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.

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.

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.

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.

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.

分布式散射体相位估计奇异值分解法

常规的分布式散射体(DS)相位估计方法需要生成全组合干涉对以构建样本协方差矩阵(SCM),然后根据SCM的统计特性估计DS相位,这一过程不但计算耗时,而且占据大量存储空间。本文提出了一种基于奇异值分解技术的DS相位快速估计方法(SVDI)。该方法分析的对象是单主影像干涉对组成的干涉相位矩阵而非全组合干涉对组成的SCM,因而可以有效提高计算效率、节省存储空间。并且,理论上证明了在一定条件下SVDI的结果与常规的特征值分解方法(EVD)是一致的。模拟数据和真实SAR数据的结果表明,SVDI算法有更高的计算效率,并且其相位估计精度以及形变解算精度与常规算法是一致的。

矩阵奇异值分解与非齐次线性方程组系数矩阵的广义逆求解

非齐次线性方程组在多数实际工程反演问题中较为常见,对其进行有效地求解是解决实际反演问题的关键,本文通过对方程组系数矩阵进行奇异值分解,推导非齐次线性方程组系数矩阵的广义逆求解过程,给出具体的求解方法和实现步骤,使得求解算法更容易进行计算机编程.

风电叶片铺层三维激光投影系统标定方案

为提高风电叶片复合材料多层铺贴的质量与效率,文中设计了一套视觉辅助自动标定的三维激光投影系统。系统采用全反射镜作为标靶,引入红外相机,建立视觉辅助的投影系统模型,利用相机与激光振镜间图像的单应性关系,标定二者之间的转换关系,并由此解算投影标靶在振镜投影坐标系的位姿,最后利用奇异值分解的方法求解激光振镜的投影矩阵。通过自主搭建三维激光投影系统实验平台,以投影幕布和网格纸作为投影面进行重投影试验,证明系统的可行性。实验结果表明,系统可以实现三维激光投影系统的自动标定,在投影距离为3 m的情况下,投影精度达到3.3 mm。

基于叶尖定时数据奇异值分解的振动事件识别

叶尖定时数据自动化测量及处理是旋转机械在线监测和智能运维的必要环节,快速、准确判断叶片振动类型,实现振动事件识别是数据自动化测量及处理的关键。提出了一种基于加窗叶尖定时数据奇异值分解的振动事件识别方法,仅需单只传感器准确识别叶片同步、异步振动事件。基于不同叶片叶尖定时数据时延特性,加窗构造了“类重构吸引子矩阵”,根据矩阵奇异值特征实现振动事件识别。开展了方法仿真及实验验证,仿真与实验结果一致性良好,压气机试验件测试数据表明,叶片发生振动事件时第1奇异值增大为7倍以上,其中发生异步振动事件时第2奇异值增大为14倍以上,提出方法能够准确识别叶片同步、异步振动事件。

基于奇异值分解张拉整体结构找形方法

张拉整体结构是由一组不连续的受压单元包含于连续受拉单元组成的稳定自平衡结构,为了寻找自平衡状态下的构型,引入奇异值分解的方法,将寻找张拉整体结构的自平衡构型问题转化为最小奇异值的判定。通过广义节点坐标和构件之间的连接关系建立结构的数学模型;引入力密度的概念,对张拉整体结构进行受力分析,列写包含平衡矩阵的平衡方程;对平衡矩阵进行奇异值分解,利用分解获得的最小奇异值判断平衡方程是否有解,对张拉整体结构是否存在自平衡构型进行初步判定,再依据获得的力密度的均匀性(同组构件力密度大小相等)和正负属性(杆的力密度小于0,索的力密度大于0)对结构自平衡状态进一步判断;通过实例分析对该找形方法的可行性进行了验证,结果表明:该方法可以找到张拉整体结构自平衡构型。本文为寻找自平衡张拉整体结构提供了一种思路。

模型误差条件下声矢量圆阵多重信号分类测向改进算法

针对非正规协方差矩阵引起声矢量圆阵多重信号分类(MUSIC)测向算法性能恶化的问题,提出了一种基于奇异值分解的声压振速联合处理MUSIC改进算法。理论分析了阵列响应误差和噪声模型误差对协方差矩阵的正规性及估计性能的影响。模型误差条件下声矢量阵声压振速联合处理的协方差矩阵不再是正规矩阵,改进的MUSIC方法通过对声压振速联合处理的协方差矩阵进行奇异值分解,利用非正规矩阵的左、右奇异向量自身正交的特性,采用奇异向量张成噪声子空间。数值仿真结果表明,改进的MUSIC方法改善了波达方向估计精度和多目标分辨能力,且具有更低、更平坦的空间背景谱。湖上试验进一步验证了改进的MUSIC方法的有效性。

基于重叠Ket增强和张量列车的非平衡频谱制图算法

近年来,基于张量补全的频谱制图得到了广泛研究.目前用于频谱制图的张量补全算法大多隐含地假设张量具有平衡特性,而对于非平衡张量,难以利用其低秩性估计完整的张量信息,导致补全算法性能受损.本文提出基于重叠Ket增强(Overlapping Ket Augmentation,OKA)和张量列车(Tensor Train,TT)的非平衡频谱制图算法,以解决非平衡张量在应用传统张量补全算法时性能下降的问题.首先使用OKA将低阶高维张量表示为高阶低维张量,在无信息损耗的情况下解决非平衡张量无法利用其低秩性进行张量补全的问题;然后使用TT矩阵化得到较平衡的矩阵,在维度较平衡条件下提高补全算法的精确度;最后利用高阶低维张量的低秩性,使用并行矩阵分解或基于F范数的无奇异值分解(Singular Value Decomposition Free,SVDFree)算法完成张量补全.仿真结果表明,针对非平衡张量,所提方案与现有的张量补全算法相比,可以获得更精确的无线电地图,同时所提SVDFree算法具有更低的计算复杂度.

