Abstract and Applied Analysis

Volume 2014 (2014), Article ID 649524, 10 pages

http://dx.doi.org/10.1155/2014/649524

Research Article

Iterative Solutions of a Set of Matrix Equations by Using the Hierarchical Identification Principle

Huamin Zhang

Department of Mathematics and Physics, Bengbu College, Bengbu 233030, China

Received 13 January 2014; Revised 29 March 2014; Accepted 30 March 2014; Published 4 May 2014

Academic Editor: Weinian Zhang

Copyright © 2014 Huamin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with iterative solution to a class of the real coupled matrix equations. By using the hierarchical identification principle, a gradient-based iterative algorithm is constructed to solve the real coupled matrix equations and . The range of the convergence factor is derived to guarantee that the iterative algorithm is convergent for any initial value. The analysis indicates that if the coupled matrix equations have a unique solution, then the iterative solution converges fast to the exact one for any initial value under proper conditions. A numerical example is provided to illustrate the effectiveness of the proposed algorithm.

1. Introduction

For systems with certain parameters, the controllability and stability are the important topics which are worth studying [1–3]. If the parameters of systems are uncertain, then how to identify these parameters is to put on the agenda. To identify the parameters of the large-scale systems, the hierarchical identification principle was proposed in [4–6]. The hierarchical gradient-based and least squares based identifications were presented for multivariable system [7]. The fruits of these effective strategies include identifications and adaptive control for dual-rate systems [8] and Hammerstein nonlinear systems [9].

Many publications have studied the solutions to matrix equations from the different points of view [10–14]. Zhou et al. proposed the positive definite solutions of the nonlinear matrix equation [15]; Li et al. discussed a class of iterative methods for the generalized Sylvester equation [16]; the Riccati equation and a class of the coupled transpose matrix equations are investigated in [17, 18].

Unlike the above methods and just like the Jacobi and Gauss-Seidel iterations, Ding and Chen proposed the gradient-based and least squares based iterations for solving and [19, 20] and a large family of iterations. These iterations include the gradient iteration, the least squares iteration, and some classical iterations as their special cases [21, 22]. By using the hierarchical identification principle, the gradient-based iterative algorithms were derived for solving different real matrix equations, such as the generalized Sylvester matrix equations and general linear matrix equations [23–25].

Ding’s strategy has received much attention. With the real representations of complex matrices as tools, Wu et al. applied the Ding’s strategy to solve the extended Sylvester-conjugate matrix equations [26], the complex conjugate and transpose matrix equations [27], and the extended coupled Sylvester conjugate matrix equations [28]; Song and Chen presented the gradient based iterative algorithm for solving the extended Sylvester-conjugate transpose matrix equations [29].

Different from the various single real matrix equations in [21–25] and the complex matrix equations in [26–29], the iterative algorithm for the real coupled matrix equations and is discussed by using the hierarchical identification principle, and the gradient-based iterative algorithm is proposed. Moreover, the gradient-based iteration is reported for solving the general real coupled matrix equations. We prove that the iterative solution always converges to the exact one for any initial value, if there exists a unique solution of these kinds of real coupled matrix equations.

The iterative methods can be applied to nonlinear system identification [30, 31]. The proposed methods of this paper can combine the iterative identification methods [32–35], the auxiliary model identification methods [36–39], the multi-innovation identification methods [40–44], and the two-stage or multistage identification methods [45] to study identification problems for other linear systems [46–50] or nonlinear systems [51–54] and other systems with colored noises [55–58].

This paper is organized as follows. Section 2 offers some notation and basic lemmas. Section 3 derives the gradient-based iterative algorithm for solving the matrix equations and . The iteration of real general coupled matrix equations is discussed in Section 4. Section 5 presents a numerical example to illustrate the effectiveness of the proposed algorithm. Finally, we offer some concluding remarks in Section 6.

2. Notations and Basic Lemmas

Some notation and lemmas are introduced first. The symbol represents the transpose of . For a square matrix , indicates the maximum eigenvalue of . denotes the norm of and is defined as . For an matrix

is defined as

If and , then their Kronecker product is defined as

The relationship between the vec-operator and the Kronecker product can be expressed as the following lemma [59].

Lemma 1. If , , and , then we have

The gradient-based iterative algorithm for solving matrix equation is listed as follows [22].

Lemma 2. For , if is a full-column rank matrix and is a full-row rank matrix, then the iterative solution given by the following gradient-based iterative algorithm converges to the exact solution (i.e., ) for any initial values [22]:

3. The Coupled Matrix Equations

In this section, we consider the following real coupled matrix equations:

where , , and , , are the given matrices, and is the unknown matrix to be determined.

3.1. The Exact Solution

According to Lemma 1, we rewrite (6) as

where is a block matrix. The exact solution of (6) can be given by the following theorem.

Theorem 3. Equation (6) has a unique solution if and only if is a full-column rank matrix; in this case, the unique solution is

where

If , then (6) has a unique solution .

