ON SINGULAR MATRIX VARIATE BETA DISTRIBUTION

In this paper the density of the matrix variate beta distribution of rank lower than itsdimensionality is obtained with respect to a suitably defined differential form under the condi-tion that the difference between the identity and this matrix has full rank. As preliminaries,the Jacobian of a transformation related to decomposing a nonnegative-definite matrix into theproduct of a matrix of full column rank and its transpose and that of the transformation of anonnegative-definite matrix into its congruent matrix are established.

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.

On Accelerated Singular Value Thresholding Algorithm for Matrix Completion

An accelerated singular value thresholding (SVT) algorithm was introduced for matrix completion in a recent paper [1], which applies an adaptive line search scheme and improves the convergence rate from O(1/N) for SVT to O(1/N2), where N is the number of iterations. In this paper, we show that it is the same as the Nemirovski’s approach, and then modify it to obtain an accelerate Nemirovski’s technique and prove the convergence. Our preliminary computational results are very favorable.

A CHARACTERIZATION OF SCHECHTER'S ESSENTIAL SPECTRUM BY MEAN OF MEASURE OF NON-STRICT-SINGULARITY AND APPLICATION TO MATRIX OPERATOR

In this paper we use the notion of measure of non-strict-singularity to give some results on Fredholm operators and we establish a fine description of the Schechter essential spectrum of a closed densely defined linear operator.

Investigation on Singularity, Signature Matrix and Spectrum of Mixed Graphs

The spectral theory of graph is an important branch of graph theory,and the main part of this theory is the connection between the spectral properties and the structural properties,characterization of the structural properties of graphs.We discuss the problems about singularity,signature matrix and spectrum of mixed graphs.Without loss of generality,parallel edges and loops are permitted in mixed graphs.Let G1 and G2 be connected mixed graphs which are obtained from an underlying graph G.When G1 and G2 have the same singularity,the number of induced cycles in Gi(i=1,2)is l(l=1,l>1),the length of the smallest induced cycles is 1,2,at least 3.According to conclusions and mathematics induction,we find that the singularity of corresponding induced cycles in G1 and G2 are the same if and only if there exists a signature matrix D such that L(G2)=DTL(G1)D.D may be the product of some signature matrices.If L(G2)=D^TL(G1)D,G1 and G2 have the same spectrum.

Study on singularity analysis with differential transform method for 3-RRUR parallel robot based on Jacobian matrix

Parallel robot is used in many different fields nowadays, but the singularity of 3-RRUR parallel robot is more complicated, so a method to analyze the singularity of the 3-RRUR parallel robot is very necessary. First, the Jacobian matrix was built based on the differential transform method through the transfer matrixes between the poles. The connection between the position parameters and singularity condition was built through the analysis of the Jacobian matrix. Second, the effect on the singularity from the position parameters was analyzed, and then the singularity condition was confirmed. The effect on the singularity condition from position parameters was displayed by the curved surface charts to provide a basic method for the designing of the parallel robot. With this method, the singularity condition could be got when the length of each link is firmed, so it can be judged that if a group of parameters are appropriate or not, and the method also provides warrant for workspace and path planning of the parallel robot.

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.

广义θ-图和广义梅花图φ的奇异性

利用奇异图的邻接矩阵行列式等于零的方法讨论广义θ-图和广义梅花图φ的奇异性,分别给出广义θ-图θ(a_(1),a_(2),…,a_(k))和广义梅花图φ(a_(1),a_(2),…,a_(k))是奇异图的充分必要条件,并计算这两类图中奇异图发生的概率值.

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。

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.

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。

稀疏阵列的鲁棒矩阵填充DOA估计算法

稀疏阵列布阵灵活,增大阵列孔径的同时还能减少阵元间耦合,但基于稀疏阵列的传统波达方向估计会导致角度模糊混叠,带来估计精度差和稳健性不足的问题。针对以上问题,提出一种适用于稀疏阵列波达方向估计的加权截断奇异值投影(weighted truncated singular value projection,WT-SVP)的鲁棒矩阵填充算法。在填充迭代过程中根据奇异值的大小分配权重,突出大奇异值包含的阵列信息,减少小奇异值中不必要的噪声信息,从而优化传统奇异值投影算法。该算法可以实现稀疏阵列的孔洞信息恢复,对不连续阵元充分利用,同时WT-SVP填充算法实现了稀疏阵列波达方向估计的高精度、高分辨以及在低信噪比、低快拍时的高鲁棒性。

