﻿ 有向图中基于扰动观测器的线性多智能体系统一致性
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 自动化学报  2018, Vol. 44 Issue (6): 1037-1044 PDF

1. 哈尔滨工业大学(深圳)机电工程与自动化学院 深圳 518055;
2. 沈阳飞机设计研究所 沈阳 110035

Disturbance Observer Based Consensus of Linear Multi-agent Systems Under a Directed Graph
YANG Dong-Yue1,2, MEI Jie1
1. School of Mechanical Engineering and Automtion, Harbin Institute of Technology, Shenzhen 518055;
2. Shenyang Aircraft Design and Research Institute, Shenyang 110035
Manuscript received : November 1, 2016, accepted: April 21, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61403094) and the Foundation Research Project of Shenzhen (JCYJ20160505175231531)
Corresponding author. MEI Jie Associate professor at the School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen. His research interest covers distributed control of multi-agent systems and its application in formation flying. Corresponding author of this paper
Recommended by Associate Editor LV Jin-Hu
Abstract: In this paper, we study the consensus problem of multi-agent systems in the existence of unknown disturbances under a general directed graph. The dynamics of each agent is a general linear system with matched disturbances. Given that the directed graph is strongly connected, we design a distributed disturbance observer to solve the consensus problem for general linear multi-agent systems with unknown external disturbances. At the end, simulation results are presented to demonstrate the effectiveness of the proposed algorithm.
Key words: Multi-agent systems     disturbance observer     consensus     directed graph

1 数学背景与问题描述 1.1 图论知识

 \begin{align} \dot{x}_i&=Ax_i-BK\chi_i+BF(w_i-\hat{w}_i) =\notag \\ &(A-BK)x_i+B[K, F]e_i \end{align} (15)
2.2 一致性分析

 \begin{align} V&=\sum\limits_{i=1}^N\alpha_i\xi_i\left[\sum\limits_{j=1}^Na_{ij}(e_i-e_j)\right]^{\rm T}Q\cdot\notag \\& \sum\limits_{j=1}^Na_{ij}(e_i-e_j) =[(\mathcal{L}_A\otimes I_p)e]^{\rm T}\cdot\notag \\ &(\Xi\Lambda\otimes Q)(\mathcal{L}_A\otimes I_p)e \end{align} (16)

 \begin{align} \dot{V}(t)=\, &\gamma^{\rm T}[\Xi\Lambda^{-1}\otimes (2H^{\rm T}H-I)-\widehat{L}\otimes H^{\rm T}H]\gamma =\notag \\ &\gamma^{\rm T}(\Xi\Lambda^{-1}\otimes 2H^{\rm T}H)\gamma-\gamma^{\rm T}(\widehat{L}\otimes H^{\rm T}H)\gamma -\notag \\ &\gamma^{\rm T}(\Xi\Lambda^{-1}\otimes I)\gamma \end{align} (18)

