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: email@example.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
LMEs have played a fundamental role in numerous problems in control, communication systems theory and power systems. They arise naturally in optimal control theory , stability analysis of dynamical systems , 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  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  , 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 . For instance, Starke and Niethammer presented an iterative method for the solutions of CT Sylvester equations by using the SOR (successive over relaxation) technique , and Mukaidani et al. discussed an iterative algorithm for generalized algebraic Lyapunov equations . MTLME and its solvability have been studied by Xu et al . 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
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 ,
AT X XA BT XB C , T T A Y YA B YB C ,
By Eq. (2) and Eq. (3), we can obtain
(the set of all have the following forms:
and the forms of symmetric matrices).
are determined by
. It is easy to know that
(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 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 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)
Substituting (6) into (5), we get: where
So the following equation should be directly discussed: (7)
. 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
Let expressed as:
, 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
the maximum and minimum eigenelement of and
, respectively. If parameter
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
, we get that
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.
convergence iterative as little as possible.
, we get
is known, so (12) is scheme
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
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 )nn
T Step 8. Let X Q
Step 9. Let X X1 Y1 ; *
Step 10. Stop.
4 NUMERICAL EXAMPLE For given coefficient matrices of
finding the solution of it. Where
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
, the convergence speed is close, but when
0 , the speed becomes slow. TABLE 1 THE NUMBER(S) AND TIME(S) FOR DIFFERENT
R 2 1010 .
1 (1 n ) 2
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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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 Xiâ€™an Jiaotong University from 2009.
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Published on Oct 16, 2013