Exponential and practical exponential stability of second-order formation control systems

TL;DR

This paper investigates exponential stability in second-order formation control systems, introducing a novel distance-only control law that achieves practical exponential stability through averaging analysis.

Contribution

It presents a new distance-only control law for formation shape control, eliminating the need for relative position measurements while ensuring practical exponential stability.

Findings

Proves local exponential stability for standard gradient-based control.

Introduces a distance-only control law based on sinusoidal perturbations.

Establishes practical exponential stability via averaging analysis.

Abstract

We study the problem of distance-based formation shape control for autonomous agents with double-integrator dynamics. Our considerations are focused on exponential stability properties. For second-order formation systems under the standard gradient-based control law, we prove local exponential stability with respect to the total energy by applying Chetaev's trick to the Lyapunov candidate function. We also propose a novel formation control law, which does not require measurements of relative positions but instead measurements of distances. The distance-only control law is based on an approximation of symmetric products of vector fields by sinusoidal perturbations. A suitable averaging analysis reveals that the averaged system coincides with the multi-agent system under the standard gradient-based control law. This allows us to prove practical exponential stability for the system under the distance-only control law.