Matrix Transformation Method Research Articles

Synchronization for a class of coupled linear partial differential systems via boundary control

The synchronization for coupled linear partial differential systems(PDSs) with boundary control is considered. Based on the nonsingular matrix transformation method, we decouple the coupled synchronization error dynamical systems and turn the synchronization control problem into the asymptotical stabilization problem. Then, using the backstepping approach, we present the boundary control strategy to guarantee the synchronization. At last, numerical example is given to illustrate the effectiveness of our result.

Zernike polynomials as a basis for modal fitting in lateral shearing interferometry: a discrete domain matrix transformation method.

A Zernike-polynomials-based wavefront reconstruction method for lateral shearing interferometry is proposed. Shear matrices are calculated using matrix transformation instead of mathematical derivation. Simulation results show that the shear matrices calculated using the proposed method are the same as those obtained from mathematical derivation. The advantage of the proposed method is that high order shear matrices can be obtained easily; thus, wavefront reconstruction can be extended to higher order Zernike terms, and reconstruction accuracy can be improved.

Planar microlens with front-face angle: design, fabrication, and characterization

This paper studies the effect of microlens front-face angle on the performance of an optical system consisting of a planar-graded refractive index (GRIN) lens pair facing each other separated by a free-space region. The planar silica microlens pairs are designed to facilitate low-loss optical signal propagation in the free-space region between the opposing optical waveguides. The planar lens is fabricated from a 38-μm-thick fluorine-doped silica layer on a silicon substrate. It has a parabolic refractive index profile in the vertical direction, which is achieved by controlled fluorine incorporation in the silica film to collimate the optical beam in the vertical direction. Horizontal beam collimation is achieved by incorporating a horizontal curvature at the front face of the lens defined by deep oxide etch. A generalized 3×3ABCDGH transformation matrix method has been derived to compute the coupling efficiency of such microlens pairs to take front-face angles that may be present due to fabrication variations or limitations and possible input/output optical fiber offset/tilt into considerations. Pairs of such planar GRIN lens with various free-space propagation distances between them ranging from 75 to 2500 μm and with front-face angles of 1.5 deg, 2 deg, and 4 deg have been fabricated and characterized. Beam propagation method simulations have been carried out to substantiate the theoretical and experimental results. The results indicate that the optical loss is reasonably low up to 1.5 deg of front-face angles and increases significantly with further increase in the front-face angle. Analysis shows that for a given system with specific microlens front-face angle, the optical loss can be significantly reduced by properly compensating the vertical position of the input and output fibers.

Improved Channel Error Calibration Algorithm for Azimuth Multichannel SAR Systems

Multichannel synthetic aperture radar systems in azimuth can effectively suppress azimuth ambiguity and are promising in high-resolution wide-swath imaging. However, unavoidable channel errors will significantly degrade the performance of ambiguity suppression. Conventional subspace calibration methods usually estimate phase error via decomposing a Doppler-variant covariance matrix from one Doppler bin, and then average these errors estimated from several Doppler bins to improve the estimation accuracy, which will result in a large computational load. This letter presents an improved channel error calibration method, which works on the undersampled data of the individual azimuth channel. By a proposed matrix transformation method, the Doppler-variant covariance matrices will be transformed into a constant covariance matrix. Therefore, the improved calibration algorithm needs to estimate and decompose the new covariance matrix only once. The computation load could be greatly reduced. Moreover, the new covariance matrix can be estimated by training samples not only from range bins but also from Doppler bins, which will improve the estimation accuracy. Theoretical analysis and experiments based on simulations and measurements showed the high accuracy, efficiency, and robustness of the improved method, particularly in low signal-to-noise ratio.

Diseño y programación de un software de transformación de matrices para el análisis de redes

Para muchos profesionales vinculados al Análisis de Redes Sociales (ARS), es de importancia transformar sus datos para realizar estudios posteriores. Es frecuente que los datos de sistemas donde intervienen conexiones se almacenen en matrices más elementales que las matrices completas o “full matrix”, frecuentemente utilizadas en programas de ARS. El objetivo principal de este trabajo fue diseñar y programar un software modificable para transformar matrices de tres columnas [A (grupo 1), B (grupo 2) y C (conexiones entre nodos de ambos grupos)] a matrices completas y simétricas, y analizar las ventajas del software con respecto a los problemas de almacenamiento y transformación de datos. Para ello se desarrolló un método de transformación de matrices y se diseñó un algoritmo (pseudocódigo) para realizar las operaciones necesarias. El algoritmo fue programado usando el lenguaje wxBasic para aumentar la compatibilidad entre sistemas operativos y facilitar su modificación por el usuario, obteniéndose un software estable y de fácil uso.

