Observers for a non-Lipschitz triangular form

TL;DR

This paper develops and analyzes observer designs for non-Lipschitz triangular systems, including high gain and homogeneous observers, ensuring convergence under certain nonlinear conditions.

Contribution

It introduces a cascaded high gain observer for systems not satisfying H{"o}lder conditions and proves convergence using Lyapunov functions.

Findings

High gain observer converges with small error under H{"o}lder-like conditions.

Cascaded high gain observer achieves convergence without H{"o}lder conditions.

Homogeneous observer and cascaded version are proven to converge with explicit Lyapunov functions.

Abstract

We address the problem of designing an observer for triangular non locally Lipschitz dynamical systems. We show the convergence with an arbitrary small error of the classical high gain observer in presence of nonlinearities verifying some H{\"o}lder-like condition. Also, for the case when this H{\"o}lder condition is not verified, we propose a novel cascaded high gain observer. Under slightly more restrictive assumptions, we prove the convergence of a homogeneous observer and of its cascaded version with the help of an explicit Lyapunov function.