Exponential Stabilization of Nonholonomic Systems by means of Oscillating Controls

TL;DR

This paper presents a method for stabilizing nonholonomic systems exponentially using oscillating controls derived from Lyapunov functions, demonstrated on the Brockett integrator.

Contribution

It introduces a novel oscillating control approach for exponential stabilization of nonlinear driftless systems with a Lyapunov-based feedback law.

Findings

Controls ensure exponential stability of the equilibrium

Method successfully stabilizes the Brockett integrator

Constructs controls via trigonometric functions approximating gradient flows

Abstract

This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.