Certifying non-existence of undesired locally stable equilibria in formation shape control problems

TL;DR

This paper addresses the challenge of certifying the non-existence of undesired locally stable equilibria in formation shape control for autonomous vehicles, using semidefinite programming to analyze polynomial conditions.

Contribution

It introduces a method employing semidefinite programming to certify the absence of undesired stable equilibria in formation control problems of any size.

Findings

Semidefinite programming can certify the non-existence of undesired equilibria.

The approach applies to formations of arbitrary size.

Provides a systematic way to analyze stability in formation control.

Abstract

A fundamental control problem for autonomous vehicle formations is formation shape control, in which the agents must maintain a prescribed formation shape using only information measured or communicated from neighboring agents. While a large and growing literature has recently emerged on distance-based formation shape control, global stability properties remain a significant open problem. Even in four-agent formations, the basic question of whether or not there can exist locally stable incorrect equilibrium shapes remains open. This paper shows how this question can be answered for any size formation in principle using semidefinite programming techniques for semialgebraic problems, involving solutions sets of polynomial equations, inequations, and inequalities.