﻿ 基于LMS的多智能体系统变步长自调优一致性 Variable Step Self-Tuning Consensus of Multi-Agent Systems with Least Mean Squares Method

Computer Science and Application
Vol.08 No.06(2018), Article ID:25620,11 pages
10.12677/CSA.2018.86097

Variable Step Self-Tuning Consensus of Multi-Agent Systems with Least Mean Squares Method

Zongting Han, Xianxiang Wu, Xiangyan Hu, Baolong Guo, Xi Chen

School of Aerospace Science and Technology, Xidian University, Xi’an Shannxi

Received: Jun. 4th, 2018; accepted: Jun. 20th, 2018; published: Jun. 27th, 2018

ABSTRACT

This paper considers the consensus problem in discrete time multi-agent systems. By using variable step particle swarm algorithm, the coupling parameters among agents are locally self-tuned by least-mean square (LMS) algorithm, without using any global information. In this process, each agent minimizes a local cost function dependent on the error between the agent state and the average of neighbors’ states. Provided that the network graph is strongly connected, it is shown that for each agent, the sequence of coupling parameters is convergent, and all agent states converge toward the same constant value. Finally, the proposed algorithm is verified by simulation under different topologies.

Keywords:Multi-Agent Systems, Consensus, Self-Tuning, Least Mean Squares Method

1. 引言

2. 预备知识以及问题描述

2.1. 图论基础

2.2. 问题描述

${x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+{\beta }_{i}{u}_{i}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,N$ (1)

${x}_{i}\left(t\right)\in R$${u}_{i}\left(t\right)\in R$ 分别表示智能体在离散时间 $t=0,1,2,\cdots$ 时的状态和控制输入。 ${\beta }_{i}$ 表示第i个智能体的输入增益。假设 ${\beta }_{i}$ 的符号已知，但是它的大小是一个未知数。公式(1)可以被视为一组连续时间多智能体系统的离散版本 [18] 。

${u}_{i}\left(t\right)=\sum _{j\in {N}_{i}}{\theta }_{ij}\left(t\right)\left({x}_{j}\left(t\right)-{x}_{i}\left(t\right)\right)$ (2)

${\theta }_{ij}\left(t\right)$ 是控制参数，也是要进行估计的数值，以便智能体的状态 ${x}_{i}\left(t\right)$ 收敛到相同的值。例如，

$\underset{t\to \infty }{\mathrm{lim}}{x}_{i}\left(t\right)={x}_{c},\forall i\in V,{x}_{c}\in R$ (3)

${x}_{c}$ 为多智能体系统的状态变量达到一致的数值。在本文中，提出了基于LMS的变步长算法自适应地调整参数 ${\theta }_{ij}\left(t\right)$ 。定义参数向量：

${\theta }_{i}{\left(t\right)}^{\text{T}}=\left[{\theta }_{i1}\left(t\right){l}_{i1},\cdots ,{\theta }_{iN}\left(t\right){l}_{iN}\right],\forall i\in v$ (4)

${l}_{ij}=\left\{\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in {N}_{i}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$

${u}_{i}\left(t\right)={\theta }_{i}{\left(t\right)}^{\text{T}}{\phi }_{i}\left(t\right)$ (5)

${\phi }_{i}\left(t\right)$ 是信号向量，定义如下：

${\phi }_{i}{\left(t\right)}^{\text{T}}=\left[{\epsilon }_{i1}\left(t\right){l}_{i1},\cdots ,{\epsilon }_{iN}\left(t\right){l}_{iN}\right]$ (6)

${\epsilon }_{ij}\left(t\right)={x}_{j}\left(t\right)-{x}_{i}\left(t\right)$ (7)

${J}_{i}\left({\theta }_{i}\right)=\frac{1}{2}{\left({x}_{i}\left(t+1\right)-{\overline{x}}_{i}\left(t+1\right)\right)}^{2}$ (8)

${\overline{x}}_{i}\left(t+1\right)=\frac{1}{1+N}\sum _{j\in {N}_{i}}{x}_{j}\left(t\right),{\overline{N}}_{i}={N}_{i}\cup \left\{i\right\}$ (9)

${\theta }_{i}\left(t+1\right)={\theta }_{i}\left(t\right)-{\mu }_{i}\mathrm{sgn}\left({\beta }_{i}\right)\frac{{\phi }_{i}\left(t\right){e}_{i}\left(t+1\right)}{{r}_{i}\left(t\right)}$ (10)

${r}_{i}\left(t\right)=1+{‖{\phi }_{i}\left(t\right)‖}^{2}$ (11)

${e}_{i}\left(t+1\right)={x}_{i}\left(t+1\right)-{\overline{x}}_{i}\left(t+1\right)$ (12)

${x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+{\beta }_{i}{\theta }_{i}{\left(t\right)}^{\text{T}}{\phi }_{i}\left(t\right)$ (13)

