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摘要: 多智能体系统是分布式人工智能的重要分支, 主要通过智能体之间的交互与协作完成复杂的任务. 一致性是协同控制的核心问题, 二分一致性作为一致性的延伸受到了广泛关注. 首先, 介绍一些符号图知识, 给出一阶、二阶、高阶及分数阶多智能体系统的模型及一些二分一致性的定义. 其次, 概述这四类多智能体系统多种二分一致性的研究进展. 此外, 分析有限时间、固定时间和预设时间二分一致性的优势及保守性, 讨论二分一致性和二分输出一致性的特点. 最后, 对多智能体系统二分一致性进行总结与展望.Abstract: As an important branch of distributed artificial intelligence, multi-agent systems achieve complex tasks mainly through interaction and collaboration among agents. Consensus is the core issue of collaborative control. Bipartite consensus has received widespread attention since it is an extension of consensus. Firstly, this paper introduces some knowledge of signed graph and provides models for first-order, second-order, high-order, and fractional-order multi-agent systems, as well as some definitions of bipartite consensus. Secondly, the research progress on multiple bipartite consensus of these four types of multi-agent systems is summarized. In addition, it analyzes the advantages and conservatism of finite-time, fixed-time, and predefined-time bipartite consensus and discusses the characteristics of bipartite consensus and bipartite output consensus. Finally, the summary and outlook on bipartite consensus of multi-agent systems are provided.
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Key words:
- Bipartite consensus /
- bipartite output consensus /
- signed graph /
- multi-agent systems
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多智能体系统的协同控制问题一直是控制领域的研究热点[1−6]. 特别是近十年来, 随着计算机和通信技术的快速发展, 相关协同控制方法被广泛应用在社会、工业和国防等领域, 其中多智能体协同编队控制作为协同控制的基本问题之一备受人们关注[7−9], 如多无人机集群[7]、多水面无人艇编队[8]、多无人机器人[9]等. 编队控制方法的分类方式是多样的, 从控制结构进行分类, 目前编队控制方法大体可以归结为集中式控制[10]和分布式控制[11−13], 以及二者混合的方法[14]等. 考虑到实际通信资源有限、环境复杂等情况, 分布式控制方法[11−13]比集中式控制方法[10]具有更强的鲁棒性、灵活性和可拓展性等, 因而更受关注.
在多智能体系统中, 分布式协同控制任务的实现依赖相应的通信网络作为支撑, 需要智能体间具有良好的通信关系, 使其能够进行有效的信息交互. 这意味着, 表征智能体间通信关系的通信拓扑结构在协同控制实现过程中起着重要作用. 其中通信拓扑的连通性条件是实现协同控制任务的必要条件. 而通信拓扑的连通性由拓扑结构中各个通信链路共同决定. 在实际协同控制设计中, 由于应用场景或环境的差异化等因素, 所选择使用的通信拓扑结构往往也不同, 比如静态拓扑[15−17]或动态拓扑[18−24]. 相对来说, 采用动态拓扑的控制方案[18−24]更具灵活性和适应性, 能够应对更加复杂的环境. 文献[18]针对带有扰动的多智能体系统, 提出基于动态事件触发机制的协议, 实现系统在切换拓扑下实际固定时间平均一致性问题; 文献[19]在时变平衡图情况下, 设计一种新型的具有压缩通信的分布式聚合梯度跟踪算法, 解决分布式聚合优化问题; 文献[20]针对通信范围较短的情况, 设计一种基于分层深度确定性策略梯度的强化学习框架和控制方案, 解决动态通信拓扑连通性问题和集群控制问题. 文献[21]针对通信链路故障导致通信链路权值具有时变性, 提出一种基于分布式状态观测器的自适应控制协议, 增强了通信链路对故障的抵抗能力. 文献[22]针对通信故障或恶意攻击等原因造成不规则的拓扑切换的情况, 提出一个事件触发的参考调控器, 实现系统无碰撞编队控制. 现有控制方案中所考虑的通信拓扑的动态性, 尤其是通信连接权重的改变, 主要是从性能优化[18−20]、故障[21−22]、通信资源受限[23−24]等因素考虑, 并基于图论建模为动态拓扑. 然而需要指出的是, 在实际通信中, 通信链路存在路径损耗的情况[25−28]. 也就是说, 通信路径损耗是通信链路的自身属性, 在通信中将难以避免. 通信路径损耗问题将导致预先规划好的网络拓扑[18−24]会出现新的不确定改变, 从而使系统中出现非线性耦合动态, 致使现有方法难以正常工作, 甚至将可能威胁整个多智能体系统的稳定性和安全性. 而目前尚未有尝试从通信技术和图论知识相结合的角度, 对路径衰减下通信拓扑进行量化建模的研究工作, 同时也鲜少涉及通信路径衰减下协同控制的研究工作.
另一方面, 在多智能体系统协同编队控制中, 碰撞以及离群问题是一类重要问题. 碰撞会使智能体物理受损, 离群可能会导致拓扑连通性受损, 进而影响整个编队任务的完成甚至损害系统的稳定性和安全性. 目前关于防碰撞防离群的研究已取得一些较好的进展[29−36]. 例如文献[29]提出的防碰策略是基于人工势场的方法, 为每一个智能体在一定距离内设计一个虚拟的斥力, 当其他智能体靠近时, 这个斥力使得智能体之间远离, 从而实现避免碰撞的目的. 文献[30]利用积分−乘正切李雅普诺夫障碍函数方法, 设计一种针对多智能体系统的有限时间控制器, 确保系统在有限时间内实现无碰撞编队任务. 同时在文献[31]中, 为了避免单独个体离群(即防离), 作者们提出一种基于矩形区域的连通性保持协议, 在维持拓扑连通性的前提下, 实现了离散时间多智能体系统的一致性. 文献[32]研究了通信范围受限下的通信维持问题和协同控制问题, 基于障碍函数技术设计一种自适应模糊规定时间控制协议, 确保在规定的时间内实现共识, 同时维持拓扑连通性. 此外, 文献[33]采用模型预测控制的方法通过考虑相对速度与位置, 为智能体生成一个无碰撞防远离的运动轨迹, 从而实现防碰防离. 文献[34]利用一类新型的广义混合势场构建一套基于分布式梯度的集群控制协议, 同时保证底层通信网络的避碰、理想距离稳定和连通性. 文献[35]则利用动态曲面技术, 采用递归设计原理, 提出基于近似的局部自适应跟踪设计方法以及相应的防碰防离策略. 需要指出的是, 在现有这些防碰防离策略[29−36] 中, 其防碰防离过程中所消耗的时间不确定且无上界, 缺乏快速性和时效性.
此外, 现实中人们对控制系统性能的收敛速度以及控制精度的要求不断提高. 因收敛时间的有界可控性, 有限时间/固定时间/预设时间协同控制方法引起众多学者的重视[37−46]. 其中, 滑模控制技术作为一种非线性控制方法, 具有较好鲁棒性、快速响应性等优点, 其结合有限时间/固定时间/预设时间控制技术, 成为实现系统快速收敛的策略之一, 并取得众多成果[47−50]. 例如, 文献[47]设计一个基于一阶滑模的观测算法, 该算法可以实现半全局固定时间收敛. 文献[48]设计一种二阶新型非奇异终端滑模固定时间控制器, 克服传统滑模控制方法常遇见的奇点问题、收敛时间与初始状态有关、抖震问题. 特别地, 针对高阶系统, 文献[49]提出一种高阶分层积分滑模面的结构, 能有效处理抖震现象, 并改善系统的动态性能. 此外, 为解决多智能体的未知非线性问题, 滑模控制常常与自适应神经网络控制[49]、自适应模糊控制[50]等相结合.
基于以上分析, 受到文献[25−28]中通信路径损耗模型的启发, 拟将通信拓扑的连接权重建模为距离衰减模型. 同时, 为确保编队任务在固定时间内完成, 一方面基于文献[35]的防碰防离思想及预设时间控制技术[46], 提出一种预定义时间防碰防离人工势场, 确保防碰防离策略所消耗的时间是有界且可预设的; 另一方面, 结合滑模控制技术, 提出一种自调节分层滑模面结构, 设计通信路径损耗下的自适应固定时间协同控制协议, 确保多智能体在固定时间内编队完成, 同时全程防碰防离. 本文所提方法的创新性和优势总结如下:
1)本文引入通信领域中的通信路径损耗模型[25−28], 并结合数学图论知识, 构建一种路径损耗下的通信拓扑权重变化的量化模型. 相比于现有结果[18−24], 所构建的路径损耗下的通信拓扑模型更具一般性和实际意义, 扩展了协同控制研究的领域和范围.
2)与现有人工势场方法[33−36]相比, 本文提出一种新的预定义时间人工势场策略, 不仅能避免智能体之间的碰撞和离群问题, 还能确保智能体在预设的碰撞和离群预警区内停留的总时间是有界的, 同时避免了通信路径损耗所引起的潜在的拓扑连通性问题.
3)与现有协同控制结果[37−40]相比, 本文基于自适应技术和反推控制方法, 设计一套通信路径损耗下多智能体系统固定时间编队协同控制方案, 解决了通信路径损耗所引起的非线性动态耦合问题, 实现了预期的固定时间编队协同任务. 此外, 与文献[49]相比, 本文提出一种具有自适应增益的新的分层滑模结构, 进一步改善了系统的动态性能.
1. 预备知识
1.1 图论
在本文中, 利用有向图$ {\cal{G}} = \left \{ {\cal{V}} ,\; {\cal{E}},\; {\cal{A}} \right \} $来描述多智能体之间信息交换的拓扑结构, 其中$ {\cal{V}} = \{ 1, 2,\; \cdots,\; m \} $代表网络节点集合; $ {\cal{E}} $代表通信链路集合; $ {\cal{A}} = [a_{il}]\in{\bf{R}} ^{m\times m} $代表邻接矩阵, 如果节点$ i $可以获得节点$ l $的信息, 称节点$ l $是节点$ i $的邻居, 同时有$ a_{il}> 0 $, 否则$ a_{il} = 0 $. 此外, $ a_{ii} = 0 $, $ d_i = \sum_{l = 1}^m a_{il} $. 如果智能体$ N_i $可以从领导者处获得信息则$ b_i = 1 $, 否则$ b_i = 0 $.
