Input-to-state stability meets small-gain theory

TL;DR

This paper explores input-to-state stability (ISS), providing Lyapunov characterizations, superposition theorems, and applications to event-based control and network stability analysis, including small-gain theorems for finite and infinite systems.

Contribution

It offers new Lyapunov-based characterizations of ISS, superposition theorems, and extends small-gain theorems to infinite networks with ISS components.

Findings

Lyapunov characterizations for ISS

ISS superposition theorems relating to robustness

Small-gain theorems for finite and infinite networks

Abstract

Input-to-state stability (ISS) unifies global asymptotic stability with respect to variations of initial conditions with robustness with respect to external disturbances. First, we present Lyapunov characterizations for input-to-state stability as well as ISS superpositions theorems showing relations of ISS to other robust stability properties. Next, we present one of the characteristic applications of the ISS framework - the design of event-based control schemes for the stabilization of nonlinear systems. In the second half of the paper, we focus on small-gain theorems for stability analysis of finite and infinite networks with input-to-state stable components. First, we present a classical small-gain theorem in terms of trajectories for the feedback interconnection of 2 nonlinear systems. Finally, a recent Lyapunov-based small-gain result for a network with infinitely many ISS components is shown.