半奇异值算法的推导及其应用

奇异值分解是将一矩阵分解为一个对角矩阵和两个正交矩阵,奇异值分解有着非常好的性质。但在其部分应用中,如秩亏损的最小二乘问题,线性方程组的最小范数解中,并没有充分利用它的所有性质。提出了半奇异值分解A=USR,其中U为正交矩阵,S为对角矩阵,R为上三角矩阵。在经过文中所述的后期数学处理后,它能够非常好地利用在各个方面,比如最小二乘问题和线性方程组中。这种分解不仅保留了奇异值分解后所应有的性质,更大大地降低了计算复杂度。因为该算法有求极值的能力,所以它将在应用领域中发挥更大的作用。     

不确定离散广义系统的D稳定鲁棒控制

研究了具有圆盘区域极点约束的一类不确定离散广义系统的鲁棒控制问题.首先,研 究了控制输入项不含扰动的不确定离散广义系统,提出了广义二次D镇定的概念,基于矩阵不 等式和广义Riccati方程,给出了一种广义二次D镇定器的设计方法,所得到的结论能够实现研 究目标;然后,讨论了控制输入项含有扰动的不确定离散广义系统,在一定的假设条件下,给出 了期望状态反馈增益阵的存在条件及其解析表达式.最后,用数值示例说明所给方法的有效性 及可行性.     

Time domain signal enhancement based on an optimized singular vector denoising algorithm

This paper presents a new time domain noise reduction approach based on Singular Value Decomposition (SVD) technique. In the proposed approach, the noisy signal is initially represented in a Hankel Matrix. Then SVD is applied on the Hankel Matrix to divide the data into signal subspace and noise subspace. Since singular vectors are the span bases of the matrix, reducing the effect of noise from the singular vectors and using them in reproducing the matrix leads to considerable enhancement of information embedded in the matrix. The noise-reduced singular vectors from the signal subspace are utilized to reconstruct the data matrix. This matrix is finally used to obtain the time-series signal. The results of applying the proposed method to different synthetic noisy signals indicate a better efficiency in noise reduction compared to the other time series methods.     

Numerical performance of hyperplane constrained method and its hybrid method for singular value decomposition

The hyperplane constrained method has been proposed in Yadani et al. (Appl Math Comp 216:779–790, 2010) computing singular value decomposition (SVD) of matrix. In the method, the SVD is replaced with solving nonlinear systems whose solutions are constrained on hyperplane, and then their solutions are computed with the help of Newton’s iterative method. In this paper, we present a new convergence theorem concerning the hyperplane constrained method in finite arithmetic. We also clarify the numerical performance of the hyperplane constrained method. In numerical experiments, we first show that the computed singular values and singular vectors are with high accuracy, even if the target matrix of SVD has small singular values, almost the same singular values, not small condition number. Though the hyperplane constrained method requires not small amount of computations, it fastens by combining other fast singular value decomposition method. We next propose a hybrid method which adopts the singular vectors computed by other fast method as the initial guess of the Newton type iteration in order to decrease the iteration number. By numerical experiments, we can see that the hybrid method runs faster than the original hyperplane constrained method with almost same accuracy.     

Exact Maximum Singular Value Calculation of an Interval Matrix

In this note, we present a method for calculating the maximum singular value of an interval matrix. First, we provide an algorithm for calculating the maximum singular value of a square interval matrix. Then, based on this algorithm, we extend the result to non-square interval matrix case and to the case of computing the minimum singular value. Through numerical examples, the validity of the suggested methods is illustrated. Particularly, we compare the newly-proposed method with an existing method to show that the new method finds the correct bound of the maximum singular value with no exception     

面向矩阵秩函数准确估计的自表示子空间聚类方法

传统子空间聚类方法通常使用矩阵核范数代替矩阵秩函数进行低秩矩阵恢复,然而在目标优化过程中主要关注低秩矩阵大奇异值的影响,容易导致矩阵秩估计不准确的问题。为此,在分析矩阵奇异值长尾分布特点基础上,提出使用基于截断Schatten-p范数的低秩子空间聚类模型。该模型充分考虑小奇异值对低秩矩阵恢复过程的贡献,利用小奇异值信息拟合矩阵奇异值的长尾分布,通过对矩阵秩函数进行准确估计以提升子空间聚类性能。实验结果表明,与现有加权核范数子空间聚类WNNM-LRR和近邻约束子空间聚类BDR算法相比,在Extended Yale B数据集上的聚类准确性分别提升了11%和8%,所提方法能够更好地拟合数据奇异值分布以及生成准确的相似度矩阵。     

