Elsevier

Systems & Control Letters

Volume 54, Issue 2, February 2005, Pages 95-107
Systems & Control Letters

Iterative least-squares solutions of coupled Sylvester matrix equations

https://doi.org/10.1016/j.sysconle.2004.06.008Get rights and content

Abstract

In this paper, we present a general family of iterative methods to solve linear equations, which includes the well-known Jacobi and Gauss–Seidel iterations as its special cases. The methods are extended to solve coupled Sylvester matrix equations. In our approach, we regard the unknown matrices to be solved as the system parameters to be identified, and propose a least-squares iterative algorithm by applying a hierarchical identification principle and by introducing the block-matrix inner product (the star product for short). We prove that the iterative solution consistently converges to the exact solution for any initial value. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.

Introduction

Lyapunov and Sylvester matrix equations play important roles in system theory [5], [6], [33], [34], [35]. Although exact solutions, which can be computed by using the Kronecker product, are important, the computational efforts rapidly increase with the dimensions of the matrices to be solved. For some applications such as stability analysis, it is often not necessary to compute exact solutions; approximate solutions or bounds of solutions are sufficient. Also, if the parameters in system matrices are uncertain, it is not possible to obtain exact solutions for robust stability results [10], [12], [16], [21], [22], [23], [24], [25], [26], [28], [29], [30], [31], [32], [37].

Alternative ways exist which transform the matrix equations into forms for which solutions may be readily computed, for example, the Jordan canonical form [15], companion-type form [2], [3], Hessenberg–Schur form [1], [13]. In this area, Chu gave a numerical algorithm for solving the coupled Sylvester equations [7]; and Borno presented a parallel algorithm for solving the coupled Lyapunov equations [4]. But, these algorithms require computing some additional matrix transformation/decomposition; moreover, they are not suitable for more general coupled matrix equations of the form:j=1pAijXjBij=Ci,i=1,2,,p,which includes the coupled Lyapunov and Sylvester equations as its special cases. In (1), XiRm×n are the unknown matrices to be solved; Aij, Bij, and Cij represent constant (coefficient) matrices of appropriate dimensions. For such coupled matrix equations, the conventional methods require dealing with matrices whose dimensions are mnp×mnp. Such a dimensionality problem leads to computational difficulty in that excessive computer memory is required for computation and inversion of large matrices of size mnp×mnp. For instance, if m=n=p=100, then mnp×mnp=106×106.

In the field of matrix algebra and system identification, iterative algorithms have received much attention [8], [10], [14], [27], [32]. For example, Starke presented an iterative method for solutions of the Sylvester equations by using the SOR technique [36]; Jonsson and Kägström proposed recursive block algorithms for solving the coupled Sylvester matrix equations [18], [19]; Kägström derived an approximate solution of the coupled Sylvester equation [20]. To our best knowledge, numerical algorithms for general matrix equations have not been fully investigated, especially the iterative solutions of the coupled Sylvester matrix equations, as well as the general coupled matrix equations in (1), and the convergence of the iterative solutions involved, which are the focus of this work.

In this paper, the problem will be tackled in a new way—we regard the unknown matrices Xj to be solved as the parameters (parameter matrices) of the system to be identified, and apply the so-called hierarchical identification principle to decompose the system into some subsystems, and derive iterative algorithms of the matrix equations involved. Our methods will generate solutions to the matrix equations which are arbitrarily close to the exact solutions.

The paper is organized as follows. In Section 2, we extend the well-known Jacobi and Gauss–Seidel iterations and present a large family of iterative methods. In Sections 3 and 4, we define the block-matrix inner product (the star product for short) and derive iterative algorithms for the coupled Sylvester matrix equations and general coupled matrix equations, respectively, and study the convergence properties of the algorithms. In Section 5, we give an example for illustrating the effectiveness of the algorithms proposed. Finally, we offer some concluding remarks in Section 6.