工业机器人奇异规避轨迹规划方法

针对激光追踪仪测量工业机器人定位误差测量中出现的六轴机器人运动学奇异问题,提出了奇异区域预测及实时优化的奇异规避轨迹规划方法。以国产配天AIR50-A机器人为研究对象,建立六自由度串联机器人运动学模型,解算机器人笛卡尔空间与关节空间映射关系。基于雅克比矩阵与奇异位形的关系模型,分析和预测机器人运动轨迹中存在的奇异区域。当机器人末端处于奇异区域或奇异预测域时,记录初始关节角序列作为奇异区域轨迹规划输入,利用五次多项式插值方法迭代更新奇异区域内的轨迹路径,机器人既定的运动路径与更新后的奇异规避路径结合形成机器人新的规划轨迹路径,保证机器人关节角运动学参数连续性的同时,规避奇异区域。实验结果表明,提出的奇异预测域及实时优化的奇异规避轨迹规划方法,能够抑制机械臂奇异区域附近关节角速度的激变,减少奇异位形对机器人运动性能的影响。

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。

Stewart型并联机器人的奇异对其运动精度的影响

以9-3 Stewart型并联机器人为研究对象,分析其在机构接近奇异位形时动平台运动精度的情况。通过计算并联机器人速度Jacobian矩阵的行列式得到奇异点的坐标。通过虚拟样机实验,得到动平台参考点位置反解和正解计算的相对误差。用转动Jacobian矩阵条件数的倒数来度量机构接近奇异的程度,并分析该程度与上述相对误差之间的映射关系。研究结果表明:当指标值大于7.027×10^(-5)时,正、反解的相对误差可忽略不计,这为奇异规避、路径规划等后续工作提供了理论参考。

基于矩阵恢复的OFDM信道估计方法

正交频分复用(Orthogonal Frequency Division Multiplexing,OFDM)中至关重要的一项技术是信道估计,本文提出一种基于矩阵恢复的OFDM信道估计方法,将连续多个OFDM信号的频域信道构造成一个信道矩阵,由于这个信道矩阵是低秩的,所以可以将信道估计问题转换为信道矩阵的加权截断核范数最小化问题,并使用改进的奇异值阈值(Singular Value Thresholding,SVT)算法对信道矩阵进行恢复。仿真结果表明,本文提出的方法和传统信道估计算法相比,使用相同导频数可以获得更高的估计精度,在获得相同估计精度时,消耗导频数更少。与基于压缩感知的信道估计方法相比,本文方法消耗相同数量的导频,但可直接获得高精度的OFDM信道的频域估计。

基于ADRC的尾座式无人机姿态控制

基于小型尾座式无人机进行自抗扰控制器的设计。无人机平台采用凤凰无人机平台:一套完整开源的尾座式无人机平台。该小型尾座式无人机采用2个电机和舵面完成无人机的垂直起降和前飞。在建模方面建立了无人机运动学、动力学和作动器的数学模型,为解决该无人机姿态表示的奇异性问题,采用旋转矩阵表示姿态。为提高无人机垂直起飞和前飞过程的抗干扰能力,设计了基于ADRC(主动抗干扰控制)的姿态控制器,通过Simulink进行了仿真验证。

基于快速最大奇异值幂正规化的全局协方差池化

近期的研究工作表明,矩阵正规化对全局协方差池化起着关键作用,有助于生成分辨能力更强的表征,从而提升图像识别任务的性能。在不同的矩阵正规化方法中,矩阵结构正规化能充分利用协方差矩阵的几何结构,因此可以获得更好的性能。然而,结构正规化一般依赖计算代价很高的奇异值分解(SVD)或者特征值分解(EIG),不能充分利用GPU的并行计算能力,从而形成计算瓶颈。迭代矩阵平方根正规化(iSQRT)通过牛顿-舒尔兹迭代对协方差矩阵进行正规化,速度比基于SVD和EIG的方法更快。但是随着迭代次数和维度的提高,iSQRT的时间和内存开销都会显著增加,而且该方法无法完成一般幂次的正规化,限制了其应用范围。为了弥补iSQRT的不足,文中提出了一种基于最大奇异值幂的协方差矩阵正规化方法。该方法通过将协方差矩阵除以其最大奇异值的幂来实现,计算过程仅需迭代幂法计算矩阵的最大奇异值。详细的消融实验的结果表明,与iSQRT相比,所提方法的速度更快并占用更少的显存,在时间复杂度和空间复杂度上都优于iSQRT方法,同时性能上与iSQRT方法相当或更好。所提方法在大规模图像分类数据库和细粒度识别数据库中取得了领先的性能,其中在Aircraft,Cars和Indoor67上分别表现为90.7%,93.3%以及83.9%,充分验证了所提方法的鲁棒性和泛化性。

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。