$\frac{\partial {J}_{i}\left({\theta }_{i}\right)}{\partial {\theta }_{i}\left(t\right)}={e}_{i}\left(t+1\right)\frac{\partial {x}_{i}\left(t+1\right)}{\partial {\theta }_{i}\left(t\right)}={\beta }_{i}{\phi }_{i}\left(t\right){e}_{i}\left(t+1\right)$ (14)

$x\left(t+1\right)=W\left(t\right)x\left(t\right)$ (15)

$x{\left(t\right)}^{\text{T}}=\left[{x}_{1}\left(t\right),{x}_{2}\left(t\right),\cdots ,{x}_{N}\left(t\right)\right]$ (16)

$W\left(t\right)\in {R}^{N×N}$ 是耦合权重矩阵，定义如下:

$W\left(t\right)=\left[{w}_{ij}\right],{w}_{ij}\left(t\right)=\left\{\begin{array}{l}{\beta }_{i}{\theta }_{ij}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in {N}_{i}\\ 1-\sum _{K\in {N}_{i}}{\beta }_{i}{\theta }_{ij}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=i\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (17)

2.3. 相关证明

${\overline{x}}_{i}\left(t+1\right)={a}_{i}^{\text{T}}x\left(t\right)$ (18)

${a}_{i}^{\text{T}}=\left[{a}_{i1},\cdots ,{a}_{iN}\right],{a}_{ij}=\left\{\begin{array}{l}\frac{1}{1+{N}_{i}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in {N}_{i},j=i\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (19)

$e{\left(t\right)}^{\text{T}}=\left[{e}_{i}\left(t\right),\cdots ,{e}_{N}\left(t\right)\right]$ (20)

$x\left(t+1\right)=Ax\left(t\right)+e\left(t+1\right)$ (21)

$A={A}_{1}+l{y}_{A}^{\text{T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}l{t}_{A}^{\text{T}}=1$ (22)

$x\left(t+1\right)={A}_{1}x\left(t\right)+l{y}_{A}^{\text{T}}x\left(t\right)+e\left(t+1\right)$ (23)

$H\left({q}^{-1}\right)={\left(I-{q}^{-1}{A}_{1}\right)}^{-1}$ (24)

$x\left(t+1\right)=l{y}_{A}^{\text{T}}x\left(t\right)+H\left({q}^{-1}\right)e\left(t+1\right)$ (25)

${\varphi }_{i}\left(t\right)=x\left(t\right)-{x}_{i}\left(t\right)l,i\in v$ (26)

${\varphi }_{i}\left(t+1\right)=l{y}_{A}^{\text{T}}\left(x\left(t\right)-l{x}_{i}\left(t\right)\right)+l{y}_{A}^{\text{T}}l{x}_{i}\left(t\right)-l{x}_{i}\left(t+1\right)+H\left({q}^{-1}\right)e\left(t+1\right)$ (27)

${\varphi }_{i}\left(t+1\right)=l{y}_{A}^{\text{T}}{\varphi }_{i}\left(t\right)+l\left[{x}_{i}\left(t\right)-{x}_{i}\left(t+1\right)\right]+H\left({q}^{-1}\right)e\left(t+1\right)$ (28)

${\overline{x}}_{i}\left(t+1\right)={x}_{i}\left(t\right)+{b}_{i}^{\text{T}}{\varphi }_{i}\left(t\right)$ (29)

${b}_{i}^{\text{T}}=\left[{b}_{i1},\cdots ,{b}_{iN}\right],{b}_{ij}=\left\{\begin{array}{l}\frac{1}{1+{N}_{i}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in {N}_{i}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (30)

${e}_{i}\left(t+1\right)={x}_{i}\left(t+1\right)-{x}_{i}\left(t\right)-{b}_{i}^{\text{T}}{\varphi }_{i}\left(t\right)$ (31)

${\varphi }_{i}\left(t+1\right)={Q}_{i}{\varphi }_{i}\left(t\right)-l{e}_{i}\left(t+1\right)+H\left({q}^{-1}\right)e\left(t+1\right)$ (32)

${Q}_{i}=l{\left({y}_{A}-{b}_{i}\right)}^{\text{T}}$ (33)

${b}_{i}^{\text{T}}=\left[{b}_{i1},\cdots ,{b}_{iN}\right],{b}_{ij}=\left\{\begin{array}{l}\frac{1}{1+{N}_{i}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in {N}_{i}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (34)

${\rho }_{1}={l}^{\text{T}}\left({y}_{A}-{b}_{i}\right)=1-{l}^{\text{T}}{b}_{i}=1-{\sum }_{k=1}^{N}{b}_{ik}=1-\frac{{N}_{i}}{1+{N}_{i}}<1$ 因此存在正常数 ${c}_{0}$ ，使得对于所有的 $k\ge 0$$‖{Q}_{i}^{k}‖\le {c}_{0}{\rho }_{1}^{k},0<{\rho }_{1}<1$

${\varphi }_{i}\left(t+1\right)={Q}_{i}^{t+1}{\varphi }_{i}\left(0\right)+\sum _{k=0}^{t}{Q}_{i}^{t-k}\left[H\left({q}^{-1}\right)e\left(k+1\right)-l{e}_{i}\left(k+1\right)\right]$ (35)

