Approximate controllability of the n -th order infinite dimensional systems with controls delayed by the control devices

The main aim of this article is to derive general conditions for a few types of controllability at once for an arbitrary order abstract differential equation and arbitrary eigenvalues multiplicities, instead of conditions for fixed order equation and single eigenvalues. Another innovation of this article is taking into account delays caused by electronic control microcontrollers. This was possible thanks to analysis of the n-th order linear system in the Frobenius form, generating Jordan transition matrix of the confluent Vandermonde form. Using the explicit analytical form of the inverse confluent Vandermonde matrix enabled us to receive general conditions of different types of controllability for the infinite dimensional systems. We derived this analytical form of the inverse confluent Vandermonde matrix using new results from the linear algebra, presented in the paper by S. Hou and W. Pang, “Inversion of confluent Vandermonde matrices”, Int. J. Comput. Math. Appl., 43, pp. 1539–1547, 2002.     

MDS矩阵和对合MDS矩阵的新构造方法

首先对Lacan等人给出的由Vandermonde矩阵构造MDS码的方法进行了研究, 指出了其中存在的问题, 给出了由两个Vandermonde矩阵构造MDS矩阵的充要条件; 然后利用矩阵乘的方法, 给出了由标量乘Vandermonde矩阵构造MDS矩阵的充要条件; 最后在Sajadieh等人给出的由两个Vandermonde矩阵构造对合MDS矩阵方法的基础之上, 给出了标量乘Vandermonde矩阵构造对合MDS矩阵的方法。对标量乘矩阵来讲, 可以通过调控标量中分量的大小来调整标量乘矩阵元素大小和元素重量大小来满足其软、硬件实现性能, 因此该构造MDS矩阵及对合MDS矩阵的方法具有实用价值。     

Constrained ALS-based tensor blind receivers for multi-user MIMO systems

Tensor factorizations has shown to be an efficient approach for symbols and/or channel estimation in multi-input multi-output (MIMO) systems, where the factor matrices of tensor that correspond to symbols, channel, code/diversity of signals, are often estimated by using alternating least squares (ALS) algorithm. Although the performance of tensor approaches strongly depend on the initializations of the factor matrices. However, due to the absence of a priori on channels, these initializations are done randomly in traditional ALS algorithm. This generally implies a slow convergence. Further, ALS does not take into account the potential orthogonal structure in the factor matrices, which can be exploited to improve the accuracy of factor matrices recovery. To address these insures, this paper proposes constrained ALS tensor blind receivers for multi-user MIMO systems. We show that the multi-user MIMO signals can be expressed as a third-order tensor model, where the matrices of users symbols, direction-of-arrival (DOA) and delay can be viewed as three factor matrices of the tensor model. Two constrained ALS blind algorithms that take into account the potential orthogonal and Vandermonde structures in the factor matrices, are proposed to learn the tensor model, where the users symbols, DOA and delay are joint estimated as three factor matrices. Besides provide the estimations for the factor matrices, the orthogonal and Vandermonde structures also give a better uniqueness results for the use of tensor model. Interestingly, these structures are the nature properties of the factor matrices in our system. This results in an efficient blind approach that has better performance and lower complexity compare with the traditional ALS.     

一个反求Bezier曲面控制点的算法

本文将反求m×n次Bezier曲面控制点问题,转化为求解m+1个n+1阶线性方程组和n+1个m+1阶线性方程组问题。这些线性方程组的系数矩阵是著名的Vandermonde矩阵。通过求解Vandermonde矩阵的逆矩阵,使CAD/CAM曲面造型中常常遇到的反求Bezier曲面控制点问题得到有效的解决。同时本文给出了一种求解Vandermonde矩阵的逆矩阵的方法。     

Computer generation of difference approximations

An algorithmic approach, based on the method of undetermined coefficients, for generating difference approximation to partial differential operators is presented here. The approach is based on inverting Vandermonde matrices of lower order. Consequently, the computational time required to obtain the symbolic coefficients is very small in all cases. The algorithm is demonstrated by generating multi-parameters approximations for several problems.     

Vandermonde and resultant matrices: An abstract approach

Vandermonde and resultant matrices: An abstract approach     

A Periodic qd-Type Reduction for Computing Eigenvalues of Structured Matrix Products to High Relative Accuracy

In this paper, we consider the product eigenvalue problem for a wide class of structured matrices containing the well-known Vandermonde and Cauchy matrices. A periodic qd-type reduction method is developed for computing eigenvalues of products of these rectangular matrices so that no subtraction of like-signed numbers occurs. Consequently, all the eigenvalues of such a product are computed to high relative accuracy in a preferable complexity. Error analysis and numerical experiments are provided to confirm the high relative accuracy.     