多输入多输出线性定常系统稳定裕度的分析与改进

针对多输入多输出(MIMO)控制系统的稳定裕度求解问题,首先分析了现有的回差阵奇异值法这一计算方法,并得到其解决单输入单输出(SISO)系统的稳定裕度结论,在此基础上,提出两种基于系统回差阵的稳定裕度改进方法;一种是在有限条件下利用矩阵的特征值代替奇异值来建立与稳定裕度关系的策略,另一种是利用系统逆回差阵的行列式,通过求其奇异值来计算系统稳定裕度;最后结合工程实例,通过数值仿真验证两种稳定裕度计算方法相比原方法都有不同程度的改进,而且三种方法可以结合起来进行分析,最大化的减小系统稳定裕度结果的保守性.     

离散奇异时滞系统时滞依赖稳定性分析与镇定

分别研究了离散奇异时滞系统时滞依赖稳定性分析与镇定问题. 首先给出了一个新的离散奇异时滞系统时滞依赖容许性充分条件. 经过证明, 所提出的方法与现有结论相比, 具有一定的优势. 然后, 运用矩阵理论技巧,设计出状态反馈控制器保证闭环离散奇异时滞系统是容许的. 最后,两个数值例子说明了所用方法的有效性.     

A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

A polynomial interpolation based on a uniform grid yields the well-known Runge phenomenon, where maximum error is unbounded for functions with complex roots in the Runge zone. In this paper, we investigate the Runge phenomenon with the finite precision operation. We first show that the maximum error is bounded because of round-off errors inherent to the finite precision operation. Then we propose a simple truncation method based on the truncated singular value decomposition. The method consists of two stages: In the first stage, a new interpolating matrix is constructed using the assumption that the function is analytic. The new interpolating matrix is preconditioned using the statistical filter method. In the second stage, a truncation procedure is applied such that singular values of the new interpolating matrix are truncated if they are equal to or lower than a certain tolerance level. We generalize the method, by analyzing the singular vectors of both the original and new interpolation matrices based on the assumption in the first stage. We show that the structure of the singular vectors makes the first stage essential for an accurate reconstruction of the original function. Numerical examples show that exponential decay of the error can be achieved if an appropriate truncation is chosen.     

快速判定几何约束奇异性的切面扰动法

针对冗余奇异和分支奇异的判定问题,提出一种新的切面扰动的判定方法.该方法将奇异的雅可比矩阵分为独立构型空间和奇异空间,变量沿独立构型空间的切面扰动,计算更新的雅克比矩阵的秩,依据秩亏的变化可以快速、稳定地判定约束奇异性.该算法克服了残量扰动法的数值迭代、计算量大和不稳定的缺点,并且在参数化特征造型系统InteSolid中得到验证.     

Robust admissibility and admissibilisation of uncertain discrete singular time-delay systems

This paper is concerned with the characterisation of robust admissibility and admissibilisation for uncertain discrete-time singular system with interval time-varying delay. Considering the norm-bounded uncertainty and the interval time-varying delay, a new comparison model is introduced to transform the original singular system into two connected subsystems. After this transformation, a singular system without uncertainty and delay can be handled by the Lyapunov–Krasovskii functional method. By virtue of the scaled small gain theorem, an admissibility condition of the original singular system is proposed in terms of linear matrix inequalities. Moreover, the problem of robust admissibilisation of uncertain discrete singular time-varying system is also studied by iterative linear matrix inequality algorithm with initial condition optimisation. Several numerical examples are used to illustrate that the results are less conservative than existing ones.     

奇异H∞控制问题的二次矩阵不等式的可解性

研究具有无穷远零点的奇异H∞控制问题的二次矩阵不等式的可解性,证明了通过 求解广义特征值问题,可以直接求得二次矩阵不等式的解,从而简化了原需通过复杂的系统 分解和变换来求解二次矩阵不等式的方法.文中建立了基于几何控制理论和基于J-无损分 解理论的两种不同的奇异H∞控制分析方法之间的联系,并揭示了具有无穷远零点的奇异 H∞控制系统的特征空间结构.     

