Journal of Applied Mathematics

Volume 2013 (2013), Article ID 792782, 9 pages

http://dx.doi.org/10.1155/2013/792782

Research Article

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

Zübeyde Ulukök and Ramazan Türkmen

Department of Mathematics, Science Faculty, Selçuk University, 42031 Konya, Turkey

Received 22 May 2013; Accepted 24 October 2013

Academic Editor: Baolin Wang

Copyright © 2013 Zübeyde Ulukök and Ramazan Türkmen. 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.

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.

1. Introduction

In many areas of optimal control [1–3], robust control [1], robust stability [4], filter design [5], stability theory and analysis [6–8], control design [9] in control theory [10, 11] including optimization stability theory, and transient performance performance nonlinear systems [12], the algebraic Riccati and Lyapunov matrix equations play an important role.

For example, consider the following linear system such that,,[13]:

with the state feedback control

and the performance index

where,is positive semidefinite matrix, andis the positive semidefinite solution to the continuous algebraic Riccati matrix equation (CARE)

Whenandis stable matrix, the CARE (4) becomes the continuous algebraic Lyapunov matrix equation (CALE)

It is assumed that the pairis stabilizable. Then the CARE (4) has a unique symmetric positive semidefinite stabilizing solution if the pairis 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, 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 [16–20] 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.

Letbe the set ofreal matrices. In this paper, we denote the eigenvalues of anreal matrix by; ifis a symmetric matrix, then its eigenvalues are arranged in the nonincreasing order. For, suppose that the singular values ofare ordered in nonincreasing form; that is,. Also, let,,, anddenote the trace, transpose, inverse, determinant, respectively. Additionally, the spectral condition number of any matrixis defined by. Write, ifis a positive semidefinite (positive definite) matrix. For the symmetric matrices of the same sizeand, ifis positive semidefinite, we writeor. Then, Weyl’s monotonicity principle means thatleads to,. The identity matrix inis shown by.

The following lemmas are used to prove the main result of this paper.

Lemma 1 (see [27, 28]). Letbe symmetric matrix. Then the following inequality holds:

Lemma 2 (see [27, 28]). For any matrixand any positive semidefinite matricessuch that, it holds that, with strict inequality ifandare positive definite andis of full rank.

Lemma 3 (see [27, 28]). For any symmetric matrices, the following inequality holds:

Lemma 4 (see [28]). Let, for, one has

Lemma 5 (see [28]). Let,, for, one has

Lemma 6 (see [29]). Let, for, then

Lemma 7 (see [30]). The following matrix inequality:

whereand, is equivalent to either

or

Lemma 8 (see [17]). The positive semidefinite solutionof the CARE (4) has the following upper bound on its maximal eigenvalue:

whereis any matrix stabilizing(i.e.,for all) and the nonsingular matrixand positive definite matrixare chosen to yield the LMI

This eigenvalue upper bound (14) is always calculated if there exists a unique positive semidefinite solution of the CARE (4).

2. 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 thatis symmetric positive definite and there exists a unique symmetric positive semidefinite solutionto the CARE (4). Thensatisfies the following inequality:

where the positive semidefinite matrixand the positive constantare defined by

whereis any positive constant such that

and positive constantis defined by

Proof. By adding and subtractingfrom (4), one gets

therefore,

Applying Lemmas 1 and 2 to (21) gives

For the partof (22), applying Lemmas 1, 6, and 5, respectively, shows that

Thus, in light of the fact (23), (22) becomes

Ifandsatisfies (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, (28) can be rewritten as

Utilizing the relations in Lemmas 1 and 3, (29) becomes

Solving (30) according togives

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

it is seen that the upper bound in Theorem 9 is always sharper than the result given by Theoremin [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.

Corollary 12. Assume thatis symmetric positive definite and there exists a unique symmetric positive semidefinite solutionto the CARE (4). Thensatisfies the following upper eigenvalue bounds:

wheresatisfies (18) andis defined by (16).

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. Letbe any symmetric positive definite matrix. Then the unique symmetric positive semidefinite solutionto the CARE (4) has the following upper bound

where the positive definite matrixis chosen so that

andis defined by (14).

Proof. By adding and subtractingto 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) ofand pre- and postmultiplyingto (40) yields

Solving this inequality forshows the upper bound (36).

This builds the proof.

Remark 14. Note that for the upper bound (36), the matricesanddon 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 matrixwhich arbitrarily is selected.

From Theorem 13, we have the following corollaries.

Corollary 15. The positive semidefinite solutionto the CARE (4) has

whereandfor the positive definite matrixare defined by (14) and (37), respectively.

Proof. Applying Lemma 1 to the right side of (41) and solving it with regard togive the upper bound.

Corollary 16. The solutionto the CARE (4) satisfies the following upper eigenvalue bounds:

whereis defined by (14) and the positive matrixis selected so as to satisfy the definition (37), respectively.

Theorem 17. Letbe the positive semidefinite solution of the CARE (4). Thenhas the upper bound

where the positive definite matrixis chosen so that

andis 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 ofin (47), since the following inequalities hold:

via the definition offrom (47), we arrive at

Applying Lemmas 1 and 3 to (49), we have

Then,

Solving (51) with respect togives

Substitutinginto (49), we get

Pre- and postmultiplyingto (53) leads to

Therefore, by the nonsingularity of, 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 solutionto the CARE (4) satisfies

where the positive definite matrices andand the positive constantare defined by (45) and (46), respectively.

Proof. Substitutinginto (50), having solved (50) regard to, we obtain the upper bound (55).

Corollary 19. The positive semidefinite solutionto the CARE (4) has the following eigenvalue upper bounds:

where the positive definite matricesandand the positive constantare defined by (45) and (46), respectively.

As considered a diverse matrix identity, in the case that the matrixis nonsingular, we can derive the following alternative upper bounds for the solution of the CARE (4).

Theorem 20. If the positive definite matrixis a unique solution matrix of the CARE (4), then

whereis a positive constant matrix such thatandis defined by (14).

Proof. When the termis added and subtracted from the CARE (4), we can write

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 termof (60), we can write

Therefore, if the above inequality is solved with respect to, we arrive at the upper bound.

Thus, the proof is established.

Theorem 21. Letbe the positive semidefinite solution of the CARE (4). Then

where the positive definite matrixis selected such that

and the nonnegative constantis 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, 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, one can reach the nonnegative constantis defined by (64). If it is substitutedinto (68), then

Thus, solving the inequality (70) derives the upper bound (62) for the solutionof the CARE (4).

This concludes the proof of the theorem.

Corollary 22. The solutionto the CARE (4) has the following eigenvalue bounds for:

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.

3. Numerical Example

In this section, we will give a numerical example to demonstrate the effectiveness of the proposed results of this paper.

Example 1. Consider the CARE (4) with

Choose, then using (16) shows the following upper matrix bound:

The upper bound (55) gives

with, and the upper bound (62) gives

with.

Using Theoremof [19], we obtain the following upper matrix bound:

The boundsandproposed in [18] are

for.

By a simple computation, we have

which means that our upper bounds give more precise solution estimates than the results given by Theoremin [19] and Theoremsandin [18] for this case.

4. 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.

Acknowledgments

The authors would like to thank the editor and the reviewers for the very helpful comments and suggestions to improve the presentation of this study. This study has been supported by the Coordinatorship of Selçuk University’s Scientific Research Projects (BAP) and The Scientific and Technical Research Council of Turkey (TUBITAK).

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