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

Abstract

This article is an overview of recent developments in the Input-to-State Stability framework, dealing in particular with the extension of the classical concept to systems with multiple invariant sets and possibly evolving on Riemannian manifolds. Lyapunov-based characterizations of the properties are discussed as well as applications to the study of cascaded nonlinear systems.

On robust stability of multistable passive systems

“…y T ϕ(y) ≥ ε|y| 2 , ∀y ∈ Y, ε > 0. (15) If the condition (15) is satisfied and the function β ∈ K∞ or |h(x)| ≥ ̺(|x|W ) for all x ∈ M, then the control (13) guarantees ISS property with respect for the set W and the disturbance input v, and V is an ISS-Lyapunov function.…”

“…Let a passive system (10), (11) admit Assumption 2. If the control (13) is applied under (15) and an additional restriction:…”

“…In fact, it was made clear in [2] that the most natural way of investigating stability properties for multistable systems is to relax the Lyapunov stability requirement on its invariant sets under relatively mild additional assumptions on the decomposition of such sets. This intuitive path has led to a new line of research which starts from the characterizations of inputto-state stability (ISS) for this class of systems in terms of usual Lyapunov dissipation inequalities, generalizing the classical ISS theory [33], [34], [6], [2], [15]. Formulated within the framework introduced in [2], the notion of detectability or output-to-state The project is supported by the ANR project WaQMoS (ANR 15 CE 04 0002).…”

Robust Global Synchronization of Brockett Oscillators

“…Following the results of [33], the authors in [45] have provided conditions for the robust synchronization of multistable systems in the presence of external inputs. Readers can consult [46] for an overview of recent developments in the ISS framework, dealing in particular with the extension of the classical concept to systems with multiple invariant sets and possibly evolving on Riemannian manifolds.…”

Universal formula for robust stabilization of affine nonlinear multistable systems

“…inequalities, generalizing the classical ISS theory [30], [31], [8], [3], [15]. In its turn, integral input-to-state stability (iISS) characterization, which is weaker than the classical one given in [29], [21], [5], was extended in [16] for systems with multiple invariant sets.…”