Finite element approximation of fractional order elliptic boundary value problems

A finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous Dirichlet type boundary conditions. It is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non-fractional elliptic operators. It is proved that this approach, which is also called the matrix transformation or matrix transfer method, delivers optimal rate of convergence in the L2-norm.

Efficient method to compute full eigenspectrum of incompressible viscous flows: Application on two-layer rectinear flow

An efficient algorithm based on the matrix transformation method (Valerio et al., 2007) is presented for solving the generalized eigenvalue problem (GEVP) derived from linear stability analysis of incompressible viscous flow. The proposed method uses the formulation based on primitive variables, i.e. velocity and pressure, instead of streamfunction used by typical Orr-Sommerfeld equation. A series of matrix operations removes non-physical eigenvalues at infinity and leads to a non-singular smaller size eigenvalue problem (EVP), which contains full eigenspectrum, than the original GEVP. Two different solution strategies for the transformed EVP are proposed, and their accuracies are discussed. The proposed procedure is used to solve the stability of two layer rectilinear flow. The computed eigenspectrum are compared to previously reported values.

Stochastic sampled-data H∞ synchronization of coupled neutral-type delay partial differential systems

This paper discusses the asymptotical synchronization and H∞ synchronization of coupled neutral-type delay partial differential systems (NDPDSs). A sampled-data controller with m stochastically varying sampling periods whose occurrence probabilities are given constants is considered. By using the method of a nonsingular matrix transformation, we decouple the coupled error dynamical systems. Then, sufficient conditions that guarantee the asymptotic stability of decoupled synchronization error dynamical systems are derived in terms of linear matrix inequalities (LMIs) by constructing a suitable Lyapunov–Krasovskii functional with triple integral terms and by using Jensen׳s inequality and reciprocally convex combination technique, which implies the asymptotical synchronization of the coupled NDPDSs. Moreover, the stability criteria for the H∞ stabilization of decoupled synchronization error dynamical systems with external disturbances are also derived in terms of LMIs, which guarantee the H∞ synchronization of the coupled NDPDSs. The equivalence between the H∞ stability of decoupled synchronization error dynamical systems and the H∞ synchronization of coupled NDPDSs is also proved by mathematical analysis. Finally, numerical examples are provided to demonstrate the effectiveness of the obtained results.

Development of variable-magnification X-ray Bragg optics.

A novel X-ray Bragg optics is proposed for variable-magnification of an X-ray beam. This X-ray Bragg optics is composed of two magnifiers in a crossed arrangement, and the magnification factor, M, is controlled through the azimuth angle of each magnifier. The basic properties of the X-ray optics such as the magnification factor, image transformation matrix and intrinsic acceptance angle are described based on the dynamical theory of X-ray diffraction. The feasibility of the variable-magnification X-ray Bragg optics was verified at the vertical-wiggler beamline BL-14B of the Photon Factory. For X-ray Bragg magnifiers, Si(220) crystals with an asymmetric angle of 14° were used. The magnification factor was calculated to be tunable between 0.1 and 10.0 at a wavelength of 0.112 nm. At various magnification factors (M ≥ 1.0), X-ray images of a nylon mesh were observed with an air-cooled X-ray CCD camera. Image deformation caused by the optics could be corrected by using a 2 × 2 transformation matrix and bilinear interpolation method. Not only absorption-contrast but also edge-contrast due to Fresnel diffraction was observed in the magnified images.

Synchronization of Coupled Neutral-Type Delay Partial Differential Systems

This paper considers the asymptotical synchronization and $$H_\infty $$H? synchronization for coupled neutral-type delay partial differential systems (NDPDSs). First, we construct a coupled synchronization error dynamic. Using the method of nonsingular matrix transformation, we decouple these coupled synchronization error dynamical systems. Then we study the asymptotical stability of the decoupled synchronization error dynamical systems through the Lyapunov---Krasovskii functional method, which implies the asymptotical synchronization of the coupled NDPDSs. Furthermore, when external disturbances enter the coupled NDPDSs, the $$H_\infty $$H? synchronization problem is also considered. The equivalence between the $$H_\infty $$H? stability of decoupled synchronization error dynamical systems and the $$H_\infty $$H? synchronization of coupled NDPDSs is proved by rigorous mathematical analysis. Then the criterion for the $$H_\infty $$H? stabilization is presented, which guarantees the $$H_\infty $$H? synchronization of the coupled NDPDSs. Moreover, as a remarkable difference between the ordinary differential systems and partial differential systems, the effect of the spatial domain on the synchronization is revealed through the obtained criteria. At last, numerical examples are given to illustrate the correctness of our results.