1.2 通信路径损耗模型
根据参考文献[25−28], 智能体间存在通信路径损耗现象. 随着智能体间通信路径距离的增大, 路径损耗随之增多. 当通信距离足够大时, 路径损耗达到最大(表征为两个智能体间不通信). 当智能体之间距离较近时, 路径损耗几乎为零(通信质量较好). 其通信路径损耗模型为:
$$ \begin{equation} L(d) = 10\mathrm{lg}\left(\frac{P_t}{P_r}\right) = K+10\Upsilon\mathrm {lg}\left(\frac{d}{d_0}\right) \end{equation} $$ (1) 其中$ P_r $是接收功率; $ P_t $是发射功率; $ \Upsilon $是路径损耗指数, 在不同环境中对应的值不同; $ K $是一个依赖于天线特性和平均信道损耗的常系数, $ K = 20\mathrm{lg}({\lambda}/({4\pi d_0})) $, $ \lambda $是波长; $ d_0 $是参考距离; $ d $是接收端和发射端的距离. 考虑实际智能体之间的尺寸, 智能体之间最短距离为$ 2r $ ($ r $是智能体半径), 在本文中参考距离设置为$ d_0 = 2r $.
定义1. 定义$ \chi = ({P_t-P_r})/{P_t} = 1-10^{-0.1K}({d_0^2}/ {d^2})^{0.5\Upsilon} $为通信路径损耗率.
通信路径损耗率表征了从发送端到接收端的功率损失百分比. 由式$ (1) $可知, 当发射功率一定时, 随着发射端与接收端距离的增大, 路径损耗逐渐增大, 接收端所接收功率逐渐减小, 对应的通信路径损耗率越大. 根据定义1可知, 通信路径损耗率$ \chi $在最短的通信距离$ d = d_0 $时最小, 并随着距离的增加呈现非线性增加, 直至$ \chi = 1 $, 即智能体间通信信号在通信路径中完全衰减, 表现为不通信.
1.3 定义和引理
定义2. 定义$ X_{ij}^{[\gamma]}=[X_{ij,\; 1}^{[\gamma]},\; X_{ij,\; 2}^{[\gamma]},\; \cdots,\; X_{ij,\; w}^{[\gamma]}]^{\mathrm{T}} $, 其中$ X_{{ij,\; h}}^{[\gamma]} $为:
$$ \begin{equation} X_{{ij,\; h}}^{[\gamma]} = \left\{ \begin{aligned} & |X_{{ij,\; h}}|^{\gamma}{{\mathrm{sign}}}(X_{{ij,\; h}}),\; \\& \quad\quad\quad\quad\quad\quad\quad\quad|X_{{ij,\; h}}|\geq\delta_1 \; \text{或}\; \gamma> 1\\ & \frac{(3-\gamma)\delta_1^{\gamma-1}}{2} X_{{ij,\; h}}+\frac{(\gamma-1)\delta_1^\gamma}{2\delta_1^3}X_{{ij,\; h}}^3,\; \quad\\& \quad\quad\quad\quad\quad\quad\quad\quad|X_{{ij,\; h}}|< \delta_1 \; \text{且}\; \gamma\leq1 \end{aligned} \right. \nonumber \end{equation} $$ 其中$ \gamma> 0 $; $ \delta_1 $是大于零的小常数.
引理1[49]. 根据径向基神经网络(Radial basis function-neural networks, RBF-NNs)的定义, 连续未知的非线性函数可以被任意精度逼近:
$$ \begin{equation} G( x) = \theta^{\mathrm T} \xi( x)+ \delta \nonumber \end{equation} $$ 其中, $ \theta\in {\bf{R}}^{q\times w} $是理想的常数权重向量, $ q $为神经元数量; $ \xi( x)\in {\bf{R}}^q $是径向基函数向量, $ || \xi( x)||^2\leq q $; $ \delta\in {\bf{R}}^w $是逼近误差.
引理2[42]. 假设$ \xi( { \overline{x}}_r) = [ \xi_1( { \overline{x}}_r),\;\xi_2( { \overline{x}}_r),\; \cdots, \xi_q( { \overline{x}}_r) ]^\mathrm T $, 其中$ { \overline{x}}_r = [ x_1,\; x_2,\; \cdots,\; x_r]^\mathrm T $是RBF-NNs的基函数向量. 对于任何正整数$ t\leq r $, 有以下不等式成立: $ || \xi( { \overline{x}}_r)||^2\leq|| \xi( { \overline{x}}_t)||^2 \nonumber $.
引理3[18]. 考虑以下的非线性系统:
$$ \begin{equation} \dot{ z}(t) = f(t,\; z(t)),\; z(0) = z_0 \end{equation} $$ (2) 其中, $ z\in {\bf{ R}}^n $; $ f: [0,\; +\infty)\times {\bf{R}}^m\xrightarrow{} {\bf{R}}^m $是一个非线性函数. 如果有一个正定、正则且径向无界的函数$ V(z): {\bf{R}}^m\xrightarrow{} {\bf{R}} $, 使得式$ (2) $的解满足:
$$ \begin{equation} \dot V(z)\leq-\alpha_1V^{\gamma_1}(z)-\alpha_2V^{\gamma_2}(z)+\eta \nonumber \end{equation} $$ 其中$ \alpha_1> 0 $; $ \alpha_2> 0 $; $ \gamma_1\in(0,\; 1) $; $ \gamma_2> 1 $; $ \eta> 0 $. 系统$ (2) $可以在固定时间收敛到一个区间内, 收敛时间为:
$$ \begin{equation} T\leq T_{\mathrm{max}}: = \frac{1}{\alpha_1\varpi(1-\gamma_1)}+\frac{1}{\alpha_2\varpi(\gamma_2-1)}\nonumber \end{equation} $$ 其中$ \varpi\in (0,\; 1) $, 残差集合为:
$$ \begin{equation} \begin{aligned} z\in& \Bigg \{ V_1(z)\leq \\& \mathrm{min}\left \{\left ( \frac{\eta }{(1-\varpi)\alpha _1} \right )^{\frac{1}{\gamma _1} },\; \left ( \frac{\eta }{(1-\varpi )\alpha _2} \right ) ^{\frac{1}{\gamma_2} } \right \} \Bigg \} \end{aligned}\nonumber \end{equation} $$ 引理4[48]. 对于任意$ x_1,\; \cdots,\; x_n\geq0 $, 下列不等式成立:
$$ \left\{\begin{aligned} &\left ( \sum\limits_{j = 1}^n x_j \right ) ^\beta \leq \sum\limits_{j = 1}^nx_j^\beta ,\; &&\beta\in(0,\; 1]\\ &n^{1-\beta }\left ( \sum\limits_{j = 1}^n x_j \right ) ^\beta \leq \sum\limits_{j = 1}^nx_j^\beta ,\;&& \beta\in(1,\; +\infty)\end{aligned}\right.$$ 2. 问题描述
2.1 系统模型
本文研究的多智能体系统包含一个虚拟领导者与$ m $个跟随者. 对于$ 1\leq i \leq m $, 跟随者$ N_i $的动态方程描述为:
$$ \begin{equation} \left\{ \begin{aligned} & \dot{ x}_{ij} = x_{{i(j+1)}}+ f_{ij}(\overline { x}_{ij}) \\& \dot{ x}_{in} = u_{i}+ f_{in}(\overline{ x}_{in}) \\ & y_{i} = x_{i1} \end{aligned} \right. \end{equation} $$ (3) 其中$ x_{ij}\in {\bf{ R}}^w $代表智能体$ N_i $的第$ j $阶状态; $ y_{i}\in {\bf{R}}^w $代表智能体$ N_i $的输出; $ u_{i}\in {\bf{ R}}^w $是系统的输入; $ \overline{ x}_{ij} = [ x_{i1},\; x_{i2},\; \cdots,\; x_{ij}]^\mathrm T $; $ f_{ij}(\cdot)\in {\bf{R}}^w $是未知连续的非线性函数, $ i = 1,\; 2,\; \cdots,\; m $.
同时, 领导者的动态方程描述为:
$$ \begin{equation} \left\{ \begin{aligned} & \dot{ x}_{0j} = x_{{0(j+1)}} \\ & \dot{ x}_{0n} = u_{0} \\ & y_{0} = x_{01} \end{aligned} \right. \end{equation} $$ (4) 其中$ x_{0j} $、$ y_{0} $、$ u_{0}\in {\bf{R}}^w $分别表示领导者的状态、输出以及输入.
队形任务分配由$ \Delta = \{\Delta_{il}\} $给出, 其中$ i = 1,\; \cdots, m $; $ l = 0,\; 1,\; \cdots,\; m $. 编队子任务为$ \Delta_{il}\in {\bf{R}}^w $, 表示智能体$ N_i $的编队任务是通过控制自身的位置与邻居$ N_l $保持的位置矢量偏差.
假设1. 有向图$ {\cal{G}} $是连通的, 至少有一个跟随者可以从领导者接收到信息.
2.2 路径损耗下通信拓扑模型
一般地, 无通信路径耗损下, 有向图表示为$ {\cal{G}} = \left \{ {\cal{V}} ,\; {\cal{E}},\; {\cal{A}} \right \} $. 考虑到通信路径损耗情况, 根据通信路径损耗模型$ (1) $和定义2, 有向图则可以表示为$ \widetilde{{\cal{G}}} = \{ {\cal{V}} ,\; \widetilde{{\cal{E}}},\; \widetilde{{\cal{A}}} \} $, 其中$ {\cal{V}} = \left \{ 1,\; 2,\; \cdots,\; m \right \} $代表网络节点集合; $ {\widetilde{\cal{E}}} $代表通信链路集合; $ \widetilde{{\cal{A}}} = [\phi_{il}(d_{il}^2)]\in{\bf{R}}^{m\times m} $代表邻接矩阵, 其中:
$$ \begin{equation} \begin{aligned} \phi_{il}(d_{il}^2) = a_{il}(1-\chi_{il}) = a_{il}10^{-0.1K}\left(\frac{d_0^2}{d_{il}^2}\right)^{0.5\Upsilon} \end{aligned} \end{equation} $$ (5) 式中, $ a_{il} $是图$ {\cal{G}} $的邻接矩阵$ {\cal{A}} $的元素. 如果节点$ i $可以获得节点$ l $的信息, 有$ \phi_{il}(d_{il}^2) = a_{il}(1-\chi_{il})> 0 $, 否则$ \phi_{il}(d_{il}^2) = a_{il}(1-\chi_{il}) = 0 $, 其中$ d_{il} = || y_i- y_l|| $. 此外, $ \phi_{ii}(d_{ii}^2) = 0 $, $ {d}_i = \sum_{l = 1}^m \phi_{il}(d_{il}^2) $. 如果智能体$ N_i $可以从领导者处获得信息则$ b_i> 0 $, 否则$ b_i = 0 $.
