A definition of column reduced proper rational matrices

The aim of this work is to develop analogue concepts of column reduced polynomial matrices for proper rational matrices. A definition of column reducedness for a class of proper rational matrices is proposed and the properties of such matrices are studied, in particular reduction to column reduced form by elementary operations over the ring of proper rational functions, and the relationship between the degrees of the invariant factors of a column reduced matrix and the so-defined column indices. The physical significance of such matrices in terms of their finite structure is explained; this interpretation completely complements the physical interpretation of a column reduced polynomial matrix. An application of the properties of column reduced proper rational matrices to the decoupling problem is also presented: the infinite structure which can be obtained while decoupling a linear multivariable system by non-regular static state feedback is completely characterized.     

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

Many of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Solving polynomial nonlinear matrix equations by a linearization method

This paper deals with some modification of a matrix linearization method. The scheme proposed makes it possible to find tuples of solutions for systems of polynomial nonlinear equations defined on a commutative matrix ring. The matrix linearization method reduces an initial polynomial nonlinear problem to a linear one with respect to matrices of solutions. Then, the method of elimination of unknowns is used to obtain a generalized eigenvalue problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 60–69, May–June 2006.     

Optimal control of linear distributed-parameter systems via polynomial series

A method for finding the optimal control of linear distributed-parameter systems using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into a Taylor polynomial by the use of a suitable transformation. In this paper, the optimal control of a distributed-parameter system is simplified into the solution of a linear two-point boundary value problem, and, as a result, the optimal control is obtained via a Taylor series. It is shown that the implementation of Taylor series for this problem involves the use of an ill-conditioned matrix commonly known as the Hilbert matrix. The optimal control of linear distributed-parameter systems using other polynomial series is then calculated by transforming the properties of the Taylor series into other polynomial series. The formulation is straightforward and convenient for digital computation. An illustrative example is given.     

Matrix method of polynomial expansion of symmetric Boolean functions

A problem of polynomial expansion of symmetric Boolean functions is considered. A matrix method for polynomial expansion of symmetric functions that can be used to calculate the working numbers of homogeneous polynomial symmetric Boolean functions is proposed.     

Robust stability of state-space models with structureduncertainties

A method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters. A method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed. The method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when cases with one or two uncertain parameters are considered. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

一种基于块共同特征值的人脸识别方法

主成分分析与线性判别分析是人脸识别的重要识别方法,它们都通过求解特征值问题实现特征提取,但由于维数灾难会导致小样本和奇异性问题。提出了一种简单的人脸识别方法,无需进行奇异值分解,能有效地降低计算代价。首先将图像划分成块,然后计算多项式系数,得到友阵用于特征提取。基于两张不同图像的多项式系数友阵来计算对称阵。最后通过计算对称阵的零空间的零化度识别相似的人脸图像。为验证提出方法的有效性,在ORL、Yale和FERET人脸数据库上进行了实验。结果表明,该方法对于有较大姿态与光照变化的人脸识别具有较高的识别性能。     

A division algorithm for polynomial matrices

A new algorithm is presented for directly determining both the quotient and the remainder associated with the division of one polynomial matrix by another. The procedure requires that the denominator matrix be column proper with nonzero column degrees. However, unlike earlier algorithms, only real matrix multiplications are employed; i.e., there is no need for matrix inversions, either explicit or implicit.     

Stabilisation of matrix polynomials

A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order. The results are illustrated through examples.     

基于她分解与多项式的无线传感器网络密钥算法

提出了基于QR分解与二元多项式的密钥建立与分配方案。该方案以二元多项式的计算结果作为无线传感器网络的密钥。二元多项式的其中一个参数由对称矩阵进行QR分解生成,节点部署后交换Q矩阵的行信息再与R矩阵的列信息相乘生成多项式的参数。多项式的另一个参数由各自生成的随机数确定。分析结果表明：该方案可以提高存储效率、网络连通性、抗捕获性能,并能提供额外的通信链路验证。     

Matrix power series method for nonlinear problems of optimal regulator construction

We propose a relatively simple and efficient method for solving the problem of analytic construction of optimal regulators for multidimensional control objects with polynomial nonlinearities based on the A.A. Krasovskii’s generalized work criterion. Our method employs an extension of the power series method which is based on using matrix theory with Kroneker (direct) product. The matrix formalism has let us establish a simple recursive relation for the matrices of coefficients of the Bellman-Lyapunov function, with which the control problem can be solved with any reasonable precision on modern computers.     

