主子阵约束下矩阵方程AX=B的对称最小二乘解

本文主要讨论主子阵约束下矩阵方程AX=B的对称最小二乘解．基于投影定理,巧妙的把最小二乘问题转化为等式问题求解,并利用奇异值分解的方法,给出了该对称最小二乘解的一般表达式．此外,文章还考虑了此对称最小二乘解集合对任一给定矩阵的最佳逼近问题,得到了最佳逼近解,并给出了相应的算法步骤和数值例子．     

Parametric Solutions to the Generalized Discrete Yakubovich‐Transpose Matrix Equation

This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich‐transpose matrix equation X − AX^TB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich‐transpose matrix equation, which is called the Karm‐Yakubovich‐transpose matrix equation, is considered. The explicit solutions to the Karm‐Yakubovich‐transpose matrix equation are also presented by the so‐called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm.     

求解矩阵方程AX=B几种算法的比较

探究了求解矩阵方程AX=B的广义共轭残量法（GCR）、正交极小化法（ORTHOMIN）、重开始的广义共轭残量法（GCR（k））、重开始的正交极小化法（ORTHOMIN（k））等四种算法的迭代思想,讨论了算法的收敛性和收敛速度;用数值实验比较四种算法的性能,得出了重开始的广义共轭残量法能更好地求解大规模矩阵方程的结论。     

Hermitian and non-negative definite reflexive and anti-reflexive solutions to AX=B

In this paper, we establish some conditions for the existence and the representations for the Hermitian reflexive and Hermitian anti-reflexive, and non-negative definite reflexive solutions to the matrix equation AX=B with respect to a generalized reflection matrix P by using the Moore–Penrose inverse. Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided.     

The reflexive solutions of the matrix equation AX B = C

In this paper, we study the existence of a reflexive, with respect to the generalized reflection matrix P, solution of the matrix equation AX B = C. For the special case when B = I, we get the result of Peng and Hu [1].     

The symmetric solutions of the matrix inequality AX≥B in least-squares sense

In this paper, we present an iterative method to compute the symmetric solutions of the matrix inequality AX≥B in least-squares sense. The presented iteration method can be also used to compute the symmetric solutions of the matrix equation AX=B and the consistent matrix inequality AX≥B. Numerical experiments are given to illustrate the usefulness of the proposed approach.     

矩阵方程AXB=C的最小二乘Hamilton解术

对于任意给定的矩阵A∈R^k×2m,B∈R^2m×n,C∈R^k×n,本文利用投影定理,矩阵对的广义奇异值分解（GSVD）,标准相关分解（CCD）,研究矩阵方程AXB=C的最小二乘Hamilton解,得到了解的表达式．并由此考虑了解集合对给定矩阵的最佳逼近问题．     

利用并行方法解AX+XB=C型线性矩阵方程

提出了一种新的递推算法用于求解AX+XB=C型线性矩阵方程,这种算法可以用脉 动阵列结构并行实现,该算法和结构还可求解其它几种类似的线性矩阵方程,特殊情况下求解 方程的阵列结构可进一步简化.仿真结果表明,这种并行方法有较高的加速比及效率.     

New Finite Algorithm for Solving the Generalized Nonhomogeneous Yakubovich‐Transpose Matrix Equation

In this paper, the development of the conjugate direction (CD) method is constructed to solve the generalized nonhomogeneous Yakubovich‐transpose matrix equation AXB + CX^TD + EYF = R. We prove that the constructed method can obtain the (least Frobenius norm) solution pair (X,Y) of the generalized nonhomogeneous Yakubovich‐transpose matrix equation for any (special) initial matrix pair within a finite number of iterations in the absence of round‐off errors. Finally, two numerical examples show that the constructed method is more efficient than other similar iterative methods in practical computation.     

Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

In this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.     

Hand to sensor calibration: A geometrical interpretation of the matrix equation AX=XB

In this paper, the matrix equation AX=XB used for hand to sensor calibration of robot‐mounted sensors is analyzed using a geometrical approach. The analysis leads to an original way to describe the properties of the equation and to find all of its solutions. It will also be highlighted why, when multiple instances A_iX=XB_i (i=1,2,...) of the equation are to be solved simultaneously, the system is overconstrained. Finally, singular cases are also discussed. © 2005 Wiley Periodicals, Inc.     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper,solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M∈Rn×n,B,T∈Rn×n,F,N∈Rp×p and the matrices N,F being in companion form,are established by a singular value decomposition of a matrix with dimensions n×(n pr).The algorithm proposed in this paper for the euqation AX-XF=BY does not require the controllability of matrix pair(A,B)andthe restriction that A,F do not have common eigenvalues.Since singular value decomposition is adopted,the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations,and can perform important functions in many design problems in control systems theory.     

Solution to Generalized Sylvester Matrix Equations

An explicit solution to the generalized Sylvester matrix equation AX+BY= EXF is established. This solution is expressed in terms of the R-controllability matrix of (E, A, B), a generalized symmetric operator matrix and an observability matrix. Moreover, based on this solution, solutions to some other matrix equations are also derived. The results may provide great convenience for the analysis and synthesis problems related to these equations.     

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     

Local input‐to‐state stabilization and ℓ ∞ ‐induced norm control of discrete‐time quadratic systems

This paper addresses the problems of local stabilization and control of open‐loop unstable discrete‐time quadratic systems subject to persistent magnitude bounded disturbances and actuator saturation. Firstly, for some polytopic region of the state‐space containing the origin, a method is derived to design a static nonlinear state feedback control law that achieves local input‐to‐state stabilization with a guaranteed stability region under nonzero initial conditions and persistent bounded disturbances. Secondly, the stabilization method is extended to deliver an optimized upper bound on the ? _∞ ‐induced norm of the closed‐loop system for a given set of persistent bounded disturbances. Thirdly, the stabilization and ? _∞ designs are adapted to cope with actuator saturation by means of a generalized sector bound constraint. The proposed controller designs are tailored via a finite set of state‐dependent linear matrix inequalities. Numerical examples are presented to illustrate the potentials of the proposed control design methods. Copyright © 2014 John Wiley & Sons, Ltd.