Characterizations of Integral Input-to-State Stability for Systems With Multiple Invariant Sets

Abstract

We extend the classical integral Input-to-State Stability (iISS) theory to systems evolving on complete Riemannian manifolds and admitting multiple disjoint invariant sets, so as to allow a much broader variety of dynamical behaviors of interest. Building upon a recent extension of the Input-to-State (ISS) theory for this same class of systems, we provide characterizations of the iISS concept in terms of dissipation inequalities and integral estimates as well as connections with the Strong iISS notion. Finally, we discuss some examples within the domain of mechanical systems.

The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview

“…Once again, the characterization will be given in terms of asymptotic properties and Lyapunov dissipation inequalities. The reader is referred to [22] for an exhaustive presentation of this subject.…”

“…Our generalized notion of ISS in [22] replaces the latter property with Assumption 1 and the 0-GATT property, and is formalized as follows.…”

“…for all x ∈ M and d ∈ D (Section C in [7] and Lemma 3 in [22]). Since ϖ might not be a proper function, dissipation at "larger" values of the state might not be able to properly counteract the injected energy.…”

“…A strong iISS system is in fact an iISS system which exhibits the ISS behavior whenever the magnitude of the input is less or equal to a certain threshold. In the following, we will be focusing on a sufficient condition for Strong iISS in the context of multistable systems [22].…”

“…In case of vanishing friction, i.e. b(ω) = bω/(1 + ω 2 ), the nonlinear pendulum is iISS and not Strong iISS (Section V A in [22]). In case of friction saturation, i.e.…”