Matrix transformation method in quadratic binary optimization

The paper deals with the binary minimization of a quadratic functional. Typically the problem is NP-hard and the organization of the quadratic functional landscape in space of multiple binary variables even without constraints is not currently understood. We tackle the problem with the classical Hopfield neural network and show theoretically the exponential dependence between the energy and the attraction area of a minimum. As a consequence we propose a new algorithm using matrix transformation to cope with the problem. The matrix that gave birth to the functional is replaced by a new mix-matrix. This change of the objective function brings about a sharp increase in the minimization algorithm efficiency: the chances to find a global minimum grow as exp(?N) where N is the dimensionality of the problem, and the energy spectrum of minima being found shifts considerably to the deep side so that the average of the spectrum differs from the energy of the global minimum by only 4---6% (subject to the type of matrix).

Static output feedback stabilization for systems with time-varying delay based on a matrix transformation method

This paper is concerned with the problem of static output feedback stabilization for linear systems with time-varying delay. A novel controller design is proposed based on a matrix transformation method with a new equation condition, which can solve the controller gain more easily and avoid complicated calculations of the non-linear matrix inequality. Then, the corresponding criteria can be obtained by combining the matrix transformation method and the ideas of non-uniformly dividing delay interval, which can guarantee the asymptotic stability of the closed-loop systems and calculate the satisfactory controller gain. Finally, two numerical examples are provided to verify the effectiveness of the proposed design scheme.

Theory on resistance of m × n cobweb network and its application

SummaryA basic theorem of equivalent resistance between two arbitrary nodes in an m × n cobweb network in both finite and infinite conditions is discovered, and two conjectures on the equivalent resistance are proved in terms of the basic theorem. We built a tridiagonal matrix equation by means of network analysis and made a diagonalization method of matrix transformation and work out its explicit expressions. The new formulae obtained here can be effectively applied in complex impedance network, especially the formulation leads to the occurrence of resonances and a series of novel results in RLC (denote resistor, inductance and capacitance) network. These curious results suggest the possibility of practical applications to resonant circuits. Copyright © 2014 John Wiley & Sons, Ltd.

The extended unsymmetric frontal solution for multiple-point constraints

Purpose – The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling. Design/methodology/approach – Re-written frontal solution method with a priori pivot and front sequence. OpenMP parallelization, nearly linear (in elimination and substitution) up to 40 threads. Constraints enforced at the local assembling stage. Findings – When compared with both standard sparse solvers and classical frontal implementations, memory requirements and code size are significantly reduced. Research limitations/implications – Large, non-linear problems with constraints typically make use of the Newton method with Lagrange multipliers. In the context of the solution of problems with large number of constraints, the matrix transformation methods (MTM) are often more cost-effective. The paper presents a complete solution, with topological ordering, for this problem. Practical implications – A complete software package in Fortran 2003 is described. Examples of clique-based problems are shown with large systems solved in core. Social implications – More realistic non-linear problems can be solved with this Frontal code at the core of the Newton method. Originality/value – Use of topological ordering of constraints. A-priori pivot and front sequences. No need for symbolic assembling. Constraints treated at the core of the Frontal solver. Use of OpenMP in the main Frontal loop, now quantified. Availability of Software.

Integrated geometric and thermal error modeling and compensation for vertical machining centers

The accuracy of machine tools is significantly affected by the geometric defect and thermal deformation of the mechanical components. This paper intends to provide a comprehensive compensation method for the integrated geometric and thermal errors of machine tools. Firstly, a synthesized volumetric model is established with homogeneous transformation matrix method, considering both the geometric and thermal effects. Then, in order to improve the modeling accuracy and efficiency of the geometric error components, an automatic modeling algorithm is proposed with the Chebyshev polynomial-based orthogonal least squares regression. Also, to improve the robustness of the thermal error models for the feed axes and the spindle system, the thermal effects caused by external ambient variation and internal heat sources are identified and modeled separately. Finally, an intelligent virtual compensation system is developed for machine tools based on the function of external machine original coordinate shift and fast Ethernet data interaction technique, and compensation tests on a vertical machining center showed that the position accuracy of the machine tool could be significantly improved after compensation.

