Iterative least-squares solutions of coupled Sylvester matrix equations☆
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:which includes the coupled Lyapunov and Sylvester equations as its special cases. In (1), are the unknown matrices to be solved; , , and represent constant (coefficient) matrices of appropriate dimensions. For such coupled matrix equations, the conventional methods require dealing with matrices whose dimensions are . Such a dimensionality problem leads to computational difficulty in that excessive computer memory is required for computation and inversion of large matrices of size . For instance, if , then .
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 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:Here, () is a given full-rank matrix with non-zero diagonal elements, is a constant vector, and 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: which satisfy . 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 equationHere, and are given constant matrices, are the unknown matrices to be solved.
First, let us introduce some notation. The notation is the identity matrix of . For two matrices M and N, is their Kronecker product. For two matrices X and Y with is an mn-dimensional vector formed by columns of X
General coupled matrix equations
In this section, we will extend the iterative method to solve more general coupled matrix equations of the formHere, and are given constant matrices, 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: [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 with Then the solutions of X and Y from (8) are Taking , we apply the algorithm in (15) and (16) to compute and . 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
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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.
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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.