一种基于奇异值分解的图像压缩方法

根据奇异值分解的基本原理及其特点,结合图像的矩阵结构,提出了用奇异值分解,然后选取部分奇异值和奇异向量重构矩阵进行图像压缩的方法,并通过对图像进行分块降低计算量,实验结果表明奇异值分解能够有效用于图像压缩。     

Finding Unmanipulable Singularities in Parallel Mechanisms Using Jacobian Decomposition

A geometric method is presented to determine the unmanipulable singular configurations of a general class of parallel mechanisms. In unmanipulable singular configurations, the composite Jacobian matrix that maps active joints velocities to end-effector velocity loses rank, indicating the loss of a task degree-of-freedom. Finding unmanipulable configurations is difficult due to the complexity of the Jacobian matrix. The problem is greatly simplified by a novel decomposition of the matrix presented in this paper. The method is used to find singularities in several example parallel machines.     

Guaranteed cost control of uncertain discrete‐time singular Markov jump systems with indefinite quadratic cost

The guaranteed cost control problem for discrete‐time singular Markov jump systems with parameter uncertainties is discussed. The weighting matrix in quadratic cost function is indefinite. For full and partial knowledge of transition probabilities cases, state feedback controllers are designed based on linear matrix inequalities method which guarantee that the closed‐loop discrete‐time singular Markov jump systems are regular, causal and robust stochastically stable, and the cost value has a zero lower bound and a finite upper bound. A numerical example to illustrate the effectiveness of the method is given in the paper. Copyright © 2010 John Wiley & Sons, Ltd.     

一种两阶段判别分析方法

为了解决LDA 对复杂分布数据的表达问题,本文提出了一种新的非参数形式的散度矩阵构造方法。该方法 能更好的刻画分类边界信息,并保留更多对分类有用的信息。同时针对小样本问题中非参数结构形式的类内散度矩阵可能奇 异,提出了一种两阶段鉴别分析方法对准则函数进行了最优化求解。该方法通过奇异值分解把人脸图像投影到混合散度矩阵 的主元空间,使类内散度矩阵在投影空间中是非奇异的,通过CS 分解,从理论上分析了同时对角化散度矩阵的求解,并证明了 得到的投影矩阵满足正交约束条件。在ORL,Yale 和YaleB 人脸库上测试的结果显示,改进的算法在性能上优于PCA+LDA, ULDA 和OLDA 等子空间方法。     

Neural data fusion algorithms based on a linearly constrained leastsquare method

Two novel neural data fusion algorithms based on a linearly constrained least square (LCLS) method are proposed. While the LCLS method is used to minimize the energy of the linearly fused information, two neural-network algorithms are developed to overcome the ill-conditioned and singular problems of the sample covariance matrix arisen in the LCLS method. The proposed neural fusion algorithms are samples for implementation using both software and hardware. Compared with the existing fusion methods, the proposed neural data fusion method has an unbiased statistical property and does not require any a priori knowledge about the noise covariance. It is shown that the proposed neural fusion algorithms converge globally to the optimal fusion solution when the sample covariance matrix is singular, and converge globally with exponential rate when the sample covariance matrix is nonsingular. We apply the proposed neural fusion method to image and signal fusion, and it is shown that the quality of the solution can be greatly enhanced by the proposed technique.     

Stochastic admissibility and stabilization of singular Markovian jump systems with multiple time-varying delays

This paper is concerned with stochastic admissibility and state feedback stabilization for a class of singular Markovian jump systems with multiple time-varying delays. The singular matrix E with both modedependent and mode-independent is considered in the system. Firstly, based on Lyapunov functional method and free-weighting matrix method, sufficient condition is presented in the form of linear matrix inequalities (LMIs) to guarantee the considered system to be stochastically admissible. Secondly, by state feedback controller, sufficient condition is derived in terms of strict LMIs to ensure the closed-loop system to be stochastically stabilizable. Finally, numerical examples are provided to illustrate the effectiveness of the proposed approaches.     

基于局部奇异值对称平均的人脸识别方法

提出了一种基于局部奇异值对称平均的人脸识别方法。该方法首先对原始图像进行线性映射处理;接着采用局部奇异值分解提取人脸特征,并对所获得的特征作对称平均处理;最后依据最近邻决策规则进行识别。基于ORL人脸数据库的实验结果表明,该方法大大降低了原始特征空间的维数,有效地消除了图像亮度和噪声的影响,并取得了较高且稳定的正确识别率,在人脸识别中是一种有效的方法。     

Determination of a transfer function matrix in multivariablesystems

A conceptually simple method is presented for determining the system transfer function matrix of linear multivariable systems described by their state-space equations. The polynomial coefficients of the transfer function matrix are computed directly by using the controllability matrix method and the transformation to canonical (phase-variable) form matrices, even if the transformation matrices are singular. The method does not require matrix inversion and calculation of eigenvalues, and requires only O(n³) operations