Skip to main content

On the estimation of upper bound for solutions of perturbed discrete Lyapunov equations

Abstract

The estimation of the positive definite solutions to perturbed discrete Lyapunov equations is discussed. Several upper bounds of the positive definite solutions are obtained when the perturbation parameters are norm-bounded uncertain. In the derivation of the bounds, one only needs to deal with eigenvalues of matrices and linear matrix inequalities, and thus avoids solving high-order algebraic equations. A numerical example is presented.

[1234567]

References

  1. 1.

    Amir-Moéz AR: Extreme properties of eigenvalues of a hermitian transformation and singular values of the sum and product of linear transformations. Duke Mathematical Journal 1956, 23: 463–476. 10.1215/S0012-7094-56-02343-2

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985:xiii+561.

    Google Scholar 

  3. 3.

    Kwon WH, Moon YS, Ahn SC: Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results. International Journal of Control 1996,64(3):377–389. 10.1080/00207179608921634

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Lu LZ: Matrix bounds and simple iterations for positive semidefinite solutions to discrete-time algebraic Riccati equations. Journal of Xiamen University. Natural Science 1995,34(4):512–516.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Moheimani SOR, Petersen IR: Optimal quadratic guaranteed cost control of a class of uncertaintime-delay systems. IEE Proceedings-Control Theory and Applications 1997,144(2):183–188. 10.1049/ip-cta:19970844

    Article  MATH  Google Scholar 

  6. 6.

    Xie L: Output feedbackcontrol of systems with parameter uncertainty. International Journal of Control 1996,63(4):741–750. 10.1080/00207179608921866

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Xu J-H, Skelton RE, Zhu G: Upper and lower covariance bounds for perturbed linear systems. IEEE Transactions on Automatic Control 1990,35(8):944–948. 10.1109/9.58509

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dong-Yan Chen.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Chen, DY., Wang, DY. On the estimation of upper bound for solutions of perturbed discrete Lyapunov equations. J Inequal Appl 2006, 58931 (2006). https://doi.org/10.1155/JIA/2006/58931

Download citation

Keywords

  • Algebraic Equation
  • Linear Matrix Inequality
  • Linear Matrix
  • Perturbation Parameter
  • Lyapunov Equation
\