Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

Randomized Generalized Singular Value Decomposition

The generalized singular value decomposition(GSVD)of two matrices with the same number of columns is a very useful tool in many practical applications.However,the GSVD may suffer from heavy computational time and memory requirement when the scale of the matrices is quite large.In this paper,we use random projections to capture the most of the action of the matrices and propose randomized algorithms for computing a low-rank approximation of the GSVD.Serval error bounds of the approximation are also presented for the proposed randomized algorithms.Finally,some experimental results show that the proposed randomized algorithms can achieve a good accuracy with less computational cost and storage requirement.

DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.

PERTURBATION METHOD FOR REANALYSIS OF THE MATRIX SINGULAR VALUE DECOMPOSITION

The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

Continuous-Time and Discrete-Time Singular Value Decomposition of an Impulse Response Function

This paper proposes the continuous-time singular value decomposition (SVD) for the impulse response function, a special kind of Green’s functions, in order to find a set of singular functions and singular values so that the convolutions of such function with the set of singular functions on a specified domain are the solutions to the inhomogeneous differential equations for those singular functions. A numerical example was illustrated to verify the proposed method. Besides the continuous-time SVD, a discrete-time SVD is also presented for the impulse response function, which is modeled using a Toeplitz matrix in the discrete system. The proposed method has broad applications in signal processing, dynamic system analysis, acoustic analysis, thermal analysis, as well as macroeconomic modeling.

Coupled Cross-correlation Neural Network Algorithm for Principal Singular Triplet Extraction of a Cross-covariance Matrix

This paper proposes a novel coupled neural network learning algorithm to extract the principal singular triplet (PST) of a cross-correlation matrix between two high-dimensional data streams. We firstly introduce a novel information criterion (NIC), in which the stationary points are singular triplet of the crosscorrelation matrix. Then, based on Newton's method, we obtain a coupled system of ordinary differential equations (ODEs) from the NIC. The ODEs have the same equilibria as the gradient of NIC, however, only the first PST of the system is stable (which is also the desired solution), and all others are (unstable) saddle points. Based on the system, we finally obtain a fast and stable algorithm for PST extraction. The proposed algorithm can solve the speed-stability problem that plagues most noncoupled learning rules. Moreover, the proposed algorithm can also be used to extract multiple PSTs effectively by using sequential method. © 2014 Chinese Association of Automation.

Promote the Compression Efficiency of Digital Images by Using Improved CUR Matrix Decomposition Algorithm

In order to overcome the problem that the CUR matrix decomposition algorithm loses a large amount of information when compressing images, the quality of reconstructed images is not high, we propose a CUR matrix decomposition algorithm based on standard deviation sampling. Because of retaining more image information, the reconstructed image quality is higher under the same compression ratio. At the same time, in order to further reduce the amount of image information lost during the sampling process of the CUR matrix decomposition algorithm, we propose the SVD-CUR algorithm. The experimental results verify that our algorithm can achieve high image compression efficiency, and also demonstrate the high precision and robustness of CUR matrix decomposition algorithm in dealing with low rank sparse matrix data.

Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.

An Approximate Linear Solver in Least Square Support Vector Machine Using Randomized Singular Value Decomposition

In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition（randomized SVD）. Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nystrom method or Lanczos process.

Least-Squares Solutions of the Matrix Equation A^TXA=B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.

Development of the Decoupled Discreet-Time Jacobian Eigenvalue Approximation for Situational Awareness Utilizing Open PDC

With the increased number of PMUs in the power grid, effective high speed, realtime methods to ascertain relevant data for situational awareness are needed. Several techniques have used data from PMUs in conjunction with state estimation to assess system stability and event detection. However, these techniques require system topology and a large computational time. This paper presents a novel approach that uses real-time PMU data streams without the need of system connectivity or additional state estimation. The new development is based on the approximation of the eigenvalues related to the decoupled discreet-time power flow Jacobian matrix using direct openPDC data in real-time. Results are compared with other methods, such as Prony’s method, which can be too slow to handle big data. The newly developed Discreet-Time Jacobian Eigenvalue Approximation (DDJEA) method not only proves its accuracy, but also shows its effectiveness with minimal computational time: an essential element when considering situational awareness.

{P,Q,k+1}-reflexive solutions to a system of matrix equations

In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above to have the{P,Q,k+1}-reflexive and anti-reflexive solutions.We also obtain the expressions of such solutions to the system by the singular value decomposition.Moreover,we consider the least squares{P,Q,k+1}-reflexive and anti-reflexive solutions to the system.Finally,we give an algorithm to illustrate the results of this paper.

