Generalized Inverse Of Matrix Research Articles

Rheological behavior analysis of emulsified cutback composite cold-mixed epoxy asphalt binder based on apparent viscosity

In this paper, a composite cold-mixed epoxy asphalt binder (CCEAB) is made using diluted epoxy resin (DER) and cold-mixed solvent (CMS). The rheological behavior of CCEAB is analyzed by rotary viscosimeter at different temperatures, curing times, shear rates, DER and CMS contents. The fluid type of the CCEAB is identified, and its flow curve is modeled. The parallel phenomenon of apparent viscosity curves is revealed, and the shift factor is analyzed. The results showed that temperature, CMS content and shear rate are negatively correlated with viscosity, while time and DER content are positively correlated with it. CCEAB has typical shear thinning behavior and is a pseudoplastic fluid without yield stress. When the shear rate is greater than 18.6 s−1, the viscosity curves trend stable, and the final viscosity is about 60 % of the initial viscosity. The 5 variables have been added to the power law model to establish the flow curve of CCEAB, whose parameters are identified by the generalized inverse matrix. The parallel of viscosity curves follows the time-temperature superposition, and the shift factor function with 4 variables describes the conversion of viscosity curves. The flow curve of CCEAB has the same form and essence as the shift factor function, meaning the flow curve also describes the parallel. If the shift factor is regarded as a viscosity prediction method, it is not as accurate as the flow curve.

Large-Area Film Thickness Identification of Transparent Glass by Hyperspectral Imaging.

This study introduces a novel method for detecting and measuring transparent glass sheets using hyperspectral imaging (HSI). The main goal of this study is to create a conversion technique that can accurately display spectral information from collected images, particularly in the visible light spectrum (VIS) and near-infrared (NIR) areas. This technique enables the capture of relevant spectral data when used with images provided by industrial cameras. The next step in this investigation is using principal component analysis to examine the obtained hyperspectral images derived from different treated glass samples. This analytical procedure standardizes the magnitude of light wavelengths that are inherent in the HSI images. The simulated spectral profiles are obtained using the generalized inverse matrix technique on the normalized HSI images. These profiles are then matched with spectroscopic data obtained from microscopic imaging, resulting in the observation of distinct dispersion patterns. The novel use of images coloring methods effectively displays the thickness of the glass processing sheet in a visually noticeable way. Based on empirical research, changes in the thickness of the glass coating in the NIR-HSI range cause significant changes in the transmission of infrared light at different wavelengths within the NIR spectrum. This phenomenon serves as the foundation for the study of film thickness. The root mean square error inside the NIR area is impressively low, calculated to be just 0.02. This highlights the high level of accuracy achieved by the technique stated above. Potential areas of investigation that arise from this study are incorporating the proposed approach into the design of a real-time, wide-scale automated optical inspection system.

Semi-blind hyperspectral and multispectral image fusion via generalized inverse matrix optimization

Combining a low spatial resolution hyperspectral image (HSI) with a high spatial resolution multispectral image (MSI) is an economical way to obtain a high spatial resolution hyperspectral image (HR-HSI). However, existing fusion methods often rely on a critical variable, the point spread function (PSF), which is difficult to obtain in practice. Additionally, these methods often require a long fusion time and have many parameters that need re-debugging on different datasets to achieve satisfactory results. Many existing methods lack robustness in handling complex scenarios, such as complex PSF, changing spatial downsampling factors, and large-scale datasets. These issues can result in degraded performance or render the methods inapplicable to practical scenarios. Therefore, we propose an efficient and parameterless semi-blind fusion method based on generalized inverse matrix optimization (GIMO), which is robust for multiple complex situations. Our method uses the definition of generalized inverse matrix and the relationship among HSI, MSI, and HR-HSI to approximate the function module of PSF. Then, we calculate the generalized inverse matrix of the spectral response function (SRF). Finally, we optimize the generalized inverse matrix of SRF using the Sylvester equation and multiply it with MSI to obtain HR-HSI. This is the first proposed method to achieve hyperspectral image super-resolution by optimizing the inverse matrix of the spectral response function. Experiments on multiple simulated datasets and one real dataset have validated that our GIMO method outperforms existing state-of-the-art methods, especially regarding fusion time and robustness for various complex scenarios.

