Set input-to-state stability under input delays for nonlinear systems with disturbances

TL;DR

This paper develops a Razumikhin-type theorem using nonlinear small-gain theory to ensure set input-to-state stability for nonlinear systems with input delays and disturbances, broadening robustness results.

Contribution

It introduces a new theoretical framework for set ISS under input delays, removing previous constraints and retaining ISS gains, with practical case studies demonstrating effectiveness.

Findings

Proves a Razumikhin-type theorem for set ISS with delays.

Ensures robustness of nonlinear systems with input delays.

Validates results through nonlinear oscillator case studies.

Abstract

The study proposes new results on the set input-to-state stability (ISS) subject to a small input time delay for compact, invariant sets that contains the origin. First, using the nonlinear small-gain theory, we prove a Razumikhin-type theorem that ensures ISS for sets in the setup of functional differential equations (FDEs) with disturbances. Next we show how this theorem can be used to ensure set ISS for nonlinear systems with input delays and disturbances. In comparison to the existing research on set ISS robustness with respect to small time delays at the input, our results are rather broad, retaining the ISS gain and without any constraints on time delayed states. The effectiveness of the method is illustrated through two case-studies addressing set stability for classes of nonlinear oscillators of practical interest.