Mixed and componentwise condition numbers for rectangular structured matrices

Abstract  We provide explicit expressions for both mixed and componentwise structured condition numbers for several classes of structured rectangular matrices: upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. Mathematics Subject Classification (2000): 15A12, 65F35     

A property of the elementary symmetric functions

Abstract We exhibit a simple relation concerning the elementary symmetric functions and present two applications concerning the inverse of a Vandermonde matrix and the spectral properties of square matrices.     

A relation between the Hilbert and Vandermonde matrices

A relation is established between two famous matrices and two simple applications are proposed. We first study the asymptotic behaviour of the spectral condition number of a Vandermonde matrix with a particular configuration of points. We then apply our relation to the problem of the computation of the Christoffel number starting from the moments of the weight function; we show that the relative asymptotic condition number of the problem is essentially the Turing condition number of the Hilbert matrix as observed numerically by other authors in the past.     

大系统的局部分散镇定

本文给出了一种计算范德蒙矩阵逆的模的方法.利用这种方法,得到了可以用局部分散 状态反馈镇定的一类大系统.与过去的结果比较容易计算,且不要求关联项中很多元素为零.     

低编码复杂度不规则准循环LDPC码的构造方法

针对不规则低密度奇偶校验码（LDPC码）误码性能好,但编码复杂度高的问题,利用重复积累码（RA码）能有效编码的特性和掩模技术,提出了一种不规则LDPC码的构造方法,该码具有线性复杂度的编码算法。该构造方法,首先对RA码的校验矩阵进行了改进,消除了RA码常产生的错误平层效应;然后基于范德蒙矩阵构造了一种新的校验矩阵,该校验矩阵具有代数结构,易于硬件实现。理论分析和实验结果表明,构造的不规则LDPC码的编码复杂度低于Mackay随机码,在加性高斯白噪声（AWGN）信道条件下,误码率为1.0×10^-4时,比Mackay随机码性能提高约0.4~0.6 dB。     

基于Volterra频域核辨识的非线性模拟电路故障诊断

基于Volterra级数时域频域混合模型,提出了辨识非线性模拟电路频域核的故障诊断方法.利用混合模型辨识算法和范德蒙特法估计各种故障状态下电路响应的前3阶频域核,提取故障特征并与相应的故障模式一起构成特征样本集,借助于支持向量机多分类器进行分类识别,实现非线性模拟电路的故障诊断.阐述了诊断原理及诊断步骤,并给出了诊断实例.仿真结果表明,该方法的故障识别率较高,便于计算机计算.     

Comments on ‘Inversion of A Generalized Vandermonde Matrix’ by M.E.A. El-Mikkawy, 80 (2003) 759–765

In this paper, we give the comments on the article ‘Inversion of a Generalized Vandermonde Matrix’ by M.E.A. El Mikkawy, Int. J. Computer Math. 80 (2003), pp. 759–765. The article gives an algorithm for the elementary symmetric function's calculation which contains a severe error. In these comments, we have proposed necessary corrections of that algorithm.     

Vandermonde and resultant matrices: An abstract approach

We propose an abstract approach to the problems of common divisors and common multiples of rational matrix functions which (in the case of matrix polynomials) have been studied before using Vandermonde and resultant matrices.Supported in part by the Office of Naval Research, Air Force Office of Scientific Research, and the National Science Foundation.The work of this author was partially supported by an NSF grant and was carried out while visiting the University of California, San Diego.     

Solving Toeplitz systems after extension and transformation

Subject of the paper are systems of linear equations with an indefinite or nonsymmetric Toeplitz coefficient matrixT=[a _i−j ]. In order to avoid instabilities which often occur during the application of Levinson and Schur type algorithms for these matrices transformation techniques combined with pivoting strategies have been proposed in earlier papers, starting with [19]. These transformations have some deficiencies. To overcome these we propose to carry out the transformation after a convenient extension. In particular, we discuss the transformation after extension into paired Vandermonde matrices. The corresponding systems admitO(n ²) complexity complete pivoting.     

Robust correlation coefficients as the initial data for the solution of problems of confluent analysis

A technology for the solution of problems of confluent analysis of noisy technological processes is developed based on the technologies used in calculation of the variance of interference, the degree of robustness of estimators of the auto-and cross-correlation functions, robust correlation matrices, and balance algorithms for the relative errors of their elements.     

Evaluation of matrix polynomials in the state companion matrix of linear time invariant systems

In this paper a new iterative algorithm is presented for the numerical evaluation of matrix polynomials in the state companion matrix of a linear time invariant system. The algorithm also enables the polynomial coefficients of the characteristic equation and the eigenvectors of the state companion matrix to be found. In particular, a truncated series approximation is given for the state transition matrix, providing the most efficient computation known for a prescribed truncation error. The adjustment recursive coefficients used in evaluating the state transition matrix are shown to be on alternative means of evaluating the elements of the inverse Vandermonde matrix of the systems' eigenvalues.     

An algorithm for inverting rational matrices

We propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices. The algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a MATLAB¹ function. Experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems. Illustrative examples are given.