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Set-Induced Stability Results for Delay Difference Equations

  • Rob H. GielenEmail author
  • Mircea Lazar
  • Sorin Olaru
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)

Abstract

This chapter focuses on the relation between stability of delay difference equations (DDEs) and the existence of \(\mathcal{D}\)-contractive sets. Such sets are of importance as they provide a region of attraction, which is difficult to obtain for delay systems. Firstly, it is established that a DDE admits a \(\mathcal{D}\)-contractive set if and only if it admits a Lyapunov-Razumikhin function. However, it is also shown that there exist stable DDEs that do not admit a \(\mathcal{D}\)-contractive set. Therefore, secondly, further necessary conditions for the existence of a \(\mathcal{D}\)-contractive set are established. These necessary conditions provide a first step towards the derivation of a notion of asymptotic stability for DDEs which is equivalent to the existence of a \(\mathcal{D}\)-contractive set.

Keywords

Lyapunov Function Asymptotic Stability Model Predictive Control Augmented System Algebraic Condition 
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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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Automatic Control Department and DISCO - INRIASUPELEC Systems Sciences (E3S)Gif-sur-YvetteFrance

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