About this Journal Submit a Manuscript Table of Contents
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

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 [13], robust control [1], robust stability [4], filter design [5], stability theory and analysis [68], 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 [1620] 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).

References

  1. M. Basin, J. Rodriguez-Gonzalez, and L. Fridman, “Optimal and robust control for linear state-delay systems,” Journal of the Franklin Institute, vol. 344, no. 6, pp. 830–845, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. Davies, P. Shi, and R. Wiltshire, “New upper solution bounds for perturbed continuous algebraic Riccati equations applied to automatic control,” Chaos, Solitons and Fractals, vol. 32, no. 2, pp. 487–495, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, NY, USA, 1972. View at MathSciNet
  4. S. S. Wang, B. S. Chen, and T. P. Lin, “Robust stability of uncertain time-delay systems,” International Journal of Control, vol. 46, no. 3, pp. 963–976, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Su, P. Shi, L. Wu, and Y. D. Song, “A novel approach to filter design for T-Sfuzzy discrete time systems with time-varying delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 6, pp. 1114–1129, 2012. View at Publisher · View at Google Scholar
  6. S. Barnett and C. Storey, Matrix Methods in Stability Theory, Barnes and Noble Inc., New York, NY, USA, 1970.
  7. T. M. Huang and W. W. Lin, “Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations,” Linear Algebra and Its Applications, vol. 430, no. 5-6, pp. 1452–1478, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 1, pp. 273–286, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. C. H. Lee, “Simple stabilizability criteria and memoryless state feedback control design for time-delay systems with time-varying perturbations,” IEEE Transactions on Circuits and Systems I, vol. 45, no. 11, pp. 1211–1215, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Mori and I. A. Derese, “A brief summary of the bounds on the solution of the algebraic matrix equations in control theory,” International Journal of Control, vol. 39, no. 2, pp. 247–256, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  11. K. Ogata, Modern Control Engineering, Prentice-Hall, Upper Saddle River, NJ, USA, 3rd edition, 1997.
  12. W. Zhang, H. Su, H. Wang, and Z. Han, “Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4968–4977, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. L. Ni, “Existence condition on solutions to the algebraic Riccati equation,” Acta Automatica Sinica, vol. 34, no. 1, pp. 85–87, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  14. W. H. Kwon, Y. S. Moon, and S. C. Ahn, “Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results,” International Journal of Control, vol. 64, no. 3, pp. 377–389, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. H. Lee, “New results for the bounds of the solution for the continuous Riccati and Lyapunov equations,” IEEE Transactions on Automatic Control, vol. 42, no. 1, pp. 118–123, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. C. Y. Chen and C. H. Lee, “Explicit matrix bounds of the solution for the continuous Riccati equation,” ICIC Express Letters, vol. 3, no. 2, pp. 147–152, 2009. View at Scopus
  17. S. W. Kim and P. G. Park, “Upper bounds of the continuous ARE solution,” IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, vol. 83, pp. 380–385, 2000.
  18. C. H. Lee, “Solution bounds of the continuous Riccati matrix equation,” IEEE Transactions on Automatic Control, vol. 48, no. 8, pp. 1409–1413, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Zhang and J. Liu, “Matrix bounds for the solution of the continuous algebraic Riccati equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 819064, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. Zhang, H. Su, and J. Wang, “Computation of upper bounds for the solution of continuous algebraic Riccati equations,” Circuits, Systems and Signal Processing, vol. 32, no. 3, pp. 1477–1488, 2013. View at Publisher · View at Google Scholar
  21. H. H. Choi and T. Y. Kuc, “Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations,” Automatica, vol. 38, no. 8, pp. 1147–1152, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. V. R. Karanam, “A note on eigenvalue bounds in algebraic Riccati equation,” IEEE Transactions on Automatic Control, vol. 28, no. 1, pp. 109–111, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. K. Yasuda and K. Hirai, “Upper and lower bounds on the solution of the algebraic Riccati equation,” IEEE Transactions on Automatic Control, vol. 24, no. 3, pp. 483–487, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. B. H. Kwon, M. J. Youn, and Z. Bien, “On bounds of the Riccati and Lyapunov matrix equations,” IEEE Transactions on Automatic Control, vol. 30, no. 11, pp. 1134–1135, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. D. Wang, T. S. Kuo, and C. F. Hsu, “Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation,” IEEE Transactions on Automatic Control, vol. 31, no. 7, pp. 654–656, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. V. Patel and M. Toda, “On norm bounds for algebraic Riccati and Lyapunov equations,” IEEE Transactions on Automatic Control, vol. 23, no. 1, pp. 87–88, 1978. View at Zentralblatt MATH · View at MathSciNet
  27. D. S. Bernstein, Matrix Mathematics: Theory, Facts and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, NJ, USA, 2005. View at MathSciNet
  28. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. View at MathSciNet
  29. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979. View at MathSciNet
  30. E. Kreindler and A. Jameson, “Conditions for nonnegativeness of partitioned matrices,” IEEE Transactions on Automatic Control, vol. 17, pp. 147–148, 1972. View at Zentralblatt MATH · View at MathSciNet