根据以上构建的路径损耗下通信拓扑模型, 所考虑的智能体间通信权重的变化是与智能体间通信距离相关的, 即$ \phi_{il}(d_{il}^2) = a_{il}(1-\chi_{il}) $. 由通信路径损耗模型$ (1) $和定义1可知, $ \phi_{il}(d_{il}^2) $函数是一个连续递减的非线性函数. 不失一般性, 将智能体间的通信距离等同于实际距离. 假设智能体$ N_i $能够接收到智能体$ N_l $信息, 为清晰展示权重$ \phi_{il}(d_{il}^2) $与智能体间距离$ d_{il} $的关系, 画出距离−变权重示意图, 如图1所示.
图1中, 当$ d_{il} $为智能体之间最短距离时, 即$ d_{il} = d_0 $时, $ \phi_{il}(d_{il}^2) = \overline{a}_{il} $; 当$ d_{il} $趋向于无穷时, $ \phi_{il}(d_{il}^2) $趋向于零.
注1. 不同于现有基于图论的通信拓扑模型[18−24], 本文从通信路径损耗的角度, 结合通信技术和图论知识建立了更具一般性的路径损耗通信拓扑模型, 解释并量化通信距离远近对于通信连接权重的影响, 扩展了协同控制研究的领域和范围.
2.3 控制目标
本文的控制目的是设计一种固定时间控制器使得: 1)在通信路径损耗情况下, 固定时间内完成编队任务, 即$ ||(y_i- y_l -\Delta_{il})||_{t\to T_{\mathrm{max}}} $收敛于原点的邻域中. 2)在编队过程中, 任意智能体之间不发生碰撞, 同时也不远离彼此的通信范围, 并且防碰防离过程中所消耗总时长为$ ({m^2}/{2})\times T^{ca}_{pr}(1)+g\times T^{lc}_{pr}(1) $, 其中$ T^{ca}_{pr}(1) $是斥力场预定时间; $T^{lc}_{pr}(1) $是引力场预定时间.
3. 控制方案设计
在本节中, 首先为防止智能体在编队过程中相撞以及避免需要通信的智能体离开通信范围, 设计一种预定义时间防碰防离人工势场策略; 然后, 针对高阶系统提出一种固定时间分层滑模的控制方法; 最后, 在理论上证明所提出的控制方法以及人工势场策略是有效的.
3.1 预定义时间防碰防离人工势场策略
防碰势场的建立一般需要依靠智能体所配备的测距传感器[29]. 本文在文献[29]的基础上, 将仅具有防碰撞功能的人工势场改进为预定义时间防碰防离的人工势场.
预定义时间防碰防离人工势场的主要设计思想是模拟现实中存在的引力和排斥力. 智能体对于避碰区域的邻居存在排斥力, 对于处于通信边界的邻居存在吸引力, 其示意图如图2.
在图2中, 灰色的圆代表实际的智能体, 依据距离不同在其周围形成五个区域命名为I、II、III、IV、V, 分别代表碰撞区、防碰区、编队区、防离区、失去联系区. 即当两智能体之间的距离$ d_{il}\in(r,\; 2d_{{{in1}}}] $表示邻居智能体处于I区, 两个智能体已经相撞; 当$ d_{il}\in(2d_{{in1}},\; d_{{in2}}) $代表邻居智能体处于II区, 两智能体之间有排斥力; 当$ d_{il}\in[d_{{in2}},\; d_{{out1}}] $代表邻居智能体处于III区, 两个智能体之间不存在排斥力与吸引力, 可以正常地完成编队任务; 当$ d_{il}\in (d_{{out1}},\; d_{{out2}}) $代表邻居智能体处于IV区, 智能体之间存在吸引力; 当$ d_{il}\in[d_{{out2}},\; +\infty) $代表邻居智能体处于V区, 智能体之间距离过大失去通信.
本文设计一种防碰防离策略, 使得智能体不进入I碰撞区以及V失联区, 同时在II防碰区和IV防离区的停留时间不超过有界预定时间$ T^*_{pr} $.
为实现以上目的, 改进文献[29]的防碰人工势场函数, 建立预定义时间防碰防离的人工势场, 其数学表达式为:
$$ \begin{equation} \varphi_{il}(d) = \varphi_{il}^{ca}(d)+\varphi_{il}^{lc}(d) \end{equation} $$ (6) 其中, $ \varphi_{il}^{ca}(d) = \int_{d_{{in1}}}^d\Phi_{r}(s)\mathrm ds,\; \varphi_{il}^{lc}(d) = \int_{{out2}}^d\Phi_{a}(s)\mathrm ds $分别代表斥力场与引力场, $ \Phi_{r}(s) $和$ \Phi_{a}(s) $分别为:
$$ \begin{equation} \Phi_{r}(s) = \left\{ \begin{aligned} & \frac{-w_i^{ca}(x-d_{{{in2}}})^2}{(T_{pr}^{ca}-t_{il}^{ca})(x-2d_{{in1}})^2}, & & x\in (2d_{{in1}},\; d_{{in2}}) \\ & 0, & & \text{其他} \end{aligned} \right. \end{equation} $$ (7) $$ \begin{equation} \Phi_{a}(s) = \left\{ \begin{aligned} & \frac{-w_i^{lc}(x-d_{{out1}})^2}{(T_{pr}^{lc}-t_{il}^{lc})(x-d_{{out2}})^2}, & & x\in (d_{{out1}},\; d_{{out2}}) \\& 0, & & \text{其他} \end{aligned} \right. \end{equation} $$ (8) 其中, $ w_{i}^{ca} $, $ w_i^{lc} $是大于零的常数, 决定人工势场的强弱; $ T_{pr}^* $是预定义的时间长度, $t_{il}^* $是智能体处于斥力场或引力场的时间, 其中$ * $对应着上标$ ca $、$ lc $ ($ ca $、$ lc $分别对应着防碰和防离势场的情况). 为确保智能体$ N_i $和邻居智能体$ N_l $尽快离开彼此约定的碰撞或离群预警区, 设置$ T_{pr}^*(1) $为总时间长度. 对$ T_{pr}^*(k) $进行更新, 其规则如下:
$$ \begin{equation} T_{pr}^{*}(k+1) = T_{pr}^{*}(k)-t_{il}^{*}(k) \end{equation} $$ (9) 其中$ k\,\; (k = 1,\; 2,\; \cdots) $用来表示智能体$ N_i $第$ k $次进入碰撞或离群预警区; $ T_{pr}^*(k) $表示第$ k-1 $次防碰防离势场的预定义时间长度; $ T_{pr}^*(k+1) $表示第$ k $次防碰防离势场的预定义时间长度; $ t_{il}^*(k) $是第$ k-1 $次在势场内停留的时长.
由人工势场形成的力表示为:
$$ \begin{equation} h_{il} = h_{il}^{ca}+ h_{il}^{lc} = -\nabla_{x_{i1}}\varphi_{il}^{ca}(d)-\nabla_{x_{i1}}\varphi_{il}^{lc}(d) \end{equation} $$ (10) 其中, $ -\nabla_{x_{i1}} $表示沿$ x_{i1} $方向的负梯度.
结合式(6) ~ (10)可以推导出整个势能场的数学表达式为:
$$ \begin{split}u_i^{apf}=\; & u_i^{ca}+u_i^{lc}=-\sum_{l\in N_i^{ca}}^{ }\Phi_r(d_{il})\frac{y_i-y_l}{d_{il}}\ + \\ & \sum_{l\in N_i^{lc}}^{ }\Phi_a(d_{il})\frac{y_i-y_l}{d_{il}}\end{split} $$ (11) 其中, $u_i^{ca} $、$u_i^{lc} $分别代表虚拟斥力、虚拟引力项; $ N_i^{ca} $、$ N_i^{lc} $分别表示智能体$ N_i $的防撞与距离约束邻居. 防碰撞邻居与距离约束邻居$ N_i^{ca} $、$ N_i^{lc} $与通信拓扑的邻居不同, 它们仅依赖于智能体之间的距离, 即$ N_i^{ca} = \{ l|d_{il}\in(2d_{{in1}},\; d_{{in2}})\} $, $ N_i^{lc} = \{ l|d_{il}\in (d_{{out1}},\; d_{{out2}}) $且$\phi_{il}(d_{il}^2)\ne 0\} $.
注2. 注意, 由$ T_{pr}^* $的更新律$ (9) $可知, 智能体$ N_i $停留在智能体$ N_l $的防碰/防离势场的总时间长度满足$ t^* = t_{il}^*(1)+t_{il}^*(2)+\cdots< T^*_{pr}(1) $. 相比于文献[33−35]防碰防离过程中消耗的时长不确定, 所提出的预定时间防碰防离策略可以预定义防碰防离过程中消耗的总时长$ T^*_{pr}(1) $.
3.2 跟踪误差及其动态方程
定义跟随者$ N_i $的局部跟踪误差如下:
$$ \begin{equation} \left\{ \begin{aligned} e_{i1} =\; & \sum_{l = 1}^m \phi_{il}({d_{il}^2})( x_{i1}- x_{l1}- \Delta_{il})\;+\\& b_i( x_{i1}- x_{01}- \Delta_{i0})\\ e_{ij} =\; & \sum_{l = 1}^{m}\phi_{il}({d_{il}^2})( x_{ij}- x_{lj})+b_i( x_{ij}- x_{0j}) \end{aligned} \right. \end{equation} $$ (12) 其中$ 1\leq i\leq m $; $ 2\leq j\leq n $; $ \phi_{il}(d_{il}^2) = a_{il}(1-\chi_{il}) = a_{il}10^{-0.1K}({d_0^2}/{d_{il}^2})^{0.5\Upsilon} $表示路径损耗下的权重($ \chi_{il} $为定义$1$中的通信路径损耗率); $ \Delta_{il} $表示编队子任务.
