Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation

We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.

For example, consider the following linear system such that ∈ R × , ∈ R × , 0 ∈ R [13]: with the state feedback control and the performance index where = ∈ R × , ∈ R × is positive semidefinite matrix, and is the positive semidefinite solution to the continuous algebraic Riccati matrix equation (CARE) When = 0 and is stable matrix, the CARE (4) becomes the continuous algebraic Lyapunov matrix equation (CALE) It is assumed that the pair ( , 1/2 ) is stabilizable. Then the CARE (4) has a unique symmetric positive semidefinite stabilizing solution if the pair ( , 1/2 ) is observable.
The problem of estimating solution bounds for the algebraic Riccati and Lyapunov matrix equations has widely been considered in the recent years, since these equations are widely used in many fields of control system analysis and design. A number of works have reported numerical algorithms to get the exact solution of the mentioned equations [7]. However, we should note that the analytical solution of these equations has some complications and computational burdens, specially, when the dimensions of the system matrices increase. Thus, for some applications such as stability analysis [8], it is the only preferred solution matrix bounds for the exact solution that can be obtained without hard and complicated burdens. Moreover, as mentioned in [12], in practice, the solution matrix bounds can also be used as approximations of the exact solution or initial guesses in the numerical algorithms for the exact solution [10].
The existing results obtained during 1974-1994 have been summarized by Kwon et al. [14] only including all eigenvalue bounds such as the extreme eigenvalues, the summation, 2 Journal of Applied Mathematics the trace, majorization inequalities, the product, and the determinant. Unfortunately, by this time, the upper matrix bounds for the solution of the CARE (4) have not been proposed in the literature. However, Lee in [15] has proposed upper and lower matrix bounds for the CARE (4) and henceforth many reports have been presented for the upper [16][17][18][19][20] and lower [18,19,21] bounds for the solution of the CARE (4). As matrix bounds include all eigenvalue bounds [14,22,23] particularly the minimum and maximum eigenvalues, trace [10,24,25], determinant [14], and norm [26] bounds, it is seen that they are the most general and useful. Therefore, this paper presents upper matrix bounds for the solution of the CARE (4) by utilizing various matrix identities and matrix inequalities.
Let R × be the set of × real matrices. In this paper, we denote the eigenvalues of an × real matrix by ( ); if ∈ R × is a symmetric matrix, then its eigenvalues are arranged in the nonincreasing order 1 ( ) ≥ 2 ( ) ≥ ⋅ ⋅ ⋅ ≥ ( ).

For
∈ R × , suppose that the singular values of are ordered in nonincreasing form; that is, 1 ( ) ≥ 2 ( ) ≥ ⋅ ⋅ ⋅ ≥ ( ). Also, let tr( ), , −1 , and det( ) denote the trace, transpose, inverse, determinant, respectively. Additionally, the spectral condition number of any matrix is defined by if is a positive semidefinite (positive definite) matrix. For the symmetric matrices of the same size and , if − is positive semidefinite, we write ≥ or ≤ . Then, Weyl's monotonicity principle means that ≤ leads to ( ) ≤ ( ), = 1, 2, . . . , . The identity matrix in R × is shown by .
The following lemmas are used to prove the main result of this paper.

Main Results
Zhang and Liu in [19] obtained the lower and upper bounds for the solution of the CARE (4) which improve the results in [21]. Also, Lee in [18] proposed upper and lower bounds for the solution of the CARE (4) by considering a different approach. In this section, we will present diverse upper matrix bounds for the solution matrix of the CARE (4) in the light of the reported results in [18,19], by utilizing the above lemmas and generating some matrix identities.

Theorem 9.
Assume that is symmetric positive definite and there exists a unique symmetric positive semidefinite solution to the CARE (4). Then satisfies the following inequality: where the positive semidefinite matrix 1 and the positive constant are defined by where is any positive constant such that and positive constant 1 is defined by Proof. By adding and subtracting (1/ ) therefore, Applying Lemmas 1 and 2 to (21) gives For the part (22), applying Lemmas 1, 6, and 5, respectively, shows that Thus, in light of the fact (23), (22) becomes If > 0 and satisfies (18), then By the application of the Schur complement formula of Lemma 7 to (25), we can say that the above inequalities are satisfied if and only if which means that Therefore, we say that (24) is equivalent to Since 1 − 1 ( − ) > 0, (28) can be rewritten as 4

Journal of Applied Mathematics
Utilizing the relations in Lemmas 1 and 3, (29) becomes Solving (30) according to 1 ( ) gives Substituting (31) into (29) results in the upper bound This completes the proof.
Remark 10. The inequality (3.5) in [19] is clearly as follows: Thus, when the inequality (28) is considered, from the facts 2 ≤ 2 1 ( ) , it is seen that the upper bound in Theorem 9 is always sharper than the result given by Theorem 3.1 in [19].
Remark 11. It is well known that most of the studies in the literature have focused to derive the bounds for the maximum and minimum eigenvalues, the trace, and the determinant for the solution of the CARE (4); however, the matrix solution bounds are quite restriction. Among the mentioned bounds, the matrix solution bounds are the most useful and efficient because other bounds that are dependent on eigenvalue can be derived directly from matrix solution bounds via monotonicity.
By using Theorem 9, we can derive the following result immediately.
By establishing the more general form than the matrix identity used in Theorem 9 for the CARE (4), one gets the following upper bounds.