Agreement between manual and automated measurements of simple QRS/T angle

Abstract The spatial QRS/T angle (QRS/T) has been identified as a strong and independent predictor of adverse cardiac events. QRS/T can be determined from the electrocardiogram (ECG) by matrix transformation methods or formula which uses a combination of net QRS and T-wave amplitudes (QRS/Tsimple). Amplitudes can be measured automatically by using dedicated software (QRS/Tauto) or can be manually measured on a computer screen (QRS/Tmanual). This latter method allows analysis of QRS/T, when digital ECGs are not available. The aim of the study was to determine the agreement in the measurements between automatically derived QRS and T amplitudes and manually measured on the computer screen amplitudes. The relative error of the QRS/T between the two methods was estimated in 73 patients. In the case of QRS/Tmanual the inter-observer as well as intra-observer variability was estimated. The relative error between QRS/Tauto vs. QRS/Tmanual was 3.51%. Inter-observer and intra-observer variability of the QRS/Tmanual was 1.19% and 1.18% respectively. Manual measurement of the QRS/T is reliable, however, the predictive value of this parameter should be tested in clinical trials, before QRS/Tmanual can be considered a useful tool in clinical practice or retrospective studies.

Kinematics Analysis of Working Edge Point for Gauge Disc Cutter in Cutterhead of Tunneling Machine

Based on the analysis of movement characteristics of gauge disc cutter in cutterhead of tunneling machine, the motion equation to describe the working behavior of the edge point of gauge disc cutter in tunneling process is deduced by using the method of space coordinate transformation matrix. The motion equation has the advantages of considering more parameters, such as position variables of disc cutter, cutter radius, cutterhead rotating speed and advance speed.

Probing microstructural information of anisotropic scattering media using rotation-independent polarization parameters.

Polarization parameters contain rich information on the micro- and macro-structure of scattering media. However, many of these parameters are sensitive to the spatial orientation of anisotropic media, and may not effectively reveal the microstructural information. In this paper, we take polarization images of different textile samples at different azimuth angles. The results demonstrate that the rotation insensitive polarization parameters from rotating linear polarization imaging and Mueller matrix transformation methods can be used to distinguish the characteristic features of different textile samples. Further examinations using both experiments and Monte Carlo simulations reveal that the residue rotation dependence in these polarization parameters is due to the oblique incidence illumination. This study shows that such rotation independent parameters are potentially capable of quantitatively classifying anisotropic samples, such as textiles or biological tissues.

A general approach for error modeling of machine tools

This paper presents a general and systematic approach for geometric error modeling of machine tools due to the geometric errors arising from manufacturing and assembly. The approach can be implemented in three steps: (1) development of a linear map between the pose error twist and source errors within machine tool kinematic chains using homogeneous transformation matrix method; (2) formulation of a linear map between the pose error twist and the error intensities of a machine tool; (3) combination of these two models for error separation. The merit of this approach lies in that it enables the source errors affecting the compensatable and uncompensatable pose accuracy of the machine tool to be explicitly separated, thereby providing designers and/or field engineers with an informative guideline for the accuracy improvement by suitable measures, i.e. component tolerancing in design, manufacturing and assembly processes, and error compensation. Two typical multi-axis machine tools are taken as examples to illustrate the generality and effectiveness of this approach.

Sensitivity Analysis of Deviation Source for Fast Assembly Precision Optimization

Assembly precision optimization of complex product has a huge benefit in improving the quality of our products. Due to the impact of a variety of deviation source coupling phenomena, the goal of assembly precision optimization is difficult to be confirmed accurately. In order to achieve optimization of assembly precision accurately and rapidly, sensitivity analysis of deviation source is proposed. First, deviation source sensitivity is defined as the ratio of assembly dimension variation and deviation source dimension variation. Second, according to assembly constraint relations, assembly sequences and locating, deviation transmission paths are established by locating the joints between the adjacent parts, and establishing each part’s datum reference frame. Third, assembly multidimensional vector loops are created using deviation transmission paths, and the corresponding scalar equations of each dimension are established. Then, assembly deviation source sensitivity is calculated by using a first-order Taylor expansion and matrix transformation method. Finally, taking assembly precision optimization of wing flap rocker as an example, the effectiveness and efficiency of the deviation source sensitivity analysis method are verified.