$\gamma_i$写成$\gamma=[\gamma_1^{\rm T}, \cdots, \gamma_N^{\rm T}]^{\rm T}\in{\bf R}^{Np}$, 其中$\gamma_i\in{\bf R}^p$.设$v_k=[v_{k1}, \cdots, v_{kN}]^{\rm T}\in{\bf R}^N$是所有S\gamma_ik个元素的堆栈向量, 根据引理3, 可得:  \begin{align} \gamma^{\rm T}(\widehat{L}\otimes H^{\rm T}H)\gamma=\sum\limits_{k=1}^{p}\lambda_k(H^{\rm T}H)v_k^{\rm T}\widehat{L}v_k \end{align} (19) S\gamma_i=\alpha_iS\sum_{j=1}^Na_{ij}(e_i-e_j), 向量Se_i的第k个元素我们记为(Se_i)_k, 那么则有:  \begin{align} v_{ki}=\alpha_i\sum\limits_{j=1}^{N}[(Se_i)_k-(Se_j)_k] \end{align} (20) 定义\theta_k:=[(Se_1)_k, \cdots, (Se_N)_k]^{\rm T}\in{\bf R}^N.那么式(20)可以写成v_k=\Lambda\mathcal{L}_A\theta_k.因此, 对于\xi_i>0, \alpha_i>0, \forall i=1, \cdots, N, 正向量\Lambda^{-1}\xi使得:  \begin{align} v_k^{\rm T}\Lambda^{-1}\xi=\theta_k^{\rm T}{\mathcal{L}_A}^{\rm T}\xi=0 \notag \end{align} 则由引理2可知:  \begin{align} v_k^{\rm T}\widehat{L}v_k&\geq\min\limits_{v_k^{\rm T}\varsigma=0}v_k^{\rm T}\widehat{L}v_k= \notag \\ &a(\mathcal{L}_A)v_{k}^{\rm T}v_k \end{align} (21) 综上所述, 可以得出:  \begin{align} \gamma^{\rm T}(\widehat{L}\otimes H^{\rm T}H)\gamma&=\sum\limits_{k=1}^{p} \lambda_k(H^{\rm T}H)v_k^{\rm T}\widehat{L}v_k \geq\notag \\ & \sum\limits_{k=1}^{p}\lambda_k(H^{\rm T}H)\min\limits_{v_k^{\rm T}\varsigma=0} v_k^{\rm T}\widehat{L}v_k =\notag\\ & a(\mathcal{L}_A)\sum\limits_{k=1}^{p} \lambda_k(H^{\rm T}H)v_k^{\rm T}v_k =\notag \\ &a(\mathcal{L}_A)\gamma^{\rm T}(I_p\otimes H^{\rm T}H)\gamma \end{align} (22) 将式(22)代回式(18)中, 可得:  \begin{align} \dot{V}\leq\, & -\sum\limits_{i=1}^N\left[a(\mathcal{L}_A)-\frac{2\xi_i}{\alpha_i}\right]\gamma_i^{\rm T}H^{\rm T}H\gamma_i-\notag \\ &\sum\limits_{i=1}^N\frac{\xi_i}{\alpha_i}\gamma_i^{\rm T}\gamma_i \end{align} (23) 由此得到本文的主要结论: 定理1. 在假设1, 2和3成立的条件下, 将观测器(9), (10)及控制输入(8)作用于系统(3), 选取适当的异质增益\alpha_i, 系统将达到状态一致性, 即: \lim_{t\rightarrow\infty}x_1(t)=\lim_{t\rightarrow\infty}x_2(t)=\cdots=\lim_{t\rightarrow\infty}x_N(t). 证明. 从式(23)可知, 当控制增益满足条件:  \begin{align} \alpha_i>\frac{2\max_k\xi_k}{a(\mathcal{L}_A)} \end{align} (24) k=1, \cdots, N时, 可知\dot{V}是半负定的, 则系统中的所有变量是有界的, 并且\gamma\in L_2\bigcap L_\infty, 因此, 有\sum_{j=1}^Na_{ij}(e_i-e_j)\in L_2\bigcap L_\infty.从式(14)可知, \dot{e}\in L_\infty, 那么根据Barbalat引理可知, \sum_{j=1}^Na_{ij}(e_i-e_j)\rightarrow 0, 即\lim_{t\rightarrow\infty}e_1(t)=\lim_{t\rightarrow\infty}e_2(t)=\cdots=\lim_{t\rightarrow\infty}e_N(t)则由式(15)可知, 对任意i, j:  \begin{align} \dot{x}_i-\dot{x}_j=(A-BK)(x_i-x_j)+B[K, F](e_i-e_j) \end{align} (25) 注意到(A-BK)是Hurwitz的, 那么将B[K, F](e_i-e_j)看作输入, (x_i-x_j)看作状态, 系统(25)是输入到状态稳定的.由于\lim_{t\rightarrow\infty}(e_i-e_j)=0, 那么\lim_{t\rightarrow\infty}(x_i-x_j)=0, 即系统达到一致性. 注4. 在分布式的背景下, 与传统的每个智能体采用相同的控制增益方法相比, 异质增益更符合算法分布式的要求.另外, 注意到\xi_ia(\mathcal{L}_A)的信息并不能通过分布式方法得到, 另一方面, 由式(24)可知, 可以通过选取足够大的$\alpha_i$使系统达到一致性.基于Lyapunov方法的分析是一个充分条件, $\alpha_i$并不需要严格满足(24), 在实际中可以从较小的$\alpha_i$开始, 如果系统不能满足要求, 则可以通过不断增加$\alpha_i$来完成系统的一致性.

3 仿真分析

 \begin{align} \begin{bmatrix} \dot{V} \\ \dot{\beta}\\ \dot{q}\\ \dot{\theta} \end{bmatrix}=\, &\begin{bmatrix} -0.284&-23.096&2.420&9.913 \\ 0 &-4.117 &0.843&0.272 \\ 0 &-33.884&-8.263&-19.543 \\ 0 &0 &1 &0 \end{bmatrix}\times\notag \\ & \begin{bmatrix} V \\ \beta \\ q \\ \theta \end{bmatrix} +\begin{bmatrix} 20.168 \\ 0.544 \\ -39.085 \\ 0 \end{bmatrix} i_{H} \end{align} (26)

 \begin{align} G=\begin{bmatrix} 0.0058 & 0.0393 & 0.0033 & -0.0412 \\ 0.0393 & 17.0846 & 2.9318 & 6.3884 \\ 0.0033 & 2.9318 & 1.6088 & 2.3888 \\ -0.0412 & 6.3884 & 2.3888 & 6.69543 \\ 0 & 0.0004 & 0.0033 & 0.0028 \\ 0 & -0.3486 & -0.2041 & -0.3025 \end{bmatrix} \notag \end{align}

 图 2 UAV纵向控制时的速度状态轨线 Figure 2 Speed of UAV longitudinal control
 图 3 UAV纵向控制时的攻角状态轨线 Figure 3 Angle of attack of UAV longitudinal control
 图 4 UAV纵向控制时的俯仰率状态轨线 Figure 4 Pitch rate of UAV longitudinal control
 图 5 UAV纵向控制时的俯仰角状态轨线 Figure 5 Pitch of UAV longitudinal control

 图 6 无人机受到的不同扰动 Figure 6 Different disturbances for UAVs

4 结论

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