Lyapunov Conditions for Uniform Asymptotic Output Stability and a Relaxation of Barbalat's Lemma

TL;DR

This paper introduces new Lyapunov-based conditions and extends Barbalat's lemma to ensure uniform asymptotic output stability, addressing limitations of classical tools in control theory for finite and infinite-dimensional systems.

Contribution

It provides a testable sufficient condition for uniform convergence and relaxes the uniform continuity requirement in Barbalat's lemma, applicable to finite and infinite-dimensional systems.

Findings

Established conditions for uniform AOS in control systems.

Extended Barbalat's lemma to relax continuity assumptions.

Illustrated results with academic examples and adaptive control applications.

Abstract

Asymptotic output stability (AOS) is an interesting property when addressing control applications in which not all state variables are requested to converge to the origin. AOS is often established by invoking classical tools such as Barbashin-Krasovskii-LaSalle's invariance principle or Barbalat's lemma. Nevertheless, none of these tools allow to predict whether the output convergence is uniform on bounded sets of initial conditions, which may lead to practical issues related to convergence speed and robustness. The contribution of this paper is twofold. First, we provide a testable sufficient condition under which this uniform convergence holds. Second, we provide an extension of Barbalat's lemma, which relaxes the uniform continuity requirement. Both these results are first stated in a finite-dimensional context and then extended to infinite-dimensional systems. We provide academic examples to illustrate the usefulness of these results and show that they can be invoked to establish uniform AOS for systems under adaptive control.