State Space for Time Varying Delay

  • Erik I. Verriest
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)


The construction of a state space for systems with time variant delay is analyzed. We show that under causality and consistency constraints a state space can be derived, but fails if the conditions are not satisfied. We rederive a known result on spectral reachability using an discretization approach followed by taking limits. It is also shown that when a system with fixed delay is modeled as one in a class with larger delay, reachability can no longer be preserved. This has repercussions in modeling systems with bounded time varying delay by embedding them in the class of delay systems with fixed delay, equal to the maximum of the delay function τ(t), or by using lossless causalization.


State Space Delay System Functional Differential Equation Reachability Condition Large Delay 
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  1. 1.
    Banks, H.T.: Necessary conditions for control problems with variable time lags. SIAM J. Contr. 6(1), 9–47 (1968)zbMATHCrossRefGoogle Scholar
  2. 2.
    Fridman, E., Shaked, U.: An Improved Stabilization Method for Linear Time-Delay Systems. IEEE Transactions on Automatic Control 47(11), 1931–1937 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers (1992)Google Scholar
  4. 4.
    Verriest, E.I.: Robust Stability of Time-Varying Systems with Unknown Bounded Delays. In: Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, FL, pp. 417–422 (1994)Google Scholar
  5. 5.
    Verriest, E.I.: Stability of Systems with Distributed Delays. In: Proceedings IFAC Workshop on System Structure and Control, Nantes, France, pp. 294–299 (1995)Google Scholar
  6. 6.
    Verriest, E.I.: Stability of Systems with State-Dependent and Random Delays. IMA Journal of Mathematical Control and Information 19, 103–114 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Verriest, E.I.: Causal Behavior of Switched Delay Systems as Multi-Mode Multi-Dimensional Systems. In: Proceedings of the 8-th IFAC International Symposium on Time-Delay Systems, Sinaia, Romania (2009)Google Scholar
  8. 8.
    Verriest, E.I.: Well-Posedness of Problems involving Time-Varying Delays. In: Proceedings of the 18-th International Symposium on Mathematical Theory of Networks and Systems, Budapest, Hungary, pp. 1985–1988 (2010)Google Scholar
  9. 9.
    Verriest, E.I.: Inconsistencies in systems with time varying delays and their resolution. To appear: IMA Journal of Mathematical Control and Information (2011)Google Scholar
  10. 10.
    Yamamoto, Y.: Minimal representations for delay systems. In: Proc. 17-th IFAC World Congress, Seoul, KR, pp. 1249–1254 (2008)Google Scholar
  11. 11.
    Willems, J.C.: The behavioral approach to open and interconnected system. IEEE Control Systems Magazine 27(6), 46–99 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlantaUSA

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