  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.
