- Research Article
- Open Access
Trace Inequalities for Matrix Products and Trace Bounds for the Solution of the Algebraic Riccati Equations
© Jianzhou Liu et al. 2009
- Received: 25 February 2009
- Accepted: 6 November 2009
- Published: 10 November 2009
By using diagonalizable matrix decomposition and majorization inequalities, we propose new trace bounds for the product of two real square matrices in which one is diagonalizable. These bounds improve and extend the previous results. Furthermore, we give some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions. Finally, numerical examples have illustrated that our results are effective and superior.
- System Matrice
- Unit Sphere
- Matrix Product
- Riccati Equation
- Symmetric Case
As we all know, the Riccati equations are of great importance in both theory and practice in the analysis and design of controllers and filters for linear dynamical systems (see [1–5]). For example, consider the following linear system (see ):
with the cost
The optimal control rate the optimal cost of (1.1) and (1.2) are
where is the initial state of system (1.1) and (1.2) and is the positive semidefinite solution of the following algebraic Riccati equation (ARE):
with and being positive definite and positive semidefinite matrices, respectively. To guarantee the existence of the positive definite solution to (1.4), we will make the following assumptions: the pair is stabilizable, and the pair is observable.
In practice, it is hard to solve the ARE, and there is no general method unless the system matrices are special and there are some methods and algorithms to solve (1.4); however, the solution can be time-consuming and computationally difficult, particularly as the dimensions of the system matrices increase. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of the ARE (see [6–16]). Moreover, in terms of [2, 6], we know that an interpretation of is that is the average value of the optimal cost as varies over the surface of a unit sphere. Therefore, considering its applications, it is important to discuss trace bounds for the product of two matrices. In symmetric case, a number of works have been proposed for the trace of matrix products ([2, 6–8, 17–20]), and  is the tightest among the parallel results.
In 1995, Lasserre showed  the following given any matrix then the following.
This paper is organized as follows. In Section 2, we propose new trace bounds for the product of two general matrices. The new trace bounds improve the previous results. Then, we present some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions in Section 3. In Section 4, we give numerical examples to demonstrate the effectiveness of our results. Finally, we get conclusions in Section 5.
In the following, let denote the set of real matrices and let denote the subset of consisting of symmetric matrices. For , we assume that , denote the trace, the inverse, the transpose, the diagonal elements, the singular values of , respectively, and define . If is an arbitrary symmetric matrix, then and denote the eigenvalues and the real part of eigenvalues of . Suppose is a real -element array such as which is reordered, and its elements are arranged in nonincreasing order; that is, . The notation ( ) is used to denote that is a symmetric positive definite (semidefinite) matrix.
Let be two real -element arrays. If they satisfy
then it is said that is controlled weakly by , which is signed by .
then it is said that is controlled by , which is signed by .
The following lemmas are used to prove the main results.
Lemma 2.1 (see [21, Page 92, H.2.c]).
Lemma 2.2 (see [21, Page 218, B.1]).
Lemma 2.3 (see [21, Page 240, F.4.a]).
Lemma 2.4 (see ).
Note that if or , obviously, (2.6) holds. If , choose , then (2.6) also holds.
( ) If , , for , choose , then and . Thus, is a convex function. As and , from the property of the convex function, we have
( ) If , without loss of generality, we may assume . Then from ( ), we have
This completes the proof.
Since applying (2.18) with in lieu of we immediately have the following corollary.
This completes the proof.
Note that if , , then from (2.34) we obtain (1.5) immediately. This implies that (2.18) improves (1.5).
Since applying (2.31) with in lieu of we immediately have the following corollary.
Komaroff (1994) in  obtained the following. Let be the positive semidefinite solution of the ARE (1.4). Then the trace of has the upper bound given by
In this section, by appling our new trace bounds in Section 2, we obtain some lower trace bounds for the solution of the algebraic Riccati equations. Furthermore, we obtain some upper trace bounds which improve (3.1) under certain conditions.
If , and is the positive semidefinite solution of the ARE (1.4).
where denotes and denotes
Solving (3.13) for yields the left-hand side of (3.2).
Since are positive semidefiniteness, is positive definiteness, then by (1.5), note that for , , and considering (2.6), (3.8), and (3.10), we have
Solving (3.15) for yields the right-hand side of (3.2).
( ) Note that when , by (1.5), (2.6), and (3.8), we have
Solving (3.17) for yields the left-hand side of (3.3).
By (1.5), (2.6), and (3.8), we have
Solving (3.19) for yields the right-hand side of (3.3).
( ) Note that when , by (3.3), we obtain (3.4) immediately. This completes the proof.
From Remark 2.6 and Theorem 3.1, we have the following results.
( ) Let in (3.2), then we have
( ) Let in (3.3), then we obtain (3.20).
( ) Let in (3.4). Note that when ,
Then we can also obtain (3.20).
Note that the right-hand side of (3.20) is (3.1), which implies that Theorem 3.1 improves (3.1).
In this section, firstly, we will give an example to illustrate that our new trace bounds are better than those of the recent results. Then, to illustrate that the application in the algebraic Riccati equations of our results will have different superiority if we choose different and , we will give two examples.
where both lower and upper bounds are better than those of the main result of , that is, (1.5).
where the upper bound is better than that of the main result of , that is, (3.1).
where the lower and upper bounds are better than those of (4.8).
In this paper, we have proposed lower and upper bounds for the trace of the product of two real square matrices in which one is diagonalizable. We have shown that our bounds for the trace are the tightest among the parallel trace bounds in symmetric case. Then, we have obtained some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions. Finally, numerical examples have illustrated that our bounds are better than those of the previous results.
The authors would like to thank Professor Jozef Banas and the referees for the very helpful comments and suggestions to improve the contents and presentation of this paper. The work was also supported in part by the National Natural Science Foundation of China (10971176).
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