Non-conservative discrete-time ISS small-gain conditions for closed sets

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Noroozi, Navid ; Geiselhart, Roman ; Grüne, Lars ; Rüffer, Björn ; Wirth, Fabian:
Non-conservative discrete-time ISS small-gain conditions for closed sets.
In: IEEE Transactions on Automatic Control. Bd. 63 (2018) Heft 5 . - S. 1231-1242.
ISSN 1558-2523
DOI: https://doi.org/10.1109/TAC.2017.2735194

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Abstract

This paper presents a unification and a generalization of the small-gain theory subsuming a wide range of existing small-gain theorems. In particular, we introduce small-gain conditions that are necessary and sufficient to ensure input-to-state stability (ISS) with respect to closed sets. Toward this end, we first develop a Lyapunov characterization of ω ISS via finite-step ω ISS Lyapunov functions. Then, we provide the small-gain conditions to guarantee ωISS of a network of systems. Finally, applications of our results to partial ISS, ISS of time-varying systems, synchronization problems, incremental stability, and distributed observers are given.