Positivity of Complete Quadratic Lyapunov-Krasovskii Functionals in Time-Delay Systems

  • Keqin Gu
  • Yashun Zhang
  • Matthew Peet
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)


This chapter discusses positivity of quadratic functionals that arise in the stability analysis of time-delay systems. When both the single and double integral terms are positive, a necessary and sufficient condition for positivity is obtained using operator theory. This is applied to the Lyapunov-Krasovskii functional and its derivative. The coupled differential-difference equations are studied using the Sum-of-Squares (SOS) method.


Polynomial Matrice Quadratic Inequality Quadratic Functional Small State Space Continuous Matrix Function 
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  1. 1.
    Gu, K.: Discretized LMI set in the stability problem of linear uncertain time-delay systems. Int. J. Control 68(4), 923–924 (1997)zbMATHCrossRefGoogle Scholar
  2. 2.
    Gu, K.: Stability problem of systems with multiple delay channels. Automatica 46, 727–735 (2010)CrossRefGoogle Scholar
  3. 3.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, New York (1966)Google Scholar
  4. 4.
    Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Translated and edited by R. A. Silverman. Dover, New York (1975)Google Scholar
  5. 5.
    Li, H., Gu, K.: Discretized Lyapunov Krasovskii functional for coupled differential difference equations with multiple delay channels. Automatica (2010), doi:10.1016/j.automatica.2010.02.007Google Scholar
  6. 6.
    Peet, M.M., Papachristodoulou, A.: Positive forms and the stability of linear time-delay systems. In: Proc. 45th IEEE Conf. Decision Control, San Diego, CA, USA, December 13–15 (2006)Google Scholar
  7. 7.
    Peet, M.M., Papachristodoulou, A.: Positive forms and stability of linear time-delay systems. SIAM J. Control Optim. 47, 3237–3258 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proc. 41th IEEE Conf. Decision Control, Las Vegas, USA, December 10–13 (2002)Google Scholar
  9. 9.
    Rǎsvan, V.: Functional differential equations of lossless propagation and almost linear behavior. Plenary Lecture. In: 6th IFAC Workshop on Time-Delay Systems, L’Aquila, Italy, July 10–12 (2006)Google Scholar
  10. 10.
    Repin, Y.M.: Quadratic Lyapunov Functionals for Systems With Delay (Russian) Prikl. Mat. Meh. 29, 564–566 (1965)MathSciNetGoogle Scholar
  11. 11.
    Zhang, Y., Peet, M.M., Gu, K.: Reducing the Complexity of the Sum-of-Squares Test for Stability of Delayed Linear Systems. IEEE Transactions on Automatic Control 56(1), 229–234 (2011)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringSouthern Illinois University EdwardsvilleEdwardsvilleUSA
  2. 2.School of AutomationNanjing University of Science and TechnologyNanjingChina
  3. 3.Department of Mechanical, Materials, and Aerospace EngineeringIllinois Institute of TechnologyChicagoUSA

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