对式$ (12) $中第一个方程求导可得:
$$ \begin{split}\dot{e}_{i1}=\; & \sum_{l=1}^m2\frac{\mathrm{d}[\phi_{il}(d_{il}^2)]}{\mathrm{d}[d_{il}^2]}(x_{i1}-x_{l1})^{\mathrm{T}}(\dot{x}_{i1}-\dot{x}_{l1})\ \times \\ & (x_{i1}-x_{l1}-\Delta_{i1})\ + \\ & \sum_{l=1}^m\phi_{il}(d_{il}^2)(x_{i2}-x_{l2}+f_{i1}(\overline{x}_{i1})-f_{l1}(\overline{x}_{l1})\ -\end{split} $$ $$ \begin{split} &\;\; \quad\dot { \Delta}_{il})+b_i( x_{i2}+ f_{i1}(\overline{ x}_{i1})- x_{02}-\dot { \Delta}_{i0}) = \\ &\;\;\quad e_{i2}+ F_{i1}(Z_{i1})+ K_{i1} \end{split} $$ (13) 其中$ F_{i1}( Z_{i1}) \;= \;\sum_{l = 1}^m2 \frac{\mathrm d[\phi_{il}(d_{il}^2)]}{\mathrm d[d_{il}^2]}( x_{i1}\; -\; x_{l1})^\mathrm T (\dot { x}_{i1}\; - $ $\dot { x}_{l1}) ( x_{i1}\, \;-\;\, x_{l1}\, \;-\;\, \Delta _{i1})\;\; + $ $\sum_{l = 1}^m\phi_{il} (d_{il}^2) ( f_{i1} (\overline{ x}_{i1})\;- $ $f_{l1}(\overline{ x}_{l1}))\,+\,b_i f_{i1}(\overline{ x}_{i1}) $是未知非线性项, $ Z_{i1} = [ {\overline{x}_1}, {\overline{\dot{x}}_1}]^\mathrm T,\; {\overline{x}_1} = [ x_{11},\; x_{21},\; \cdots,\; x_{m1}]^\mathrm T$, $ {\overline{\dot{x}}_1} = [\dot { x}_{11},\; \dot { x}_{21},\; \cdots , $$\dot { x}_{m1}]^\mathrm T; $ $ K_{i1} = -\sum_{l = 1}^m\phi_{il}(d_{il}^2)\dot { \Delta}_{il}-b_i\dot { \Delta}_{i0} $是已知项.
当$ 2\leq j\leq n $时, 对式$ (12) $中第二个方程求导可得:
$$ \begin{equation} \dot { e}_{ij} = \left\{ \begin{aligned} & e_{{i(j+1)}}+ F_{ij}(Z_{ij})+ K_{ij},\; & & 2\leq j\leq n-1 \\ & (d_i+b_i) u_{i}+ F_{ij}(Z_{ij})+ K_{ij},\; & & j = n \end{aligned} \right. \end{equation} $$ (14) 其中$F_{ij}( Z_{ij}) = \sum_{l = 1}^m 2\frac{\mathrm d[\phi_{il}(d_{il}^2)]}{\mathrm d[d_{il}^2]}( x_{i1} - x_{l1})^\mathrm T ( f_{i1}(\overline{ x}_{i1})\,- $ $ f_{l1}(\overline{ x}_{l1}))( x_{ij}- x_{lj})\,+$ $\sum_{l = 1}^m \phi_{il}(d_{il}^2)( f_{ij}(\overline{ x}_{ij}) - f_{lj}(\overline{ x}_{lj})) $ $+\;b_i f_{ij}(\overline{ x}_{ij}) $是未知项, $ Z_{ij} = [ {\overline{x}_1},\; {\overline{x}_2},\; \cdots, \; {\overline{x}_j}]^\mathrm T,\; {\overline{x}_j}\,= [ x_{1j},\; x_{2j},\; \cdots,\; x_{mj}]^\mathrm T $; $ K_{ij} \;=\; \sum_{l = 1}^m 2\frac{{\mathrm d} [\phi_{il} (d_{il}^2)]} {\mathrm d[d_{il}^2]}( x_{i1}\, - $ $x_{l1})^\mathrm T( x_{i2}- x_{l2})( x_{ij}- x_{lj})-\kappa_1 (\sum_{l = 1}^m u_{l} +b_i u_{0}) $是已知项, 其中当$ j = n $时$ \kappa_1 = 1 $, 否则$ \kappa_1 = 0 $.
通过式(13) ~ (14), 可得编队误差的方程:
$$ \begin{equation} \left\{ \begin{aligned} & \dot{ e}_{ij} = e_{{i(j+1)}}+ F_{ij}(Z_{ij})+ K_{ij} \\ & \dot{ e}_{in} = (d_i+b_i) u_{i}+ F_{in}(Z_{in})+ K_{in} \end{aligned} \right. \end{equation} $$ (15) 其中$ 1\leq j\leq n-1 $. 为方便书写, 后文将$ F_{ij}(Z_{ij}) $简写为$ F_{ij} $.
3.3 分层滑模面结构设计
首先, 定义以下一组新的变量:
$$ \begin{equation} \left\{ \begin{aligned} & z_{i1} = e_{i1} \\ & z_{ij} = e_{ij}- \alpha_{ij} \end{aligned} \right. \end{equation} $$ (16) 其中$ \alpha_{ij} $是后续设计的虚拟控制项, $ 2\leq j\leq n $.
针对高阶非线性系统的每一阶都设计了一个类似递归的分层滑模面$ (j = 1,\; \cdots,\; n $, 其中$ S_{i0} = 0) $定义为:
$$ \begin{equation} S_{ij} = S_{i(j-1)}+ s_{ij} \end{equation} $$ (17) 其中每一个子层滑模面为:
$$ \begin{equation} s_{ij} = c_{ij}(t)\int_0^t (z_{ij}^{[\gamma_1]}(\theta)+ z_{ij}^{[\gamma_2]}(\theta)\mathrm) d\theta+ z_{ij} \label{第一层滑模面'} \nonumber \end{equation} $$ 为更清晰展示所提分层滑模的结构, 给出图3.
从图3可知, 第$ j $层滑模面包含第$ j-1 $层滑模面与第$ j $个子系统的信息. 由此递推可知, 当$ j = n $时, $ S_{in} $包含每个子系统的分层滑模面信息.
注3. $ z_{ij}^{[\gamma_1]} $、$ z_{ij}^{[\gamma_2]} $设计为定义2形式, 其主要目的是避免奇异值问题(分母为零的情况), 即, $ s_{ij} $为一种非奇异终端滑模面. 同时, 与现有分层滑模结构的文献[49] 相比, 本文所设计的滑模面增益可以根据系统的状态动态调节, 在不影响系统稳定性的前提下, 加快系统的收敛速度, 从而改善系统的动态性能.
3.4 控制器设计和稳定性分析
针对高阶非线性多智能体系统, 在控制设计中, 利用自适应RBF-NNs逼近非线性项, 提出一种固定时间分层滑模控制方案, 其控制方案如图4.
注4. 本文中, 编队目标、人工势场策略、通信路径损耗三者同时与智能体彼此之间距离有关, 出现三者彼此耦合现象, 给控制系统带来复杂的非线性耦合问题, 使得控制设计和稳定分析充满挑战性.
步骤一: 对第一层滑模面$ (17) $求导:
$$ \begin{equation} \begin{aligned} \dot { S}_{i1} = & c_{i1} z_{i1}^{[\gamma_1]}+c_{i1} z_{i1}^{[\gamma_2]}+ F_{i1}+ K_{i1}+ z_{i2}+ \alpha_{i2} \end{aligned} \end{equation} $$ (18) 考虑$ G_{i1}( Z_{i1}) = F_{i1}( Z_{i1}) $是未知连续的非线性函数, 无法直接应用在控制器设计中, 因此采用RBF-NNs进行逼近可知: $ G_{i1}( Z_{i1}) = \theta_{i1}^\mathrm T \xi( Z_{i1})+ \delta_{i1} $, 其中$ || \delta_{i1}||< \varepsilon_{i1} $, $ \varepsilon_{i1}> 0 $. 根据引理2, 可以得到:
$$ \begin{split} S_{i1}^\mathrm T \theta_{i1}^\mathrm T \xi( Z_{i1})\leq\;& || S_{i1}||\left(|| \theta_{i1}||^2+\frac{1}{4}|| \xi( Z_{i1})||^2 \right)\leq\\& || S_{i1}||\left(\Theta_{i1}+\frac{q}{4}\right)\\[-1pt] \end{split} $$ (19) 其中$ || \xi( Z_{i1})||^2\leq ||\xi( \check{Z}_{i1})||^2\leq q $, $ \check{Z}_{i1} = [ x_{11},\; x_{21},\; \cdots, x_{m1}]^\mathrm T $, $ \Theta_{i1} = || \theta_{i1}||^2 $. 设计虚拟控制器$ \alpha_{i2} $为
$$ \begin{split} \alpha_{i2} =\; & -\left(\hat{\Theta}_{i1}+\frac{q}{4}+\hat{\varepsilon}_{i1}\right)\frac{ S_{i1}}{||S_{i1}||+\tau} - K_{i1}- S_{i1}^{[\beta_1]}\;-\\ & S_{i1}^{[\beta_2]}-c_{i1} z_{i1}^{[\gamma_1]}-c_{i1} z_{i1}^{[\gamma_2]}+ u_{i}^{ca}+ u_{i}^{lc} \\[-1pt]\end{split} $$ (20) 其中, $\tau $是一个大于零的小常数; $ \beta_1\in(0,\; 1] $; $ \beta_2\in(1, 2) $; $ \hat{\Theta}_{i1}^\mathrm T $是$ \Theta_{i1}^{\mathrm{T}} $的估计; $ \hat{\varepsilon}_{i1} $是$ \varepsilon_{i1} $的估计. 另外, 本文中如果不另外说明, 则$ \hat{(\cdot)} $是$ (\cdot) $的估计, 估计误差为$ \tilde{(\cdot)} = (\cdot)-\hat{(\cdot)} $.