基于Matrix Pencil的低频振荡辨识及PSS优化配置研究

在互联系统中,低频振荡问题越来越成为影响电力系统稳定的重要因素。广域测量系统(WAMS)的发展和应用使得低频振荡的在线辨识和PSS配置及参数优化成为可能。本文探讨了在已知干扰情况下,利用测量数据构造矩阵,基于Matrix Pencil算法来进行电力系统低频振荡模式分析,建立多输入输出系统的低阶近似的辨识传递函数,并在此基础上利用奇异值分解(SVD)的方法来分析PSS优化配置问题。通过经4机2区域系统的仿真实验研究表明,基于Matrix Pencil辨识算法能建立准确的低阶近似传递函数,奇异值分解的方法能有效分析出PSS最佳配置地点。

Correction of failure in antenna array using matrix pencil technique

In this paper a non-iterative technique is developed for the correction of faulty antenna array based on matrix pencil technique（MPT）. The failure of a sensor in antenna array can damage the radiation power pattern in terms of sidelobes level and nulls. In the developed technique, the radiation pattern of the array is sampled to form discrete power pattern information set. Then this information set can be arranged in the form of Hankel matrix（HM） and execute the singular value decomposition（SVD）. By removing nonprincipal values, we obtain an optimum lower rank estimation of HM. This lower rank matrix corresponds to the corrected pattern. Then the proposed technique is employed to recover the weight excitation and position allocations from the estimated matrix. Numerical simulations confirm the efficiency of the proposed technique, which is compared with the available techniques in terms of sidelobes level and nulls.

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.

Hand-eye calibration with a new linear decomposition algorithm

To solve the homogeneous transformation equation of the form AX=XB in hand-eye calibration, where X represents an unknown transformation from the camera to the robot hand, and A and B denote the known movement transformations associated with the robot hand and the camera, respectively, this paper introduces a new linear decomposition algorithm which consists of singular value decomposition followed by the estimation of the optimal rotation matrix and the least squares equation to solve the rotation matrix of X. Without the requirements of traditional methods that A and B be rigid transformations with the same rotation angle, it enables the extension to non-rigid transformations for A and B. The details of our method are given, together with a short discussion of experimental results, showing that more precision and robustness can be achieved.

Mobility and equilibrium stability analysis of pin-jointed mechanisms with equilibrium matrix SVD

Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new crite- rion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.

Expansion of the Decoupled Discreet-Time Jacobian Eigenvalue Approximation for Model-Free Analysis of PMU Data

This paper proposes an extension of the algorithm in [1], as well as utilization of the wavelet transform in event detection, including High Impedance Fault (HIF). Techniques to analyze the abundant data of PMUs quickly and effectively are paramount to increasing response time to events and unstable parameters. With the amount of data PMUs output, unstable parameters, tie line oscillations, and HIFs are often overlooked in the bulk of the data. This paper explores model-free techniques to attain stability information and determine events in real-time. When full system connectivity is unknown, many traditional methods requiring other bus measurements can be impossible or computationally extensive to apply. The traditional method of interest is analyzing the power flow Jacobian for singularities and system weak points, attained by applying singular value decomposition. This paper further develops upon the approach in [1] to expand the Discrete-Time Jacobian Eigenvalue Approximation (DDJEA), giving values to significant off-diagonal terms while establishing a generalized connectivity between correlated buses. Statistical linear models are applied over large data sets to prove significance to each term. Then the off diagonal terms are given time-varying weights to account for changes in topology or sensitivity to events using a reduced system model. The results of this novel method are compared to the present errors of the previous publication in order to quantify the degree of improvement that this novel method imposes. The effective bus eigenvalues are briefly compared to Prony analysis to check similarities. An additional application for biorthogonal wavelets is also introduced to detect event types, including the HIF, for PMU data.

Several Generalized Matrix Versions of Kantorovich Inequalities

Kantorovich inequalities are old results. In this paper we give several Kantorovich-type matrix inequalities.

THE POSITIVE SEMIDEFINITE SOLUTION OF THE MATRIX EQUATION (A^TXA, B^TXB) = (C, D)

In this paper, we consider the positive semidefinite solution of the matrix equation (AT X A, BT X B) - (C, D). A necessary and sufficient condition for the existence of such solution is derived using the generalized singular value decomposition.The general forms of positive semidefinite solution are given.