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

Zhiyong Sun; Toshiharu Sugie

doi:10.3934/naco.2019020

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2019, Volume 9, Issue 3: 297-318. Doi: 10.3934/naco.2019020

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

Zhiyong Sun^1,2, , and
Toshiharu Sugie^3,

1.
Department of Automatic Control, Lund University, Sweden
2.
Research School of Engineering, The Australian National University, Canberra ACT, Australia
3.
Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501 Japan

^* Corresponding author: Zhiyong Sun
^* Corresponding author: Zhiyong Sun

Received: April 2018

Revised: April 2019

Published: May 2019

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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.

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Mathematics Subject Classification: Primary: 93C15, 15A99; Secondary: 93A15.

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