${‖{\varphi }_{i}\left(t+1\right)‖}^{2}\le {c}_{1}{\rho }_{1}^{k}+{c}_{2}\sum _{k=0}^{t}{\rho }_{1}^{t-k}{‖e\left(k+1\right)‖}^{2}$ (36)

$\sum _{t=0}^{n}{‖{\varphi }_{i}\left(t+1\right)‖}^{2}\le {c}_{1}+{c}_{2}\sum _{t=0}^{n}{‖e\left(t+1\right)‖}^{2},\forall n\ge 0$ (37)

${e}_{i}\left(t+1\right)={\stackrel{˜}{\theta }}_{i}\left(t\right){\phi }_{i}\left(t\right)$ (38)

${e}_{i}\left(t+1\right)$ 由公式(12)定义，并且：

${\stackrel{˜}{\theta }}_{i}\left(t\right)={\beta }_{i}{\theta }_{i}\left(t\right)-{a}_{i}$ (39)

${\stackrel{˜}{\theta }}_{i}\left(t+1\right)={\stackrel{˜}{\theta }}_{i}\left(t\right)-\frac{{\mu }_{i}|{\beta }_{i}|}{{r}_{i}\left(t\right)}{\phi }_{i}\left(t\right){e}_{i}\left(t+1\right)$ (40)

${V}_{i}\left(t+1\right)={V}_{i}\left(t\right)-\frac{2{\mu }_{i}|{\beta }_{i}|}{{r}_{i}\left(t\right)}{\stackrel{˜}{\theta }}_{i}\left(t\right){\phi }_{i}\left(t\right)e{}_{i}\left(t+1\right)+\frac{{\left({\mu }_{i}{\beta }_{i}\right)}^{2}}{{r}_{i}\left(t\right)}\cdot \frac{{\phi }_{i}{\left(t\right)}^{2}}{{r}_{i}\left(t\right)}{e}_{i}{\left(t+1\right)}^{2}$ (41)

${V}_{i}\left(t+1\right)\le {V}_{i}\left(t\right)-2{\mu }_{i}|{\beta }_{i}|\left(1-\frac{{\mu }_{i}|{\beta }_{i}|}{2}\right)\sum _{t=0}^{n}\frac{{e}_{i}{\left(t+1\right)}^{2}}{{r}_{i}\left(t\right)}$ (42)

${V}_{i}\left(n+1\right)+2{\mu }_{i}|{\beta }_{i}|\left(1-\frac{{\mu }_{i}|{\beta }_{i}|}{2}\right)\sum _{t=0}^{n}\frac{{e}_{i}{\left(t+1\right)}^{2}}{{r}_{i}\left(t\right)}\le {V}_{i}\left(0\right)$ (43)

$\sum _{t=0}^{n}\frac{{e}_{i}{\left(t+1\right)}^{2}}{{r}_{i}\left(t\right)}\le {c}_{3}<\infty ,i\in v$ (44)

$\overline{r}\left(t\right)=\sum _{i=1}^{N}\underset{1\le \tau \le t}{\mathrm{max}}{r}_{i}\left(\tau \right)$ (45)

$\sum _{t=0}^{n}\frac{{‖e\left(t+1\right)‖}^{2}}{\overline{r}\left(t\right)}\le {c}_{4}\le \infty$ (46)

$\sum _{t=0}^{n}{‖e\left(t+1\right)‖}^{2}\le {c}_{5}<\infty$ (47)

$\underset{n\to \infty }{\mathrm{lim}}\frac{1}{\overline{r}\left(n\right)}{‖e\left(t+1\right)‖}^{2}=0$ (48)

$\overline{r}\left(n\right)\le {c}_{6}+{c}_{7}\sum _{t=0}^{n}{‖e\left(t+1\right)‖}^{2}$ (49)

$\sum _{t=0}^{n}{‖e\left(t+1\right)‖}^{2}\le {c}_{8}<\infty$ (50)

$\sum _{t=0}^{n}{‖{\phi }_{i}\left(t\right)‖}^{2}\le \sum _{t=0}^{n}{‖{\varphi }_{i}\left(t\right)‖}^{2}\le {c}_{9}\le \infty$ (51)

$\underset{t\to \infty }{\mathrm{lim}}\left(x\left(t\right)-{x}_{i}\left(t\right)l\right)=0,i\in v$ (52)

3. 仿真与分析

Figure 1. A topology with a span ning tree

Table 1. The initial positions and velocities of the agents

Figure 2. The amplitude of velocities

Figure 3. Positions and directions of agents at different iterations

Figure 4. A topology without a span ning tree

Figure 5. The amplitude of velocities

Figure 6. Positions and directions of agents at different iterations

4. 结论

Variable Step Self-Tuning Consensus of Multi-Agent Systems with Least Mean Squares Method[J]. 计算机科学与应用, 2018, 08(06): 877-887. https://doi.org/10.12677/CSA.2018.86097

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