Finite-Time Time-Varying Formation Tracking for Heterogeneous Nonlinear Multiagent Systems Using Adaptive Output Regulation.

The finite-time output time-varying formation tracking (TVFT) problem for heterogeneous nonlinear multiagent system (MAS) is investigated in this article, where the dynamics of the agents can be nonidentical, and leader's input is unknown. The target of this article is that the outputs of followers need to track leader's output and realize the desired formation in finite time. First, for removing the assumption that all agents are required to know the information of leader's system matrices and the upper boundary of its unknown control input in previous studies, a kind of finite-time observer is constructed by exploiting the neighboring information, which can estimate not only the leader's state and system matrices but also can compensate for the effects of unknown input. On the basis of the developed finite-time observers and adaptive output regulation method, a novel finite-time distributed output TVFT controller is proposed with the help of the technique of coordinate transformation by introducing an extra variable, which removes the assumption that the generalized inverse matrix of follows' input matrix needs to be found in the existing results. By means of the Lyapunov and finite-time stability theory, it is proven that the expected finite-time output TVFT can be realized by the considered heterogeneous nonlinear MASs within a finite time. Finally, simulation results demonstrate the efficacy of the proposed approach.

Robust fault estimation for T‐S fuzzy systems with intermittently sampled data based on finite information learning observer

SummaryIn this article, we study the problem of robust fault estimation for a class of T‐S fuzzy systems with time delays and external disturbances based on the finite information learning observer. The learning observer with intermittent sampling is constructed and the change rate of output estimation error is introduced to estimate the system fault effectively. At the same time, by introducing a generalized inverse matrix to the designed observer, a sufficient condition for the stability of the error estimation system is given by using fuzzy Lyapunov function, which not only solves the problem of system fault and external disturbance estimation, but also reduces the conservatism of the obtained conclusion. Finally, the results described by LMI are solved numerically and simulated to demonstrate the effectiveness of the proposed method.

Toward Reducing Undesired Rotation Torque in Maglev Permanent Magnet Synchronous Linear Motor

The dq current transformation method can be used to achieve the drive control of the maglev permanent magnet synchronous linear motor (MPMSLM). Specifically, the control of the suspension force Fz can be achieved by controlling the d-axis current, and the control of the thrust force Fx can be achieved by controlling the q-axis current. However, the direct use of the dq current transformation method on traditional MPMSLM structures produces an additional rotation torque around the y-axis. In order to reduce this additional torque, a new MPMSLM structure is proposed in this work. First, the characteristics of additional torque through finite element analysis are analyzed. Second, the Halbach permanent magnet array and coil size are optimized, and the topology of the MPMSLM coil is designed to reduce the additional torque. The decoupling performance and current of the proposed MPMSLM are experimentally verified through open-loop experiments using finite element simulation software. Finally, the decoupling algorithm of the generalized inverse matrix is used to achieve the decoupling between forces Fx and Fz and torque Ty and the d-axis and q-axis currents in each drive unit. Based on this, a three-degree-of-freedom closed-loop control system of the MPMSLM is designed. The LabVIEW 2018 software is used for the simulation analysis of the three-degree-of-freedom MPMSLM motion control system, and the results show that the proposed motor structure has superior closed-loop control performance.

Improving Newton–Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory

Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary m×n complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.

State Estimation for a Class of Discrete-Time BAM Neural Networks With Multiple Time-Varying Delays

For a class of discrete-time bidirectional associative memory neural networks (DTBAMNNs) with multiple time-varying delays, the issue of state estimation is studied. By propose a mathematical induction method, we first investigate novel delay-dependent and -independent global exponential stability (GES) criteria of the error system. The obtained GES criteria are described by linear scalar inequalities. Then, a state observer is derived via the theory of generalized matrix inverses. These exponential stability conditions are very simple, which is convenient to verify based on the standard software tools (for example, YALMIP). Finally, we present two illustrative examples to present the effectiveness of the theoretical results.

Generalized inverse matrix - long short-term memory neural network data processing algorithm for multi-wavelength pyrometry.