Section snippets

Extension of the Jacobi and Guass–Seidel iterations

Consider the following linear equation:Ax=b.Here, A=[aij] (i,j=1,2,,n) is a given full-rank n×n matrix with non-zero diagonal elements, bRn is a constant vector, and xRn an unknown vector to be solved. Let D be the diagonal part of A, and L and U be the strictly lower and upper triangular parts of A: D=diag[a11,a22,,ann]Rn×n,L=000a210a31a3200an1an2an,n-10Rn×n,U=0a12a13a1n00a23a2nan-1,n000Rn×n,which satisfy L+D+U=A. Then both the Jacobi and Gauss–Seidel iterations can

Coupled Sylvester matrix equations

In this section, we study least squares iterative algorithms to solve the coupled Sylvester matrix equationAX+YB=C,DX+YE=F.Here, A,DRm×m,B,ERn×n and C,FRm×n are given constant matrices, X,YRm×n are the unknown matrices to be solved.

First, let us introduce some notation. The notation In is the identity matrix of n×n. For two matrices M and N, MN is their Kronecker product. For two m×n matrices X and Y with X=[x1,x2,,xn]Rm×n,col[X] is an mn-dimensional vector formed by columns of X col[X]=x

General coupled matrix equations

In this section, we will extend the iterative method to solve more general coupled matrix equations of the formA11X1B11+A12X2B12++A1pXpB1p=C1,A21X1B21+A22X2B22++A2pXpB2p=C2,Ap1X1Bp1+Ap2X2Bp2++AppXpBpp=Cp.Here, AijRm×m,BijRn×n and CiRm×n are given constant matrices, XiRm×n are the unknown matrix to be solved.

The general coupled matrix equations (22) include the following matrix equations as the special cases: (i) the discrete-time Sylvester equation: AXBT+X=C [18], [19]; (ii) the

Example

In this section, we give an example to illustrate the performance of the proposed algorithms.

Suppose that the coupled matrix equations are AX+YB=C,DX+YE=F with A=2.001.001.002.00,B=1.000.200.201.00,D=2.000.500.502.00,E=1.003.002.00-4.00,C=13.2010.600.608.40,F=9.5018.0016.003.50.Then the solutions of X and Y from (8) are X=x11x12x21x22=4.003.003.004.00,Y=y11y12y21y22=2.001.002.003.00.Taking X(0)=Y(0)=10-612×2, we apply the algorithm in (15) and (16) to compute X(k) and Y(k). The

Conclusions

A family of iterative methods for linear systems is presented and a least-squares iterative solution to coupled matrix equations are studied by using the hierarchical identification principle and the star product. The analysis indicates that the algorithms proposed can achieve a good convergence property for any initial values. How to use the conjugate gradient method to solve the coupled matrix equation requires further research. Although the algorithms are presented for linear coupled matrix

References (38)

  • S. Xiang

    On an inequality for the Hadamard product of an M-matrix or an H-matrix and its inverse

    Linear Algebra Appl.

    (2003)
  • A. Barraud

    A numerical algorithm to solve ATXA-X=Q

    IEEE Trans. Automat. Control

    (1977)
  • R. Bitmead

    Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman–Yakubovich equations

    IEEE Trans. Automat. Control

    (1981)
  • R. Bitmead et al.

    On the solution of the discrete-time Lyapunov matrix equation in controllable canonical form

    IEEE Trans. Automat. Control

    (1979)
  • T. Chen et al.

    Optimal Sampled-data Control Systems

    (1995)
  • T. Chen et al.

    H design of general multirate sampled-data control systems

    Automatica

    (1994)
  • Y. Fang et al.

    New estimates for solutions of Lyapunov equations

    IEEE Trans. Automat. Control

    (1997)
  • J. Garloff

    Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equation and the continuous Lyapunov equation

    Internat. J. Control

    (1986)
  • G.H. Golub et al.

    A Hessenberg–Schur method for the matrix problem AX+XB=C

    IEEE Trans. Automat. Control

    (1979)
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    This research was supported by the Natural Sciences and Engineering Research Council of Canada and the National Natural Science Foundation of China.

    1

    Feng Ding was with the Department of Automation, Tsinghua University, Beijing, China, and is currently a Visiting Professor at the University of Alberta, Edmonton, Canada.

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