Design of semi-tensor product-based kernel function for SVM nonlinear classification

The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.     

二维多项式非线性系统的镇定控制研究

针对二维多项式非线性系统,提出了基于特征根负定配置的镇定控制方法.引入自由多项式,克服系统状态矩阵描述的不惟一性,进而降低控制综合问题求解的保守性.将特征根负定配置问题转化成多项式正定性验证问题,控制器设计问题通过多项式分解最终可由半定规划工具数值求解.在所提出的处理方法的基础上,讨论了4类二维多项式非线性系统的镇定控制问题.仿真结果验证了所提出方法的有效性.     

An analytical approach to one-parameter MIMO systems passivity enforcement

Stable linear time-invariant systems can be made passive by a feedforward action. In this article, an analytical approach to obtain the matrix which allows to enforce passivity in the system is proposed. This matrix depends only on one parameter, namely α. The introduced method is based on the calculation of the characteristic polynomial of the Hamiltonian matrix associated to the Positive Real problem. This polynomial is then used to derive a finite set of values of the parameter α, in which the value assuring passivity enforcement with minimum dissipation can be selected. Numerical examples are reported.     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

Domain decomposition PCG methods for serial and parallel processing

In this paper two domain decomposition formulations are presented in conjunction with the preconditioned conjugate gradient method (PCG) for the solution of large-scale problems in solid and structural mechanics. In the first approach, the PCG method is applied to the global coefficient matrix, while in the second approach it is applied to the interface problem after eliminating the internal degrees of freedom. For both implementations, a subdomain-by-subdomain (SBS) polynomial preconditioner is employed, based on local information of each subdomain. The approximate inverse of the global coefficient matrix or the Schur complement matrix, which acts as the preconditioner, is expressed by a truncated Neumann series resulting in an additive type local preconditioner. Block type preconditioning, where full elimination is performed inside each block, is also studied and compared with the proposed polynomial preconditioning.     

Polynomial filtering for nonlinear stochastic systems with state‐ and disturbance‐dependent noises

This article is concerned with the polynomial filtering problem for a class of nonlinear stochastic systems governed by the Itô differential equation. The system under investigation involves polynomial nonlinearities, unknown‐but‐bounded disturbances, and state‐ and disturbance‐dependent noises ((x,d)‐dependent noises for short). By expanding the polynomial nonlinear functions in Taylor series around the state estimate, a new polynomial filter design method is developed with hope to reduce the conservatism of the existing results. In virtue of stochastic analysis and inequality technique, sufficient conditions in terms of parameter‐dependent linear matrix inequalities (PDLMIs) are derived to guarantee that the estimation error system is input‐to‐state stable in probability. Moreover, the desired polynomial matrix can be obtained by solving the PDLMIs via the sum‐of‐squares approach. The effectiveness and applicability of the proposed method are illustrated by two numerical examples with one concerning the permanent magnet synchronous motor.     

New solution to the inverse regulator problem by the polynomial matrix method

This paper gives a new criterion for the inverse regulator problem in terms of polynomial matrix fractions. The generalized Kalman equation in polynomial matrix form leads to a necessary and sufficient condition for a stable feedback law to be optimal for some (unknown) quadratic performance index. This criterion consists of (i) a symmetric factorizability of a polynomial matrix, and (ii) a polynomial-type criterion that guarantees detectability. The relationship between the Riccati and Kalman equations is also established. Finally, an example is given to illustrate the new result in contrast with the existing result.