考虑以下的李雅普诺夫候选函数:
$$ \begin{equation} \begin{aligned} V_1 = \frac{1}{2}\sum_{i = 1}^m S_{i1}^\mathrm T S_{i1}+\frac{1}{2}\sum_{i = 1}^m \tilde{\Theta}_{i1}^2+\frac{1}{2}\sum_{i = 1}^m\tilde\varepsilon_{i1}^2 \end{aligned} \end{equation} $$ (21) 对上式求导并代入式$ (18)\sim(20) $可得:
$$ \begin{split} \dot V_1 =\; & \sum_{i = 1}^m S_{i1}^\mathrm T\Bigg( \theta_{i1}^\mathrm T \xi( Z_{i1})+ \delta_{i1}+ z_{i2}-\Bigg(\hat{ \Theta}_{i1}+\frac{q}{4}\;+\\& \hat{\varepsilon}_{i1}\Bigg)\frac{ S_{i1}}{||S_{i1}||+\tau}- S_{i1}^{[\beta_1]} - S_{i1}^{[\beta_2]}+ u_{i}^{ca}+ u_{i}^{lc}\Bigg)\;-\\& \sum_{i = 1}^m\tilde{\Theta}_{i1}\dot{\hat{\Theta}}_{i1}-\sum_{i = 1}^m\tilde{\varepsilon}_{i1}\dot{\hat{\varepsilon}}_{i1} \\[-1pt]\end{split} $$ (22) 根据定义2有:
$$ \begin{equation} \sum\limits_{i = 1}^{m} S_{i1}^\mathrm T(- S_{i1}^{[\beta_1]}- S_{i1}^{[\beta_2]})\leq-\sum\limits_{i = 1}^{m}(| S_{i1}|^{\beta_1+1}+| S_{i1}|^{\beta_2+1}) \end{equation} $$ (23) 考虑如下的自适应律:
$$ \begin{equation} \left\{\begin{aligned} &\dot{ \hat{\Theta}}_{i1} = || S_{i1}||-\frac{\tau|| S_{i1}||}{|| S_{i1}||+\tau}-2\iota_{i1}\hat{\Theta}_{i1}\\ &\dot {\hat{\varepsilon}}_{i1} = || S_{i1}||-\frac{\tau|| S_{i1}||}{|| S_{i1}||+\tau}-2\jmath_{i1}\hat{\varepsilon}_{i1} \end{aligned}\right. \end{equation} $$ (24) 其中$ \iota_{i1}> 0 $、$ \jmath_{i1}> 0 $. 令$ H = \sum_{i = 1}^m[ S_{i1}^\mathrm T( \theta_{i1}^\mathrm T \xi( Z_{i1})\;- ( \hat{\Theta}_{i1}+\frac{q}{4}){ S_{i1}}/({|| S_{i1}||+\tau}))-\tilde{\Theta}_{i1}\dot{\hat{\Theta}}_{i1}] $, 结合式(19)、(20)与(24)第一个等式, 可以得到:
$$ \begin{equation} \begin{aligned} H\leq& \sum_{i = 1}^m\left(-\iota_{i1}\tilde{\Theta}_{i1}^2+\iota_{i1}{\Theta}_{i1}^2+\tau\left({\Theta}_{i1}+\frac{q}{4}\right)\right) \end{aligned} \end{equation} $$ (25) 令$ E=\sum_{i=1}^m(S_{i1}^{\mathrm{T}}\delta_{i1}\, -\hat{\varepsilon}_{i1}S_{i1}^{\mathrm{T}}S_{i1}/(||S_{i1}||\; +\; \tau)\; - \tilde{\varepsilon}_{i1}\dot{\hat{\varepsilon}}_{i1}) $, 结合式$ (24) $第二个等式, 可以得到:
$$ \begin{equation} \begin{aligned} E\leq\sum_{i = 1}^m (-\jmath_{i1}\tilde\varepsilon_{i1}^2+\jmath_{i1}\varepsilon_{i1}^2+\tau{\varepsilon}_{i1}) \end{aligned} \end{equation} $$ (26) 结合式$ (22)、(23)、(25)、(26) $将$ \dot V_1 $重写为:
$$ \begin{split} \dot V_1 \leq\;& -\sum_{i = 1}^{m}(|| S_{i1}||^{\beta_1+1}+|| S_{i1}||^{\beta_2+1})\;+\\& \sum_{i = 1}^m\left(-\iota_{i1}\tilde{\Theta}_{i1}^2+\iota_{i1}{\Theta}_{i1}^2+\tau\left({\Theta}_{i1}+\frac{q}{4}\right)\right)\;+\\& \sum_{i = 1}^m \left(-\jmath_{i1}\tilde\varepsilon_{i1}^2+\jmath_{i1}\varepsilon_{i1}^2+\tau{\varepsilon}_{i1}\right)\;+\\& \sum_{i = 1}^m S_{i1}^\mathrm T( z_{i2}+ u_i^{ca}+ u_{i}^{lc}) \nonumber \end{split} $$ 步骤$ k $ $ (2\leq k \leq n) $: 对第$ k $层的滑模面$ (17) $求导:
$$ \begin{equation} \left \{\begin{aligned} \dot { S}_{ik} = \;& \dot{ S}_{i(k-1)}+ z_{{i(k+1)}}+ \alpha_{{i(k+1)}}+ F_{ik}+\dot c_{ik}D_{ik}\;+\\& c_{ik}z_{ik}^{[\gamma_1]}+ c_{ik}z_{ik}^{[\gamma_2]}+ K_{ik}-\dot{ \alpha}_{ik} ,\; 2\leq k \leq n-1\\ \dot { S}_{in} =\; & \dot { S}_{i(n-1)}+\dot c_{in}D_{in}+c_{in}z_{in}^{[\gamma_1]}+ c_{in}z_{in}^{[\gamma_2]} \;+\\& (d_i+b_i)u_i+ F_{in}+ K_{in}-\dot { \alpha}_{in}\\ \end{aligned}\right. \label{dsk'}\nonumber \end{equation} $$ 其中$ D_{ik}=\int_0^t(z_{ik}^{[\gamma_1]}(\theta)+z_{ik}^{[\gamma_2]}(\theta)\mathrm){d}\theta,\; 2\le k\le n $, 由于$ G_{ik} = \dot { S}_{i(k-1)}+ F_{ik}+\dot{ \alpha}_{ik} $是未知的, 不可以直接应用在虚拟控制器(控制器)的设计中, 因此采用RBF-NNs进行非线性逼近: $ G_{ik}( Z_{ik}) = \theta_{ik}^\mathrm T \xi( Z_{ik})\;+ \delta_{ik} $, 其中$ || \delta_{ik}||< \varepsilon_{ik} $, $ \varepsilon_{ik}> 0 $.
$$ \begin{equation} S_{ik}^\mathrm T \theta_{ik}^\mathrm T \xi( Z_{ik})\leq|| S_{ik}||\left( \Theta_{ik}+\frac{q}{4}\right) \end{equation} $$ (27) 其中$ \xi^\mathrm T( Z_{ik}) \xi( Z_{ik})\leq q $, $ \Theta_{ik} = || \theta_{ik}||^2 $.
当$ 2\leq k \leq n-1 $, 设计虚拟控制器$ \alpha_{{i(k+1)}} $为:
$$ \begin{split} \alpha_{{i(k+1)}} =\; & - K_{ik}- z_{ik}- S_{ik}^{[\beta_1]}- S_{ik}^{[\beta_2]}\;-\\& \left(\hat{ \Theta}_{ik}+\frac{q}{4}+\hat{\varepsilon}_{ik}\right)\frac{ S_{ik}}{||S_{ik}||+\tau}\;-\\& \dot c_{ik}D_{ik}-c_{ik}z_{ik}^{[\gamma_1]}-c_{ik}z_{ik}^{[\gamma_2]} \end{split} $$ (28) 当$ k = n $时, 控制器$ u_{i} $为:
$$ \begin{split} u_{i} =\; & \frac{1}{d_i+b_i}\Bigg(- K_{in}- z_{in}- S_{in}^{[\beta_1]}- S_{in}^{[\beta_2]}\;-\\& \left(\hat{ \Theta}_{in}+\frac{q}{4}+\hat{\varepsilon}_{in}\right)\frac{ S_{in}}{||S_{in}||+\tau}\;-\\& \dot c_{in}D_{in}-c_{in}z_{in}^{[\gamma_1]}-c_{in}z_{in}^{[\gamma_2]}\Bigg) \end{split} $$ (29) 考虑以下的李雅普诺夫(候选)函数:
$$ \begin{equation} \begin{aligned} V_k =\; & V_{k-1}+\frac{1}{2}\sum_{i = 1}^m S_{ik}^\mathrm T S_{ik}+\frac{1}{2}\sum_{i = 1}^m(\tilde\varepsilon_{ik}^2+\tilde{\Theta}_{ik}^2) \end{aligned} \end{equation} $$ (30) 对上式求导可得:
$$ \begin{split} \dot V_k =\; & \sum_{i = 1}^m S_{ik}^\mathrm T( \delta_{ik}+ K_{ik}+ \theta_{ik}^\mathrm T \xi( Z_{ij})+\dot c_{ik}D_{ik}\;+\\& c_{ik}z_{ik}^{[\gamma_1]}+c_{ik}z_{ik}^{[\gamma_2]}+\kappa_2 \alpha_{{i(k+1)}}+\kappa_2 z_{{i(k+1)}}\;+\\& (1-\kappa_2) u_i)- \sum_{i = 1}^m(\tilde{\Theta}_{in}\dot{\hat{\Theta}}_{in}+\tilde\varepsilon_{ik}\dot{\hat{\varepsilon}}_{ik})+\dot V_{k-1} \nonumber \end{split} $$ 其中当$ 2\leq k \leq n-1 $时, $ \kappa_2 = 1 $; 当$ k = n $时, $ \kappa_2 = 0 $, 考虑自适应律:
$$ \begin{equation} \left\{\begin{aligned} &\dot{ \hat{\Theta}}_{ik} = || S_{ik}||-\frac{\tau|| S_{ik}||}{|| S_{ik}||+\tau}-2\iota_{ik}\hat{\Theta}_{ik}\\&\dot {\hat{\varepsilon}}_{ik} = || S_{ik}||-\frac{\tau|| S_{ik}||}{|| S_{ik}||+\tau}-2\jmath_{ik}\hat{\varepsilon}_{ik} \end{aligned}\right. \end{equation} $$ (31) 其中$ \iota_{ik}> 0 $、$ \jmath_{ik}> 0 $. 与第一步类似, 代入式(27) ~ (29), 最后可以得到:
$$ \begin{split} \dot V_k\leq\;& \sum_{j = 1}^{k}\sum_{i = 1}^m\Bigg(-|| S_{ij}||^{\beta_1+1}-|| S_{ij}||^{\beta_2+1}-\iota_{ij}\tilde{\Theta}_{ij}^2\;+\\& \iota_{ij}{\Theta}_{ij}^2+\tau\left({\Theta}_{ij}+\frac{q}{4}\right)+\jmath_{ij}\varepsilon_{ij}^2+\tau{\varepsilon}_{ij}-\jmath_{ij}\tilde\varepsilon_{ij}^2\Bigg)\;+\\ & \sum_{i = 1}^m S_{i1}^\mathrm T( u_i^{ca}+ u_i^{lc})+\kappa_2\sum_{i = 1}^m S_{ik}^\mathrm T z_{{i(k+1)}}\;+\\& \sum_{j = 2}^{k}\sum_{i = 1}^m(S_{i(j-1)}-S_{ij})^\mathrm Tz_{ij} \\[-1pt]\end{split} $$ (32) 令$ M_{ij} = (S_{i(j-1)}-S_{ij})^\mathrm Tz_{ij}\;(j\geq 2) $, 代入式$ (17) $可得 $ M_{ij} = -c_{ij}D_{ij}^\mathrm Tz_{ij}-z_{ij}^\mathrm Tz_{ij} $, 考虑自适应律:
$$ \begin{equation} \dot c_{ij} = \left\{ \begin{aligned} & -{\mathrm{arctan}}(M_{ij})c_{ij},&& M_{ij}\geq0 \\ & -{\mathrm{arctan}}(M_{ij})(\overline{c}-c_{ij}) ,&& M_{ij}< 0 \end{aligned} \right. \end{equation} $$ (33) 其中$ \overline{c}> c_{ij} $, 只要初始给定$ c_{ij}> 0 $, 则$ c_{ij}\geq0 $. 当$ M_{ij}\geq 0 $时, $ -c_{ij}D_{ij}^\mathrm Tz_{ij}> 0 $, 通过自适应律调整$ c_{ij} $变小, 使得$ z_{ij}^\mathrm Tz_{ij}\geq -c_{ij}D_{ij}^\mathrm Tz_{ij} $, 即$ M_{ij}\leq 0 $; 当$ M_{ij}< 0 $时, 通过自适应律调整$ c_{ij} $的值, 加快收敛速度, 但不会超过给定的上界$ \overline{c} $.
