric series will be completed.

Liouville-Green formulas for the fourth order matrix differential operators

In this paper, Liouville-Green analytical approximations for the solutions of the fourth order matrix differential equation are constructed. Using the Gronwall’s inequality, we have also determined the error bounds for the Liouville-Green approximations.

Performance analysis of multilayered transversely isotropic saturated media under temperature and horizontally circular loads

AbstractThis study constructs a multilayered transversely isotropic saturated model under thermal and horizontally circular loads, and further investigates the model's thermo‐hydro‐mechanical coupling response. Firstly, the ordinary differential matrix equations of thermoelastic saturated media in the integral transformed domain are derived. Secondly, the solution for multilayered thermoelastic saturated media is developed using the extended precise integration method (EPIM), along with the boundary conditions at both ends of the foundation and the continuity conditions between adjacent layers. After that, the solution in the physical domain is further attained with the use of Laplace‐Hankel integral transform inversion. Finally, the accuracy of the proposed theory is confirmed through numerical examples, and the influences of anisotropic parameters, the soil's stratification and porosity on the thermo‐hydro‐mechanical coupling response of the media are studied.

Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative

In this manuscript, we introduce the concept of first-time controllability and observability concerning fractional differential Sylvester matrix equations employing the Caputo fractional derivative. Our work establishes necessary and sufficient conditions for controllability and observability, wherein controllability equates to having a controllability matrix with full rank and observability aligns with an observability Gramian matrix that is nonsingular. Furthermore, we provide several theorems addressing the observability of fractional differential Sylvester matrix equations. Finally, we offer two illustrative examples to demonstrate the efficacy and application of our established results.

A Joint State and Fault Estimation Scheme for State-Saturated System with Energy Harvesting Sensors.

In this article, the issue of joint state and fault estimation is ironed out for delayed state-saturated systems subject to energy harvesting sensors. Under the effect of energy harvesting, the sensors can harvest energy from the external environment and consume an amount of energy when transmitting measurements to the estimator. The occurrence probability of measurement loss is computed at each instant according to the probability distribution of the energy harvesting mechanism. The main objective of the addressed problem is to construct a joint state and fault estimator where the estimation error covariance is ensured in some certain sense and the estimator gain is determined to accommodate energy harvesting sensors, state saturation, as well as time delays. By virtue of a set of matrix difference equations, the derived upper bound is minimized by parameterizing the estimator gain. In addition, the performance evaluation of the designed joint estimator is conducted by analyzing the boundedness of the estimation error in the mean-squared sense. Finally, two experimental examples are employed to illustrate the feasibility of the proposed estimation scheme.

OSCILLATORY BEHAVIOR OF SOLUTIONS OF FRACTIONAL MATRIX DIFFERENTIAL EQUATIONS

In this article, new oscillation criteria for the second-order self-adjoint Matrix differential equations by using the Riccatti technique are obtained. A suitable example is given to illustrate the significance and effectiveness of the result.

A differential scheme for the effective conductivity of microinhomogeneous materials with the Hall effect

A differential scheme is proposed for the calculation of the components of the effective electrical conductivity tensor of a microinhomogeneous material taking into account the Hall effect. The presence of the Hall effect results in an appearance of asymmetry of the components of the conductivity tensor and a dependence of these components on the magnitude of the magnetic field applied to the material. In this case, the differential scheme leads to a system of matrix differential equations that were solved numerically in the current work. This solution was obtained for materials containing spherical or cylindrical inclusions. In the case of cylindrical inclusions, the results were obtained for inclusions with the symmetry axes parallel or orthogonal to the magnetic field. A comparison of the results obtained by the proposed methodology to the low concentration approximation was carried out.

Numerical Covariance Evaluation for Linear Structures Subject to Non-Stationary Random Inputs

Random vibration analysis is a mathematical tool that offers great advantages in predicting the mechanical response of structural systems subjected to external dynamic loads whose nature is intrinsically stochastic, as in cases of sea waves, wind pressure, and vibrations due to road asperity. Using random vibration analysis is possible, when the input is properly modeled as a stochastic process, to derive pieces of information about the structural response with a high quality (if compared with other tools), especially in terms of reliability prevision. Moreover, the random vibration approach is quite complex in cases of non-linearity cases, as well as for non-stationary inputs, as in cases of seismic events. For non-stationary inputs, the assessment of second-order spectral moments requires resolving the Lyapunov matrix differential equation. In this research, a numerical procedure is proposed, providing an expression of response in the state-space that, to our best knowledge, has not yet been presented in the literature, by using a formal justification in accordance with earthquake input modeled as a modulated white noise with evolutive parameters. The computational efforts are reduced by considering the symmetry feature of the covariance matrix. The adopted approach is applied to analyze a multi-story building, aiming to determine the reliability related to the maximum inter-story displacement surpassing a specified acceptable threshold. The building is presumed to experience seismic input characterized by a non-stationary process in both amplitude and frequency, utilizing a general Kanai–Tajimi earthquake input stationary model. The adopted case study is modeled in the form of a multi-degree-of-freedom plane shear frame system.

An event‐triggered method to distributed filtering for nonlinear multi‐rate systems with random transmission delays

SummaryIn this article, an event‐triggered recursive filtering problem is studied for a class of nonlinear multi‐rate systems (MRSs) with random transmission delays (RTDs). The RTDs are described by utilizing random variables with a known probability distribution and the Kronecker function. To facilitate further study, the MRS is converted into a single‐rate one by adopting an iteration equation approach. To address the challenge of filter design caused by different measurement sampling periods, a modified prediction method of measurements is given. Moreover, an event‐triggered mechanism (ETM) is introduced to regulate the innovation transmission frequency. The objective of the addressed filtering problem is to design a recursive distributed filtering method for MRSs subject to ETM and RTDs, where a minimum upper bound on the filter error covariance is obtained. Moreover, the filter gain matrix is formulated by resorting to the solutions to matrix difference equations. Besides, the boundedness in the mean‐square sense of the filtering error is analyzed and a sufficient condition is provided. Finally, simulations with comparison experiments are presented to demonstrate the effectiveness of the newly proposed theoretical results.