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Mathematical Computation September 2013, Volume 2, Issue 3, PP.57-61

The Anti-symmetric Solution for the Mixedtype Lyapunov Matrix Equation by Parameter Iterative Method Xindong Zhang1, Juan Liu1, Leilei Wei2 1. College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang, 830054, P.R. China 2. College of Science, Henan University of Technology, Zhengzhou 450001, P.R. China E-mail address: liaoyuan1126@163.com

Abstract Lyapunov matrix equations (LMEs) have played a fundamental role in numerous problems in control, communication systems theory and power systems. As one of LMEs, mixed-type Lyapunov matrix equation (MTLME) also has a wide range of T T applications in practice. In this paper, the anti-symmetric solution of the MTLME A X  XA  B XB  C is solved by using

an iterative algorithm with a parameter. The steps and the conditions of convergence for this algorithm are given. Choice of the parameter is discussed. Finally, the results are illustrated by numerical example. Keywords: Mixed-type Lyapunov Matrix Equation; Anti-symmetric Solution; Iterative Algorithm

1

INTRODUCTION

LMEs have played a fundamental role in numerous problems in control, communication systems theory and power systems. They arise naturally in optimal control theory [1], stability analysis of dynamical systems [2], and model reduction of linear time-invariant systems [3,4]. LMEs have been widely studied from different perspectives [5, 6, 7]. It is well known that there have been many methods for the solution of the LMEs. For example, the GMRES algorithm [8] for the large Lyapunov equations has been proposed. Many direct methods are based on matrix transformations into forms for which solutions may be readily computed; and examples of such forms include the Jordan canonical form [9] , the companion form [10,11], and the Hessenberg-Schur form[5,12]. Iterative methods are popular in the areas of matrix algebra and systems identification [13]. For instance, Starke and Niethammer presented an iterative method for the solutions of CT Sylvester equations by using the SOR (successive over relaxation) technique [14], and Mukaidani et al. discussed an iterative algorithm for generalized algebraic Lyapunov equations [15]. MTLME and its solvability have been studied by Xu et al [16]. In this paper, the anti-symmetric solution of MTLME has been investigated by using an iterative algorithm with a parameter.

2 BASIC IDEA In this section, the anti-symmetric solution of the following matrix equation is studied (1) It is difficult to find the solution of matrix equation Eq. (1) directly, so the following form can be obtained by equivalence transformation,

 AT X 1  X 1 A  BT X 1 B  C1 ,  T T  A Y1  Y1 A  B Y1 B  C2 , - 57 www.ivypub.org/MC

(2)


then we can get

 AT X 1T  X 1T A  BT X 1T B  C1T ,  T T T T T T  A Y1  Y1 A  B Y1 B  C2 ,

(3)

 AT X  XA  BT XB  C ,  T T  A Y  YA  B YB  C ,

(4)

By Eq. (2) and Eq. (3), we can obtain

where

(the set of all have the following forms:

.

and the forms of symmetric matrices).

are determined by

anti-symmetric matrices),

. It is easy to know that

and

(the set of all

From above discussion, it can be seen that the solution of Eq. (1) can be obtained from Eq. (4). So if the solution of (4) is determined, then let

and

, where

,

and for , therefore, the solution of Eq. (1) will be expressed as only one equation of it need to be done, and the other may be acquired easily.

for . For Eq. (4),

By matrix operator, the necessary and sufficient conditions for consistence and uniqueness of solution for Eq. (4) are: where

is defined as

the line space of matrix

,

and

denote

and the Kronecker product, respectively.

If Eq. (1) is consistent, we discuss how to get anti-symmetric solution by its symmetric solution. An iterative algorithm with a parameter is given.

3 MAIN RESULT The symmetric and anti-symmetric matrices have practical application in many fields. The anti-symmetric problem can be transformed into symmetric problem. It is assumed that and the following MTLME is taken into consideration (5) The key of solving the MTLME is that how to create a high convergence iterative scheme. The purpose of parameter is to change the spectral radius of coefficient matrices. Spectral radius is expected to be as little as possible in order to get a high convergence iterative scheme. If

,

is partitioned into the following form: (6)

where

.

Substituting (6) into (5), we get: where

So the following equation should be directly discussed: (7)

where form:

. By parameter

(7) may be transformed into the following (8)

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which is equivalent to the following form: (9)

( A   I )  I  B  B is inverse, let x  vec( X ) , M  [( A   I )  I  B  B] [ I  ( A   I )] , c  [( A   I )  I  B  B]1 vec(C ) , then (9) can be expressed as , thus we can get the following iterative scheme: (10) Lemma 1. If (5) is consistent, the sequence by iterative scheme (10) converges to one solution of (5) if and only if where denotes the spectral radius of matrix . 1

If

Let expressed as:

,

and

, then (8) can be (11)

We can get the following iterative scheme by (11):

( A   I  i B) xi( k 1)  ci  X k ai (  ),  i  1, 2, , n; k  1, 2,  Theorem 2. If are

(12)

the maximum and minimum eigenelement of and

, respectively. If parameter

and

satisfies (13)

where , the sequence solution of (7) for every initial matrix .

acquired by iterative scheme (12) converges to one

Proof. It is well know that (10) is equivalent to (12). (10) is convergent if and only if needed to show that as (13) is held.

