Speed-Gradient Control of the Brockett Integrator

TL;DR

This paper introduces a nonsmooth speed-gradient control algorithm for the Brockett integrator, achieving almost global stabilization with continuous control and minimal control effort, advancing nonsmooth control methods.

Contribution

It presents a novel nonsmooth speed-gradient control law that stabilizes the Brockett integrator almost globally with continuous trajectories and minimal control effort.

Findings

Achieves almost global stabilization of the Brockett integrator.

Ensures control law is continuous along system trajectories.

Stabilizes from almost all initial conditions except on the x3-axis.

Abstract

A nonsmooth extension of the speed-gradient algorithms in finite form is proposed. The conditions ensuring control goal (convergence of the goal function to zero) are established. A new algorithm is applied to almost global stabilization of the Brockett integrator that has become a popular benchmark for nonsmooth and discontinuous algorithms. It is proved that the designed control law stabilizes the Brockett integrator for any initial point that does not lie on the x3-axis. Besides, it is shown that the speed-gradient algorithm ensures stabilization with an arbitrarily small control level. An important feature of the proposed control is the fact that it is continuous along trajectories of the closed-loop system.