Observer design for piecewise smooth and switched systems via contraction theory

TL;DR

This paper extends contraction theory to analyze and design observers for piecewise smooth and switched systems, including nonlinear cases, using regularization and non-Euclidean norms, demonstrated through examples.

Contribution

It introduces a novel approach applying contraction theory to nondifferentiable systems via regularization, enabling observer design with non-Euclidean norms for a broad class of systems.

Findings

Extended contraction theory to nondifferentiable vector fields.

Designed observers for piecewise smooth systems using non-Euclidean norms.

Validated methodology through representative examples.

Abstract

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of piecewise smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.