A note on combined generalized Sylvester matrix equations

The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.     

PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.     

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.     

进化策略算法在矩阵特征值求解中的应用

工程实践中多种振动问题的求解常常归纳为求矩阵特征值问题,另外一些稳定性分析问题及相关分析问题也可以转化为求矩阵特征值问题.为了有效求解此类问题,提出了一种新的求解矩阵特征值的进化策略算法,该算法可用于求解任意矩阵的特征值.实验结果表明,这种基于进化策略算法求解矩阵特征值的方法,与传统方法相比,表现出求解精度高,收敛速度快等优点.     

Solutions to the matrix equation

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BY, with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B), generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.     

An explicit solution to polynomial matrix right coprime factorization with application in eigenstructure assignment

In this paper, an explicit solution to polynomial matrix right coprime factorization of input-state transfer function is obtained in terms of the Krylov matrix and the Pseudo-controllability indices of the pair of coefficient matrices. The proposed approach only needs to solve a series of linear equations. Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated. These new solutions are simple, they possess better structural properties and are very convenient to use. An example shows the effect of the proposed results.     

An explicit solution to polynomial matrix right coprime factorization with application in eigenstructure assignment

In this paper, an explicit solution to polynomial matrix right coprime factorization of input-state transfer function is obtained in terms of the Krylov matrix and the Pseudo-controllability indices of the pair of coefficient matrices. The proposed approach only needs to solve a series of linear equations. Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated. These new solutions are simple, they possess better structural properties and are very convenient to use. An example shows the effect of the proposed results.     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.     

A computational solution for the matrix riccati equation using laplace transforms

The solution for the finite-time matrix Riccati equation is presented in this paper. The solution to the Riccati equation is obtained in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and solved using Laplace transforms and the computation is done by the digital computer.

A numerical example for the proposed method is given.     

矩阵方程AXB＋CX^T D=F自反最小二乘解的迭代算法

建立了求矩阵方程AXB＋CX^TD=F的自反最小二乘解的迭代算法,证明了迭代算法的收敛性,该算法能够在有限步迭代计算之后得到矩阵方程的一个自反最小二乘解,或者极小范数自反最小二乘解。另外,还给出了在解集合中对给定矩阵的最佳逼近。     

改进的正则化模型在图像恢复中的应用

目的 由拟合项与正则项组成的海森矩阵,如果不具有特殊结构,其逆矩阵计算比较困难,为克服此缺点,提出一种海森矩阵可分块对角化的牛顿投影迭代算法。方法 首先,用L₂范数描述拟合项,用自变量是有界变差函数的复合函数刻画正则项,建立能量泛函正则化模型。其次,引入势函数,将正则化模型转化为增广能量泛函。再次,构造预条件矩阵,使得海森矩阵可分块对角化。最后,为防止牛顿投影迭代算法收敛到局部最优解,采用回溯线性搜索算法和改进的Barzilai-Borwein步长更新准则使得算法全局收敛。结果 针对图像去模糊正则化模型容易使边缘平滑和产生阶梯效应“两难”问题,提出一种新的正则化模型和牛顿投影迭代算法。仿真结果表明,“两难”问题通过本文算法得到了很好的解决。结论 与其他正则化图像去模糊模型相比,本文算法明显改善图像的质量,如有效地保护图像的边缘,抑制阶梯效应,相对偏差和误差较小,较高的峰值信噪比和结构相似测度。     

A symmetric preserving iterative method for generalized Sylvester equation

The generalized Sylvester matrix equation AX + YB = C is encountered in many systems and control applications, and also has several applications relating to the problem of image restoration, and the numerical solution of implicit ordinary differential equations. In this paper, we construct a symmetric preserving iterative method, basing on the classic Conjugate Gradient Least Squares (CGLS) method, for AX + YB = C with the unknown matrices X, Y having symmetric structures. With this method, for any arbitrary initial symmetric matrix pair, a desired solution can be obtained within finitely iterate steps. The unique optimal (least norm) solution can also be obtained by choosing a special kind of initial matrix. We also consider the matrix nearness problem. Some numerical results confirm the efficiency of these algorithms. It is more important that some numerical stability analysis on the matrix nearness problem is given combined with numerical examples, which is not given in the earlier papers. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A Nonrecursive Solution Method for the Linear-Quadratic Optimal Control Problem with a Singular Transition Matrix

In the optimal linear regulator problem the control vector is usually determined by solving the algebraic matrix Riccati equation using successive substitutions. This, however, can be rather inefficient from a computational point of view. A nonrecursive method which requires that the transition matrix is nonsingular has been proposed by Vaughan (1970). In the present paper we present a nonrecursive solution to the matrix Riccati equation for the case that the transition matrix may be singular. We show that this procedure leads to the same numerical results as the standard iteration of the matrix Riccati equation.     

Explicit solutions to the quaternion matrix equations X−AXF=C and X−A[Xtilde] F=C

In the present paper, we investigate the quaternion matrix equation X?AXF=C and X?A[Xtilde] F=C. For convenience, we named the quaternion matrix equations X?AXF=C and X?A[Xtilde] F=C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method.     

具有次优保性能滑动模态的静态输出反馈滑模控制

针对一类不确定系统,提出了一种具有次优保性能滑模面的静态输出反馈滑模控制方法.首先将滑模面的设计问题等价为一个对称矩阵的求解问题,基于等效控制法,推导了保性能滑模面存在的充分条件.然后基于迭代线性矩阵不等式(iterative linear matrix inequality,ILMI)方法,给出了次优保性能线性滑模面的求解算法,最后基于线性矩阵不等式(linearmatrix inequality,LMI)方法,设计了输出反馈滑模控制器,使得闭环系统渐近稳定且切换函数能在有限时间内到达零.该方法首次实现了输出反馈滑模面的优化,且具有保守性小、无需对被控系统模型进行坐标变换的优点.仿真结果验证了本文方法的优越性.     

Least-squares symmetric solution to the matrix equation AXB = C with the norm inequality constraint

In this paper, an iterative method to compute symmetric least-squares solution of the matrix equation AXB = C with the norm inequality constraint is proposed. For this method, without the error of calculation, a desired solution can be obtained with finitely iterate steps. Numerical experiments are performed to illustrate the efficiency and application of the algorithm.