The data processing of multi-wavelength pyrometry (MWP) is faced with the problem of solving N equations and N+1 unknown underdetermined equations. The traditional iterative optimization methods are difficult to meet the actual measurement requirements in terms of accuracy and efficiency. With the development of artificial intelligence technology in the field of data processing, it is expected to solve this problem. A generalized inverse matrix (GIM) is combined with a long short-term memory (LSTM) neural network algorithm for data processing of MWP is proposed, which emissivity influence is dispensed completely. Firstly, GIM is used for classification of the emissivity. Furthermore, inputting to the LSTM network not only ensures the accuracy of temperature measurement but also greatly improves the efficiency. The simulation results demonstrated that the accuracy of the GIM-LSTM algorithm was superior to that of the GIM-EPF and BP methods. After random noise was added, the relative error was still less than that for the GIM-EPF and BP methods, and the algorithm exhibited excellent anti-noise performance. Publicly available temperature data for the exhaust plume of a rocket engine were processed by the GIM-LSTM method, and the average relative error was less than the traditional method. Especially, in terms of inversion speed, the operational time of the GIM-LSTM algorithm was at the millisecond level, which is of great significance for the real-time monitoring of rocket exhaust plumes. The proposed GIM-LSTM data processing algorithm affords high accuracy and speed and is suitable for practical measurement of high-emissivity objects in real-time via MWP.

Unified Solution of Different-Kind Future Matrix Equations Using New Nine-Instant Discretization Formula and Zeroing Neural Dynamics

In this article, time-varying matrix equation problems, including the Lyapunov equation, matrix inversion, and generalized matrix inversion are investigated in a future (or say, discrete time-varying) perspective. Then, in order to develop a unified solution model for the above three future problems, a future matrix equation (FME) is investigated. The discrete-time unified solution (DTUS) model, which is based on the zeroing neural dynamics (ZND) method and a new nine-instant Zhang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> discretization (ZeaD) formula, is thus proposed and termed the nine-instant DTUS (9IDTUS) model. Meanwhile, theoretical analyses on the stability and precision of the 9IDTUS model are provided. In addition, conventional DTUS models obtained from the Euler forward formula, Taylor–Zhang discretization formula, and a seven-instant discretization formula are also presented for comparisons. Furthermore, numerical experiments including the robot motion generation, are conducted and analyzed to substantiate the efficacy and superiority of the proposed 9IDTUS model.

INVERS TERGENERALISASI MOORE PENROSE

The generalized inverse is a concept for determining the inverse of a singular matrix and and matrix which has the characteristic of the inverse matrix. There are several types of generalized inverse, one of which is the Moore-Penrose inverse. The matrix is called Moore Penrose inverse of a matrix if it satisfies the four penrose equations and is denoted by . Furthermore, if the matrix satisfies only the first two equations of the Moore-Penrose inverse and , then is called the group inverse of and is denoted by . The purpose of this research was to determine the group inverse of a non-diagonalizable square matrix using Jordan’s canonical form and Moore Penrose’s inverse of a singular matrix, also a non-square matrix using the Singular Value Decomposition (SVD) method. The results of this study are the sufficient condition for a matrix to have a group inverse, i.e., a matrix has an index of 1 if and only if the product of two matrices forming is a full rank factorization and is invertible. Whereas for a singular matrix and a non-square , the Moore-Penrose inverse can be determined using Singular Value Decomposition (SVD). Keywords: generalized matrix inverse, Moore Penrose inverse, group inverse, Jordan canonical form, Singular Value Decomposition.

Parallel Efficiency Analysis of Large Increment Method Based on OpenMP

Large Increment Method (LIM) is a novel force-based method based on the generalized inverse matrix theory for solving material nonlinear problems in small deformation and small displacement conditions. For the spatial parallel computation, LIM can compute the constitutive equations of each element in parallel. By analyzing the parallel-calculating characteristics of LIM, the OpenMP is used for the parallel computation in spatial domain. Through the parallelism analysis of the spatial and time domain for LIM, it shows that LIM has an excellent parallel efficiency in space parallel computation.

