Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems
For researchers analyzing stability in multi-agent coordination systems, this work provides a more efficient method for Hessian identification, though it is incremental.
The paper addresses the tedious and error-prone task of identifying Hessian matrices in gradient-based multi-agent coordination systems. It presents general and fast approaches using matrix differentials and calculus rules to derive compact Hessian forms, demonstrated on edge-tension and triangular-area potential functions.
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.