Mathematics of Control, Signals, and Systems
, Volume 24, Issue 1, pp 3354
First online:
Inputtostate 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
Abstract
Inputtostate 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 smallgain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delayindependent smallgain condition holds. Furthermore, also using smallgain 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 largescale interconnections of difference equations with delay.
Keywords
Largescale systems Time delay Difference equations Lyapunov methods Smallgain theorem Title
 Inputtostate stability analysis for interconnected difference equations with delay
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Mathematics of Control, Signals, and Systems
Volume 24, Issue 12 , pp 3354
 Cover Date
 201204
 DOI
 10.1007/s0049801200804
 Print ISSN
 09324194
 Online ISSN
 1435568X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Largescale systems
 Time delay
 Difference equations
 Lyapunov methods
 Smallgain theorem
 Industry Sectors
 Authors

 Rob H. Gielen ^{(1)}
 Mircea Lazar ^{(1)}
 Andrew R. Teel ^{(2)}
 Author Affiliations

 1. Electrical Engineering Department, Eindhoven University of Technology, Eindhoven, The Netherlands
 2. Electrical and Computer Engineering Department, University of California, Santa Barbara, USA