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A Geometric Approach to Feedback Stabilization of Nonlinear Systems with Drift

Hannah Michalska, Miguel Torres-Torriti

TL;DR

The paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems, which includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback.

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.

A Geometric Approach to Feedback Stabilization of Nonlinear Systems with Drift

TL;DR

The paper presents an approach to the construction of stabilizing feedback for strongly nonlinear systems, which includes systems with drift which are affine in control and which cannot be stabilized by continuous state feedback.

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.
Paper Structure (9 sections, 57 equations, 1 figure)

This paper contains 9 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Definition and Assumptions
  3. Basic Facts and Preliminary Results
  4. Stabilizing Feedback Control Design
  5. The Stabilizing Feedback and its Analysis
  6. Stabilization of an Underactuated Rigid Body in Space
  7. Conclusions
  8. Proof of Proposition \ref{['prop:2']}
  9. Proof of Proposition \ref{['prop:3']}

Figures (1)

  • Figure 1: Results for the stabilization of the rigid body obtained with the concatenated control $u^c(x,\tau)$.