A Geometric Approach to Feedback Stabilization of Nonlinear Systems with Drift

TL;DR

This paper introduces a geometric method for designing feedback controls to stabilize strongly nonlinear systems with drift, avoiding Lyapunov functions and point-to-point steering by using nonlinear programming in logarithmic coordinates.

Contribution

It proposes a novel geometric approach that constructs stabilizing feedback for nonlinear systems with drift, independent of Lyapunov functions and not relying on point-to-point steering.

Findings

Successfully stabilizes nonlinear systems with drift.

Avoids the need for Lyapunov functions and point-to-point steering.

Uses nonlinear programming in logarithmic coordinates for control design.

Abstract

The paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems. The class of systems of interest includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback. The approach is independent of the selection of a Lyapunov type function, but requires the solution of a nonlinear programming 'satisficing problem' stated in terms of the logarithmic coordinates of flows. As opposed to other approaches, point-to-point steering is not required to achieve asymptotic stability. Instead, the flow of the controlled system is required to intersect periodically a certain reachable set in the space of the logarithmic coordinates.