Immersion and Invariance Stabilization of Nonlinear Systems: A Horizontal Contraction Approach

TL;DR

This paper introduces a contraction theory-based method for stabilizing nonlinear systems through immersion and invariance, providing a more systematic approach with explicit design parameters, demonstrated on feedback systems.

Contribution

It presents a novel contraction theory approach to enhance the immersion and invariance stabilization method, making the design process more systematic and flexible.

Findings

Ensures attractivity of the manifold via horizontal contraction.

Provides a systematic procedure with explicit degrees of freedom.

Illustrates the approach on feedback form systems.

Abstract

The main objective of this paper is to propose an alternative procedure to carry out one of the key steps of immersion and invariance stabilising controller design. Namely, the one that ensures attractivity of the manifold whose internal dynamics contains a copy of the desired system behaviour. Towards this end we invoke contraction theory principles and ensure the attractivity of the manifold rendering it horizontally contractive. The main advantage of adopting this alternative approach is to make more systematic the last step of the design with more explicit degrees of freedom to accomplish the task. The classical case of systems in feedback form is used to illustrate the proposed controller design.