二次矩阵方程X~2-bX-C=O的正定解

研究二次矩阵方程X2-bX-C=O(b>0,C为n×n阶正定阵)的正定解,证明了解的存在唯一性并且给出了求解方法.     

基于交替投影算法求解单变量线性约束矩阵方程问题

研究如下线性约束矩阵方程求解问题:给定A∈R~(m×n),B∈R~(n×p)和C∈R~(m×p),求矩阵X∈R(?)R~(n×n)"使得A×B=C以及相应的最佳逼近问题,其中集合R为如对称阵,Toeplitz阵等构成的线性子空间,或者对称半(ε)正定阵,(对称)非负阵等构成的闭凸集.给出了在相容条件下求解该问题的交替投影算法及算法收敛性分析.通过大量数值算例说明该算法的可行性和高效性,以及该算法较传统的矩阵形式的Krylov子空间方法(可行前提下)在迭代效率上的明显优势,本文也通过寻求加速技巧进一步提高算法的收敛速度.     

关于矩阵方程X+A*X-nA＝I的正定解

本文对任意正整数n界定了矩阵方程X A*X-nA=I的正定解的特征值的范围,给出了它的极大正定解一个充分条件.     

矩阵方程AX—XB=C的最小多项式解法

关于矩阵方程AX—XB=C的解法有不少的论文,大部分是采用矩阵的拉直运算或拉直运算的变形方法求解,文献[1]给出了连分式解法,本文利用矩阵A,B的最小多项式求解此方程,使得方程的解比目前已见的结果较简洁,同时当B=-A~T稳定、C为任意正定矩阵时所构造的正定二次型Liapunov函数的表达式较目前的结果更明确、简单.     

正定矩阵的子式阵仍是正定矩阵的一个充要条件

文[2]证明了实对称正定矩阵的子式阵仍然是实对称正定矩阵,文[3]给出了一般的正定矩阵的的概念,本文利用标准型给出了一般正定矩阵的子式阵仍然是正定矩阵的充要条件.     

非负定阵元素的扰动和正定完备化

在保持非负定性不变的前提下,本文对矩阵每一元素容许多大的扰动作了进一步的研究, 将本文的结论和C．R．Johnson提出的部分正定阵的正定完备化进行比较,容易发现对已知的正定矩阵求扰动,本文的结论比用C．R.Johnson的正定完备化计算扰动形式上更简单,同时也给出了不同于C．R．Johnson的部分正定阵的正定完备化表示的另外一个公式,推出了这些正定完备化矩阵应具有的若干性质．     

特征值问题的预变换方法(I): 杨辉三角阵变换与二阶PDE 特征多项式

本文提出一类求解特征值问题的下三角预变换方法, 目标是通过相似变换后矩阵下三角元素平方和明显减少、且变换后的特征值及其特征向量较易求解, 使变换后的对角线可作为全体特征值很好的一组初值, 其作用如同对于解方程组找到好的预条件子, 加速迭代收敛. 以二阶PDE 数值计算为例,对于以Laplace 方程为代表的特征波向量组及正交多项式组有广泛的应用前景.
杨辉三角是我国古代数学家的一项重要成就. 本文引入杨辉三角矩阵作为预变换子, 给出一般矩阵用杨辉三角矩阵作为左、右预变换子时变为上三角矩阵的充要条件, 给出了元素为行指标二次多项式的两个矩阵类(三对角线阵与五对角线阵) 中特征值何时保持二次多项式的充要条件, 并应用于构造新的二元PDE 正交多项式.     

正定矩阵的子式阵仍是正定矩阵的一个充要条件

文[2]证明了实对称正定矩阵的子式阵仍然是实对称正定矩阵,文[3]给出了一般的正定矩阵的的概念,本文利用标准型给出了一般正定矩阵的子式阵仍然是正定矩阵的充要条件．     

矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

广义正定矩阵与稳定矩阵的关系

本文首先修正了文[1]中有关广义正定矩阵与稳定矩阵关系的一系列错误,进而给出了广义正定矩阵与稳定矩阵的关系,最后,给出了广义正定矩阵的加性复合阵的有关性质.     

非线性中立型控制系统的镇定与控制

本文利用Riccati矩阵方程的对称正定解构造正定二次型V函数,运用L yapunov分解等价法,给出了非线性中立型定常控制系统的镇定与次优控制。同时得到了滞后量与非线性项的界限的估计公式。     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Solution of the matrix Riccati equation for the linear quadratic control problems

This paper is concerned with the solution of the matrix Riccati differential equation with a terminal boundary condition. The solution of the matrix Riccati equation is given by using the solution of the algebraic form of the Riccati equation. An illustrative example for the proposed method is given.     

Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

Let be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution to the discrete-time algebraic Riccati equation (DARE) where , is Hermitian positive definite, , the pair is stabilizable, and the pair is detectable. Assume that is nonsingular, and is stable. Let , and let be the residual of the DARE with respect to . Define the linear operator by The main result of this paper is: If where denotes any unitarily invariant norm, and then Received June 7, 1995 / Revised version received February 28, 1996     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

Error bounds for numerical solutions of ordinary differential equations

Summary An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.     

一类不确定区间时变时滞系统的时滞相关鲁棒H_∞控制

研究了一类不确定区间时变状态时滞系统的鲁棒H_∞控制问题.基于Lyapunov稳定性理论和线性矩阵不等式,采用自由权矩阵方法,得到使得相应闭环系统渐近稳定且具有H_∞性能的时滞相关充分条件,并给出状态反馈鲁棒H_∞控制律的设计方法.仿真实例表明了该方法的有效性.     

A computational solution for a Matrix Riccati differential equation

Summary This paper is concerned with the solution of the finite time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using computational methods. Examples illustrating the method are presented and the computational algorithms are given.     

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

We propose a generalization of the structured doubling algorithm to compute invariant subspaces of structured matrix pencils that arise in the context of solving linear quadratic optimal control problems. The new algorithm is designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because the stable and unstable invariant subspaces are not well separated (because of eigenvalues near or on the imaginary axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence of the method and compare the new method with the classical structured doubling algorithm as well as some structured QR methods. Copyright © 2012 John Wiley & Sons, Ltd.     

一类Riccati矩阵方程广义自反解的双迭代算法

本文研究了一类Riccati矩阵方程广义自反解的数值计算问题.利用牛顿算法将Riccati矩阵方程的广义自反解问题转化为线性矩阵方程的广义自反解或者广义自反最小二乘解问题,再利用修正共轭梯度法计算后一问题,获得了求Riccati矩阵方程的广义自反解的双迭代算法.拓宽了求解非线性矩阵方程的迭代算法.数值算例表明双迭代算法是有效的.