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.
Highlights
IntroductionIn order to determine stability of different equilibrium points of gradient systems, Hessian matrix of potential functions are necessary and should be identified
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 derive a compact form of Hessian matrix for multi-agent coordination systems
In this paper we present fast and convenient approaches for identifying Hessian matrix for several typical potentials in distributed multi-agent coordination control
Summary
In order to determine stability of different equilibrium points of gradient systems, Hessian matrix of potential functions are necessary and should be identified. With the help of matrix differentials and calculus rules, we discuss Hessian identification for several typical potentials commonly-used in gradient-based multi-agent coordination control. For potential functions involving both distance functions and triangular-area functions, we will show, by using two representative examples, how a compact form of Hessian matrix can be obtained by following basic matrix calculus rules Note it is not the aim of this paper to cover all different types of potentials in multi-agent coordination and identify their Hessian formulas. Our focus will be on the Hessian analysis of a distributed gradientbased coordination control system (10) associated with an overall potential function, with the aim of providing some unified formulas of Hessian matrix. The identification of Hessian formulas will aid the stability analysis of different equilibriums in gradient-distributed multi-agent systems
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