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| Optimization of formation for multi-agent systems based on LQR |
| Chang-bin Yu, Yin-qiu Wang, Jin-liang Shao |
| 1School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China; 2Research School of Engineering, The Australian National University, Canberra ACT 0200, Australia; 3School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, China |
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Abstract In this paper, three optimal linear formation control algorithms are proposed for first-order linear multi-agent systems from a linear quadratic regulator (LQR) perspective with cost functions consisting of both interaction energy cost and individual energy cost, because both the collective object (such as formation or consensus) and the individual goal of each agent are very important for the overall system. First, we propose the optimal formation algorithm for first-order multi-agent systems without initial physical couplings. The optimal control parameter matrix of the algorithm is the solution to an algebraic Riccati equation (ARE). It is shown that the matrix is the sum of a Laplacian matrix and a positive definite diagonal matrix. Next, for physically interconnected multi-agent systems, the optimal formation algorithm is presented, and the corresponding parameter matrix is given from the solution to a group of quadratic equations with one unknown. Finally, if the communication topology between agents is fixed, the local feedback gain is obtained from the solution to a quadratic equation with one unknown. The equation is derived from the derivative of the cost function with respect to the local feedback gain. Numerical examples are provided to validate the effectiveness of the proposed approaches and to illustrate the geometrical performances of multi-agent systems.
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Received: 30 December 2015
Published: 02 February 2016
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基于线性二次最优化的多智能体编队控制
目的：随着空间技术和计算机技术的发展，空间飞行器协作控制越来越受到重视。多智能体编队控制是研究这一类问题的基础。本文研究了三种情况下单积分器多智能体系统基于线性二次最优性能指标的编队控制问题，并设计相应的控制算法保证多智能体系统在完成编队的基础上使所定义的性能指标达到最优。
创新点：针对三种不同的单积分器多智能体最优编队情况，分别提出相应的网络连接拓扑以及局部反馈矩阵；不同于其他论文不能给出网络拓扑以及局部最优反馈矩阵的具体解析解，本文给出相应的解析解，并且证明解析解与实际物理系统完全相符。
方法：应用代数图论以及矩阵理论的相关知识，针对无物理耦合的多智能体系统，通过求解代数里卡蒂方程，设计智能体之间的网络连接拓扑以及局部反馈矩阵，保证多智能体系统在完成编队的同时相应的LQR指标最优。针对有物理耦合的多智能体系统，同样通过求解代数里卡蒂方程，得到相应的网络连接拓扑以及局部反馈矩阵，保证多智能体系统在完成编队的基础上使相应的LQR指标最优；针对有物理耦合但无法设计网络拓扑的多智能体系统，将最优指标写成局部反馈增益的函数，通过求最优指标的导数，得到最优局部反馈增益。
结论：对于无物理耦合单积分器多智能体的编队问题与有物理耦合单积分器多智能体的编队问题，分别设计网络连接拓扑以及局部反馈矩阵，在多智能体系统完成编队的基础上保证相应的性能指标达到最优。对于有物理耦合但无法改变通讯网络拓扑的单积分器多智能体系统编队问题，设计最优局部反馈增益，在多智能体系统完成编队的同时保证性能指标最优。
关键词：
线性二次最优,
编队控制,
代数里卡蒂方程,
最优控制,
多智能体系统
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