Original Article

Mathematics of Control, Signals, and Systems

, Volume 24, Issue 1, pp 33-54

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Input-to-state stability analysis for interconnected difference equations with delay

  • Rob H. GielenAffiliated withElectrical Engineering Department, Eindhoven University of Technology Email author 
  • , Mircea LazarAffiliated withElectrical Engineering Department, Eindhoven University of Technology
  • , Andrew R. TeelAffiliated withElectrical and Computer Engineering Department, University of California


Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.


Large-scale systems Time delay Difference equations Lyapunov methods Small-gain theorem