Small-Gain Stability Conditions

  • Jing ZhuEmail author
  • Tian Qi
  • Dan Ma
  • Jie Chen
Part of the Advances in Delays and Dynamics book series (ADVSDD, volume 8)


In this chapter, we show that a variety of stability conditions, both existing and new, can be derived for linear systems subject to time-varying delays in a unified manner in the form of scaled small-gain conditions. From a robust control perspective, the development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis, requiring essentially no more than a straightforward application of the small-gain theorem. The conditions are conceptually appealing and can be checked using standard robust control toolboxes. Both the \(\mathcal {L}_2\)- and \(\mathcal {L}_\infty \)-stability criteria are presented.


Small Gain Condition Time-varying Delays Robust Stability Analysis Robust Control Theory Frequency-dependent Scaling 
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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.College of Automation EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.Center for Control and OptimizationSouth China University of TechnologyGuangzhouChina
  3. 3.College of Information Science and EngineeringNortheastern UniversityShenyangChina
  4. 4.Department of Electronic EngineeringCity University of Hong KongHong KongHong Kong

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