接下来, 将本文的主要结论总结为以下定理.
定理1. 考虑高阶非线性多智能体系统(3) ~ (4)并满足假设1, 构建路径损耗下通信拓扑模型$ (5) $和预定义时间防碰防离人工势场策略(7) ~ (8)以及滑模面$ (17) $, 设计虚拟控制信号$ (20) $和$ (28) $、分布式固定时间控制器$ (29) $以及自适应律(24)、(31)和(33), 那么以下控制目标可以实现.
1) 通信路径损耗下多智能体系统在固定时间内完成编队任务, 且固定时间$ T_{\mathrm{max}} $满足$ T_{\mathrm{max}}\leq T_1\;+ T_2+({m^2}/{2})\times T^{ca}_{pr}(1)+g\times T^{lc}_{pr}(1) $. 即编队误差$ ||y_i\;- y_l - \Delta_{il}|| $固定时间收敛于原点邻域内, 其他闭环系统信号最终一致有界.
2) 编队过程中, 预定义时间防碰防离策略始终有效. 即任意智能体不会发生碰撞, 彼此通信的智能体不会远离彼此的通信范围, 执行防碰防离策略的过程所消耗的总时长小于$ ({m^2}/{2})\times T^{ca}_{pr}(1)+g\times T^{lc}_{pr}(1) $.
证明. 在多智能体系统编队过程中, 根据多智能体所处的位置(处于防碰区、防离区、编队区), 如图2, 将其分为三种情况: 1)所有智能体处于编队状态; 2)所有智能体处于防碰防离状态; 3)部分智能体处于编队状态, 部分智能体处于防碰防离状态. 接下来将对三种情况进行讨论:
情况1. 所有智能体均处于编队区, 这意味着所有智能体只进行编队任务, 不存在防碰防离任务. 根据防碰防离策略以及$ u_i^{ca} $和$ u_i^{lc} $的表达式(7) ~ (9), 可知$ u_{i}^{ca} $与$ u_{i}^{lc} $等于零, 结合李雅普诺夫函数$ (30) $及其导数$ (32) $可知:
$$ \begin{split} &\dot V_n \leq \sum_{j = 1}^n\sum_{i = 1}^m\Bigg(-|| S_{ij}^\mathrm T S_{ij}||^{\frac{\beta_1+1}{2}}-|| S_{ij}^\mathrm T S_{ij}||^{\frac{\beta_2+1}{2}}\;-\\& \left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_1+1}{2}}-\;\left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}+\; \left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_1+1}{2}}\;+\\& \left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}-{\iota_{ij}\tilde{\Theta}_{ij}^2}+\iota_{ij}\Theta_{ij}^2+\tau\left({\Theta}_{ij}+\frac{q}{4}\right)\;-\\ & \left(\frac{\jmath_{ij} \tilde\varepsilon_{ij} ^2}{2}\right)^{\frac{\beta_1+1}{2}}-\;\left(\frac{\jmath_{ij} \tilde\varepsilon_{ij} ^2}{2}\right)^{\frac{\beta_2+1}{2}} +\;\left(\frac{\jmath_{ij} \tilde\varepsilon_{ij} ^2}{2}\right)^{\frac{\beta_1+1}{2}}\;+\\& \left(\frac{\jmath_{ij} \tilde\varepsilon_{ij} ^2}{2}\right)^{\frac{\beta_2+1}{2}}-\;{\jmath_{ij} \tilde\varepsilon_{ij} ^2}+\jmath_{ij} \varepsilon_{ij} ^2+\tau{\varepsilon}_{ij}\Bigg) \\[-1pt]\end{split} $$ (34) 根据数学知识$ (\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2})^{\frac{\beta_1+1}{2}}-\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}< 1 $, 此外, 存在一个未知的正常数$ \varrho_{ij} $, 满足$ |\tilde{\Theta}_{ij}|\leq\varrho_{ij} $, 如果$ \varrho_{ij}< \sqrt{\frac{2}{\iota_{ij}}} $, 可以得到$ (\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2})^{\frac{\beta_2+1}{2}}-\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}< 0 $; 如果$ \varrho_{ij}\geq \sqrt{\frac{2}{\iota_{ij}}} $, 可以得到$ (\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2})^{\frac{\beta_2+1}{2}}-\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\leq (\frac{\iota_{ij}\varrho_{ij}^2}{2})^{\frac{\beta_2+1}{2}}\;- \frac{\iota_{ij}\varrho_{ij}^2}{2} $. 对于$ \tilde\varepsilon_{ij} $, 同样存在一个未知常数$ \ell_{ij} $, 满足$ |\tilde\varepsilon_{ij}\leq\ell_{ij}| $, 其余的分析类似于$ \tilde\Theta_{ij} $. 综上分析, 可以将式$ (34) $重写为:
$$ \begin{split} &\dot V_n\leq \sum_{j = 1}^n\sum_{i = 1}^m\Bigg(-|| S_{ij}^\mathrm T S_{ij}||^{\frac{\beta_1+1}{2}}-|| S_{ij}^\mathrm T S_{ij}||^{\frac{\beta_2+1}{2}}\;-\\ &\;\; \left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_1+1}{2}}-\left(\frac{\iota_{ij}\tilde{\Theta}_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}-\left(\frac{\jmath_{ij}\tilde\varepsilon_{ij}^2}{2}\right)^{\frac{\beta_1+1}{2}}\;-\\ &\;\; \left(\frac{\jmath_{ij}\tilde\varepsilon_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}+\left(\frac{\iota_{ij}\varrho_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}+\iota_{ij}\Theta_{ij}^2-\frac{\iota_{ij}\varrho_{ij}^2}{2}\;+\\ &\;\; \jmath_{ij}\varepsilon_{ij}^2+\left(\frac{\jmath_{ij}\ell_{ij}^2}{2}\right)^{\frac{\beta_2+1}{2}}-\frac{\jmath_{ij}\ell_{ij}^2}{2}+2+\tau{\varepsilon}_{ij}\;+\\ &\;\; \tau\left({\Theta}_{ij}+\frac{q}{4}\right)\Bigg)\nonumber \end{split} $$ 根据引理4, 可以存在两个正常数满足:
$$ \begin{equation} \dot V_n\leq-\nu_1V_n^{\frac{\beta_1+1}{2}}-\nu_2V_n^{\frac{\beta_2+1}{2}}+\eta \end{equation} $$ (35) 其中$ \nu_1 = \mathrm{min}\{ 2^{\frac{\beta_1+1}{2}},\; \iota_{ij}^{\frac{\beta_1+1}{2}},\; \jmath_{ij}^{\frac{\beta_1+1}{2}}\} $, $ \nu_2 = (mn)^{\frac{1-\beta_2}{2}}\;\times \mathrm{min}\{ 2^{\frac{\beta_2+1}{2}},\; \iota_{ij}^{\frac{\beta_2+1}{2}},\; \jmath_{ij}^{\frac{\beta_2+1}{2}}\} $, 其中$ i = 1,\; 2,\; \cdots,\; m $, $ j = 1,\; 2,\; \cdots,\; n $, $ \eta = \sum_{j = 1}^n\sum_{i = 1}^m[(\frac{\iota_{ij}\varrho_{ij}^2}{2})^{\frac{\beta_2+1}{2}}+\iota_{ij}\Theta_{ij}^2\;- \frac{\iota_{ij}\varrho_{ij}^2}{2}\;+\;\jmath_{ij}\varepsilon_{ij}^2\;+$ $(\frac{\jmath_{ij}\ell_{ij}^2}{2})^{\frac{\beta_2+1}{2}}\;-\;\frac{\jmath_{ij}\ell_{ij}^2}{2}\;+\;2\;+\;\tau{\varepsilon}_{ij}\;+ \tau({\Theta}_{ij}+ \frac{q}{4})]. $
根据引理3, $ S_{ij} $将会收敛到:
$$ \begin{split} \Omega_{ij}& = \Bigg \{S_{ij}|V_n(x)\leq \\& \mathrm{min}\left \{(\frac{\eta}{(1-\varpi_1)\nu_1})^{\frac{2}{\beta_1+1}},\; (\frac{\eta}{(1-\varpi_1)\nu_2})^{\frac{2}{\beta_2+1}}\right \} \Bigg \} \end{split} $$ (36) 在固定时间
$$ \begin{equation} T_{1}\leq\frac{2}{\nu_1\varpi_1(1-\beta_1)}+\frac{2}{\nu_1\varpi_1(\beta_2-1)} \end{equation} $$ (37) 其中$ \varpi_1\in(0,\; 1) $.
