Input-to-state stability meets small-gain theory
Andrii Mironchenko
TL;DR
This paper unifies Input-to-State Stability (ISS) with small-gain theory for nonlinear ODE networks. It develops ISS Lyapunov characterizations and ISS superposition results, and demonstrates how ISS facilitates the robust, event-based stabilization of nonlinear systems. It then extends small-gain analysis from a pair of ISS subsystems to finite networks and to countably infinite networks, providing explicit Lyapunov constructions (via a gain operator and a path of strict decay) that certify ISS of large interconnected systems. The work delivers practical tools for designing robust controllers and observers in networked settings, including efficient event-triggered control schemes and scalable stability criteria for both 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.