Theorem 13. Let be any symmetric positive definite matrix. Then the unique symmetric positive semidefinite solution to the CARE (4) has the following upper bound
where the positive definite matrix 1 is chosen so that and is defined by (14). to the CARE (4), we can get which is equivalent to Introducing Lemmas 1, 2, 4, 5, and 8, respectively, to (39) gives By the definition (37) of 1 and pre-and postmultiplying 1 to (40) yields Solving this inequality for shows the upper bound (36). This builds the proof.

Remark 14.
Note that for the upper bound (36), the matrices and don not have to be nonsingular. This means that the upper bound proposed by Theorem 13 can always be computed without any condition for positive definite matrix 1 which arbitrarily is selected.
From Theorem 13, we have the following corollaries.

Corollary 15. The positive semidefinite solution to the CARE (4) has
where and 1 for the positive definite matrix 1 are defined by (14) and (37), respectively.
Proof. Applying Lemma 1 to the right side of (41) and solving it with regard to give the upper bound 1 2 .

Theorem 17. Let be the positive semidefinite solution of the CARE (4). Then has the upper bound
where the positive definite matrix 2 is chosen so that and 2 is defined by Proof. By the use of the equality (39), from Lemmas 1 and 2, we can write Having applied Lemmas 1, 6, and 5, respectively, to the part of 2 + 2 in (47), since the following inequalities hold: Journal of Applied Mathematics via the definition of 2 from (47), we arrive at (49) Applying Lemmas 1 and 3 to (49), we have Then, Solving (51) with respect to 1 ( ) gives Substituting 2 into (49), we get Pre-and postmultiplying 1/2 2 to (53) leads to Therefore, by the nonsingularity of 2 , the upper matrix bound (44) is directly obtained by solving (54) with respect to . The proof is finished.
According to Theorem 17, we can propose the following corollaries.

Corollary 18. The positive semidefinite solution to the CARE (4) satisfies
where the positive definite matrices 2 , and 2 and the positive constant 2 are defined by (45) and (46), respectively.

Corollary 19.
The positive semidefinite solution to the CARE (4) has the following eigenvalue upper bounds: where the positive definite matrices 2 and 2 and the positive constant 2 are defined by (45) and (46), respectively.
As considered a diverse matrix identity, in the case that the matrix is nonsingular, we can derive the following alternative upper bounds for the solution of the CARE (4). Theorem 20. If the positive definite matrix is a unique solution matrix of the CARE (4), then where 3 is a positive constant matrix such that 3 ≡ + −1 3 > 0 and is defined by (14).
Proof. When the term −1 3 + 3 is added and subtracted from the CARE (4), we can write Journal of Applied Mathematics 7 which is equivalent to By the use of Lemmas 1, 2, 4, and 8 for the right side of the above equation, respectively, we obtain and by the application of Lemma 1 to the term ( + −1 3 ) of (60), we can write Therefore, if the above inequality is solved with respect to , we arrive at the upper bound 4 . Thus, the proof is established.

Theorem 21.
Let be the positive semidefinite solution of the CARE (4). Then where the positive definite matrix 4 is selected such that and the nonnegative constant 3 is defined by Proof. Consider (58). From Lemma 1, we can easily write and then via the inequality obtained by using Lemmas 1, 6, and 5, respectively, and the definition (63) of 4 , from (65), we have By the use of Lemmas 1 and 2, it is obtained that and thus applying Lemma 3 to (68) yields As solving (69) according to 1 ( ), one can reach the nonnegative constant 3 is defined by (64). If it is substituted 3 into (68), then Thus, solving the inequality (70) derives the upper bound (62) for the solution of the CARE (4). This concludes the proof of the theorem.

Corollary 22.
The solution to the CARE (4) has the following eigenvalue bounds for = 4, 5: Remark 23. Chen and Lee in [16] indicated in it is hard or impossible to determine the best matrix bound among the parallel results. Since we find that it is difficult to compare the tightness of our results to the parallel result in [18], we will only make the comparisons on an example.

Numerical Example
In this section, we will give a numerical example to demonstrate the effectiveness of the proposed results of this paper.
The bounds (18) and (23) which means that our upper bounds give more precise solution estimates than the results given by Theorem 3.1 in [19] and Theorems 2 and 3 in [18] for this case.

Conclusion
In this paper, new upper matrix bounds for the solution of the CARE are improved by using some linear algebraic techniques and matrix inequalities. A numerical example is given to show that the solution upper bounds presented in this paper are sharper than some results in the literature.