Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Banks, H.T.: Necessary conditions for control problems with variable time lags. SIAM J. Contr. 6(1), 9–47 (1968)
Fridman, E., Shaked, U.: An Improved Stabilization Method for Linear Time-Delay Systems. IEEE Transactions on Automatic Control 47(11), 1931–1937 (2002)
Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers (1992)
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)
Verriest, E.I.: Stability of Systems with Distributed Delays. In: Proceedings IFAC Workshop on System Structure and Control, Nantes, France, pp. 294–299 (1995)
Verriest, E.I.: Stability of Systems with State-Dependent and Random Delays. IMA Journal of Mathematical Control and Information 19, 103–114 (2002)
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)
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)
Verriest, E.I.: Inconsistencies in systems with time varying delays and their resolution. To appear: IMA Journal of Mathematical Control and Information (2011)
Yamamoto, Y.: Minimal representations for delay systems. In: Proc. 17-th IFAC World Congress, Seoul, KR, pp. 1249–1254 (2008)
Willems, J.C.: The behavioral approach to open and interconnected system. IEEE Control Systems Magazine 27(6), 46–99 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag GmbH Berlin Heidelberg
About this chapter
Cite this chapter
Verriest, E.I. (2012). State Space for Time Varying Delay. In: Sipahi, R., Vyhlídal, T., Niculescu, SI., Pepe, P. (eds) Time Delay Systems: Methods, Applications and New Trends. Lecture Notes in Control and Information Sciences, vol 423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25221-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-25221-1_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25220-4
Online ISBN: 978-3-642-25221-1
eBook Packages: EngineeringEngineering (R0)