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Acta Mathematicae Applicatae Sinica 2024年第1期40卷 17-34页

作者： Qing-qing Zheng Department of Mathematics College of ScienceChina University of Petroleum-BeijingBeijing102249China

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.

关键词： sylvester matrix equation coupled matrix equation minimum residual gradient descent convergence analysis

Extending GCR Algorithm for the Least Squares Solutions on a Class of sylvester matrix equations

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Numerical Mathematics(Theory,Methods and Applications) 2018年第1期11卷 140-159页

作者： Baohua Huang Changfeng Ma College of Mathematics and Informatics Fujian Key Laborotary of Mathematical Analysis and ApplicationsFujian Normal UniversityFuzhou 350117China

The purpose of this paper is to derive the generalized conjugate residual(GCR)algorithm for finding the least squares solution on a class of sylvester matrix *** prove that if the system is inconsistent,the least squares solution can be obtained within finite iterative steps in the absence of round-off ***,we provide a method for choosing the initial matrix to obtain the minimum norm least squares solution of the ***,we give some numerical examples to illustrate the performance of GCR algorithm.

关键词： sylvester matrix equation Least squares solution Generalized conjugate residual algorithm Numerical experiments

Parameterized Solution to a Class of sylvester matrix equations

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International Journal of Automation and computing 2010年第4期7卷 479-483页

作者： Yu-Peng Qiao Hong-Sheng Qi Dai-Zhan Cheng Center for Control and Optimization College of Automation Science and Engineering South China University of Technology Guangzhou 510641 PRC Key Laboratory of Systems and Control Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 PRC

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.

关键词： sylvester matrix equation parameterized solution Kronecker product linear matrix equation Luenberger observers

Parametric control of quasi-linear second-order systems with partitioned eigenstructure assignment by output feedback

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Science China(Information Sciences) 2023年第4期66卷 228-243页

作者： Weizhen LIU Guangren DUAN Dake GU Center for Control Theory and Guidance Technology Harbin Institute of Technology Center for Control Science and Technology Southern University of Science and Technology School of Automation Engineering Northeast Electric Power University

In this paper, a parametric design approach for stabilizing a quasi-linear second-order system with partitioned eigenstructure assignment(PESA) is investigated through output feedback control. The PESA approach is established by partitioning the desired eigenvalue matrix into two parts to separate the associated right and left eigenvectors into a subset of the generalized eigenvectors simultaneously. A parametric controller is established by solving two second-order generalized sylvester matrix equations, and a certain form with the desired eigenstructure can be derived with the established quasi-linear output feedback controller. Unlike the prevailing approach that assigns the entire set of generalized eigenvectors, which is difficult to satisfy a large number of complicated constraints in practical systems by the normalized pair of right and left eigenvector matrices, a subset of the generalized eigenvectors is considered. In addition, the proposed PESA approach provides less computational load and is easy to use. A numerical example and application in spacecraft rendezvous are provided to verify the numerical economy and high efficiency of the proposed approach.

关键词： quasi-linear second-order systems partitioned eigenstructure assignment parametric approach sylvester matrix equation output feedback

A System of Periodic Discrete-time Coupled sylvester Quaternion matrix equations

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Algebra Colloquium 2017年第1期24卷 169-180页

作者： Zhuoheng He Qingwen Wang Department of Mathematics Shanghai University Shanghai 200444 China Department of Mathematics and Statistics Auburn University AL 36849-5310 USA

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled sylvester matrix equations AkXk ＋ YkBk = Mk, CkXk＋l ＋ YkDk ---- Nk （k = 1, 2） over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

关键词： periodic discrete-time equation sylvester matrix equation quaternion alge-bra generalized inverse rank

Developing Bi-CG and Bi-CR Methods to Solve Generalized sylvester-transpose matrix equations

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International Journal of Automation and computing 2014年第1期11卷 25-29页

作者： Masoud Hajarian Department of Mathematics Faculty of Mathematical SciencesShahid Beheshti UniversityGeneral Campus

The bi-conjugate gradients(Bi-CG)and bi-conjugate residual(Bi-CR)methods are powerful tools for solving nonsymmetric linear systems Ax=*** using Kronecker product and vectorization operator,this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized sylvester-transpose matrix equationp i=1(Ai X Bi+Ci XTDi)=E(including Lyapunov,sylvester and sylvester-transpose matrix equations as special cases).Numerical results validate that the proposed algorithms are much more efcient than some existing algorithms.

关键词： Linear systems iterative method bi-conjugate gradients(Bi-CG) method bi-conjugate residual(Bi-CR) method sylvester matrix equation

An iterative algorithm for solving a class of matrix equations

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控制理论与应用（英文版） 2009年第1期7卷 68-72页

作者： Minghui WANG Yan FENG School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China College of Information Science ＆ Technology Qingdao University of Science ＆ Technology Qingdao Shandong 266061 China

In this paper, an iterative algorithm is presented to solve the sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].

关键词： Iterative algorithm Conjugate gradient method Lyapunov matrix equation sylvester matrix equation