基于强跟踪的移动机器人CQKF-SLAM方法

针对容积正交卡尔曼滤波(CQKF)在同时定位与地图构建(SLAM)中系统状态驱动模型与观测数据存在突变,以及协方差分解引起系统不稳定,导致移动机器人定位精度降低的问题,提出一种基于多重渐消因子强跟踪的SVDCQKF-SLAM方法。采用奇异值分解(SVD)代替CQKF算法中的乔列斯基分解,抑制状态误差协方差矩阵负定性;引入多重渐消因子强跟踪滤波器调节状态预测协方差矩阵。通过仿真实验,将所提SLAM方法与其它SLAM方法进行对比,其结果表明,该方法能够有效降低SLAM过程中的定位误差,对移动机器人同时定位与地图构建有一定参考价值。

基于奇异值分解的γ能谱弱峰识别

针对低放射性核素样品的γ能谱分析中的弱峰识别问题,提出了一种基于奇异值分解的γ能谱寻峰新方法。该方法通过改进矩阵的构造方式,将γ能谱升维为双向循环矩阵后进行奇异值分解,选择第二个奇异值进行矩阵重构,并根据重构后的矩阵寻峰。以放射源152Eu的γ能谱为实验对象,与一阶导数寻峰、对称零面积寻峰和其他矩阵的奇异值分解寻峰方法进行了对比。实验结果表明:双向循环矩阵奇异值分解寻峰方法,在检测弱峰时拥有更高的查全率、查准率和F1值,分别达到了100%、87%和0.94,优化了弱峰检测的效果,为寻峰方法提供更多的选择。

基于奇异值分解的城轨网络关键站点识别及其演变研究

为识别城市轨道交通网络关键站点并研究其多年演化,构建基于截断奇异值分解(truncated singular value decomposition,TSVD)的关键站点识别方法,选取北京市2011—2019年早高峰时段的OD数据,通过关键特征向量分析网络客流演变并对城轨网络中关键站点进行识别,将其与复杂网络方法的识别结果进行对比。分析表明:TSVD法能很好地应用于考虑OD分布的网络关键站点识别,识别结果能更好代表网络客流的空间分布。从识别结果看,北京轨道交通关键站点空间布局呈现多中心发展趋势,如西北西二旗,西南丰台科技园等站点逐步形成网络客流中心并相互联系;东南土桥、东北俸伯等站点也初步呈现网络客流中心的特征。

基于SVD-Schmidt正交化的压缩感知测量矩阵的优化

压缩感知(CS)理论中测量矩阵的性能优劣直接影响信号重构性能。为了优化测量矩阵提高其重构性能,本文提出了一种基于奇异值分解-施密特(SVD-Schmidt)正交化的CS测量矩阵优化方法。首先对测量矩阵进行奇异值分解(SVD)并选择最大的奇异值替换原来的奇异值形成新的矩阵,同时对其进行施密特正交化,对矩阵的列进行单位化,通过行和列不断循环交替自适应迭代优化得到优化后的测量矩阵。通过一维信号和二维图像的仿真实验验证所提方法的优越性。一方面,本文方法优化的测量矩阵互相关性明显降低;另一方面,实验仿真结果证明了测量矩阵经过优化之后提高了信号重构性能,本文方法重构性能优于现有的SVD法和特征值分解法。

对偶P-分解及偏序

文章应用对偶奇异值分解建立对偶复矩阵的P-分解,得到该分解的若干性质,应用该分解建立D-GL二元关系,并证明该二元关系是一类偏序。

基于前后向矩阵束方法的线性排布无人机集群目标规模估算

利用前后向矩阵束方法,提出了一种有效的“蜂群”无人机群目标个数估计方法用于目标检测。与传统的参数估算方法相比,所提方法对于稀疏信号也能准确地进行估算。首先,在一定的角度或频率下,获取线阵排列群目标的雷达散射截面;然后,根据已知的雷达回波构造Hankel-Toeplitz矩阵,并对该矩阵进行奇异值分解;最后,通过设置合适的阈值选取比例较大的奇异值,以此准确地估算线阵排列群目标的个数。此外,文中还讨论了不同信噪比对估算结果的影响。为了提高精度,文中采用多组不同方位角度的回波信号来联合估算群目标的个数。仿真结果表明,该方法能够有效、准确地估算线阵排列群目标的个数,而且估计结果不受噪声的影响,具有较好的鲁棒性。