由式$ (36) $可知$ S_{ij} $、$ s_{ij} $、$ z_{ij} $均是有界的, 结合$ \dot S_{i1} $的表达式$ (18) $可知$ \dot S_{i1} $是有界的, 即存在一个常数$ \Xi> 0 $使得$ |\dot S_{i1}|< \Xi $. 考虑一个正定、正则以及径向无界的函数 $ V_{z1} = \sum_{i = 1}^m|| z_{i1}|| $, 并对其求导:
$$ \begin{split} \dot V_{z1} =\; & \sum_{i = 1}^m\frac{ z_{i1}^\mathrm T\dot{ z}_{i1}}{|| z_{i1}||} = \\& \sum_{i = 1}^m\frac{ z_{i1}^\mathrm T(-c_{i1} z_{i1}^{[\gamma_1]}-c_{i1} z_{i1}^{[\gamma_2]}+\dot{ S}_{i1})}{|| z_{i1}||}\leq \\& \sum_{i = 1}^m(-c_{i1}|| z_{i1}||^{\gamma_1}-c_{i1}m^{1-\gamma_2}|| z_{i1}||^{\gamma_2}+\Xi) \end{split} $$ (38) 结合引理4, 将式$ (38) $改写为:
$$ \begin{equation} \dot V_{z1}\leq -\nu_3V_{z1}^{\gamma_1}-\nu_4V_{z1}^{\gamma_2}+\Xi \nonumber \end{equation} $$ 其中$ \nu_3 = \mathrm{min}\{c_{i1}\} $, $ \nu_4 = m^{1-\gamma_2}\mathrm{min}\{m^{1-\gamma_2}c_{i1}\} $, 其中$ i = 1,\; 2,\; \cdots,\; m $. 根据引理3, $ z_{i1} $将会收敛到:
$$ \begin{split} \Omega _{ z_{i1}} &= \Bigg \{ z_{i1}|| z_{i1}| \leq \\& \mathrm{min} \left \{ \left(\frac{\Xi}{\nu_3(1-\varpi_2 )} \right)^{\frac{1}{\gamma_1} } ,\; \left(\frac{\Xi}{\nu_4(1-\varpi_2 )}\right)^{\frac{1}{\gamma _2} }\right \} \Bigg \} \nonumber \end{split} $$ 在固定时间
$$ \begin{equation} T_2\leq T_{\mathrm{max}}: = \frac{1}{\nu_3\varpi_2(1-\gamma _1)}+\frac{1}{\nu_4\varpi_2(\gamma _2-1)} \end{equation} $$ (39) 其中$ \varpi_2\in(0,\; 1) $.
情况2. 当智能体处于防碰防离区域时, 设计的人工势场发挥作用, 此时虚拟斥力项$ u_{i}^{ca} $与虚拟引力项$ u_{i}^{lc} $不为零.
1) 防碰: 在执行避碰任务时, 我们只关心在避碰距离内的智能体, 类似于通信邻居, 对于避碰邻居进行如下定义:
$$ \begin{equation} r_{ij}^{ca} = \left\{ \begin{aligned} &1 ,&& ||d_{ij}||\in (2d_{{in1}},\; d_{{in2}}) \\& 0 ,&& \text{其他} \end{aligned} \right.\nonumber \end{equation} $$ 其中$ r_{ij}^{ca} = 1 $表明智能体$ N_j $是智能体$ N_i $的避碰邻居. 考虑能量函数 $ P_{ik}^{ca} = \frac{1}{2}r_{ik}^{ca}( y_i- y_k)^\mathrm T ( y_i- y_k) $, 并对其求导:
$$ \begin{split} \dot P_{ik}^{ca} =\; & r_{ik}^{ca}( y_i- y_k)^\mathrm T(\dot { x}_{i1}-\dot { x}_{k1})=\\ & r_{ik}^{ca}( y_i- y_k)^\mathrm T( x_{i2}+ f_{i}(\overline{ x}_{i2})-\dot { x}_{k1}) \end{split} $$ (40) 结合式$ (12) $和$ (16) $可得:
$$ \begin{equation} x_{i2} = \frac{1}{d_i+b_i}\left(\alpha_{i2}+z_{i2}+b_ix_{02}+\sum_{l = 1}^ma_{il}x_{il}\right) \end{equation} $$ (41) 将式$ (20) $和$ (41) $代入式$ (40) $中:
$$ \begin{equation} \begin{aligned} \dot P_{ik}^{ca} = -\frac{k_ir_{ik}^{ca}}{d_i+b_i}( y_i- y_k)^\mathrm T\Phi_{r}(d_{ik})\frac{ y_i- y_k}{d_{ik}}+Y_1 \end{aligned} \end{equation} $$ (42) 其中$ Y_1 = \sum_{i = 1}^m\frac{r_{ik}^{ca}( y_i- y_k)^\mathrm T}{d_i+b_i}(-(\hat{\Theta}_{i1}+\frac{q}{4}+\hat{\varepsilon}_{i1})\frac{ S_{i1}}{||S_{i1}||+\tau} - S_{i1}^{[\beta_1]}+ u_{i}^{lc} - S_{i1}^{[\beta_2]} + z_{i2} + b_i x_{02}$ $+ \sum_{l = 1}^ma_{il} x_{il} + f_{i2}(\overline{ x}_{i2}) $ $-\dot { x}_{k1}) $, 由于系统是连续的, 并且在防碰区域的停留时间是有限的, 因此$ Y_1 $是有界的. 随着距离减小$ -\Phi_{r}(d_{ik}) $逐渐增大, 当满足$ -\Phi_{r}(d_{ik})> |(d_i+ b_i)Y_1/ ({d_{ik}k_ir_{ik}^{ca}})| $时, 有下列不等式满足:
$$ \begin{equation} \dot P_{ik}^{ca} > 0 \end{equation} $$ (43) 此时智能体之间的距离增大, 值得注意的是, 由于$ \Phi_{r}(2d_{{in1}})\to \infty $, 因此条件$ -\Phi_{r}(d_{ik})> |(d_i\;+ b_i)Y_1/({d_{ik}k_ir_{ik}^{ca}})| $必然会在$ d_{ik}> 2d_{in1} $时满足, 故智能体之间不会发生碰撞. 并且由于智能体个数是有限的, 存在潜在碰撞风险的智能体组为$ {m^2}/{2} $. 所提出的避碰策略可以保证编队的避碰性能, 并且整个系统中智能体处于斥力场的时间小于$ ({m^2}/{2})\times T^{ca}_{pr}(1) $.
2) 防离: 类似于避碰邻居, 针对防离邻居做出如下定义:
$$ \begin{split} r_{ij}^{lc} = \left\{ \begin{aligned} &1 ,&& ||d_{ij}||\in (d_{{out1}},\; d_{{out2}}) \text{且} a_{ij}\neq0 \\&0,&& \text{其他} \end{aligned} \right.\nonumber \end{split} $$ 其中$ r_{ij}^{lc} = 1 $表明智能体$ N_j $是智能体$ N_i $的防离邻居. 考虑能量函数 $ P_{ik}^{lc} = \frac{1}{2}r_{ik}^{lc}( y_i- y_k)^\mathrm T( y_i- y_k) $, 并对其求导可得:
$$ \begin{split} \dot P_{ik}^{lc} =\; & r_{ik}^{lc}( y_i- y_k)^\mathrm T(\dot { x}_{i1}-\dot { x}_{k1})=\\ & r_{ik}^{lc}( y_i- y_k)^\mathrm T( x_{i2}+ f_{i2}(\overline{ x}_{i2})-\dot { x}_{k1}) \end{split} $$ (44) 将式$ (20) $和$ (41) $代入式$ (44) $中, 得:
$$ \begin{equation} \begin{aligned} \dot P_{ik}^{lc} = \frac{k_ir_{ik}^{lc}}{d_i+b_i}( y_i- y_k)^\mathrm T\Phi_{a}(d_{ik})\frac{y_i- y_k}{d_{ik}}+Y_2 \end{aligned} \end{equation} $$ (45) 其中$ Y_2 = \sum_{i = 1}^m\frac{r_{ik}^{lc}( y_i- y_k)^\mathrm T}{d_i+b_i}(-(\hat{\Theta}_{i1}+\frac{q}{4}+\hat{\varepsilon}_{i1})\frac{ S_{i1}}{||S_{i1}||+\tau} + u_{i}^{ca} - S_{i1}^{[\beta_1]} - S_{i1}^{[\beta_2]} + z_{i2} + b_i x_{02}+ $ $\sum_{l = 1}^ma_{il} x_{il} + f_{i2}(\overline{ x}_{i2})$ $ -\dot { x}_{k1}) $, 由于系统连续以及在防离势场中停留时间是有限的, 因此$ Y_2 $是有界的. 随着需要防离的智能体之间的距离增大, $ \Phi_a(d_{ik}) $减小, 当满足$ \Phi_a(d_{ik})< -|\frac{(d_i+b_i)Y_2}{d_{ik}}k_ir_{ik}^{lc}| $时, 有下列不等式满足:
$$ \begin{equation} \dot P_{ik}^{lc} < 0 \end{equation} $$ (46) 此时智能体之间的距离减小, 值得注意的是$ \Phi_a(d_{{out2}})\to \infty $, 因此条件$ \Phi_a(d_{ik})< -|{(d_i+b_i)Y_2}/ {d_{ik}}k_ir_{ik}^{lc}| $必然会在$ d_{ik}< d_{{out2}} $时满足, 因此所提出的距离约束策略可以保证需要通信的智能体不会远离. 并且, 由于通信拓扑图中存在的通信链路是有限的, 并假设有$ g\;(g\leq m^2/2) $条通信链路, 则整个系统中存在引力场的时间不会超过$ g\times T^{lc}_{pr}(1) $.
综合1)防碰以及2)防离分析可知, 智能体防碰防离可以确保预定义时间内完成. 同时, 由于系统的连续性以及防碰防离时间有限, 闭环系统状态是有界的. 特别地, $ ||y_i - y_k || $是有界的(因处于防碰区、编队区、防离区), 并且由于编队信号$ \Delta_{ik} $有界, 可知编队误差$ ||y_i - y_k - \Delta_{ik}|| $也是有界的.
