奇异值分解解广义Sylvester矩阵方程

需积分: 8 97 浏览量更新于2024-08-12 收藏 715KB PDF 举报

"该资源是一篇2007年的工程技术论文，发表在《控制理论与应用》期刊上，由Bin ZHOU和Guangren DUAN撰写，主要探讨了使用奇异值分解（Singular Value Decomposition, SVD）求解广义 Sylvester 矩阵方程的方法。" 在该论文中，作者研究了广义Sylvester矩阵方程的解法，这些方程包括 AX - XF = BY 和 MXN - X = TY，其中A, M是n×n的矩阵，B, T是n×r的矩阵，而F, N是p×p的矩阵。特别地，矩阵N和F被假设为同伴矩阵形式。同伴矩阵是一种特殊的矩阵形式，常用于表示线性动态系统的状态空间模型。 Sylvester矩阵方程在控制系统理论、信号处理和许多其他工程领域中有着广泛的应用。通常情况下，解决这类方程需要考虑矩阵对(A, B)的可控性，以及A和F不共享特征值的条件。然而，该论文提出的方法不再需要这些限制，这扩大了其适用范围。论文的核心在于利用奇异值分解来求解这些方程。奇异值分解是线性代数中的一个重要工具，它将矩阵分解为三个更简单的矩阵的乘积：UΣV^T，其中U和V是正交矩阵，Σ是对角矩阵，包含了原矩阵的奇异值。通过SVD，可以有效地处理数值稳定性问题，避免计算中的误差积累，这对于大型矩阵尤其重要。该算法提供了一种数值稳定且计算便捷的途径来求解广义Sylvester矩阵方程。由于SVD的特性，这种方法对于控制系统的许多设计问题具有重要的实用价值。论文关键词包括“广义Sylvester矩阵方程”、“一般解”、“同伴矩阵”和“奇异值分解”。这篇论文为解决一组特定形式的广义Sylvester矩阵方程提供了一种新颖的数值方法，特别是对于那些在传统方法下难以处理或者要求严格条件的情况，这种方法展现出显著的优势。这种方法不仅在理论上具有重要贡献，而且在实际工程问题的求解中也有很大的潜力。

Journal of Control Theory and Applications 2007 5 (4) 397–403 DOI 10.1007/s11768-006-6113-0

Solutions to the generalized Sylvester matrix

equations by a singular value decomposition

Bin ZHOU, Guangren DUAN

(Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin Heilongjiang 150001, China)

Abstract: In this paper, solutions to the generalized Sylvester matrix equations AX − XF = BY and MXN −X =

T Y with A, M ∈ R

n×n

, B, T ∈ R

n×r

, F, N ∈ R

p×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) and the restriction that A, F

don’t 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.

Keywords: Generalize Sylvester matrix equations; General solutions; Companion matrix; Singular value decompo-

sition

1 Symbols and notations

In this paper, we use B

and rank(B) to denote the trans-

pose and the rank of matrix B, respectively, and b

is the

i-th row and j-th column of matrix B. I

is the p × p iden-

tity matrix, and 0 will be used as a r × s matrix when the

dimensions are evident from the context. The symbol ⊗ is

to denote the Kronecker product. We use A ∈ F

r×p

which

means that A is an r × p matrix in the ﬁeld F .

We use col[A

]

i=p

to denote a matrix in the form of

col[A

]

i=p

= [

p+1

· · · A

q−1

and use row[A

]

i=p

to denote

row[A

]

i=p

= [col[A

]

i=p

]

Further, let A ∈ R

n×n

and B ∈ R

n×r

, we deﬁne the

so-called Krylov matrix with matrix pair (A, B) as follows:

(A, B, k) = col[A

k−1

i=0

2 Introduction

The general solution to the generalized Sylvester matrix

equation

AX − XF = BY, (1)

where A ∈ R

n×n

, B ∈ R

n×r

, F ∈ R

p×p

are known, is

closely related with many problems in linear control sys-

tems theory, such as eigenvalue assignment [1, 2], observer

design [3], eigenstructure assignment design [4, 5], con-

strained control [6], etc, and has been studied by many au-

thors (see [3, 5, 7] and the references therein). When the

matrix F is in Jordan form, an attractive analytical and

restriction-free solution with explicit freedom is presented

in [3]. To obtain this solution, one needs to carry out an

orthonormal transformation, compute a matrix inverse, and

solve a series of linear equation groups. Reference [5] pro-

poses two solutions to the matrix equation, also for the

case that the matrix F is in Jordan form. The ﬁrst one

is in an iterative form, while the second is in an explicit

parametric form. To obtain the explicit solution proposed

in [5], one needs to carry out a right coprime factorization of

(sI − A)

−1

B (when the eigenvalues of the Jordan matrix F

are undetermined) or a series of singular value decomposi-

tion (when the eigenvalues of F are known). Generalization

of this explicit solution to a more general case is considered

in [4] and [7].

When the matrix F is in companion form, a very neat and

elegant general complete parametric solution to the gener-

alized Sylvester matrix equation (1) is proposed in [8]. The

solution is expressed in terms of the controllability matrix

of the matrix pair (A, B), a symmetric operator matrix, and

a parametric matrix in the Hankel matrix form. Such a result

can provide great convenience for many analysis and design

problems associated with the matrix equation (1). However,

this proposed solution has a disadvantage that when A and

F have common eigenvalues, the given solution may be not

complete. This is not convenient to use in some problems

Received 16 June 2006; revised 22 November 2006.

This work was supported by the Chinese Outstanding Youth Foundation (No.69925308) and Program for Changjiang Scholars and Innovative Research

Team in University.

下载后可阅读完整内容，剩余6页未读，立即下载

weixin_38544075

粉丝: 10
资源: 931

奇异值分解解广义Sylvester矩阵方程

Best Approximate Solution of Matrix EquationAXB CYD=E

Lions-Generalized_Solutions_of_Hamilton-Jacobi_Equations.djvu

On the Riemann Boundary Value Problem for Null Solutions to Iterated Generalized Cauchy–Riemann Operator in Clifford Analysis

Model selection of the generalized von Mises distribution based on empirical mode decomposition with data analyses

Cauchy problem for a class of generalized Zakharov equations

An Iterative Method for the Generalized Bisymmetric Solution of Matrix Equation AXB=C

The generalized symmetric procrustes problems associated with matrix equation AX=B

Generalized Collocation Methods - Solutions to Nonlinear Problems

Working CorrelationStructure Selection in Generalized Estimating Equations

On the generalized bi(skew-)symmetric solutions of a linearmatrix equation and its procrust problems

最新资源