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Lyapunov Functionals and Matrices for Neutral Type Time Delay Systems

  • Vladimir L. Kharitonov
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)

Abstract

In this chapter we present some basic results concerning the computation of quadratic functionals with prescribed time derivatives for linear neutral type time delay systems. The functionals are defined by special matrix valued functions. These functions are called Lyapunov matrices. Basic results with respect to the existence and uniqueness of the matrices are included. Some important applications of the functionals and matrices are pointed out. A brief historical survey ends the chapter.

Keywords

Delay System Functional Differential Equation Time Delay System Exponential Estimate Special Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York (1963)zbMATHGoogle Scholar
  2. 2.
    Castelan, W.B., Infante, E.F.: A Liapunov functional for a matrix neutral difference-differential equation with one delay. Journal of Mathematical Analysis and Applications 71, 105–130 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gorecki, H., Fuksa, S., Grabowski, P., Korytowski, A.: Analysis and Synthesis of Time-Delay Systems. John Willey and Sons (PWN), Warsaw (1989)zbMATHGoogle Scholar
  4. 4.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)zbMATHGoogle Scholar
  5. 5.
    Jarlebring, E., Vanbiervliet, J., Michiels, W.: Characterizing and computing the H_2 norm of time delay systems by solving the delay Lyapunov equation. In: Proceedings 49th IEEE Conference on Decision and Control (2010)Google Scholar
  6. 6.
    Kharitonov, V.L.: Lyapunov functionals and Lyapunov matrices for neutral type time-delay systems: a single delay case. International Journal of Control 78, 783–800 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kharitonov, V.L.: Lyapunov matrices for a class of neutral type time delay systems. International Journal of Control 81, 883–893 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kharitonov, V.L.: Lyapunov matrices: Existence and uniqueness issues. Automatica 46, 1725–1729 (2010)zbMATHCrossRefGoogle Scholar
  9. 9.
    Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic Press, New York (1986)zbMATHGoogle Scholar
  10. 10.
    Louisell, J.: A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Transactions on Automatic Control 46, 2008–2012 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Marshall, J.E., Gorecki, H., Korytowski, A., Walton, K.: Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood, New York (1992)zbMATHGoogle Scholar
  12. 12.
    Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  13. 13.
    Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Velazquez-Velazquez, J., Kharitonov, V.L.: Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Systems and Control Letters 58, 17–25 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of Applied Mathematics and Control ProcessesSt.-Petersburg State UniversitySt.-PetersburgRussia

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