Transverse exponential stability and applications

TL;DR

This paper explores the relationships between different notions of transverse exponential stability, introduces conditions for their equivalence, and applies these concepts to nonlinear observer design and synchronization with exponential convergence.

Contribution

It establishes the equivalence of various transverse stability properties and provides necessary and sufficient conditions for designing nonlinear observers and synchronizers with exponential convergence.

Findings

Transverse exponential stability properties are equivalent under certain conditions.

Necessary and sufficient conditions for exponential convergence in nonlinear observer design.

Conditions for exponential synchronization in nonlinear systems.

Abstract

We investigate how the following properties are related to each other: i)-A manifold is "transversally" exponentially stable; ii)-The "transverse" linearization along any solution in the manifold is exponentially stable; iii)-There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property.