Fast Bayesian Compressed Sensing Algorithm via Relevance Vector Machine for LASAR 3D Imaging

Because of the three-dimensional (3D) imaging scene’s sparsity, compressed sensing (CS) algorithms can be used for linear array synthetic aperture radar (LASAR) 3D sparse imaging. CS algorithms usually achieve high-quality sparse imaging at the expense of computational efficiency. To solve this problem, a fast Bayesian compressed sensing algorithm via relevance vector machine (FBCS–RVM) is proposed in this paper. The proposed method calculates the maximum marginal likelihood function under the framework of the RVM to obtain the optimal hyper-parameters; the scattering units corresponding to the non-zero optimal hyper-parameters are extracted as the target-areas in the imaging scene. Then, based on the target-areas, we simplify the measurement matrix and conduct sparse imaging. In addition, under low signal to noise ratio (SNR), low sampling rate, or high sparsity, the target-areas cannot always be extracted accurately, which probably contain several elements whose scattering coefficients are too small and closer to 0 compared to other elements. Those elements probably make the diagonal matrix singular and irreversible; the scattering coefficients cannot be estimated correctly. To solve this problem, the inverse matrix of the singular matrix is replaced with the generalized inverse matrix obtained by the truncated singular value decomposition (TSVD) algorithm to estimate the scattering coefficients correctly. Based on the rank of the singular matrix, those elements with small scattering coefficients are extracted and eliminated to obtain more accurate target-areas. Both simulation and experimental results show that the proposed method can improve the computational efficiency and imaging quality of LASAR 3D imaging compared with the state-of-the-art CS-based methods.

A shape-finding approach of tree-like structure based on grouping strategy and generalized inverse matrix theory

Unstable link models in equilibrium transfer their loads only through axial forces and the generalized inverse matrix theory can be used in the shape-finding analysis of unstable link structures. According to this principle and the generalized inverse matrix theory, a shape-finding approach for tree-like structure is proposed, which forms a progressive process to update the shape of branching model until the potential energy of the model reaches minimum. In the proposed approach, members of the branching model are grouped and the total length of the members in the same group are constrained. The total length can be redistributed among the members in the same group in the shape-finding process. The redistribution of length is particularly useful for the shape-finding of tree-like structure as it can avoid the generation of inefficient members in the shape-found structure. Changing the grouping mode of members is a way to generate multiple schemes of structural shapes. Numerical examples illustrate the features of the approach and validate its effectiveness.

Research on Spiral Bevel Gear Detection Based on Environmental Tooth Surface Error Correction Method

This paper studies the influence of the machine tool adjustment parameter error on the tooth surface error of the spiral bevel gear wheel processed by the double helix method. Based on the gear meshing principle, MATLAB software was used to establish the error tooth surface equation, and the analytical formula of the discrete point error of the tooth surface containing the machine tool adjustment parameter errors was derived. The topological map of the concave and convex tooth surface errors corresponding to the machine tool adjustment parameter errors was obtained and analyzed. The mapping relationship between machine tool adjustment parameter errors and tooth surface errors is determined, and the adjustment parameters that have a greater influence on the tooth surface errors are determined. With the help of the approximate expression of the topological diagram of the tooth surface error by the second-order surface, the influence weight of the machine tool adjustment parameter error on the tooth surface error is obtained. Based on the tooth surface error and the two-tooth surface error sensitive matrices, a tooth surface error correction model was established. The least square method of the generalized inverse matrix was used to transcend the equations to obtain the correction amount of the machine tool adjustment parameters. Finally, an example is used to verify the correction effect of the error tooth surface. The results show that the error of the tooth surface on the concave and convex sides after the correction is greatly reduced.

A Geometric View on Inner Transformation between the Variables of a Linear Regression Model

In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from a visible geometrical view. As an example, the roadmap of such inner transformation is presented based on a simple multiple linear regression model in this work. By applying the matrix algorithms like singular value decomposition (SVD) and Moore-Penrose generalized matrix inverse, the dependent variable vector lands into the right space of the independent variable matrix and is metamorphosed into regression coefficient estimator vector through the three-step of inner transformation. This work explores the geometrical relationship between the dependent variable vector and regression coefficient estimator vector as well as presents a new approach for vector rotating.

Sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and application to [formula omitted] filtering for linear systems with time-varying delays

This paper is concerned with sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and their application to H∞ filtering for linear systems with time-varying delays. First, a class of higher degree matrix-valued polynomial inequalities are transformed into the first degree matrix-valued polynomial inequalities by introducing some slack matrices. As a result, a convex property is applied to derive sufficient conditions for a class of higher degree matrix-valued polynomial inequalities on closed intervals. Second, by choosing a Lyapunov–Krasovskii functional with a quadratic matrix-valued polynomial on a time-varying delay, and estimating its time-derivative as a third-degree matrix-valued polynomial on the time-varying delay, the proposed sufficient conditions are utilized to formulate a novel bounded real lemma on the existence of H∞ filters for time-delay systems. Third, some algorithms to filter design are provided and discussed in detail. It is pointed out that, if the filter gain is derived through a generalized inverse matrix method, it may be a fake solution, which is demonstrated through a liquid monopropellant rocket motor with a pressure feeding system.

Seismic Inversion for the Calculation of Velocities Using the Generalized Inverse Linear Matrix, the Wave Equation, and a Non-Reflective-Boundaries Condition

Objective: this work presents the results obtained in the development of a seismic velocity inversion mo­del. The reference times recorded on the surface are taken and using the inversion model to obtain the initial reference model (hypocenters and velocities), starting from an unknown model.&#x0D; Methodology: A hypothetical reference model is pro­posed containing 64 blocks with interval velocity, 16 recording stations on the surface, and 64 earth­quakes in the center of each block. With this model, the reference arrival times are generated for each ear­thquake registered in each station. The inversion mo­del is made up of two parts: the direct model, which allows calculating the arrival times of the signal regis­tered on the surface according to the hypo-central lo­cation of the earthquake and the velocity of the P and S wave of the medium; and the inverse model, which estimates a model of the velocity of the environment and hypo-central locations of the earthquakes that are the input variables of the direct model. The direct model was developed with the wave equation, while the inverse model was developed by modifying the generalized inverse matrix by introducing a factor called “damping.”&#x0D; Results: The discretization model is based on the finite difference method.. When estimating the values of velocity and hypo-central location with the inverse algorithm, the propagation of the wave is simulated with the direct model, and they are compared with the data of reference times measured on the surface. Depending on the value of the mean square error, we proceed to modify the mean velocities and hypo­centers of the earthquakes. This process is repeated iteratively until the calculated error is less than a to­lerance of 2x10-3s2.&#x0D; Conclusions: It was found that the estimated values of velocity and hypo central locations coincide well for regions closer to the surface, while for deep regions the error is greater compared to the hypothetical re­ference model.

Moore–Penrose generalized inverse mixture design applied in low‐density dispersive liquid–liquid microextraction

AbstractThe modification in the nutritional composition of the ruminant diet causes significant alterations in the fatty acids (FAs) structure supplemented because of the action of rumen microorganisms. The modification in the FAs structure alters the role that these play in the ruminant metabolism. The Folch method is the most often used to determine fatty acids in these animals' tissues and presents certain disadvantages such as the great volume of solvent and low mass transfer from the analyte to the extracting phase. Thus, we tested the low‐density dispersive liquid–liquid microextraction (LD‐DLLME) as an alternative method to determine these substances. In this paper, a simples‐augmented mixture design was used. The Scheffé's polynomial was applied in that design, and Moore–Penrose generalized matrix inverse was used because of the possibility of concurrently determining value estimates of coefficients of the parameters that represent cubic terms. The application of the modeling allowed the chemical interpretation of the LD‐DLLME best extraction condition for linoleic acid in ruminant serum samples.

Representations of generalized inverses via full-rank QDR decomposition

In this paper, novel representations of generalized inverses of rational matrices are developed. Therefore, a unified approach for the computation of {1,2,3} and {1,2,4} inverses and Moore-Penrose inverse of a given matrix A is considered. Full-rank QDR decomposition of a rational matrix is utilized to avoid the square roots of rational expressions in the evaluations, making the given algorithm very suitable for symbolic computations of generalized matrix inverses. Furthermore, we developed an algorithm for symbolic computation of the Moore-Penrose inverse of a polynomial matrix using the full-rank QDR decomposition, therefore maximizing the potential of using square root–free polynomial entries. Introduced algorithms are illustrated via numerical examples.