On the generalized Sylvester mapping and matrix equations

General parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the so-called generalized Sylvester mapping which has some elegant properties. The solution consists of some polynomial matrices satisfying certain conditions and a parametric matrix representing the degree of freedom in the solution. The results provide great convenience to the computation and analysis of the solutions to this family of equations, and can perform important functions in many analysis and design problems in linear system theory. It is also expected that this so-called generalized Sylvester mapping tool may have some other applications in control system theory.     

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

广义预测控制中Diophantine矩阵多项式方程的显式解

直接利用被控对象的离散差分方程推导出多变量广义预测控制中Diophantine矩阵多项式方程的显式解,从而避免了其递推求解或迭代求解,使广义预测控制的应用更加方便．     

Weighted steepest descent method for solving matrix equations

This paper describes iterative methods for solving the general linear matrix equation including the well-known Lyapunov matrix equation, Sylvester matrix equation and some related matrix equations encountered in control system theory, as special cases. We develop the methods from the optimization point of view in the sense that the iterative algorithms are constructed to solve some optimization problems whose solutions are closely related to the unique solution to the linear matrix equation. Actually, two optimization problems are considered and, therefore, two iterative algorithms are proposed to solve the linear matrix equation. To solve the two optimization problems, the steepest descent method is adopted. By means of the so-called weighted inner product that is defined and studied in this paper, the convergence properties of the algorithms are analysed. It is shown that the algorithms converge at least linearly for arbitrary initial conditions. The proposed approaches are expected to be numerically reliable as only matrix manipulation is required. Numerical examples show the effectiveness of the proposed algorithms.     

A simple approach to solving the Diophantine equation

Based on the state-space concepts, a simple approach to finding all polynomial matrix solutions of the Diophantine equation is proposed. The procedure presented is very simple in comparison to earlier ones. Unlike earlier ones, it is not necessary to solve any equation. Only two constant matrices which could be selected at random are required. All solutions are expressed in an explicit formula form     

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.     

Application of implicit system feedback to solving the diophantine equation

Based on the concepts of implicit system feedback, all polynomial matrix solutions to the polynomial dophantine equation are solved. The solutions are expressed in terms of a generalized state-space realization and a feedback gain. The explicit formulae for the solutions are derived in the closed form.     

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.     

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.     

Iterative procedure for solution of sylvester generalized matrix equation

An iteration procedure for solving Sylvester generalized matrix equation is proposed in this paper. The sufficient conditions of stability of the iteration procedure for solving this equation are obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 183–186, May–June, 2000.     

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.     

Solving the generalized Sylvester matrix equation AV +BW = VF via Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV ＋ BW = VF with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.     

Parameterized solution to a class of sylvester matrix equations

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.     

Solution to the second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW

This note provides a complete general parametric solution (V,W) to the generalized second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW, with F being an arbitrary square matrix. The primary feature of this solution is that the matrix F does not need to be in any canonical form, or may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems involving second-order dynamical systems.