. Thus, it is only

By the relation between eigenvalues of symmetric matrix and the characteristic of Kronecker product, we get that where

For

, we get that

and

Then, it can be obtained that

Therefore, as (13) is held, it can be obtained that

which means (12) is convergent.

By Theorem 2, it is known that the choice of

is controlled by (13). Especially, when

, by (13) convergent.

If

a

high

convergence iterative as little as possible.

, we get

is known, so (12) is scheme

is

The steps of iterative scheme as following: Step 1. Input A, B, C* and initial matrix ; Step 2. By the analysis in section 1 we may get (4) from (1); Step 3. Get Q by (6); - 59 www.ivypub.org/MC

expected,

the

choice

of

must

make


Step 4. Let A  QAQT X  QXQT , C  QCQT , B   B , we get (7); Step 5. If the coefficient matrices of (7) content the conditions in Theorem 2, choosing  by Theorem 2; (1) 1 1 Step 6. Compute X  ( x1 , x2 ,

, xn1 ) by (12);

Step 7. Let R(1)  AX (1)  X (1) A  BX (1) B  C , if contents (precision require), let otherwise, replace by and repeat Step 6, until the precision require is content; Q  ( xij )nn

T Step 8. Let X  Q

and

for

and

where

,

,

for

;

Step 9. Let X  X1  Y1 ; *

Step 10. Stop.

4 NUMERICAL EXAMPLE For given coefficient matrices of

finding the solution of it. Where

B=

with

and

,

,

.

It is learnt that this equation has the form of (7), and if is large enough , then and . Based on the analysis above, this equation has unique solution. Let initial matrix and by Matlab 6.5, we get the result (in Table 1), where number(s) means the iterative number and time(s) means the CPU time for given . In the table, it is found that when

and

, the convergence speed is close, but when

  0 , the speed becomes slow. TABLE 1 THE NUMBER(S) AND TIME(S) FOR DIFFERENT

n  10

n

WITH

R 2  1010 .

n  30

n  50

number(s)

time(s)

number(s)

time(s)

number(s)

time(s)

1 (1  n ) 2

3

0.0320

4

15.7190

4

306.6560

1

4

0.0470

4

15.6250

4

306.7650

n

3

0.0310

4

15.6100

4

306.0940

0

6

0.0630

6

23.4690

6

459.4380

5 CONCLUSIONS In this paper, we have discussed how to solve mixed-type Laypunov matrix equation. At first, anti-symmetric matrix solution has been transformed into symmetric matrix solution and then how to deal with the symmetric problem has - 60 www.ivypub.org/MC


been investigated. Further, an iterative scheme with parameter has been created and the way to select parameter was also illustrated. At the end, the efficiency of the iterative scheme has been verified by numerical example.

ACKNOWLEDGMENT This work is supported by the NSF of Xinjiang Uigur Autonomous Region (No.2013211B12).

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[10] Bitmead R, Weiss H. On the solution of the discrete-time Lyapunov matrix equation in controllable canonical form[J]. IEEE Trans. Automat. Control, 1979, 24: 481-482. [11] Bitmead R. Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman-Yakubovich equations[J] IEEE Trans. Automat. Control, 1981, 26: 1291-1294. [12] Barraud A. A numerical algorithm to solve

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[15] Mukaidani M, Xu H, Mizukami K. New iterative algorithm for algebraic Riccati equation related to H control problem of singularly perturbed systems[J]. IEEE Trans. Automat. Control, 2001, 46: 1659-1666. [16] Xu S F, Cheng M S. On the solvability for the mixed-Lyapunov equation[J]. IMA Journal of Applied Mathmatics, 2006, 71(2): 287-294

AUTHORS Xindong Zhang, Male, Han Nationality,

Juan Liu, Female, Han Nationality, PhD, Aassociate Professor,

PhD, Lecturer Current Research Interests:

Current Research Interests: Graph Theory and its Application.

Numerical solutions of partial differential

Study for doctoral degree in Xinjiang University from 2006.

equations. Study for doctoral degree in

Leilei Wei, Male, Han Nationality, PhD, Lecturer, Current

Xinjiang University from 2010.

Research Interests: Numerical solutions of partial differential equations. Study for doctoral degree in Xi’an Jiaotong University from 2009.

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The anti symmetric solution for the mixed type lyapunov matrix equation by parameter iterative metho