情况3. 部分智能体处于编队状态, 部分智能体处于防碰防离状态. 由于系统中依然存在防碰防离任务, 故$ \sum_{i = 1}^mS_{i1}^\mathrm T(u_i^{ca}+u_i^{lc})\ne0 $. 从式$ (32) $可知:
$$ \begin{equation} \dot V_n\leq-\nu_1V_n^{\frac{\beta_1+1}{2}}-\nu_2V_n^{\frac{\beta_2+1}{2}}+\eta+\sum\limits_{i = 1}^mS_{i1}^\mathrm T(u_i^{ca}+u_i^{lc}) \end{equation} $$ 与情况1推导类似, $ \nu_1 $、$ \nu_2 $、$ \eta $的定义与式$ (35) $一致, 在情况2中, 我们证明了任意智能体之间都不会发生碰撞, 以及需要通信的智能体之间不会远离, 因此$ \Phi_r(d_{ik}) $、$ \Phi_a(d_{ik}) $都是有界的, 即$ u_i^{ca} $、$ u_i^{lc} $也是有界的, 我们可以得到:
$$ \begin{equation} \dot V_n\leq-\nu_1V_n^{\frac{\beta_1+1}{2}}-\nu_2V_n^{\frac{\beta_2+1}{2}}+\overline\eta \nonumber \end{equation} $$ 其中$ \overline{\eta} = \eta+\sum_{i = 1}^mS_{i1}^\mathrm T(u_i^{ca}+u_i^{lc}) $, $ \overline{\eta} $是有界的, 故我们可以得出结论在情况3时系统不会发散. 又因为, 根据预定义时间防碰防离策略(6) ~ (11), 每个智能体处于防碰防离区的总时间具有上界$ T^*_{pr} $, 这意味着每个智能体会很快离开防碰防离区, 进入正常编队, 之后类似情况1, 固定时间内完成编队任务.
□ 4. 模拟仿真
在本节中, 通过进行数值模拟, 验证所提出方法的有效性. 我们考虑由一个领导者与四个跟随者组成的二阶非线性多智能体系统, 通信拓扑结构如图5所示, 跟随者的动力学方程采用文献[51]中的无人机模型, 考虑其质量为1 kg, 智能体的动力学方程如下:
$$ \begin{equation} \left\{ \begin{aligned} &\dot{ p}_i = v_i \\ &\dot{ v}_{i} = u_{i}+ f_{i}(p_i,\; v_i) \end{aligned} \right.\nonumber \end{equation} $$ 其中$ i = 0 $代表虚拟领导者智能体; $ f_0 = 0 $; $ i = 1, 2,\; 3,\; 4 $代表跟随者智能体.
$$ \left\{ \begin{aligned} p_0(0) =\; & [0,\; 0,\; 0]^\mathrm T\\ p_1(0) =\; & [2,\; -2,\; -1]^\mathrm T \\ p_2(0) =\; & [-2,\; -3,\; -3]^\mathrm T \\ p_3(0) =\; & [3,\; 2,\; 1]^\mathrm T \\ p_4(0) =\; & [-3,\; -2,\; -1]^\mathrm T \end{aligned} \right. \qquad \left\{ \begin{aligned} v_0(0) =\; & [1,\; 1,\; 1]^\mathrm T\\ v_1(0) =\; & [2,\; 1,\; 1 ]^\mathrm T \\ v_2(0) =\; & [1,\; 2,\; 1 ]^\mathrm T \\ v_3(0) =\; & [2,\; 2,\; 1 ]^\mathrm T \\ v_4(0) =\; & [-1,\; -1,\; 1 ]^\mathrm T \end{aligned} \right. $$ 仿真实验设计的队形为一个边长以$(3\;+ $ $ 0.5\mathrm{cos}(0.5t))\times\sqrt{2} $变化的正方形, 智能体之间期望的相对位置为:
$$\left\{ \begin{aligned} \Delta_{10} = \;& [ 3+0.5\mathrm{cos}(0.5t),\; 0 ,\; -5]^\mathrm T \\ \Delta_{20} =\; & [-3-0.5\mathrm{cos}(0.5t),\; 0 ,\; -5 ]^\mathrm T \\ \Delta_{30} = \;& [ 0 ,\; 3+0.5\mathrm{cos}(0.5t),\; -5]^\mathrm T \\ \Delta_{40} =\; & [0 ,\; -3-0.5\mathrm{cos}(0.5t) ,\; -5]^\mathrm T \end{aligned} \right. $$ 其余未给出的智能体期望的相对位置可以通过上述给出的队形计算得到.
系统的非线性为$ f_i = -0.012v_i $, 其余的参数分别为$ K = 1 $, $ \Upsilon = 0.4 $, $ a_{il} = 10 $, $ d_{{in1}} = 2 $, $ d_{{in2}} = 3 $, $ d_{{{out1}}} = 8 $, $ d_{{out2}} = 10 $, $ w_{i}^{ca} = w_i^{lc} = 100 $, $ \delta_1 = \delta_2 = \tau = 0.1 $, $ c_{i1} = 0.01 $, $ \gamma_1 = \frac{9}{11} $, $ \gamma_2 = \frac{11}{9} $, $ \beta_1 = \frac{9}{11} $, $ \beta_2 = \frac{11}{9} $, $ \iota_{i1} = \iota_{i2} = 0.1 $, $ \jmath_{i1} = \jmath_{i2} = 1 $, 其中$ i = 1,\; 2,\; 3,\; 4 $.
图6是多智能体编队的队形, 在经过初始的调整后组成一个正方形队伍; 图7是智能体每一维的编队误差, 根据图显示编队误差在15 s时收敛, 小于根据式$ (37)\ 和\ (39) $计算的固定时间; 图8是通信损耗率随着时间的变化规律. 由于智能体间距离不断变化并结合图8, 可以得出结论, 通信损耗率随着智能体之间距离的改变也在不断地变化. 由图6 ~ 图8、图9(a)可以得到, 在考虑通信路径损耗下, 所设计的通信路径损耗算法与防碰防离策略是有效的, 可以实现在固定时间内误差一致有界. 同时在编队过程中任意智能体间不会发生碰撞, 彼此通信的智能体间不会远离彼此的通信范围, 并且执行防碰防离策略的过程中所消耗的总时长具有上界.
图 9 两种编队方法在考虑路径损耗时智能体之间的距离Fig. 9 Distance between the agents for two formation methods with considering path loss
图 10 两种编队方法在不考虑路径损耗时智能体之间的距离Fig. 10 Distance between the agents for two formation methods without considering path loss为突出本文算法针对路径损耗情况以及防碰防离策略的有效性, 设计了3组对比试验.
1)当考虑通信路径损耗时
采用本文所设计算法的智能体之间距离仿真效果如图9(a)所示, 采用文献[52]所设计算法的智能体之间距离仿真效果如图9(b)所示. 结果表明, 从编队的完成角度上看, 本文所设计的算法在考虑路径损耗, 即通信权重发生改变时, 智能体之间的距离符合预设值, 编队完成. 但是对比文献[52], 智能体之间的距离不符合预设值, 并没有完成编队任务.
2)当不考虑通信路径损耗时
采用本文所设计算法的智能体之间距离仿真效果如图10(a)所示, 采用文献[52]所设计算法的智能体之间距离仿真效果如图10(b)所示. 根据图10(a)和图10(b)显示, 当编队队形完成后智能体之间的距离满足所预设的值. 因此可以得出结论, 本文所设计的算法与文献[52]所设计的算法在不考虑路径损耗情况时, 均可以顺利地完成编队任务.
结合以上两个对比仿真结果可知, 实际中通信路径损耗问题对系统的影响是存在且重要的. 本文所设计的控制算法无论理想情况下(无通信路径损耗)还是实际情况下(存在通信路径损耗), 均可以实现多智能体系统的编队任务. 而对比文献[52]只能在理想情况下(无通信路径损耗)实现预期编队任务, 对于实际情况下(存在通信路径损耗)的编队任务则无法完成. 相较于已有结果[52], 本文所提方法更具一般性和先进性.
3)有/无防碰防离策略
图11分别是有/无防碰防离策略下采用本文算法智能体之间的距离. 从图11(a)中可看出, 采用所设计的防碰防离策略时, 智能体之间的距离可以一直处于2 (智能体间允许的最小距离)到10 (智能体间允许的最大距离), 这表明在编队过程中, 多智能体之间没有发生碰撞, 同时需要通信的智能体之间没有由于路径损耗过大而失去通信. 从图11(b)中可看出, 在不存在防碰防离策略时, 智能体2, 4之间的距离有小于2的时刻, 这意味着智能体2, 4之间发生碰撞, 而智能体3由于距离太远失去与其他智能体通信的可能, 因此智能体3脱离了整个系统, 其他智能体与智能体3的距离不再遵从预先设定的编队规则. 通过这个对比试验, 可以验证本文所提出防碰防离算法的有效性.
图 11 包含/去除防碰防离策略下采用本文算法智能体之间的距离Fig. 11 Distance between the agents using the proposed algorithm with/without collision and isolation avoidance strategy5. 结束语
本文研究了通信路径损耗下高阶非线性多智能体系统的固定时间编队问题. 首先所考虑的智能体间通信权重的变化与它们之间通信路径损耗相关, 距离越长路径损耗越大. 其次, 设计一个自适应增益的固定时间分层滑模面, 并开发一种自适应固定时间编队控制协议. 为解决编队中存在的防碰防离问题, 引入预定义时间人工势场, 并融入所设计的编队协议中. 基于李雅普诺夫理论, 证明了编队协议可以保证所有智能体可以在固定时间完成编队任务, 并且在过程中不会出现相碰与相离情况. 最后通过仿真验证编队协议的有效性. 事实上, 影响智能体之间的通信权重不仅取决于它们之间的距离, 网络延时、丢包、攻击都可能会产生影响, 我们将在未来的工作中继续考虑这些问题.
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图 2 $w_{i1}(t)$的状态轨迹, $i=1,\;2,\;\cdots,\;7$
Fig. 2 State trajectories of $w_{i1}(t)$, $i=1,\;2,\;\cdots,\;7$
图 4 $w_{i2}(t)$的状态轨迹, $i=1,\;2,\;\cdots,\;7$
Fig. 4 State trajectories of $w_{i2}(t)$, $i=1,\;2,\;\cdots,\;7$
表 1 多智能体系统二分一致性文献总结
Table 1 Summary of literature on bipartite consensus of multi-agent systems
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