Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems

TL;DR

This paper introduces efficient methods for identifying the Hessian matrix in multi-agent coordination control systems, simplifying stability analysis and reducing calculation errors.

Contribution

It proposes general, fast approaches based on matrix differentials for deriving Hessian matrices in gradient-based multi-agent systems, with practical examples.

Findings

New methods simplify Hessian identification process

Applications demonstrated in formation control scenarios

Reduces calculation errors in stability analysis

Abstract

Multi-agent coordination control usually involves a potential function that encodes information of a global control task, while the control input for individual agents is often designed by a gradient-based control law. The property of Hessian matrix associated with a potential function plays an important role in the stability analysis of equilibrium points in gradient-based coordination control systems. Therefore, the identification of Hessian matrix in gradient-based multi-agent coordination systems becomes a key step in multi-agent equilibrium analysis. However, very often the identification of Hessian matrix via the entry-wise calculation is a very tedious task and can easily introduce calculation errors. In this paper we present some general and fast approaches for the identification of Hessian matrix based on matrix differentials and calculus rules, which can easily derive a compact form of Hessian matrix for multi-agent coordination systems. We also present several examples on Hessian identification for certain typical potential functions involving edge-tension distance functions and triangular-area functions, and illustrate their applications in the context of distributed coordination and formation control.