自适应解卷积与二次型共轭梯度算法

提出一种新的解卷积算法，它是将卷积运算关系ｙｉ＝ｈｉ＊ｆｉ转换为解矩阵方程Ｙ＝ＡＨ．运算这个矩阵方程可实现卷积与解卷积。文中也给出解矩阵方程的优化算法、ＱＣＧ（二次型共轭梯度）算法。计算机模拟表明，新算法具有精度高、速度快、数据短和计算稳定等优点．     

求解对偶Riccati方程的矩阵符号函数法

本文介绍一种求解对偶代数Riccati方程正定(负定)稳定(反稳定)解的方法——矩阵符号函数法,给出这些解的唯一存在的充分必要条件和算法实现。     

基于秩一摄动理论配置Lyapunov指数的新方法

本文提出一种混沌控制与反控制的新算法。利用秩一摄动理论精确摄动离散受控系统矩阵的特征值,来配置Lyapunov指数。新算法完全吻合混沌的Lyapunov指数判据。既能更准确地配置正的Lyapunov指数,又能配置传统算法无法获得的负Lyapunov指数。仿真结果显示了新算法的有效性。     

离散时间Itˆo 型跳变系统Lyapunov 方程的有限次迭代求解算法

针对离散时间Itˆo 型马尔科夫跳变系统Lyapunov 方程的求解给出一种迭代算法. 经证明, 在误差允许的范围内, 该算法可以在确定的有限次数内收敛到系统的精确解, 收敛速度较快, 具有良好的数值稳定性, 并且该算法为显式迭代, 可避免迭代过程中求解其他矩阵方程对结果精度产生的影响. 最后通过一个数值算例对该算法的有效性进行了验证.

     

基于Delta算子的统一代数Lyapunov方程解的上下界

基于Delta算子描述,统一研究了连续代数Lyapunov方程(CALE)和离散代数Lyapunov方程(DALE)的定界估计问题.采用矩阵不等式方法,给出了统一的代数Lyapunov方程(UALE)解矩阵的上下界估计,在极限情形下可分别得到CALE和DALE的估计结果.计算实例表明了本文方法的有效性.     

基于秩一摄动理论配置Lyapunov指数的新算法

本文提出一种为混沌控制与反控制配置Lyapunov指数的新算法.该算法利用秩一摄动理论精确摄动离散受控系统矩阵的特征值来配置Lyapunov指数,使其完全吻合混沌判据.该算法既能更准确地配置正的Lyapunov指数,又能配置传统算法无法获得的负Lyapunov指数.仿真结果显示了新算法的有效性.     

一类广义Riccati矩阵方程对称解的双迭代算法

研究了一类广义系统控制理论导出的Riccati矩阵方程对称解的数值计算方法．运用牛顿算法将Riccati矩阵方程的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,采用修正共轭梯度法解决导出的线性矩阵方程的对称解问题,可建立求Riccati矩阵方程对称解的双迭代算法．数值算例表明,双迭代算法是有效的．     

不确定性问题中逻辑关系方程的置换矩阵解法

本文给出了在不确定性问题中逻辑关系方程有解 ,有唯一解的充分必要条件 ,并把求解逻辑关系方程的问题转化为求解一些系数矩阵是置换矩阵的逻辑方程组问题 ,从而给出一种求解逻辑关系方程的新算法     

多矩阵变量线性矩阵方程的广义自反解的迭代算法

基于求线性矩阵方程约束解的修正共轭梯度法的思想方法,通过修改某些矩阵的结构,建立了求特殊类型的多矩阵变量线性矩阵方程的广义自反解的迭代算法,证明了迭代算法的收敛性,解决了给定矩阵在该矩阵方程的广义自反解集合中的最佳逼近计算问题.当矩阵方程相容时,该算法可以在有限步计算后得到其一组广义自反解;选取特殊的初始矩阵,能够求得其极小范数广义自反解.数值算例表明,迭代算法是有效的.     

Riccati和Lyapunov矩阵代数方程解的一种简便快速算法

本文讨论了Riccati和Lyapunov矩阵代数方程解的矩阵符号函数算法。给出了算式、程序框图和算例。     

Iterative algorithm for minimal norm least squares solution to general linear matrix equations

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory.     

Gain margin and Lyapunov analysis of discrete-time network synchronization via Riccati design

This paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete-time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single-input controllable system is analyzed. The solution matrix entries are represented using unknown closed-loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi-input and single-input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete-time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.     

Solution of the Lyapunov matrix differential equations by the frequency method

Methods for solving the Lyapunov matrix differential and algebraic equations in the time and frequency domains are considered. The solutions of these equations are finite and infinite Gramians of various forms. A feature of the proposed new approach to the calculation of Gramians is the expansion of the Gramians in a sum of matrix bilinear or quadratic forms that are formed using Faddeev’s matrices, where each form is a solution of the linear differential or algebraic equation corresponding to an eigenvalue of the matrix or to a combination of such eigenvalues. An example illustrating the calculation of finite and infinite Gramians is discussed.     

An imbedded initialization of Newton's algorithm for matrix Riccati equation

An effective numerical computation of the steady-state Riccati matrix is based on the successive solutions of a Lyapunov equation using Newton's method. The requirements of this algorithm are an initial stabilizing matrix and the numerical solution of the associated Lyapunov equation. Computationally, the first requirement is the more influencing factor in solving the Riccati equation with reasonable accuracy and speed. In this paper an initial matrix, based on the parameter imbedded solution of the Riccati equation, is introduced for the Newton's algorithm. The imbedding Newton algorithm has been applied to a variety of system, both stable and unstable as well as high-dimensional, A matrices, one of which is reported here. The proposed modification has improved the required CPU time of previous initialization schemes by as much as a factor of 6 times for the same order of accuracy.     

On the Lyapunov matrix differential equation

A lower bound for the determinant of the solution to the Lyapunov matrix differential equation is derived. It is shown that this bound is obtained as a solution to a simple scalar differential equation. In the limiting case where the solution to the Lyapunov differential equation becomes stationary, the result reduces to one of the existing bounds for the algebraic equation.     

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.     

Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equations and the continuous Lyapunov equation

We present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapunov matrix equation. Nearly all of our bounds for the discrete Riccati equation are new. The bounds for the discrete and continuous Lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation.     

Matrix bounds of the discrete ARE solution

This paper provides new lower and upper matrix bounds of the solution to the discrete algebraic Riccati equation. The lower bound always works if the solution exists. The upper bounds are presented in terms of the solution of the discrete Lyapunov equation and its upper matrix bound. The upper bounds are always calculated if the solution of the Lyapunov equation exists. A numerical example shows that the new bounds are tighter than previous results in many cases.     

摄动离散矩阵Lyapunov方程解的估计

研究摄动离散矩阵Lyapunov方程解的估计问题,利用矩阵运算性质及Lyapunov稳定性理论,给出在结构不确定性假设下方程解的存在条件及解的上下界估计,估计结果由一个线性矩阵不等式(LMI)和两个矩阵代数Riccati方程确定．针对几种不确定性假设,进一步给出矩阵代数Riccati方程的具体形式．最后通过一个算例说明了所得结果的有效性．