3.2. The Gradient Iterative Algorithm

The hierarchical identification principle implies that the related system can be decomposed into several subsystems, so we introduce four intermediate matrices:

Using these four intermediate matrices, we decompose (6) into four subsystems

According to Lemma 2, it is not hard to get the iterative solutions , , , and of these four subsystems , , , and as follows:

We will determine the convergence factor later. Using (10), we have

Because the unknown matrix appears in the right-hand side, this algorithm is impossible to realize. By using the hierarchical identification principle, we replace the unknown matrix with its estimate at iteration . Hence, we obtain

In fact, only an iterative solution is needed in this algorithm; we take the average of , , , and as and obtain the gradient-based iterative algorithm:

Theorem 4. If (6) has a unique solution , then the iterative solution given by (15)–(20) converges to , that is, , or the error matrix converges to zero for any initial value .

Proof. We define estimation error matrices:

From this definition and (15)–(19), we have the following error formulas:

Set

By the trace formula and from (22), we obtain

Adding (25) to (26) gives

Similarly, adding (27) to (28) yields

Using (29) and (30), we have

Taking the norm of both sides of (23) gives

Substituting (31) into (32) yields

Thus, we have

Since

that is,

if the convergence factor is chosen to satisfy , then we have

It follows that and , as or as , we obtain

From Theorem 3, we have , as . This completes the proof of Theorem 4.

4. General Real Coupled Matrix Equations

In this section, we study the general real coupled matrix equations of form

where , , , , and are the given matrices, and is the unknown matrix to be determined.

4.1. The Exact Solution

According to Lemma 1, (39) can be rewritten as

where is a block matrix. Set

The exact solution of (39) can be given by the following theorem.

Theorem 5. Equation (39) has a unique solution if and only if is a full-column rank matrix; in this case, the unique solution is

If , then (39) has a unique solution .

4.2. The Gradient-Based Iterative Algorithm

We define the intermediate matrices:

By using the hierarchical identification principle, we decompose (39) into

According to Lemma 2, we have

where and denote the iterative solutions of equations and at iteration , respectively. Substituting into (45) and into (46) gives

To realize the above algorithm, we replace the unknown matrix in the right-hand side of the above two equations with and , respectively, and obtain

In fact, only an iterative solution is needed; taking the average of and as , we obtain the gradient iterative algorithm for solving (39) as follows:

Theorem 6. If (39) has a unique solution , then the iterative solution given by (49)–(52) converges to , that is, , or the error matrix converges to zero for any initial value .

Proof. The estimation error matrices are defined as

By using (49)–(51), we have

Set

According to the trace formula and using (54), for and , we obtain

For and , adding (57) to (58) gives

Taking the norm of both sides of (55) and using the norm inequality gives

Dividing (59) by and using the inequality (60) gives

In the second “,” the assumption that is used. Since , we have

If the convergence factor is chosen to satisfy , then we have

It follows that , as or as , we obtain

According to Theorem 5, we have , as . This completes the proof of Theorem 6.

5. A Numerical Example

This section offers a numerical example to illustrate the performance of the proposed algorithm. Consider (6) with

From Theorem 3, the exact solution is

Taking as initial iterative value, we apply the gradient-based iterative algorithm in (15)–(19) to compute . The iterative solutions are shown in Table 1, where is the relative error and the convergence factor is . The relation of the relative error with different convergence factors is shown in Figure 1. From Table 1 and Figure 1, we can find that becomes smaller and smaller and tends to zero as the iterative times increase. This demonstrates that the algorithm proposed in this paper is effective.

Table 1: The gradient iterative solution ().

(%)

2
−1.60864
−7.04633
6.47331
5.28946
10.98102

4
−1.89505
−7.89255
6.97696
5.89765
1.48121

6
−1.96855
−7.98772
7.00523
5.98269
0.30968

8
−1.99026
−7.99846
7.00259
5.99669
0.08667

10
−1.99696
−7.99976
7.00084
5.99931
0.02617

12
−1.99905
−7.99995
7.00025
5.99985
0.00805

14
−1.99970
−7.99999
7.00007
5.99996
0.00249

16
−1.99991
−8.00000
7.00002
5.99999
0.00077

18
−1.99997
−8.00000
7.00001
6.00000
0.00024

20
−1.99999
−8.00000
7.00000
6.00000
0.00007

Solution
−2.00000
−8.00000
7.00000
6.00000

Figure 1: The relative error versus   (dots: ).

A simple calculation indicates that the range of the convergence factor is conservative. Based on the deep analysis of Figure 1, we find that the rate of convergence increases when enlarges from to . However, if we keep enlarging from to , the rate of convergence will drop. This shows that the best convergence factor is uncovered in this algorithm. How to find the best convergence factor subtly is our work in the future.

6. Conclusions

This paper has proposed a gradient-based iterative algorithm for solving a class of the real coupled matrix equations. By using the hierarchical identification principle, we prove that the iterative solution is convergent if the unique solution exists. An example demonstrates that the algorithm is effective and indicates that the best convergence factor is existent. We will find the best convergence factor of this algorithm in the future.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China.

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